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Formulations for Estimating Spatial Variations of Analysis Error Variance to Improve Multiscale and Multistep Variational Data Assimilation

Formulations for Estimating Spatial Variations of Analysis Error Variance to Improve Multiscale... Hindawi Advances in Meteorology Volume 2018, Article ID 7931964, 17 pages https://doi.org/10.1155/2018/7931964 Research Article Formulations for Estimating Spatial Variations of Analysis Error Variance to Improve Multiscale and Multistep Variational Data Assimilation 1 2 Qin Xu and Li Wei NOAA/National Severe Storms Laboratory, Norman, OK, USA Cooperative Institute for Mesoscale Meteorological Studies, University of Oklahoma, Norman, OK, USA Correspondence should be addressed to Qin Xu; qin.xu@noaa.gov Received 24 April 2017; Revised 24 November 2017; Accepted 3 December 2017; Published 7 February 2018 Academic Editor: Shaoqing Zhang Copyright © 2018 Qin Xu and Li Wei. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. When the coarse-resolution observations used in the first step of multiscale and multistep variational data assimilation become increasingly nonuniform and/or sparse, the error variance of the first-step analysis tends to have increasingly large spatial variations. However, the analysis error variance computed from the previously developed spectral formulations is constant and thus limited to represent only the spatially averaged error variance. To overcome this limitation, analytic formulations are constructed to efficiently estimate the spatial variation of analysis error variance and associated spatial variation in analysis error covariance. First, a suite of formulations is constructed to efficiently estimate the error variance reduction produced by analyzing the coarse- resolution observations in one- and two-dimensional spaces with increased complexity and generality (from uniformly distributed observations with periodic extension to nonuniformly distributed observations without periodic extension). en, Th three different formulations are constructed for using the estimated analysis error variance to modify the analysis error covariance computed from the spectral formulations. The successively improved accuracies of these three formulations and their increasingly positive impacts on the two-step variational analysis (or multistep variational analysis in first two steps) are demonstrated by idealized experiments. 1. Introduction steps of a multistep approach) in which broadly distributed coarse-resolution observations are analyzed rfi st and then Multiple Gaussians with different decorrelation length scales locally distributed high-resolution observations are analyzed have been used at NCEP to model the background error in the second step, an important issue is how to objectively covariance in variational data assimilation (Wu et al. [1], estimate or efficiently compute the analysis error covariance Purser et al. [2]), but mesoscale features are still poorly for the analyzed efi ld that is obtained in the rfi st step and resolved in the analyzed incremental efi lds even in areas used to update the background field in the second step. To covered by remotely sensed high-resolution observations, address this issue, spectral formulations were derived by Xu et such as those from operational weather radars (Liu et al. al. [8] for estimating the analysis error covariance. As shown [3]). This problem is common for the widely adopted single- in Xu et al. [8], the analysis error covariance can be computed step approach in operational variational data assimilation, very ecffi iently from the spectral formulations with very (or especially when patchy high-resolution observations, such fairly) good approximations for uniformly (or nonuniformly) as those remotely sensed from radars and satellites, are distributed coarse-resolution observations and, by using the assimilated together with coarse-resolution observations into approximately computed analysis error covariance, the two- a high-resolution model. To solve this problem, multiscale step analysis can outperform the single-step analysis under and multistep approaches were explored and proposed by thesamecomputational constraint (thatmimicsthe opera- severalauthors (Xieetal.[4],Gao etal.[5],Lietal.[6], and Xu et al. [7, 8]). For a two-step approach (or the first two tional situation). 2 Advances in Meteorology 2 2 2 2 2 eTh analysis error covariance functions computed from where 𝛾 =𝜎 /(𝜎 +𝜎 ), 𝜎 (or 𝜎 )isthebackground 𝑏 𝑏 𝑏 𝑜 𝑏 𝑜 the spectral formulations in Xu et al. [8] are spatially (or observation) error variance, 𝐶 (𝑥) is the background homogeneous, so their associated error variances are spatially error correlation function, 𝑥 denotes the 𝑖th point in the constant. Although such a constant error variance can rep- discretized analysis space 𝑅 ,and 𝑁 is the number of grid resent the spatial averaged value of the true analysis error points over the analysis domain. eTh length of the analysis variance, it cannot capture the spatial variation in the true domain is 𝐷=𝑁Δ𝑥 ,where Δ𝑥 is the analysis grid spacing analysis error variance. eTh true analysis error variance can and 𝐷 is assumedtobemuchlargerthanthe background have significant spatial variations, especially when the coarse- error decorrelation length scale 𝐿. resolution observations become increasingly nonuniform Note that 𝐶 (𝑥) is a continuous function of 𝑥,so(2) can 2 2 2 and/or sparse. In this case, the spatial variation of analysis be written into 𝜎 (𝑥) ≡ 𝜎 −Δ𝜎 (𝑥) also as a continuous 𝑚 𝑏 𝑚 error variance and associated spatial variation in analysis function of 𝑥,where error covariance need to be estimated based on the spatial distribution of the coarse-resolution observations in order (3) Δ𝜎 (𝑥 ) ≡𝛾 [𝜎 𝐶 (𝑥 −𝑥 )] to further improve the two-step analysis. This paper aims to 𝑏 𝑏 𝑏 𝑖 𝑚 explore and address this issue beyond the preliminary study reported in Xu and Wei [9]. In particular, as will be shown in is the error variance reduction produced by analyzing a single this paper, analytic formulations for efficiently estimating the observation at 𝑥=𝑥 . eTh error variance reduction in spatial variation of analysis error variance can be constructed (3) decreases rapidly as |𝑥−𝑥 | increases, and it becomes by properly combining the error variance reduction produced 2 2 much smallerthanitpeakvalueof 𝛾 𝜎 C at 𝑥=𝑥 as 𝑏 𝑏 𝑏 𝑚 by analyzing each and every coarse-resolution observation |𝑥−𝑥 | increases to 𝐿. This implies that the error variance as asingleobservation,andtheestimated analysiserror reduction produced by analyzing 𝑀 sparsely distributed variance canbeusedtofurther estimatetherelatedvariation coarse-resolution observations can be estimated by properly in analysis error covariance. eTh detailed formulations are combining the error variance reduction computed by (3) for presented for one-dimensional cases first in the next section each coarse-resolution observation as a single observation. and then extended to two-dimensional cases in Section 3. This idea is explored in the following three subsections Idealized numerical experiments are performed for one- for one-dimensional cases with successively increased com- dimensional cases in Section 4 and for two-dimensional cases plexity and generality: from uniformly distributed coarse- in Section 5 to show the eeff ctiveness of these formulations resolution observations with periodic extension to nonuni- for improving the two-step analysis. Conclusions follow in formly distributed coarse-resolution observations without Section 6. periodic extension. 2. Analysis Error Variance Formulations for 2.2. Uniform Coarse-Resolution Observations with Periodic One-Dimensional Cases Extension. Consider that there are 𝑀 coarse-resolution observations uniformly distributed in the above analysis 2.1. Error Variance Reduction Produced by a Single Observa- domain of length 𝐷 with periodic extension, so their res- tion. When observations are optimally analyzed in terms of olution is Δ𝑥 ≡𝐷/𝑀 . In this case, the error variance co the Bayesian estimation (see chapter 7 of Jazwinski [10]), the reduction produced by each observation can be considered background state vector b is updated to the analysis state as an additional reduction to the reduction produced by its vector a with the following analysis increment: neighboring observations, and this additional reduction is always smaller than the reduction produced by the same −1 T T (1a) Δa ≡ a − b = BH (HBH + R) d, observationbuttreatedasasingle observation. Thisimplies that the error variance reduction produced by analyzing the and the background error covariance matrix B is updated to 𝑀 coarse-resolution observations, denoted by Δ𝜎 (𝑥),is the analysis error covariance matrix A according to bounded above by ∑ Δ𝜎 (𝑥);thatis, 𝑚 𝑚 −1 T T (1b) A = B − BH (HBH + R) HB, 2 2 Δ𝜎 (𝑥 ) ≤ ∑Δ𝜎 (𝑥 ) , (4) 𝑀 𝑚 where R is the observation error covariance matrix, d = y − 𝑚 h(b) is the innovation vector (observation minus background in the observation space), y is the observation vector, h() where ∑ denotes the summation over 𝑚 for the 𝑀 observa- denotes the observation operator, and H is the linearized h(). tions. The equality in (4) is for the limiting case of Δ𝑥 /𝐿 → co For a single observation, say, at 𝑥 in the one-dimensional ∞ only. eTh inequality in (4) implies that the domain- −1 space of 𝑥,the inversematrix (HBH + R) reduces to 2 averaged value of ∑ Δ𝜎 (𝑥) is larger than the true averaged −1 𝑚 𝑚 2 2 2 2 2 2 (𝜎 +𝜎 ) ,sothe 𝑖th diagonal element of A in (1b) is simply 𝑏 𝑜 reduction estimated by Δ𝜎 ≡𝜎 −𝜎 ,where 𝜎 is the 𝑏 𝑒 𝑒 given by domain-averaged analysis error variance estimated by the spectral formulation for one-dimensional cases in Section 2.2 2 2 (2) 𝜎 (𝑥 )≡𝜎 −𝛾 [𝜎 𝐶 (𝑥 −𝑥 )] , 𝑚 𝑖 𝑏 𝑏 𝑏 𝑏 𝑖 𝑚 of Xu et al. [8]. 𝑏𝑒 Advances in Meteorology 3 The domain-averaged value of ∑ Δ𝜎 (𝑥) can be com- 𝑚 𝑚 puted by ∫ ∑ Δ𝜎 (𝑥 ) 𝑚 𝑚 2 𝐷 Δ𝜎 ≡ 2 2 𝛾 𝜎 ∑ ∫ 𝐶 (𝑥 − 𝑥 ) 𝑏 𝑚 𝑚 𝑏 𝑏 (5a) 2 2 𝛾 𝜎 ∑ ∑ 𝐶 (𝑥 −𝑥 ) 𝑏 𝑖 𝑚 𝑏 𝑚 𝑖 𝑏 6 ≈ , where ∫ denotes the integration over the analysis −60 −40 −20 04 20 0 60 domain, ∑ denotes the summation over 𝑖 for the 𝑁 grid x (km) points, and 𝐷=𝑁Δ𝑥 is used in the last step. By extending 2 2 2 2 𝐶 (𝑥−𝑥 ) with the analysis domain periodically, Δ𝜎 can 𝑏 𝑚  (x)  a e be also estimated analytically as follows: 2 2  (x) ∗  a o Figure 1: Benchmark analysis error variance 𝜎 (𝑥) plotted by red ∫ ∑ Δ𝜎 (𝑥 ) 𝑚 𝑚 2 𝐷 Δ𝜎 ≡ solid curve and estimated analysis error variance 𝜎 (𝑥) in (7) plotted by blue dotted curve. eTh green dashed line shows the 2 2 constant analysis error variance 𝜎 estimated from the spectral 𝛾 𝜎 ∑ ∑ ∫ 𝐶 (𝑥 − 𝑥 −𝑘𝐷) 𝑏 𝑚 𝑘 𝑚 𝑏 𝑏 (5b) formulation. The purple + signs show the observation error variance 2 2 2 −2 (𝜎 = 2.5 m s )atthe locationsof 𝑀 (=10) uniformly distributed coarse-resolution observations with Δ𝑥 =𝐷/𝑀 (=11.04 km). The 2 2 2 co 𝛾 𝜎 𝑀∫𝑑𝑥𝐶 (𝑥 ) 𝛾 𝜎 𝐼 𝐿 2 𝑏 𝑏 𝑏 𝑏 𝑏 1 background error covariance 𝜎 𝐶 (𝑥) has the double-Gaussian form = = , 𝑏 2 2 2 2 2 2 2 −2 𝐷 Δ𝑥 with 𝐶 (𝑥) = 0.6 exp(−𝑥 /2𝐿 ) + 0.4 exp(−2𝑥 /𝐿 ), 𝜎 =5 m s co 𝑏 𝑏 and 𝐿=10 km. eTh analysis domain length is 𝐷 = 𝑁Δ𝑥 = 110.4 km with 𝑁 = 260 and Δ𝑥 = 0.24 km, and the number of coarse- where ∫𝑑𝑥 denotes the integration over the infinite space resolution observations is 𝑀=10 . 2 2 of 𝑥, ∑ ∑ ∫ 𝐶 (𝑥−𝑥 −𝑘𝐷) = ∑ ∫𝑑𝑥𝐶 (𝑥−𝑥 ) 𝑚 𝑘 𝑏 𝑚 𝑚 𝑏 𝑚 2 2 = ∑ ∫𝑑𝑥𝐶 (𝑥) = 𝑀∫𝑑𝑥𝐶 (𝑥) is used in the second to 𝑏 𝑏 last step, and 𝐼 ≡∫𝐶 (𝑥)𝑑𝑥/𝐿 is used with Δ𝑥 ≡ 1 𝑏 co 2 monotonically toward its asymptotic upper limit of 𝛾 𝜎 = 𝑏 𝑏 𝐷/𝑀 in the last step. For the double-Gaussian form of 2 −2 2 2 2 2 20 m s )as Δ𝑥 /𝐿 decreasesto0.5andthento Δ𝑥/𝐿 = 0.1 𝐶 (𝑥) = 0.6 exp(−𝑥 /2𝐿 ) + 0.4 exp(−2𝑥 /𝐿 ) used in (5) co 1/2 1/2 1/2 (or increases toward ∞),andthisdecrease(orincrease)ofthe of Xuet al.[8],wehave 𝐼 =(2)𝜋 (0.44/2 + 0.48/5 ). amplitude of spatial variation of 𝜎 (𝑥) with Δ𝑥 /𝐿 is closely co The analytically derived value in (5b) is very close to (slightly captured by the amplitude of spatial variation of the estimated larger than) the numerically computed value from (5a). With 2 2 𝜎 ∗(𝑥) as a function of Δ𝑥 /𝐿. co the domain-averaged value of ∑ Δ𝜎 (𝑥) adjusted from Δ𝜎 𝑚 𝑚 Using the estimated 𝜎 ∗(𝑥) in (7), the previously esti- 2 2 𝑎 to Δ𝜎 , Δ𝜎 (𝑥) canbeestimated by mated analysis error covariance matrix, denoted by A with its th element 𝐴 ≡𝜎 𝐶 (𝑥 −𝑥 ) obtained from the 𝑗𝑒𝑖 𝑎 𝑖 𝑗 2 2 2 2 Δ𝜎 (𝑥 ) = ∑Δ𝜎 (𝑥 ) −Δ𝜎 +Δ𝜎 . spectral formulations, can be modiefi d into A , A ,or A with (6) 𝑀 𝑚 𝑎 𝑏 𝑐 its th element given by eTh analysis error variance, 𝜎 (𝑥),isthenestimatedby 𝐴 ≡𝜎 ∗ (𝑥 )𝜎 ∗ (𝑥 )𝐶 (𝑥 −𝑥 ) 𝑗𝑎𝑖 𝑎 𝑖 𝑎 𝑗 𝑎 𝑖 𝑗 2 2 2 2 𝜎 (𝑥 ) ≈𝜎 (𝑥 ) ≡𝜎 −Δ𝜎 (𝑥 ) . (7) 𝑎 𝑎 𝑏 𝑀 =𝐴 (8a) 𝑗𝑒𝑖 As shown by the example in Figure 1 (in which 𝐷 = 110.4 km and 𝑀 =10so Δ𝑥 = 𝐷/𝑀 = 11.04 km +[𝜎 ∗ (𝑥 )𝜎 ∗ (𝑥 )−𝜎 ]𝐶 (𝑥 −𝑥 ), co 𝑎 𝑖 𝑎 𝑗 𝑎 𝑖 𝑗 is close to 𝐿=10 km), the estimated 𝜎 ∗(𝑥) in (7) has nearly the same spatial variation as the benchmark 𝜎 (𝑥) 2 2 𝐴 ≡𝐴 +{𝜎 ( + )−𝜎 }𝐶 (𝑥 −𝑥 ) 𝑗𝑏𝑖 𝑗𝑒𝑖 𝑎 𝑒 𝑎 𝑖 𝑗 that is computed precisely from (1b), although the amplitude 2 2 2 2 (8b) of spatial variation of 𝜎 (𝑥), defined by max 𝜎 (𝑥) − ∗ ∗ 𝑎 𝑎 𝑥 𝑗 2 2 2 𝑖 min 𝜎 (𝑥), is slightly smaller than that of the true 𝜎 (𝑥), ∗ =𝜎 ( + )𝐶 (𝑥 −𝑥 ), 𝑎 𝑎 𝑎 𝑎 𝑖 𝑗 2 2 2 2 defined by max 𝜎 (𝑥) − min 𝜎 (𝑥). As shown in Figure 2, the 𝑎 𝑎 amplitudeofspatialvariationofbenchmark 𝜎 (𝑥) decreases 𝑎 2 𝑖 2 (8c) or 𝐴 ≡𝐴 +{𝜎 ∗ ( + )−𝜎 }𝐶 (𝑥 −𝑥 ). 𝑗𝑐𝑖 𝑗𝑒𝑖 𝑏 𝑖 𝑗 rapidly to virtually zero and then exactly zero (or increases 𝑎 𝑒 2 2 2 2 (Ｇ /Ｍ ) 𝑖𝑗 𝑏𝑒 𝑏𝑠 𝑖𝑗 𝑏𝑒 𝑏𝑠 𝑑𝑥 𝑑𝑥 𝑏𝑠 𝑑𝑥 𝑏𝑠 𝑑𝑥 𝑑𝑥 𝑏𝑠 𝑑𝑥 4 Advances in Meteorology 4(b), 4(c), and 4(d), respectively. As shown, the deviation becomes increasingly small when A is modified successively to A , A ,and A . Note that the correction term in (8a) is 𝑎 𝑏 𝑐 15 𝐶 (𝑥 −𝑥 ) modulated by 𝜎 ∗(𝑥 )𝜎 ∗(𝑥 )−𝜎 .Thismodulation 𝑎 𝑖 𝑗 𝑎 𝑖 𝑎 𝑗 𝑒 has a chessboard structure, while the desired modulation revealed by the to-be-corrected deviation of A in Figure 4(a) has a banded structure (along the direction of 𝑥 + 𝑥 = 𝑖 𝑗 constant, perpendicular to the diagonal line). This explains why the correction term in (8a) offsets only a part of the deviation as revealed by the deviation of A in Figure 4(b). On the other hand, the correction term in (8b) is modulated 2 2 (𝑥 /2 + 𝑥 /2) − 𝜎 . This modulation not only retains the by 𝜎 ∗ 𝑎 𝑖 𝑗 𝑒 self-adjointness butalsohasthedesired banded structure, so thecorrectiontermin(8b)isanimprovement over thatin 0.5 1 1.5 2 2.5 3 x /L (8a), as shown by the deviation of A in Figure 4(c) versus that ＝Ｉ 𝑏 of A in Figure 4(b). However, as revealed by Figure 4(c), the Figure 2: Amplitude of spatial variation of benchmark 𝜎 (𝑥), deviation of A still has two significant maxima (or minima) 2 2 defined by max 𝜎 (𝑥) − min 𝜎 (𝑥), plotted by red solid curve as 𝑎 𝑎 along each band on the two sides of the diagonal line of 𝑥 = a function of Δ𝑥 /𝐿 . Amplitude of spatial variation of estimated co 2 2 2 𝑥 , while the to-be-corrected deviation of A in Figure 4(a) 𝑗 𝑒 𝜎 (𝑥), defined by max 𝜎 (𝑥) − min 𝜎 (𝑥) and plotted by blue ∗ ∗ ∗ 𝑎 𝑎 𝑎 has a single maximum (or minimum) along each band. This dotted curve as a function of Δ𝑥 /𝐿 . co implies that the function form of 𝐶 (𝑥 −𝑥 ) is not sufficiently 𝑎 𝑖 𝑗 wide for the correction. As a further improvement, this function form is widened to 𝐶 (𝑥 −𝑥 ) for the correction 𝑏 𝑖 𝑗 term in (8c), so the deviation of A in Figure 4(d) is further reduced from that of A in Figure 4(c). 150 𝑏 When an estimated A is used to update the background 100 error covariance in the second step for analyzing the high- resolution observations in the nested domain, the accuracy of the second-step analysis depends not only, to a certain extent, on the number of iterations performed by the minimization algorithm but also on the accuracy of the estimated A over the nested domain plus its extended vicinities within the −50 distance of 2𝐿 outside the nested domain. Here, 𝐿 is 𝑎 𝑎 −100 the decorrelation length scale of 𝐶 (𝑥) defined by 𝐿 ≡ 𝑎 𝑎 [−𝐶 (𝑥)/𝑑 𝐶 (𝑥)]| according to (4.3.10) of Daley [12], and 𝑎 𝑥 𝑎 𝑥=0 −150 𝐿 (=4.45kmfor thecaseinFigures 1and 3)canbeeasily computed as a by-product from the spectral formulation. −200 Over this extended nested domain, the relative error (RE) of the estimated A with respect to the benchmark A can be −200 −150 −100 −50 0 50 100 150 200 𝑒 measured by Figure 3: Structure of benchmark A plotted by color contours every 󵄩 󵄩 󵄩 󵄩 2 −2 I (A − A) I 󵄩 󵄩 𝑠 𝑒 𝑠 1m s for the case in Figure 1. 󵄩 󵄩 𝐹 RE (A )≡ , (9) 𝑒 󵄩 󵄩 󵄩 󵄩 I AI 󵄩 󵄩 󵄩 𝑠 𝑠 󵄩 The formulation in (8a) is conventional, as in (2.1) of Purser where I denotes the unit matrix in the subspace associated et al. [2] or originally (11) of Rutherford [11], in which the with the grid points in the extended nested domain and covariance is modified by applying 𝜎 ∗(𝑥) separately to each thus I (A − A)I (or I AI )isthe submatrixof A − A 𝑎 𝑠 𝑒 𝑠 𝑠 𝑠 𝑒 entry (indexed by 𝑖 and 𝑗)of 𝐶 (𝑥 −𝑥 ) to retain the (or A) associated only with the grid points in the extended 𝑎 𝑖 𝑗 self-adjointness. eTh second equation in (8a) shows that the nested domain and ‖()‖ denotes the Frobenius norm of conventional approach can be viewed alternatively as 𝐴 () den fi ed by the square root of the sum of the squared 𝑗𝑒𝑖 plus a correction term, the last term in (8a). Ideally, the absolute values of the elements of the matrix in () according correction term should completely offset the deviation of 𝐴 to (2.2–4) of Golub and Van Loan [13]. eTh REs of A , A , 𝑗𝑒𝑖 𝑎 𝑏 from the true covariance, but the correction term in (8a) and A can be measured by the same form of Frobenius norm offsets only a part of the deviation. ratioasthatdenfi edfor A in (9). The REs of A , A , A , 𝑒 𝑒 𝑎 𝑏 For the case in Figure 1, the benchmark analysis error and A are computed for the case in Figure 1 and listed in covariance matrix, denoted by A, is computed precisely from the first column of Table 1. As shown by the listed values, (1b) and is plotted in Figure 3, while the deviations of A , A , the RE becomes increasingly small when A is modified 𝑒 𝑎 𝑒 A ,and A from the benchmark A are shown in Figures 4(a), successively to A , A ,and A , and this is consistent with and 𝑏 𝑐 𝑎 𝑏 𝑐 2 2 (Ｇ /Ｍ ) Advances in Meteorology 5 200 200 150 150 100 100 50 50 j 0 j 0 −50 −50 −100 −100 −150 −150 −200 −200 −200 −150 −100 −50 0 50 100 150 200 −200 −150 −100 −50 0 50 100 150 200 i i (a) (b) 200 200 150 150 100 100 50 50 j 0 j 0 −50 −50 −100 −100 −150 −150 −200 −200 −200 −150 −100 −50 0 50 100 150 200 −200 −150 −100 −50 0 50 100 150 200 i i (c) (d) 2 −2 Figure 4: (a) Deviation of A from benchmark A in Figure 3 plotted by color contours every 0.5 m s .Deviationsof A , A ,and A from 𝑒 𝑎 𝑏 𝑐 2 −2 benchmark A are plotted by color contours every 0.2 m s in panels (b), (c), and (d), respectively. Here, A is the previously estimated analysis error covariance matrix with its th element 𝐴 ≡𝜎 𝐶 (𝑥 −𝑥 ) obtained from the spectral formulation, while A , A ,and A are the newly 𝑗𝑒𝑖 𝑒 𝑎 𝑖 𝑗 𝑎 𝑏 𝑐 modified estimates of A as shown in (8a), (8b), and (8c), respectively. also quantifies the successively reduced deviation shown in First, we consider the case of Δ𝑥 →0 with Δ𝑥 = co𝑚+ co𝑚− Figures 4(a)–4(d). Δ𝑥 (or Δ𝑥 →0 with Δ𝑥 =Δ𝑥 ). In this co co𝑚− co𝑚+ co case, the concerned 𝑚th observation collapses onto the same point with its right (or left) adjacent observation, that is, the 2.3. Nonuniform Coarse-Resolution Observations with Peri- (𝑚 + 1)th [or (𝑚 − 1)th] observation. eTh two collapsed odic Extension. Consider that the 𝑀 coarse-resolution obser- observations should be combined into one superobservation vations are now nonuniformly distributed in the analysis 2 2 with a reduced error variance from 𝜎 to 𝜎 /2.Theerror domain of length 𝐷 with periodic extension, so their averaged 𝑜 𝑜 variance reduction produced by this superobservation still resolution canbedenfi edby Δ𝑥 ≡𝐷/𝑀 .Thespacing co can be estimated by (3) but with of a concerned coarse-resolution observation, say the 𝑚th observation, from its right (or left) adjacent observation can be denoted by Δ𝑥 (or Δ𝑥 ). Now we can consider the 𝛾 = . (10a) co𝑚+ co𝑚− 2 2 (𝜎 +𝜎 /2) following two limiting cases. 𝑏 𝑖𝑗 6 Advances in Meteorology −1 Table 1: Entire-domain averaged RMS errors (in ms ) for the analysis increments obtained from SE, TEe, TEa, TEb, and TEc applied to the first set of innovations with periodic extension and consecutively increased 𝑛,where 𝑛 is the number of iterations. All the RMS errors are evaluated with respect to the benchmark analysis increment. The relative error (RE) of the estimated analysis error covariance for updating the background error covariance in the second step of the two-step analysis is listed with the experiment name in the first column for each two-step experiment. Experiment 𝑛 = 20 𝑛 = 50 𝑛 = 100 Final SE 0.671 0.365 0.187 0.013 at 𝑛 = 481 TEe with RE(A ) = 0.229 0.171 0.150 0.142 0.135 at 𝑛 = 210 TEa with RE(A ) = 0.156 0.169 0.142 0.144 0.144 at 𝑛 = 116 TEb with RE(A ) = 0.101 0.147 0.098 0.090 at 𝑛=67 TEc with RE(A ) = 0.042 0.145 0.063 0.062 0.032 at 𝑛 = 176 On the other hand, without super-Obbing, the error variance the result derived from (10c)-(10d). Note also that 𝛽 =1 reduction produced by the two collapsed observations will be for Δ𝑥 =0 and Δ𝑥 =Δ𝑥 (or Δ𝑥 =0 and co𝑚+ co𝑚− co co𝑚− overestimated by (3) with Δ𝑥 =Δ𝑥 ), so the adjusted error variance is 𝜎 = co𝑚+ co 𝑜𝑚 2 2 𝜎 +𝜎 which recovers the result derived from (10a)-(10b). 2 2 𝑜 𝑏 2𝜎 𝜎 𝑏 𝑏 Clearly, for Δ𝑥 = Δ𝑥 = Δ𝑥 , 𝛽 =0,so 𝜎 is not 𝛾 = = . (10b) 𝑏 co𝑚− co𝑚+ co 𝑚 𝑜 2 2 2 2 (𝜎 +𝜎 ) (𝜎 /2 + 𝜎 /2) 𝑜 𝑜 𝑏 𝑏 adjusted which recovers the result for uniformly distributed coarse-resolution observations. By comparing (10b) with (10a), it is easy to see that this 2 2 2 The above results suggest that 𝛾 =𝜎 /(𝜎 +𝜎 ) should 𝑏 𝑏 𝑏 𝑜 overestimation can be corrected if the error variance is 2 2 2 be modiefi d into inflated from 𝜎 to 𝜎 +𝜎 for each of the two collapsed 𝑜 𝑜 𝑏 observations. 𝛾 = Then, we consider the case of Δ𝑥 →0 and Δ𝑥 → co𝑚+ co𝑚− 2 2 2 (𝜎 +𝛽 𝜎 +𝜎 ) 𝑏 𝑏 𝑜 (11b) 0.Inthiscase,theconcerned 𝑚th observation collapses with its two adjacent observations, that is, the (𝑚 + 1)th and (𝑚 − 2 for the den fi ition of Δ𝜎 (𝑥 ) in (3) . 1)thobservations.Thethreecollapsedobservationsshould be combined into one superobservation with a reduced error This modification can improve the similarity of the spatial 2 2 variance from 𝜎 to 𝜎 /3. eTh error variance reduction 𝑜 𝑜 variation of ∑ Δ𝜎 (𝑥) to that of the true error variance produced by this superobservation still can be estimated by 2 2 2 reduction, denoted by Δ𝜎 (𝑥) ≡ 𝜎 −𝜎 (𝑥),butthe 𝑏 𝑎 (3) but with 2 2 maximum (or minimum) of ∑ Δ𝜎 (𝑥), denoted by Δ𝜎 𝑚 𝑚 emx 2 2 2 𝜎 (or Δ𝜎 ), is usually not very close to that of Δ𝜎 (𝑥). 𝑏 emn 𝛾 = . (10c) 2 eTh maximum (or minimum) of Δ𝜎 (𝑥) can be closely (𝜎 +𝜎 /3) 𝑏 𝑜 2 2 estimated by Δ𝜎 (or Δ𝜎 ), the maximum (or minimum) mx mn On the other hand, without super-Obbing, the error variance 2 of Δ𝜎 (𝑥) computed by (6) for uniform coarse-resolution reduction produced by the three collapsed observations will observations but with Δ𝑥 decreased to Δ𝑥 (or increased co omn be overestimated by (3) with to Δ𝑥 ), where Δ𝑥 (or Δ𝑥 )istheminimum(ormax- omx omn omx 2 2 imum) spacing between two adjacent observations among all 3𝜎 𝜎 𝑏 𝑏 𝛾 = = . (10d) nonuniformly distributed coarse-resolution observations in 2 2 2 2 (𝜎 +𝜎 ) (𝜎 /3 + 𝜎 /3) 𝑜 𝑜 𝑏 𝑏 the one-dimension analysis domain. By adjusting Δ𝜎 to emx 2 2 2 Δ𝜎 and Δ𝜎 to Δ𝜎 ,theerrorvariancereduction can By comparing (10d) with (10c), it is easy to see that this mx emn mn be estimated by overestimation can be corrected if the error variance is 2 2 2 inflated from 𝜎 to 𝜎 +2𝜎 for each of the three collapsed 𝑜 𝑜 𝑏 2 2 2 2 observations. Δ𝜎 𝑥 =𝐹 𝑥 ≡[ ∑Δ𝜎 𝑥 −Δ𝜎 ]𝜌 + Δ𝜎 , (12a) ( ) ( ) ( ) 𝑀 𝑚 emn mn Basedonthe aboveanalyses,when theerror variance 𝑚 reduction produced by the 𝑚th observation is estimated by 2 2 2 2 where 𝜌=[Δ𝜎 −Δ𝜎 ]/[Δ𝜎 −Δ𝜎 ]. (3), the error variance should be adjusted for this observation mx mn emx emn eTh analysis error variance is then estimated by 𝜎 (𝑥) ≈ unless Δ𝑥 =Δ𝑥 =Δ𝑥 .Inparticular, itserror co𝑚+ co𝑚− co 𝑎 2 2 2 2 2 2 2 2 𝜎 (𝑥) ≡ 𝜎 −Δ𝜎 (𝑥) as in (7), except that Δ𝜎 (𝑥) is variance can be adjusted from 𝜎 to 𝜎 =𝜎 +𝛽 𝜎 with ∗ 𝑎 𝑏 𝑀 𝑀 𝑜 𝑜𝑚 𝑜 𝑚 𝑏 computed by (12a) instead of (6). As shown by the example 𝛽 given empirically by in Figure 5, the estimated 𝜎 (𝑥) captures closely not only 2 2 2 [𝐶 (Δ𝑥 )+𝐶 (Δ𝑥 )−2𝐶 (Δ𝑥 )] 𝑏 co𝑚+ 𝑏 co𝑚− 𝑏 co the maximum and minimum but also the spatial variation (11a) 𝛽 = . 2 of the benchmark 𝜎 (𝑥) computed from (1b). Using this [1 − 𝐶 (Δ𝑥 )] co estimated 𝜎 ∗(𝑥),the previously estimated A from the 𝑎 𝑒 Note that 𝛽 =2 for Δ𝑥 =Δ𝑥 =0, so the spectral formulation can be modiefi d into A , A ,or A with 𝑚 co𝑚+ co𝑚− 𝑎 𝑏 𝑐 2 2 2 adjusted error variance is 𝜎 =𝜎 +2𝜎 which recovers its th element given by the same formulation as shown in 𝑜𝑚 𝑜 𝑏 𝑖𝑗 𝑏𝑎 𝑏𝑎 𝑏𝑎 Advances in Meteorology 7 small when A is modified successively to A , A ,and A , 𝑒 𝑎 𝑏 𝑐 which quantifies the successively reduced deviation shown in Figures 7(a)–7(d). 2.4. Nonuniform Coarse-Resolution Observations without Periodic Extension. Consider that the 𝑀 coarse-resolution observations are still nonuniformly distributed in the one- dimensional analysis domain of length 𝐷 but without peri- odic extension. In this case, their produced error variance reduction Δ𝜎 (𝑥) still can be estimated by (12a) except for 6 𝑀 the following three modica fi tions. 4 2 (i) eTh maximum (or minimum) of ∑ Δ𝜎 (𝑥),thatis, 2 2 Δ𝜎 (or Δ𝜎 ), should be found in the interior domain emx emn −60 −40 −20 04 20 0 60 between the leftist and rightist observation points. x (km) (ii) For the leftist (or rightist) observation that has only one adjacent observation to its right (or le)ft in the  (x)  2 2 one-dimensional analysis domain, its error variance is still  (x) ∗  a o 2 2 2 2 adjusted from 𝜎 to 𝜎 =𝜎 +𝛽 𝜎 but 𝛽 is calculated 𝑜 𝑜𝑚 𝑜 𝑚 𝑏 𝑚 2 2 Figure 5: As in Figure 1 but for 𝑀 (=10) nonuniformly distributed by setting 𝐶 (Δ𝑥 )=0 [or 𝐶 (Δ𝑥 )=0]in(11a)for co𝑚− co𝑚+ 𝑏 𝑏 coarse-resolution observations. calculating 𝛾 in (11b). (iii) Note from (12a) that ∑ Δ𝜎 (𝑥)→0 and thus 2 2 𝐹(𝑥)→Δ𝜎 −𝜌Δ𝜎 as 𝑥 moves outward far away from the mn emn leftist (or rightist) measurement point and thus far away from 2 2 all the observations points. In this case, if Δ𝜎 −𝜌Δ𝜎 <0 mn emn (as for the case in this section), then Δ𝜎 (𝑥) estimated by 𝐹(𝑥) in (12a) may become unrealistically negative as 𝑥 moves outward beyond the leftist (or rightist) measurement point, denoted by 𝑥 . To avoid this problem, (12a) is modified into 𝑚𝑏 j 0 Δ𝜎 𝑥 =𝐹(𝑥 )−[𝐹 (𝑥 )−𝐹 𝑥 ]𝑅 ( ) ( ) 𝑀 𝑚𝑏 𝑚𝑏 1 (12b) −50 for 𝑥 beyond 𝑥 , 𝑚𝑏 −100 where 𝑅 is a factor defined by −150 𝐹(𝑥 ) 𝑚𝑏 −200 𝑅 = min {1, }. (13) 2 2 [𝐹 (𝑥 )+𝜌Δ𝜎 −Δ𝜎 ] 𝑚𝑏 emn mn −200 −150 −100 −50 0 50 100 150 200 2 2 It is easy to see from (12b) that for Δ𝜎 −𝜌Δ𝜎 <0 and mn emn thus 𝑅 =1, Δ𝜎 (𝑥) = 𝐹(𝑥 ) − [𝐹(𝑥 ) − 𝐹(𝑥)]𝑅 →0 as 1 𝑀 𝑚𝑏 𝑚𝑏 1 Figure 6: As in Figure 3 but for the case in Figure 5. |𝑥| → ∞,sothe estimated Δ𝜎 (𝑥) in (12b) can never become unrealistically negative. eTh analysis error variance is estimated by 𝜎 (𝑥) ≈ (8a), (8b), or (8c). For the case in Figure 5, the benchmark 2 2 2 2 𝜎 ∗(𝑥) ≡ 𝜎 −Δ𝜎 (𝑥) as in (7), except that Δ𝜎 (𝑥) is A isplottedinFigure6,whilethedeviationsof A , A , A , 𝑎 𝑏 𝑀 𝑀 𝑒 𝑎 𝑏 computed by (12a) [or (12b)] for 𝑥 within (or beyond) the and A from the benchmark A are shown in Figures 7(a), interior domain. As shown by the example in Figure 8, the 7(b), 7(c), and 7(d), respectively. As shown, the deviation estimated 𝜎 ∗(𝑥) captures closely the spatial variation of the becomes increasingly small when the estimated analysis error covariance matrix is modified successively to A , A ,and A . benchmark 𝜎 (𝑥) not only within but also beyond the interior 𝑎 𝑏 𝑐 As explained in Section 2.2, the accuracy of the second- domain. Using this estimated 𝜎 ∗(𝑥), A can be modified 𝑎 𝑒 step analysis depends on the accuracy of the estimated A over into A , A ,or A with its th element given by the same 𝑎 𝑏 𝑐 the extended nested domain (i.e., the nested domain plus its formulation as shown in (8a), (8b), or (8c). For the case in extended vicinities within the distance of 2𝐿 on each side Figure 8, the benchmark A (not shown) has the same interior outside the nested domain), while the latter can be measured structure (for interior grid points 𝑖 and 𝑗)asthatforthecase by the smallness of the RE of the estimated A with respect to with periodic extension in Figure 6, but significant differences the benchmark A, as defined for A in (9). The REs of A , A , are seen in the following two aspects around the four corners 𝑒 𝑒 𝑎 A ,and A computed for the case in Figure 5 are listed in the (similar to those seen from Figures 7(a) and 11(a) of Xu et 𝑏 𝑐 first column of Table 2. As listed, the RE becomes increasingly al. [8]). (i) eTh element value becomes large toward the two 2 2 (Ｇ /Ｍ ) 𝑖𝑗 8 Advances in Meteorology 200 200 150 150 100 100 50 50 j 0 j 0 −50 −50 −100 −100 −150 −150 −200 −200 −200 −150 −100 −50 0 50 100 150 200 −200 −150 −100 −50 0 50 100 150 200 i i (a) (b) 200 200 150 150 100 100 50 50 j 0 j 0 −50 −50 −100 −100 −150 −150 −200 −200 −200 −150 −100 −50 0 50 100 150 200 −200 −150 −100 −50 0 50 100 150 200 i i (c) (d) Figure 7: As in Figure 4 but for the case in Figure 5. cornersalong thediagonalline(whichisconsistentwiththe 3. Analysis Error Variance Formulations for increased analysis error variance toward the two ends of the Two-Dimensional Cases analysis domain as shown in Figure 8 in comparison with 3.1. Error Variance Reduction Produced by a Single Obser- that in Figure 5). (ii) The element value becomes virtually vation. For a single observation, say, at x ≡(𝑥 ,𝑦 ) in 𝑚 𝑚 𝑚 zero toward the two off-diagonal corners (because there is the two-dimensional space of x =(𝑥,𝑦) ,the inversematrix no periodic extension). The deviations of A , A , A ,and A 𝑒 𝑎 𝑏 𝑐 −1 −1 𝑇 2 2 from the benchmark A are shown in Figures 9(a), 9(b), 9(c), (HBH + R) in (1b) also reduces to (𝜎 +𝜎 ) ,sothe 𝑖th 𝑏 𝑜 diagonal element of A is given by the same formulation as and 9(d), respectively, for the case in Figure 8. As shown, in (2) except that 𝑥 (or 𝑥 )isreplacedby x (or x ). Here, the deviation becomes increasingly small when the estimated 𝑖 𝑚 𝑖 𝑚 analysis error covariance matrix is modiefi d successively to x denotes the 𝑖th point in the discretized analysis space 𝑅 A , A ,and A .TheREs of A , A , A ,and A are listed in the with 𝑁=𝑁 𝑁 , 𝑁 (or 𝑁 )isthenumberof analysis grid 𝑎 𝑏 𝑐 𝑒 𝑎 𝑏 𝑐 𝑥 𝑦 𝑥 𝑦 first column of Table 3. As listed, the RE becomes increasingly points along the 𝑥 (or 𝑦) direction in the two-dimensional small when A is modified successively to A , A ,and A , analysis domain. eTh length (or width) of the analysis domain 𝑒 𝑎 𝑏 𝑐 which quantifies the successively reduced deviation shown in is 𝐷 =𝑁 Δ𝑥 (or 𝐷 =𝑁 Δ𝑦) and is assumed to be much 𝑥 𝑥 𝑦 𝑦 Figures 9(a)–9(d). larger than the background error decorrelation length scale Advances in Meteorology 9 spectral formulation for two-dimensional cases in Section 2.3 of Xu et al. [8]. The domain-averaged value of ∑ Δ𝜎 (x) can be com- 𝑚 𝑚 puted by 12 2 ∬ 𝑑x ∑ Δ𝜎 (x) 𝑚 𝑚 2 𝐷 Δ𝜎 ≡ (𝐷 𝐷 ) 𝑥 𝑦 2 2 𝛾 𝜎 ∑ ∬ 𝑑x 𝐶 (x − x ) 𝑏 𝑚 𝑚 𝑏 𝑏 (15a) (𝐷 𝐷 ) 𝑥 𝑦 2 2 𝛾 𝜎 ∑ ∑ 𝐶 (x − x ) 𝑏 𝑏 𝑚 𝑖 𝑏 𝑖 𝑚 ≈ , −60 −40 −20 020 40 60 x (km) where ∬ 𝑑x denotes the integration over the two- 2 2  (x)  a e dimensional analysis domain, ∑ denotes the summation 2 2  (x)  a o over 𝑖 for the 𝑁 grid points, and 𝐷 𝐷 =𝑁 Δ𝑥𝑁 Δ𝑦 = 𝑥 𝑦 𝑥 𝑦 𝑁Δ𝑥Δ𝑦 is used in the last step. By extending 𝐶 (x − x ) Figure 8: As in Figure 5 but without periodic extension. 𝑚 with the analysis domain periodically in both the 𝑥 and 𝑦 directions, Δ𝜎 canbeestimated analytically as follows: 𝐿 in x,where Δ𝑥 (or Δ𝑦) is the grid spacing in the 𝑥 (or 𝑦) ∬ 𝑑x ∑ Δ𝜎 (x) 2 𝑚 Δ𝜎 ≡ direction and Δ𝑥 = Δ𝑦 is assumed for simplicity. (𝐷 𝐷 ) 𝑥 𝑦 Since 𝐶 (x) is a continuous function of x, the aforemen- 2 2 tioned formulation for the 𝑖th diagonal element of A can 𝛾 𝜎 ∑ ∑ ∑ ∬ 𝑦𝐶𝑑 (𝑥 − 𝑥 −𝑝𝐷 ,𝑦 − 𝑦 −𝑞𝐷 ) 𝑏 𝑏 𝑚 𝑝 𝑞 𝑏 𝑚 𝑥 𝑚 𝑦 2 2 2 = (15b) be written into 𝜎 (x)≡𝜎 −Δ𝜎 (x) also as a continuous 𝑚 𝑏 𝑚 (𝐷 𝐷 ) 𝑥 𝑦 function of x,where 2 2 2 2 𝛾 𝜎 𝑀∬𝑑 x𝐶 (x) 𝛾 𝜎 𝐼 𝐿 𝑏 𝑏 𝑏 𝑏 2 2 = = , (14) Δ𝜎 (x) ≡𝛾 [𝜎 𝐶 (x − x )] Δ𝑥 (𝐷 𝐷 ) 𝑏 𝑏 𝑏 𝑚 co 𝑚 𝑥 𝑦 is the error variance reduction produced by analyzing a single where ∬𝑑 x =∬𝑥𝑑𝑦𝑑 denotes the integration over the observation at x = x . This reduction decreases rapidly and 2 entire space of x, ∑ ∑ ∑ ∬ 𝐶 (𝑥 − 𝑥 −𝑝𝐷 ,𝑦 − 𝑚 𝑥 𝑚 𝑝 𝑞 𝑏 2 2 𝐷 becomes much smaller than it peak value of 𝛾 𝜎 𝐶 at x = x 2 2 𝑏 𝑏 𝑏 𝑚 𝑦 −𝑞𝐷 )=∑ ∬𝑑 x𝐶 (x − x )=∑ ∬𝑑 x𝐶 (x)= 𝑚 𝑦 𝑚 𝑏 𝑚 𝑚 𝑏 as |x − x | increases to 𝐿 and beyond. 𝑀∬𝑑 x𝐶 (x) is used in the second to last step, and 2 2 𝐼 ≡∬𝑑 x𝐶 (x)/𝐿 is used with Δ𝑥 ≡𝐷 𝐷 /𝑀 in 2 𝑏 co 𝑥 𝑦 3.2. Uniform Coarse-Resolution Observations with Periodic the last step. For the double-Gaussian form of 𝐶 (x)= Extension. Consider that there are 𝑀 coarse-resolution 2 2 2 2 0.6 exp(−|x| /2𝐿 ) + 0.4 exp(−2|x| /𝐿 ) used in Section 4 of observations uniformly distributed in the above analysis Xuet al.[8],wehave 𝐼 = 2𝜋(0.2 + 0.48/5) . eTh derived value domain of length 𝐷 and width 𝐷 with periodic extension 𝑥 𝑦 in (15b) is very close to the numerically computed value from 1/2 1/2 along 𝑥 and 𝑦,sotheir resolution is Δ𝑥 ≡(𝐷 𝐷 ) /𝑀 , co 𝑥 𝑦 (15a). where 𝑀=𝑀 𝑀 , 𝑀 (or 𝑀 ) denotes the number of 2 𝑥 𝑦 𝑥 𝑦 With the domain-averaged value adjusted from Δ𝜎 to observations uniformly distributed along the 𝑥 (or 𝑦)direc- 2 2 Δ𝜎 , Δ𝜎 (x) can be estimated by the same formulation as in tion in the two-dimensional analysis domain, and 𝐷 /𝑀 = 𝑥 𝑥 (6) except that 𝑥 is replaced by x.Theanalysiserror variance 𝐷 /𝑀 is assumed (so Δ𝑥 =𝐷 /𝑀 =𝐷 /𝑀 ). In this 𝑦 𝑦 co 𝑥 𝑥 𝑦 𝑦 is then estimated by case, as explained for the one-dimensional case in Section 2.2, the error variance reduction produced by each observation 2 2 2 2 𝜎 (x) ≈𝜎 (x) ≡𝜎 −Δ𝜎 (x) . (16) 𝑎 𝑎 𝑏 𝑀 can be considered as an additional reduction to the reduction produced by its neighboring observations. This additional As shown by the example in Figure 10, the estimated 𝜎 ∗(x) in reduction is smaller than the reduction produced by a single (16) is very close to the benchmark 𝜎 (x) computed precisely observation, so the error variance reduction produced by from (1b), and the deviation of 𝜎 (x) from the benchmark analyzing the 𝑀 coarse-resolution observations is bounded 𝑎 2 2 −2 𝜎 (x) is within (−0.21, 0.35) m s . On the other hand, the above by ∑ Δ𝜎 (x),which is similartothatfortheone- 𝑎 2 2 −2 constant analysis error variance (𝜎 =6.7m s )estimated by dimensionalcasein(4).Forthesamereasonasexplained for the one-dimensional case in (4), this implies that the the spectral formulation deviates from the benchmark 𝜎 (x) 2 −2 domain-averaged value of ∑ Δ𝜎 (x) is larger than the true widely from −1.91 to 2.22 m s . 2 2 2 2 averaged reduction estimated by Δ𝜎 ≡𝜎 −𝜎 ,where 𝜎 is Using the estimated 𝜎 ∗(x) in (16), the previously esti- 𝑏 𝑒 𝑒 the domain-averaged analysis error variance estimated by the mated analysis error covariance matrix, denoted by A with 2 2 (Ｇ /Ｍ ) 𝑏𝑒 𝑏𝑒 𝑏𝑠 𝑑𝑥 𝑏𝑠 𝑏𝑠 𝑏𝑠 10 Advances in Meteorology 200 200 150 150 100 100 50 50 j 0 j 0 −50 −50 −100 −100 −150 −150 −200 −200 −200 −150 −100 −50 0 50 100 150 200 −200 −150 −100 −50 0 50 100 150 200 i i (a) (b) 200 200 150 150 100 100 50 50 j 0 j 0 −50 −50 −100 −100 −150 −150 −200 −200 −200 −150 −100 −50 0 50 100 150 200 −200 −150 −100 −50 0 50 100 150 200 i i (c) (d) Figure 9: As in Figure 7 but for the case in Figure 8. its th element 𝐴 ≡𝜎 𝐶 (x − x ) obtained from the benchmark A computed precisely from (1b) can be den fi ed 𝑗𝑒𝑖 𝑒 𝑎 𝑖 𝑗 spectral formulation, can be modified into A , A ,or A in the same way as that for A in (9), except that the extended 𝑎 𝑏 𝑐 𝑒 with its th element given by the same formulation as in nested domain is two-dimensional here. The REs of A , A , 𝑒 𝑎 (8a), (8b), or (8c) except that 𝑥 (or 𝑥 )isreplacedby A ,and A computed forthecase inFigure 10 arelistedinthe 𝑖 𝑗 𝑏 𝑐 x (or x ). Again, as explained in Section 2.2 but for the first column of Table 4. As listed, the RE becomes increasingly 𝑖 𝑗 two-dimensional case here, the accuracy of the second-step small when A is modified successively to A , A ,and 𝑒 𝑎 𝑏 analysis depends on the accuracy of the estimated A over A . the extended nested domain, that is, the nested domain plus its extended vicinities within the distance of 2𝐿 outside the 3.3. Nonuniform Coarse-Resolution Observations with Peri- odic Extension. Consider that the 𝑀 coarse-resolution obser- nested domain. Here, 𝐿 is the decorrelation length scale 2 2 vations are now nonuniformly distributed in the analy- of 𝐶 (x) defined by 𝐿 ≡[−2𝐶 (x)/∇ 𝐶 (x)]| according 𝑎 𝑎 𝑎 x=0 sis domain of length 𝐷 and width 𝐷 with periodic to (4.3.12) of Daley [12], and 𝐿 (=4.52 km for the case in 𝑥 𝑦 extension, so their averaged resolution can be defined by Figure 10) can be easily computed as a by-product from the 1/2 1/2 spectral formulation. Over this extended nested domain, the Δ𝑥 ≡(𝐷 𝐷 ) /𝑀 . eTh spacing of a concerned coarse- co 𝑥 𝑦 relative error (RE) of each estimated A with respect to the resolution observation, say the 𝑚th observation, from its 𝑖𝑗 𝑖𝑗 Advances in Meteorology 11 30 30 20 20 10 10 0 0 −10 −10 −20 −20 −30 −30 −60 −40 −20 020 40 60 −60 −40 −20 020 40 60 x (km) x (km) (a) (b) 2 2 −2 2 Figure 10: (a) Benchmark analysis error variance 𝜎 (x) plotted by red contours of every 2 m s and estimated analysis error variance 𝜎 ∗ (x) 𝑎 𝑎 2 2 2 −2 in (16) plotted by blue contours. (b) Deviation of 𝜎 (x) from 𝜎 (x) plotted by colored contours of every 0.2 m s . eTh black + signs show the 𝑎 𝑎 locations of 𝑀 (= M M =12 × 6) uniformly distributed coarse-resolution observations with periodic extension along 𝑥 and 𝑦.Thecoarse- x y 2 2 2 −2 resolution observation resolution is Δ𝑥 =𝐷 /𝑀 = 𝐷 /𝑀 = 10 km, and the observation error variance is 𝜎 = 2.5 m s . eTh background co 𝑥 𝑥 𝑦 𝑦 2 2 2 2 2 2 2 2 −2 error covariance 𝜎 𝐶 (x) has the double-Gaussian form with 𝐶 (x) = 0.6 exp(−|x| /2𝐿 ) + 0.4 exp(−2|x| /𝐿 ), 𝜎 =5 m s ,and 𝐿=10 km. 𝑏 𝑏 𝑏 𝑏 The analysis domain length (or width) is 𝐷 =𝑁 Δ𝑥 = 120 km (or 𝐷 =𝑁 Δ𝑦=60 km), and the number of coarse-resolution observations 𝑥 𝑥 𝑦 𝑦 is M = M M =12 × 6. x y 𝑘th adjacent observation (among the total 4 adjacent obser- where ∑ denotes the summation over 𝑘 for the four adjacent vations), can be denoted by Δ𝑥 . Now we can consider observations nearest to the concerned 𝑚th observation. With co𝑚𝑘 2 2 2 the limiting case of Δ𝑥 →0 for 𝐾 (≤4) adjacent 𝛽 givenby(18a),theadjusted 𝜎 =𝜎 +𝛽 𝜎 recovers co𝑚𝑘 𝑚 𝑜𝑚 𝑜 𝑚 𝑏 observations with Δ𝑥 =Δ𝑥 for the remaining 4−𝐾 not only the inflated observation error variance derived above co𝑚𝑘 co (≥0) adjacent observations. In this case, the concerned 𝑚th for each limiting case [with Δ𝑥 →0 for 𝑘 = 1,2,...,𝐾 co𝑘 observationcollapses ontothesamepointwithits 𝐾 adjacent (≤4) and Δ𝑥 =Δ𝑥 for the remaining 4−𝐾 (≥0) co𝑘 co observations. The 𝐾+1 collapsed observations should be observations] but also the original observation error variance combined into one superobservation with a reduced error 𝜎 for uniformly distributed coarse-resolution observations. 2 2 2 2 2 variance from 𝜎 to 𝜎 /(𝐾+1) . eTh error variance reduction 𝑜 𝑜 The above results suggest that 𝛾 =𝜎 /(𝜎 +𝜎 ) should 𝑏 𝑏 𝑜 produced by this superobservation still can be estimated by be modiefi d into (14) but with 𝛾 = 2 𝑚 2 2 2 (𝜎 +𝛽 𝜎 +𝜎 ) 𝑚 𝑜 𝑏 𝑏 (18b) 𝛾 = . (17a) 2 2 [𝜎 +𝜎 / (𝐾+1 )] 𝑏 𝑜 2 for the den fi ition of Δ𝜎 (x) in (14) . On the other hand, without super-Obbing, the error variance This modification can improve the similarity of the spatial reduction produced by the 𝐾+1 collapsed observations will variation of ∑ Δ𝜎 (x) to that of the true error variance 𝑚 𝑚 be overestimated by (14) with 2 2 2 reduction, denoted by Δ𝜎 (x)≡𝜎 −𝜎 (x),butthe 𝑏 𝑎 2 2 2 2 maximum (or minimum) of ∑ Δ𝜎 (x), denoted by Δ𝜎 (𝐾+1 ) 𝜎 𝜎 𝑚 emx 𝑏 𝑏 2 2 𝛾 = = . (17b) 𝑏 (or Δ𝜎 ), is usually not very close to that of Δ𝜎 (x).The 2 2 2 2 emn (𝜎 +𝜎 ) [𝜎 / (𝐾+1 ) +𝜎 / (𝐾+1 )] 𝑏 𝑜 𝑏 𝑜 2 maximum (or minimum) of Δ𝜎 (x) canbeestimated by 2 2 2 Δ𝜎 (or Δ𝜎 ), the maximum (or minimum) of Δ𝜎 (x) By comparing (17b) with (17a), it is easy to see that this mx mn 𝑀 computed for uniformly distributed observations but with overestimation can be corrected if the error variance is 2 2 2 2 Δ𝑥 decreased to Δ𝑥 (or increased to Δ𝑥 ), where inflated from 𝜎 to 𝜎 =𝜎 +𝐾𝜎 for each of the (𝐾 + 1) co omn omx 𝑜 𝑜𝑚 𝑜 𝑏 Δ𝑥 (or Δ𝑥 ) is the minimum (or maximum) spacing collapsed observations. omn omx of adjacent observations among all nonuniformly distributed Basedonthe aboveanalyses,when theerror variance coarse-resolution observations in the two-dimension analysis reduction produced by the concerned 𝑚th observation is domain. Specifically, Δ𝑥 (or Δ𝑥 )isestimatedby estimatedby(14),theerror variance should be adjusted for omn omx min (∑ |x − x |)/𝐾 with 𝐾=2 and Δ𝑥 is estimated this observation unless Δ𝑥 =Δ𝑥 for 𝑘 = 1,2,3 ,and 4. 𝑚 𝑘 𝑚 𝑚𝑘 omx co𝑚𝑘 co 2 2 2 2 by max (∑ |x − x |)/𝐾 with 𝐾=4 ,where x denotes In particular, 𝜎 canbeadjustedto 𝜎 =𝜎 +𝛽 𝜎 with 𝛽 𝑚 𝑘 𝑚 𝑚𝑘 𝑚 𝑜 𝑜𝑚 𝑜 𝑚 𝑏 𝑚 the 𝑚th observation point, x denotes the observation point given empirically by 𝑚𝑘 that is 𝑘th nearest to x ,min (or max ) denotes the 𝑚 m m 2 2 minimum (or maximum) over index 𝑚 for all the coarse- [∑ 𝐶 (Δ𝑥 )−4𝐶 (Δ𝑥 )] 𝑘 𝑏 co𝑚𝑘 𝑏 co (18a) 𝛽 = , resolution observation points in the two-dimension analysis [1 − 𝐶 (Δ𝑥 )] co 𝑏 domain, ∑ denotes the summation over 𝑘 from 1 to 𝐾,and y (km) y (km) 𝑏𝑎 𝑏𝑎 𝑏𝑎 12 Advances in Meteorology 30 30 20 20 10 10 0 0 −10 −10 −20 −20 −30 −30 −60 −40 −20 0 20 40 60 −60 −40 −20 020 40 60 x (km) x (km) (a) (b) Figure 11: As in Figure 10 but for the second set of innovations with nonuniformly distributed coarse-resolution observations, and the colored 2 −2 2 2 contours are plotted every 1 m s for the deviation of 𝜎 (x) from 𝜎 (x) in panel (b). 𝑎 𝑎 𝐾 is the total number of adjacent observation points (nearest where 𝑀 (or 𝑀 ) is estimated by the nearest integer to 𝑥 𝑦 to x ) used for estimating Δ𝑥 (with 𝐾=2 )or Δ𝑥 (or 𝐷 /Δ𝑥 (or 𝐷 /Δ𝑥 ). eTh total number of near-boundary 𝑚 omn omx 𝑥 co 𝑦 co 2 2 2 2 observations is thus given by 2(𝑀 +𝑀 )−8. To identify 𝐾=4 ). By adjusting Δ𝜎 to Δ𝜎 and Δ𝜎 to Δ𝜎 ,the 𝑥 𝑦 emx mx emn mn these near-boundary observations, we need to divide the 2D error variance reduction can be estimated by domain uniformly along the 𝑥-direction and 𝑦-direction into 𝑀 𝑀 boxes, so there are 2(𝑀 +𝑀 )−8 boundary boxes 2 2 2 2 𝑥 𝑦 𝑥 𝑦 Δ𝜎 (x) =𝐹 (x) ≡[ ∑Δ𝜎 (x) −Δ𝜎 ]𝜌 + Δ𝜎 , (19a) 𝑀 𝑚 emn mn (not including the four corner boxes). If a boundary box containsnocoarse-resolutionobservation,thenitisanempty 2 2 2 2 box and should be substituted by its adjacent interior box (as where 𝜌=[Δ𝜎 −Δ𝜎 ]/[Δ𝜎 −Δ𝜎 ]. mx mn emx emn 2 a substituted boundary box). From each nonempty boundary eTh analysis error variance is then estimated by 𝜎 (x)≈ box (including substituted boundary box), we can n fi d one 2 2 2 2 𝜎 (x)≡𝜎 −Δ𝜎 (x) as in (16), except that Δ𝜎 (x) is 𝑎 𝑏 𝑀 𝑀 near-boundary observation that is nearest to the associated computed by (19a). As shown by the example in Figure 11, boundary. A closed loop of observation boundary can be 2 2 the estimated 𝜎 ∗(x) is fairly close to the benchmark 𝜎 (x), 𝑎 𝑎 constructed by piece-wise linear segments with every two 2 2 and the deviation of 𝜎 ∗(x) from the benchmark 𝜎 (x)is 𝑎 𝑎 neighboring near-boundary observation points connected by 2 −2 within (−2.40, 4.20) m s . On the other hand, the constant alinearsegmentandwitheachnear-cornerobservationpoint 2 2 −2 analysis error variance (𝜎 = 6.7 m s )estimated bythe connectedbyalinear segmenttoeachofits twoneighboring spectral formulation deviates from the benchmark 𝜎 (x) near-boundary observation points. 2 −2 After the above preparations, the error variance reduction widely from −9.98 to 3.83 m s .Using this estimated 𝜎 ∗(x), Δ𝜎 (x) can be estimated by (19a) with the following three the previously estimated A from the spectral formulation can modifications: be modiefi d into A , A ,or A with its th element given 𝑎 𝑏 𝑐 (i) eTh maximum (or minimum) of ∑ Δ𝜎 (x),thatis, by thesametwo-dimensional versionof(8a),(8b),or(8c) 𝑚 𝑚 2 2 as explained in Section 3.2. eTh REs of A , A , A ,and A Δ𝜎 (or Δ𝜎 ) should be found in the interior domain of 𝑒 𝑎 𝑏 𝑐 emx emn computed for the case in Figure 11 are listed in the first column |𝑥| < 𝐷 /2 − Δ𝑥 and |𝑦| < 𝐷 /2 − Δ𝑥 . 𝑥 co 𝑦 co of Table 5. As listed, the RE becomes increasingly small when (ii) For each above defined near-boundary (or near- A is modified successively to A , A ,and A . corner) observation that has only three (or two) adjacent 𝑒 𝑎 𝑏 𝑐 observations, its error variance is still adjusted from 𝜎 to 2 2 2 3.4. Nonuniform Coarse-Resolution Observations without 𝜎 +𝛽 𝜎 but 𝛽 is calculated by setting 𝐶 (Δ𝑥 )=0 in 𝑜 𝑚 𝑏 𝑚 𝑏 co𝑘 Periodic Extension. Consider that the 𝑀 coarse-resolution (18a) for 𝑘=4 (or 𝑘=3 and 4). observations are still nonuniformly distributed in the analysis (iii) Note from (19a) that ∑ Δ𝜎 (x)→0 and thus 𝑚 𝑚 2 2 2 domain of length 𝐷 and width 𝐷 but without periodic Δ𝜎 (x)→Δ𝜎 −𝜌Δ𝜎 <0 as x moves outward far 𝑥 𝑦 𝑀 mn emn extension. In this case, their averaged resolution is still away from all the observations points. In this case, if Δ𝜎 − mn 1/2 2 2 defined by Δ𝑥 ≡(𝐷 𝐷 /𝑀) . To estimate their produced co 𝑥 𝑦 𝜌Δ𝜎 <0 (as for the case in this section), then Δ𝜎 (x) emn 𝑀 error variance reduction, we need to modify the formulations estimated by (19a) may become unrealistically negative as x constructed in the previous subsection with the following moves outward beyond the above constructed observation preparations. First, we need to identify four near-corner boundary loop. To avoid this problem, (19a) is modified into observations among all the coarse-resolution observations. Each near-corner observation is defined as the one that near- est to one of the four corners of the analysis domain. eTh n, Δ𝜎 (x) =𝐹( x )−[𝐹 ( x )−𝐹 (x)]𝑅 𝑀 𝑚𝑏 𝑚𝑏 2 we need to identify 𝑀 −2 (or 𝑀 −2) near-boundary obser- 𝑥 𝑦 (19b) vations associated with each 𝑥-boundary (or 𝑦-boundary), for x outside the observation boundary loop, y (km) y (km) 𝑖𝑗 Advances in Meteorology 13 30 30 20 20 10 10 0 0 −10 −10 −20 −20 −30 −30 −60 −40 −20 0 20 40 60 −60 −40 −20 0 20 40 60 x (km) x (km) (a) (b) Figure 12: As in Figure 11 but without periodic extension. where 𝑅 is a factor defined by analyzing the high-resolution innovations in the second step with the background error covariance updated by A , A , A , 𝑒 𝑎 𝑏 𝐹( x ) 𝑚𝑏 and A , respectively, aer ft the coarse-resolution innovations 𝑅 = min {1, }, (20) 2 2 [𝐹 (x )+𝜌Δ𝜎 −Δ𝜎 ] 𝑚𝑏 emn mn areanalyzedinthefirststep.TheTEeissimilartothefirsttype of two-step experiment (named TEA) in Xu et al. [8], but the x is the projection of x on the observation boundary loop 𝑚𝑏 TEa, TEb, and TEc are new here. As in Xu et al. [8], a single- and the projection from x is along the direction normal to step experiment, named SE, is also designed for analyzing the x-associated domain boundary (nearest to x). However, if all the innovations in a single step. In each of the above vfi e x is closer to a corner observation point than the remaining types of experiments, the analysis increment is obtained by part of the observation boundary loop, then x is simply that 𝑚𝑏 using the standard conjugate gradient descent algorithm to near-corner observation point. It is easy to see from (19b) 2 2 2 minimize the cost-function (formulated as in (7) of Xu et al. that, for Δ𝜎 −𝜌Δ𝜎 <0 and thus 𝑅 =1, Δ𝜎 (x)= mn emn 2 𝑀 [8]) with the number of iterations limited to𝑛=20 ,50, or 100 𝐹(x )−[𝐹(x )−𝐹( x)]𝑅 =𝐹( x )→0 as |x|→−∞,sothe 𝑚𝑏 𝑚𝑏 2 𝑚𝑏 2 before the final convergence to mimic the computationally estimated Δ𝜎 (x) in (19b) can never become unrealistically constrained situations in operational data assimilation. Three negative. sets of simulated innovations are generated for the above eTh analysis error variance is estimated by 𝜎 (x)≈ vfi e types of experiments. eTh first set consists of 𝑀 (=10) 2 2 2 2 𝜎 ∗(x)≡𝜎 −Δ𝜎 (x) as in (16), except that Δ𝜎 (x) is com- 𝑎 𝑏 𝑀 𝑀 uniformly distributed coarse-resolution innovations over the puted by (19a) [or (19b)] for x inside (or outside) the closed analysis domain (see Figure 1) with periodic extension and observation boundary loop. As shown by the example in 󸀠 𝑀 (=74) high-resolution innovations in the nested domain Figure 12, the estimated 𝜎 ∗(x) is fairly close to the benchmark of length 𝐷/6 (similar to those shown by the purple × signs 2 2 2 𝜎 (x), and the deviation of 𝜎 ∗(x) from the benchmark 𝜎 (x) in Figure 1 of Xu et al. [8] but generated at the grid points 𝑎 𝑎 𝑎 2 −2 is within (−4.08, 5.54) m s . On the other hand, the constant not covered by the coarse-resolution innovations within the 2 2 −2 analysis error variance (𝜎 = 6.7 m s )estimated bythe nested domain). The second (or third) set is the same as the rfi st set except that the coarse-resolution innovations spectral formulation deviates from the benchmark 𝜎 (x) very 2 −2 are nonuniformly distributed with (or without) periodic widely from −16.1 to 3.82 m s .Using theestimated 𝜎 ∗(x), extension as shown in Figure 5 (or Figure 8). All the innova- the previously estimated A from the spectral formulation tions are generated by simulated observation errors subtract- can be modified into A , A ,or A with its th element 𝑎 𝑏 𝑐 ing simulated background errors at observation locations. given by the same two-dimensional version of (8a), (8b), or Observation errors are sampled from computer-generated (8c) as explained in Section 3.2. eTh REs of A , A , A ,and 𝑒 𝑎 𝑏 −1 uncorrelated Gaussian random numbers with 𝜎 =2.5 ms A computed for the case in Figure 12 are listed in the first for both coarse-resolution and high-resolution observations. column of Table 6. As listed, the RE becomes increasingly Background errors are sampled from computer-generated small when A is modified successively to A , A ,and A . 𝑒 𝑎 𝑏 𝑐 −1 spatially correlated Gaussian random efi lds with 𝜎 =5 ms and 𝐶 (𝑥) modeled by the double-Gaussian form given in 4. Numerical Experiments for Section2.2 (alsosee thecaption of Figure 1).Thecoarse- One-Dimensional Cases resolution innovations in the rfi st, second, and third sets are thus generated in consistency with the three cases in Figures 4.1. Experiment Design and Innovation Data. In this sec- 1, 5, and 8, respectively. tion, idealized one-dimensional experiments are designed and performed to examine to what extent the successively improved estimate of A in (8a), (8b), and (8c) can improve the 4.2. Results from the First Set of Innovations. The first set of two-step analysis. In particular, four types of two-step exper- innovations is used here to perform each of the vfi e types iments, named TEe, TEa, TEb, and TEc, are designed for of experiments with the number of iterations limited to y (km) y (km) 𝑖𝑗 14 Advances in Meteorology Table 2: As in Table 1 but for the second set of innovations with periodic extension. Experiment 𝑛 = 20 𝑛 = 50 𝑛 = 100 Final SE 0.711 0.334 0.276 0.018 at 𝑛 = 404 TEe with RE(A ) = 0.355 0.482 0.439 0.442 at 𝑛=76 TEa with RE(A ) = 0.238 0.418 0.388 0.348 0.353 at 𝑛 = 108 TEb with RE(A ) = 0.197 0.318 0.288 0.257 0.243 at 𝑛 = 179 TEc with RE(A ) = 0.148 0.213 0.151 0.155 at 𝑛=52 Table 3: As in Table 2 but for the third set of innovations without periodic extension. Experiment 𝑛 = 20 𝑛 = 50 𝑛 = 100 Final SE 0.499 0.328 0.194 0.012 at 𝑛 = 451 TEe with RE(A ) = 0.355 0.463 0.424 0.399 at 𝑛=73 TEa with RE(A ) = 0.238 0.394 0.358 0.385 at 𝑛=54 TEb with RE(A ) = 0.196 0.281 0.273 0.248 at 𝑛=77 TEc with RE(A ) = 0.147 0.215 0.149 0.123 at 𝑛=77 𝑛=20 , 50, or 100 before the n fi al convergence. eTh accuracy butmuchlesssignicfi antthanthatofTEAoverSE in Table of the analysis increment obtained from each experiment 2 of Xu et al. [8]. This reduced improvement can be largely with each limited 𝑛 is measured by its domain-averaged RMS explained by the fact that the coarse-resolution innovations error (called RMS error for short hereaer) ft with respect to are generated here not only more sparsely but also more the benchmark analysis increment computed precisely from nonuniformly than those in Section 3.3 of Xu et al. [8] and (1a). Table 1 lists the RMS errors of the analysis increments the deviation of A from the benchmark A becomes much obtained from the SE, TEe, TEa, TEb, and TEc with the larger in Figure 7(a) here than that in Figure 7(b) of Xu et number of iterations increased from 𝑛=20 to 50, 100, and/or al.[8].TheTEa outperformsTEe for 𝑛=20 and 50 but the n fi al convergence. still underperforms SE for 𝑛 increased to 50 and beyond. The As showninTable 1, theTEeoutperformsSEfor 𝑛=20 , improvement of TEa over TEe is consistent with the improved 50, and 100 but not for 𝑛 increased to the final convergence. accuracy of A [RE(A ) = 0.238] over A [RE(A ) = 0.355]. 𝑎 𝑎 𝑒 𝑒 The improved performance of TEe over SE is similar to but The TEb outperforms TEa for each listed value of 𝑛 and less signica fi nt than that of TEA over SE in Table 1 of Xu et al. also outperforms SE for 𝑛 up to 100. The improvement of [8]. The reduced improvement can be largely explained by the TEb over TEa is consistent with the improved accuracy of fact that the coarse-resolution innovations are generated here A [RE(A ) = 0.197] over A .TheTEc outperforms TEb 𝑏 𝑏 𝑎 more sparsely and the deviation of A from the benchmark for each listed value of 𝑛, and the improvement is consistent A is thus increased (as seen from Figure 4(a) in comparison with the improved accuracy of A [RE(A ) = 0.148] 𝑐 𝑐 with Figure 5(b) of Xu et al. [8]). The TEa outperforms TEe over A . for 𝑛=20 and 50 before 𝑛 increased to 100 (which is very closetothe nfi alconvergenceat 𝑛 = 116 for TEa). eTh 4.4. Results from the iTh rd Set of Innovations. The third set of improvement of TEa over TEe is consistent with and can be innovations is used here to perform each of the vfi e types of largely explained by the improved accuracy of A [RE(A )= experiments with the number of iterations limited to 𝑛=20 , 𝑎 𝑎 0.156] over A [RE(A ) = 0.229].TheTEb outperforms 50,or100 before thefinalconvergence.Thedomain-averaged 𝑒 𝑒 TEa for 𝑛=20 and50(before thefinalconvergence at RMS errors of the analysis increments obtained from the four 𝑛=67 ). The improvement of TEb over TEa is consistent two-step experiments are shown in Table 3 versus those from with the improved accuracy of A [RE(A ) = 0.101] over the SE. As shown, the TEe outperforms SE for 𝑛=20 but not 𝑏 𝑏 A . eTh TEc outperforms TEb for each listed value of 𝑛,and so for 𝑛=50 . eTh improvement of TEe over SE is much less the improvement is consistent with the improved accuracy of signica fi nt than that of TEA over SE in Table 3 of Xu et al. [8], A [with RE(A ) = 0.042] over A . and this reduced improvement can be explained by the same 𝑐 𝑐 𝑏 fact as stated for the previous case in Section 4.3. eTh TEa 4.3. Results from the Second Set of Innovations. The second set outperforms TEe for 𝑛=20 and 50, and the improvement of innovations is used here to perform each of the ve fi types of is consistent with the improved accuracy of A [RE(A )= 𝑎 𝑎 experiments with the number of iterations limited to 𝑛=20 , 0.238] over A [RE(A ) = 0.355].TheTEb outperforms 𝑒 𝑒 50,or100 beforethefinalconvergence.Thedomain-averaged TEa for each listed value of 𝑛, which is consistent with the RMS errors of the analysis increments obtained from the four improved accuracy of A [RE(A ) = 0.196] over A .TheTEc 𝑏 𝑏 𝑎 two-step experiments are shown in Table 2 versus those from outperforms TEb for each listed value of 𝑛, which is consistent the SE. As shown, the TEe outperforms SE for 𝑛=20 but not with the improved accuracy of A [RE(A ) = 0.147] 𝑐 𝑐 so for 𝑛=50 .Theimprovement ofTEeoverSEissimilarto over A . 𝑏 Advances in Meteorology 15 Table 4: As in Table 1 but for the two-dimensional case in Figure 10 in which the first set of two-dimensional innovations is used with periodic extension. Experiment 𝑛 = 20 𝑛 = 50 𝑛 = 100 Final SE 0.742 0.364 0.154 0.071 at 𝑛 = 201 TEe with RE(A ) = 0.233 0.394 0.185 0.108 0.104 at 𝑛 = 140 TEa with RE(A ) = 0.181 0.397 0.186 0.102 0.097 at 𝑛 = 145 TEb with RE(A ) = 0.130 0.403 0.183 0.089 0.085 at 𝑛 = 133 TEc with RE(A ) = 0.038 0.397 0.160 0.064 0.059 at 𝑛 = 183 Table 5: As in Table 4 but for the two-dimensional case in Figure 11 where the second set of innovations is used with periodic extension. Experiment 𝑛 = 20 𝑛 = 50 𝑛 = 100 Final SE 0.757 0.364 0.175 0.069 at 𝑛 = 241 TEe with RE(A ) = 0.462 0.416 0.202 0.149 0.144 at 𝑛 = 140 TEa with RE(A ) = 0.274 0.411 0.202 0.143 0.144 at 𝑛 = 215 TEb with RE(A ) = 0.244 0.402 0.195 0.136 0.132 at 𝑛 = 191 TEc with RE(A ) = 0.165 0.403 0.179 0.106 0.103 at 𝑛 = 287 5. Numerical Experiments for value of 𝑛 before the final convergence, which is similar to the improved performance of TEA over SE shown in Table 4 Two-Dimensional Cases ofXu et al.[8].TheTEa outperformsTEeas 𝑛 increases to 100 5.1. Experiment Design and Innovation Data. In this section, and beyond, which is consistent with the improved accuracy idealized two-dimensional experiments are designed and of A [RE(A ) = 0.181] over A [RE(A ) = 0.233].The 𝑎 𝑎 𝑒 𝑒 namedsimilarly to thoseinSection 4exceptthatsimu- TEb outperforms TEa as 𝑛 increases to 50 and beyond, which lated innovations are generated in three sets for the two- is consistent with the improved accuracy of A [RE(A )= 𝑏 𝑏 dimensional cases in Figures 10, 11, and 12, respectively. 0.130] over A . eTh TEc outperforms TEb for each listed In particular, the first set consists of 𝑀 (= M M =12 value of 𝑛, which is consistent with the improved accuracy x y × 6) uniformly distributed coarse-resolution innovations of A [RE(A ) = 0.038] over A . 𝑐 𝑐 𝑏 over the periodic analysis domain (as shown in Figure 10) and 𝑀 (=66) high-resolution innovations generated at the 5.3. Results from the Second Set of Innovations. The second grid points not covered by the coarse-resolution innovations set of innovations is used here to perform each of the five within the nested domain. eTh nested domain ( 𝐷 /6 = 20 km types of experiments with the number of iterations limited to long and 𝐷 /6 = 10 km wide) is the same as that shown 𝑛=20 , 50, or 100 before the final convergence. The domain- in Figure 16 of Xu et al. [8]. Again, all the innovations averaged RMS errors of the analysis increments obtained are generated by simulated observation errors subtract- from the four two-step experiments are shown in Table 5 ing simulated background errors at observation locations. versus those from the SE. As shown, the TEe outperforms SE Observation errors are sampled from computer-generated for each listed value of 𝑛 before the n fi al convergence. eTh −1 uncorrelated Gaussian random numbers with 𝜎 =2.5ms TEa outperforms TEe slightly, and the improved performance for both coarse-resolution and high-resolution observations. is consistent with the improved accuracy of A [RE(A )= 𝑎 𝑎 Background errors are sampled from computer-generated 0.274] over A [RE(A ) = 0.462].TheTEb outperforms 𝑒 𝑒 −1 TEA for each listed value of 𝑛, which is consistent with the spatially correlated Gaussian random efi lds with 𝜎 =5ms and 𝐶 (x) modeled by the double-Gaussian form given in improved accuracy of A [RE(A ) = 0.244] over A .TheTEc 𝑏 𝑏 𝑎 outperforms TEb for 𝑛>20 ,andtheimprovedperformance Section 3.2 (also see the caption of Figure 10). The second is consistent with the improved accuracy of A [RE(A )= (or third) set is the same as the first set except that the 𝑐 𝑐 coarse-resolution innovations are nonuniformly distributed 0.165] over A . with (or without) periodic extension as shown in Figure 11 (or Figure 12). 5.4. Results from the Third Set of Innovations. The third set of innovations is used here to perform each of the vfi e types of 5.2. Results from the First Set of Innovations. The first set of experiments with the number of iterations limited to 𝑛=20 , innovations is used here to perform each of the vfi e types of 50,or100 before thefinalconvergence.Thedomain-averaged experiments with the number of iterations limited to 𝑛=20 , RMS errors of the analysis increments obtained from the 50,or100 beforethefinalconvergence.Thedomain-averaged four two-step experiments are shown in Table 6 versus those RMS errors of the analysis increments obtained from the four from the SE. As shown, the TEe outperforms SE for each two-step experiments are shown in Table 4 versus those from listed value of 𝑛 before the n fi al convergence. eTh improved the SE. As shown, the TEe outperforms SE for each listed performance of TEe over SE is similar to but less significant 16 Advances in Meteorology Table 6: As in Table 5 but for the two-dimensional case in Figure 12 where the third set of innovations is used without periodic extension. Experiment 𝑛 = 20 𝑛 = 50 𝑛 = 100 Final SE 0.808 0.453 0.176 0.078 at 𝑛 = 235 TEe with RE(A ) = 0.462 0.509 0.179 0.136 0.134 at 𝑛 = 161 TEa with RE(A ) = 0.305 0.497 0.184 0.146 0.140 at 𝑛 = 213 TEb with RE(A ) = 0.258 0.495 0.179 0.135 0.127 at 𝑛 = 170 TEc with RE(A ) = 0.240 0.473 0.157 0.103 0.101 at 𝑛 = 201 than that ofTEAoverSEinTable 5ofXuetal.[8],andthe simplest type, the total error variance reduction is estimated reason is mainly because the coarse-resolution innovations in two steps. First, the error variance reduction produced are generated more sparsely and nonuniformly than those by analyzing each coarse-resolution observation as a single in Section 4.3 of Xu et al. [8]. eTh TEa outperforms TEe observation is equally weighted and combined into the total. for 𝑛=20 but not so as 𝑛 increases to 50 and beyond, Then, the combined total error variance reduction is adjusted although A has an improved accuracy [RE(A ) = 0.305] by a constant to match to the domain-averaged total error 𝑎 𝑎 over A [RE(A ) = 0.462]. eTh TEb outperforms TEa for variance reduction estimated by the spectral formulation 𝑒 𝑒 each listed value of 𝑛,andtheimprovedperformance is [see (5a), (5b), (15a), and (15b)]. eTh estimated analysis consistent with the improved accuracy of A [RE(A )= error variance (i.e., the background error variance minus the 𝑏 𝑏 0.258] over A . eTh TEc outperforms TEb for each listed adjusted total error variance reduction) captures not only the value of 𝑛, which is consistent with the improved accuracy domain-averaged value but also the spatial variation of the of A [RE(A ) = 0.240] over A . benchmark truth (see Figures 1, 2, and 10). 𝑐 𝑐 𝑏 (ii) The second type consists of nonuniformly distributed coarse-resolution observations with periodic extension. For 6. Conclusions this more general type, the total error variance reduction is also estimated in two steps: The rfi st step is similar to In this paper, the two-step variational method developed that for the first type but the combination into the total is in Xu et al. [8] for analyzing observations of different weighted based on the averaged spacing of each concerned spatial resolutions is improved by considering and efficiently estimating the spatial variation of analysis error variance observation from its neighboring observations [see (11a), (11b), (18a), and (18b)]. In the second step, the combined produced by analyzing coarse-resolution observations in the total error variance reduction is adjusted and scaled to first step. eTh constant analysis error variance computed from the spectral formulations in Xu et al. [8] can represent the match the maximum and minimum of the true total error variance reduction estimated from the spectral formulation spatial averaged value of the true analysis error variance but for uniformly distributed coarse-resolution observations but itcannotcapturethespatialvariation inthetrueanalysis with the observation resolutions set, respectively, to the error variance. As revealed by the examples presented in minimum spacing and maximum spacing of the nonuni- this paper (see Figures 1, 2, 5, and 8 for one-dimensional formly distributed coarse-resolution observations [see (12a) cases and Figures 10–12 for two-dimensional cases), the true and (19a)]. The estimated analysis error variance captures analysis error variance tends to have increasingly large spatial notonlythe maximumandminimumbutalsothespatial variations when the coarse-resolution observations become increasingly nonuniform and/or sparse, and this is especially variation of the benchmark truth (see Figures 5 and 11). (iii) The third type consists of nonuniformly distributed true and serious when the separation distances between coarse-resolution observations without periodic extension. neighboring coarse-resolution observations become close to or even locally larger than the background error decorrelation For this most general type, the total error variance reduction is estimated with the same two steps as for the second type, length scale. In this case, the spatial variation of analysis error except that three modica fi tions are made to improve the variance and associated spatial variation in analysis error estimation near and at the domain boundaries [see (i)–(iii) covariance need to be considered and estimated efficiently in in Sections 2.4 and 3.4]. The analysis error variance na fi lly order to further improve the two-step analysis. eTh analysis error variance can be viewed equivalently estimated captures the spatial variation of the benchmark truth not only in the interior domain but also near and at the and conveniently as the background error variance minus domain boundaries (see Figures 8 and 12). the total error variance reduction produced by analyzing all the coarse-resolution observations. To ecffi iently estimate the The above estimated spatially varying analysis error latter, analytic formulations are constructed for three types of variance is used to modify the analysis error covariance coarse-resolution observations in one- and two-dimensional computed from the spectral formulations of Xu et al. [8] spaces with successively increased complexity and generality. in three different forms [see (8a), (8b), and (8c)]. eTh The main results and major ndings fi are summarized below rfi st is a conventional formulation in which the covariance for each type of coarse-resolution observations: is modulated by the spatially varying standard deviation (i) eTh rst fi type consists of uniformly distributed coarse- separately via each entry of the covariance to retain the resolution observations with periodic extension. For this self-adjointness. This modulation has a chessboard structure Advances in Meteorology 17 but the desired modulation has a banded structure (along References the direction perpendicular to the diagonal line) as revealed [1] W.-S. Wu, R. J. Purser, and D. F. Parrish, “Three-dimensional by the to-be-corrected deviation from the benchmark truth variational analysis with spatially inhomogeneous covariances,” (see Figure 4(a)), so the deviation is only partially reduced Monthly Weather Review,vol.130,no. 12,pp.2905–2916, 2002. (see Figure 4(b)). The second formulation is new, in which [2] R.J.Purser, W.-S.Wu, D.F.Parrish,andN.M.Roberts, the modulation is realigned to capture the desired banded “Numerical aspects of the application of recursive filters to structure and yet still retain the self-adjointness. The devia- variational statistical analysis. Part II: Spatially inhomogeneous tion from the benchmark truth is thus further reduced (see and anisotropic general covariances,” Monthly Weather Review, Figure 4(c)), but the deviation is reduced not broadly enough vol. 131, no. 8, pp. 1536–1548, 2003. along each band. By properly broadening the reduction [3] S.Liu,M.Xue,J.Gao,and D.Parrish,“Analysis andimpactof distribution in the third formulation, the deviation is much super-obbed Doppler radial velocity in the NCEP grid-point further reduced (see Figure 4(d)). statistical interpolation (GSI) analysis system,” in Proceedings of The successive improvements made by the above three the 21st Conference on on Weather Analysis and Forecasting/17th formulations are demonstrated for all the three types of Conference on Numerical Weather Prediction,AmericanMeteo- rological Society, Washington, DC, USA, 2005. coarse-resolution observations in one- and two-dimensional spaces. eTh improvements are quantiefi d by the successively [4] Y. Xie, S. Koch, J. McGinley et al., “A space-time multiscale reduced relative errors [REs, measured by the Frobenius analysis system: a sequential variational analysis approach,” Monthly Weather Review,vol.139,no.4, pp.1224–1240,2011. norm defined in (9)] of their modified analysis error covari- ance matrices with respect to the benchmark truths (see [5] J.Gao,T.M.Smith,D.J.Stensrudetal.,“Areal-timeweather- REs listed in the first columns of Tables 1–6). The impacts adaptive 3DVAR analysis system for severe weather detections and warnings,” Weather and Forecasting,vol.28, no.3,pp.727– of the improved accuracies of the modified analysis error 745, 2013. covariance matrices on the two-step analyses are examined [6] Z.Li,J. C.McWilliams,K.Ide,andJ. D.Farrara,“A multiscale with idealized experiments that are similar to but extend variational data assimilation scheme: Formulation and illustra- those in Xu et al. [8]. As expected, the impacts are found tion,” Monthly Weather Review,vol.143,no.9, pp.3804–3822, to be mostly positive (especially for the third formulation) and largely in consistency with the improved accuracies of [7] Q. Xu, L. Wei, K. Nai, S. Liu, R. M. Rabin, and Q. Zhao, “A the modiefi d analysis error covariance matrices (see Tables radar wind analysis system for nowcast applications,” Advances 1–6). As new improvements to the conventional formulation, in Meteorology, vol. 2015, Article ID 264515, 13 pages, 2015. the second and third formulations may also be useful in [8] Q.Xu, L.Wei,J. Gao, Q. Zhao,K.Nai,andS. Liu,“Multistep constructing covariance matrices with nonconstant variances variational data assimilation: Important issues and a spectral for general applications beyond this paper. approach,” Tellus Series A: Dynamic Meteorology and Oceanog- eTh formulations constructed in this paper for estimating raphy, vol. 68, no. 1, Article ID 31110, 2016. the spatial variation of analysis error variance and associated [9] Q. Xu and L. Wei, “Multistep and multi-scale variational data spatial variation in analysis error covariance are effective for assimilation: Spatial variations of analysis error variance,” in further improving the two-step variational method devel- Proceedings of the 17th Conference on Mesoscale Processes, opedin Xuetal.[8],especiallywhenthe coarse-resolution American Meteorological Society,SanDiego,CA,USA,2017. observations become increasingly nonuniform and/or sparse. [10] A. H. Jazwinski, Stochastic Processes and Filtering eo Th ry , These formulations will be extended together with the spec- Academic Press, New York, NY, USA, 1970. tral formulations of Xu et al. [8] for real-data applications [11] I. D. Rutherford, “Data assimilation by statistical interpolation in three-dimensional space with the variational data assim- of forecast error fields,” Journal of the Atmospheric Sciences,vol. ilation system of Gao et al. [5], in which the analyses are 29,no.5, pp.809–815,1972. univariate and performed in two steps. Such an extension is [12] R. Daley, Atmospheric Data Analysis, Cambridge University currently being developed. Press, Cambridge, UK, 1991. [13] G.H.GolubandC.F.Van Loan, Matrix computations,vol.3of Conflicts of Interest Johns Hopkins University Press,Baltimore,MD,USA, 1983. eTh authors declare that there are no conflicts of interest regarding the publication of this paper. Acknowledgments eTh authors are thankful to Dr. Jindong Gao for their constructive comments and suggestions that improved the presentation of the paper. eTh research work was supported by the ONR Grants N000141410281 and N000141712375 to the University of Oklahoma (OU). Funding was also provided to CIMMS by NOAA/Office of Oceanic and Atmospheric Research under NOAA-OU Cooperative Agreement no. NA11OAR4320072, US Department of Commerce. 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Formulations for Estimating Spatial Variations of Analysis Error Variance to Improve Multiscale and Multistep Variational Data Assimilation

Xu, Qin; Wei, Li
Advances in Meteorology , Volume 2018: 17 – Feb 7, 2018
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Hindawi Publishing Corporation
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Copyright © 2018 Qin Xu and Li Wei. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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10.1155/2018/7931964
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Hindawi Advances in Meteorology Volume 2018, Article ID 7931964, 17 pages https://doi.org/10.1155/2018/7931964 Research Article Formulations for Estimating Spatial Variations of Analysis Error Variance to Improve Multiscale and Multistep Variational Data Assimilation 1 2 Qin Xu and Li Wei NOAA/National Severe Storms Laboratory, Norman, OK, USA Cooperative Institute for Mesoscale Meteorological Studies, University of Oklahoma, Norman, OK, USA Correspondence should be addressed to Qin Xu; qin.xu@noaa.gov Received 24 April 2017; Revised 24 November 2017; Accepted 3 December 2017; Published 7 February 2018 Academic Editor: Shaoqing Zhang Copyright © 2018 Qin Xu and Li Wei. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. When the coarse-resolution observations used in the first step of multiscale and multistep variational data assimilation become increasingly nonuniform and/or sparse, the error variance of the first-step analysis tends to have increasingly large spatial variations. However, the analysis error variance computed from the previously developed spectral formulations is constant and thus limited to represent only the spatially averaged error variance. To overcome this limitation, analytic formulations are constructed to efficiently estimate the spatial variation of analysis error variance and associated spatial variation in analysis error covariance. First, a suite of formulations is constructed to efficiently estimate the error variance reduction produced by analyzing the coarse- resolution observations in one- and two-dimensional spaces with increased complexity and generality (from uniformly distributed observations with periodic extension to nonuniformly distributed observations without periodic extension). en, Th three different formulations are constructed for using the estimated analysis error variance to modify the analysis error covariance computed from the spectral formulations. The successively improved accuracies of these three formulations and their increasingly positive impacts on the two-step variational analysis (or multistep variational analysis in first two steps) are demonstrated by idealized experiments. 1. Introduction steps of a multistep approach) in which broadly distributed coarse-resolution observations are analyzed rfi st and then Multiple Gaussians with different decorrelation length scales locally distributed high-resolution observations are analyzed have been used at NCEP to model the background error in the second step, an important issue is how to objectively covariance in variational data assimilation (Wu et al. [1], estimate or efficiently compute the analysis error covariance Purser et al. [2]), but mesoscale features are still poorly for the analyzed efi ld that is obtained in the rfi st step and resolved in the analyzed incremental efi lds even in areas used to update the background field in the second step. To covered by remotely sensed high-resolution observations, address this issue, spectral formulations were derived by Xu et such as those from operational weather radars (Liu et al. al. [8] for estimating the analysis error covariance. As shown [3]). This problem is common for the widely adopted single- in Xu et al. [8], the analysis error covariance can be computed step approach in operational variational data assimilation, very ecffi iently from the spectral formulations with very (or especially when patchy high-resolution observations, such fairly) good approximations for uniformly (or nonuniformly) as those remotely sensed from radars and satellites, are distributed coarse-resolution observations and, by using the assimilated together with coarse-resolution observations into approximately computed analysis error covariance, the two- a high-resolution model. To solve this problem, multiscale step analysis can outperform the single-step analysis under and multistep approaches were explored and proposed by thesamecomputational constraint (thatmimicsthe opera- severalauthors (Xieetal.[4],Gao etal.[5],Lietal.[6], and Xu et al. [7, 8]). For a two-step approach (or the first two tional situation). 2 Advances in Meteorology 2 2 2 2 2 eTh analysis error covariance functions computed from where 𝛾 =𝜎 /(𝜎 +𝜎 ), 𝜎 (or 𝜎 )isthebackground 𝑏 𝑏 𝑏 𝑜 𝑏 𝑜 the spectral formulations in Xu et al. [8] are spatially (or observation) error variance, 𝐶 (𝑥) is the background homogeneous, so their associated error variances are spatially error correlation function, 𝑥 denotes the 𝑖th point in the constant. Although such a constant error variance can rep- discretized analysis space 𝑅 ,and 𝑁 is the number of grid resent the spatial averaged value of the true analysis error points over the analysis domain. eTh length of the analysis variance, it cannot capture the spatial variation in the true domain is 𝐷=𝑁Δ𝑥 ,where Δ𝑥 is the analysis grid spacing analysis error variance. eTh true analysis error variance can and 𝐷 is assumedtobemuchlargerthanthe background have significant spatial variations, especially when the coarse- error decorrelation length scale 𝐿. resolution observations become increasingly nonuniform Note that 𝐶 (𝑥) is a continuous function of 𝑥,so(2) can 2 2 2 and/or sparse. In this case, the spatial variation of analysis be written into 𝜎 (𝑥) ≡ 𝜎 −Δ𝜎 (𝑥) also as a continuous 𝑚 𝑏 𝑚 error variance and associated spatial variation in analysis function of 𝑥,where error covariance need to be estimated based on the spatial distribution of the coarse-resolution observations in order (3) Δ𝜎 (𝑥 ) ≡𝛾 [𝜎 𝐶 (𝑥 −𝑥 )] to further improve the two-step analysis. This paper aims to 𝑏 𝑏 𝑏 𝑖 𝑚 explore and address this issue beyond the preliminary study reported in Xu and Wei [9]. In particular, as will be shown in is the error variance reduction produced by analyzing a single this paper, analytic formulations for efficiently estimating the observation at 𝑥=𝑥 . eTh error variance reduction in spatial variation of analysis error variance can be constructed (3) decreases rapidly as |𝑥−𝑥 | increases, and it becomes by properly combining the error variance reduction produced 2 2 much smallerthanitpeakvalueof 𝛾 𝜎 C at 𝑥=𝑥 as 𝑏 𝑏 𝑏 𝑚 by analyzing each and every coarse-resolution observation |𝑥−𝑥 | increases to 𝐿. This implies that the error variance as asingleobservation,andtheestimated analysiserror reduction produced by analyzing 𝑀 sparsely distributed variance canbeusedtofurther estimatetherelatedvariation coarse-resolution observations can be estimated by properly in analysis error covariance. eTh detailed formulations are combining the error variance reduction computed by (3) for presented for one-dimensional cases first in the next section each coarse-resolution observation as a single observation. and then extended to two-dimensional cases in Section 3. This idea is explored in the following three subsections Idealized numerical experiments are performed for one- for one-dimensional cases with successively increased com- dimensional cases in Section 4 and for two-dimensional cases plexity and generality: from uniformly distributed coarse- in Section 5 to show the eeff ctiveness of these formulations resolution observations with periodic extension to nonuni- for improving the two-step analysis. Conclusions follow in formly distributed coarse-resolution observations without Section 6. periodic extension. 2. Analysis Error Variance Formulations for 2.2. Uniform Coarse-Resolution Observations with Periodic One-Dimensional Cases Extension. Consider that there are 𝑀 coarse-resolution observations uniformly distributed in the above analysis 2.1. Error Variance Reduction Produced by a Single Observa- domain of length 𝐷 with periodic extension, so their res- tion. When observations are optimally analyzed in terms of olution is Δ𝑥 ≡𝐷/𝑀 . In this case, the error variance co the Bayesian estimation (see chapter 7 of Jazwinski [10]), the reduction produced by each observation can be considered background state vector b is updated to the analysis state as an additional reduction to the reduction produced by its vector a with the following analysis increment: neighboring observations, and this additional reduction is always smaller than the reduction produced by the same −1 T T (1a) Δa ≡ a − b = BH (HBH + R) d, observationbuttreatedasasingle observation. Thisimplies that the error variance reduction produced by analyzing the and the background error covariance matrix B is updated to 𝑀 coarse-resolution observations, denoted by Δ𝜎 (𝑥),is the analysis error covariance matrix A according to bounded above by ∑ Δ𝜎 (𝑥);thatis, 𝑚 𝑚 −1 T T (1b) A = B − BH (HBH + R) HB, 2 2 Δ𝜎 (𝑥 ) ≤ ∑Δ𝜎 (𝑥 ) , (4) 𝑀 𝑚 where R is the observation error covariance matrix, d = y − 𝑚 h(b) is the innovation vector (observation minus background in the observation space), y is the observation vector, h() where ∑ denotes the summation over 𝑚 for the 𝑀 observa- denotes the observation operator, and H is the linearized h(). tions. The equality in (4) is for the limiting case of Δ𝑥 /𝐿 → co For a single observation, say, at 𝑥 in the one-dimensional ∞ only. eTh inequality in (4) implies that the domain- −1 space of 𝑥,the inversematrix (HBH + R) reduces to 2 averaged value of ∑ Δ𝜎 (𝑥) is larger than the true averaged −1 𝑚 𝑚 2 2 2 2 2 2 (𝜎 +𝜎 ) ,sothe 𝑖th diagonal element of A in (1b) is simply 𝑏 𝑜 reduction estimated by Δ𝜎 ≡𝜎 −𝜎 ,where 𝜎 is the 𝑏 𝑒 𝑒 given by domain-averaged analysis error variance estimated by the spectral formulation for one-dimensional cases in Section 2.2 2 2 (2) 𝜎 (𝑥 )≡𝜎 −𝛾 [𝜎 𝐶 (𝑥 −𝑥 )] , 𝑚 𝑖 𝑏 𝑏 𝑏 𝑏 𝑖 𝑚 of Xu et al. [8]. 𝑏𝑒 Advances in Meteorology 3 The domain-averaged value of ∑ Δ𝜎 (𝑥) can be com- 𝑚 𝑚 puted by ∫ ∑ Δ𝜎 (𝑥 ) 𝑚 𝑚 2 𝐷 Δ𝜎 ≡ 2 2 𝛾 𝜎 ∑ ∫ 𝐶 (𝑥 − 𝑥 ) 𝑏 𝑚 𝑚 𝑏 𝑏 (5a) 2 2 𝛾 𝜎 ∑ ∑ 𝐶 (𝑥 −𝑥 ) 𝑏 𝑖 𝑚 𝑏 𝑚 𝑖 𝑏 6 ≈ , where ∫ denotes the integration over the analysis −60 −40 −20 04 20 0 60 domain, ∑ denotes the summation over 𝑖 for the 𝑁 grid x (km) points, and 𝐷=𝑁Δ𝑥 is used in the last step. By extending 2 2 2 2 𝐶 (𝑥−𝑥 ) with the analysis domain periodically, Δ𝜎 can 𝑏 𝑚  (x)  a e be also estimated analytically as follows: 2 2  (x) ∗  a o Figure 1: Benchmark analysis error variance 𝜎 (𝑥) plotted by red ∫ ∑ Δ𝜎 (𝑥 ) 𝑚 𝑚 2 𝐷 Δ𝜎 ≡ solid curve and estimated analysis error variance 𝜎 (𝑥) in (7) plotted by blue dotted curve. eTh green dashed line shows the 2 2 constant analysis error variance 𝜎 estimated from the spectral 𝛾 𝜎 ∑ ∑ ∫ 𝐶 (𝑥 − 𝑥 −𝑘𝐷) 𝑏 𝑚 𝑘 𝑚 𝑏 𝑏 (5b) formulation. The purple + signs show the observation error variance 2 2 2 −2 (𝜎 = 2.5 m s )atthe locationsof 𝑀 (=10) uniformly distributed coarse-resolution observations with Δ𝑥 =𝐷/𝑀 (=11.04 km). The 2 2 2 co 𝛾 𝜎 𝑀∫𝑑𝑥𝐶 (𝑥 ) 𝛾 𝜎 𝐼 𝐿 2 𝑏 𝑏 𝑏 𝑏 𝑏 1 background error covariance 𝜎 𝐶 (𝑥) has the double-Gaussian form = = , 𝑏 2 2 2 2 2 2 2 −2 𝐷 Δ𝑥 with 𝐶 (𝑥) = 0.6 exp(−𝑥 /2𝐿 ) + 0.4 exp(−2𝑥 /𝐿 ), 𝜎 =5 m s co 𝑏 𝑏 and 𝐿=10 km. eTh analysis domain length is 𝐷 = 𝑁Δ𝑥 = 110.4 km with 𝑁 = 260 and Δ𝑥 = 0.24 km, and the number of coarse- where ∫𝑑𝑥 denotes the integration over the infinite space resolution observations is 𝑀=10 . 2 2 of 𝑥, ∑ ∑ ∫ 𝐶 (𝑥−𝑥 −𝑘𝐷) = ∑ ∫𝑑𝑥𝐶 (𝑥−𝑥 ) 𝑚 𝑘 𝑏 𝑚 𝑚 𝑏 𝑚 2 2 = ∑ ∫𝑑𝑥𝐶 (𝑥) = 𝑀∫𝑑𝑥𝐶 (𝑥) is used in the second to 𝑏 𝑏 last step, and 𝐼 ≡∫𝐶 (𝑥)𝑑𝑥/𝐿 is used with Δ𝑥 ≡ 1 𝑏 co 2 monotonically toward its asymptotic upper limit of 𝛾 𝜎 = 𝑏 𝑏 𝐷/𝑀 in the last step. For the double-Gaussian form of 2 −2 2 2 2 2 20 m s )as Δ𝑥 /𝐿 decreasesto0.5andthento Δ𝑥/𝐿 = 0.1 𝐶 (𝑥) = 0.6 exp(−𝑥 /2𝐿 ) + 0.4 exp(−2𝑥 /𝐿 ) used in (5) co 1/2 1/2 1/2 (or increases toward ∞),andthisdecrease(orincrease)ofthe of Xuet al.[8],wehave 𝐼 =(2)𝜋 (0.44/2 + 0.48/5 ). amplitude of spatial variation of 𝜎 (𝑥) with Δ𝑥 /𝐿 is closely co The analytically derived value in (5b) is very close to (slightly captured by the amplitude of spatial variation of the estimated larger than) the numerically computed value from (5a). With 2 2 𝜎 ∗(𝑥) as a function of Δ𝑥 /𝐿. co the domain-averaged value of ∑ Δ𝜎 (𝑥) adjusted from Δ𝜎 𝑚 𝑚 Using the estimated 𝜎 ∗(𝑥) in (7), the previously esti- 2 2 𝑎 to Δ𝜎 , Δ𝜎 (𝑥) canbeestimated by mated analysis error covariance matrix, denoted by A with its th element 𝐴 ≡𝜎 𝐶 (𝑥 −𝑥 ) obtained from the 𝑗𝑒𝑖 𝑎 𝑖 𝑗 2 2 2 2 Δ𝜎 (𝑥 ) = ∑Δ𝜎 (𝑥 ) −Δ𝜎 +Δ𝜎 . spectral formulations, can be modiefi d into A , A ,or A with (6) 𝑀 𝑚 𝑎 𝑏 𝑐 its th element given by eTh analysis error variance, 𝜎 (𝑥),isthenestimatedby 𝐴 ≡𝜎 ∗ (𝑥 )𝜎 ∗ (𝑥 )𝐶 (𝑥 −𝑥 ) 𝑗𝑎𝑖 𝑎 𝑖 𝑎 𝑗 𝑎 𝑖 𝑗 2 2 2 2 𝜎 (𝑥 ) ≈𝜎 (𝑥 ) ≡𝜎 −Δ𝜎 (𝑥 ) . (7) 𝑎 𝑎 𝑏 𝑀 =𝐴 (8a) 𝑗𝑒𝑖 As shown by the example in Figure 1 (in which 𝐷 = 110.4 km and 𝑀 =10so Δ𝑥 = 𝐷/𝑀 = 11.04 km +[𝜎 ∗ (𝑥 )𝜎 ∗ (𝑥 )−𝜎 ]𝐶 (𝑥 −𝑥 ), co 𝑎 𝑖 𝑎 𝑗 𝑎 𝑖 𝑗 is close to 𝐿=10 km), the estimated 𝜎 ∗(𝑥) in (7) has nearly the same spatial variation as the benchmark 𝜎 (𝑥) 2 2 𝐴 ≡𝐴 +{𝜎 ( + )−𝜎 }𝐶 (𝑥 −𝑥 ) 𝑗𝑏𝑖 𝑗𝑒𝑖 𝑎 𝑒 𝑎 𝑖 𝑗 that is computed precisely from (1b), although the amplitude 2 2 2 2 (8b) of spatial variation of 𝜎 (𝑥), defined by max 𝜎 (𝑥) − ∗ ∗ 𝑎 𝑎 𝑥 𝑗 2 2 2 𝑖 min 𝜎 (𝑥), is slightly smaller than that of the true 𝜎 (𝑥), ∗ =𝜎 ( + )𝐶 (𝑥 −𝑥 ), 𝑎 𝑎 𝑎 𝑎 𝑖 𝑗 2 2 2 2 defined by max 𝜎 (𝑥) − min 𝜎 (𝑥). As shown in Figure 2, the 𝑎 𝑎 amplitudeofspatialvariationofbenchmark 𝜎 (𝑥) decreases 𝑎 2 𝑖 2 (8c) or 𝐴 ≡𝐴 +{𝜎 ∗ ( + )−𝜎 }𝐶 (𝑥 −𝑥 ). 𝑗𝑐𝑖 𝑗𝑒𝑖 𝑏 𝑖 𝑗 rapidly to virtually zero and then exactly zero (or increases 𝑎 𝑒 2 2 2 2 (Ｇ /Ｍ ) 𝑖𝑗 𝑏𝑒 𝑏𝑠 𝑖𝑗 𝑏𝑒 𝑏𝑠 𝑑𝑥 𝑑𝑥 𝑏𝑠 𝑑𝑥 𝑏𝑠 𝑑𝑥 𝑑𝑥 𝑏𝑠 𝑑𝑥 4 Advances in Meteorology 4(b), 4(c), and 4(d), respectively. As shown, the deviation becomes increasingly small when A is modified successively to A , A ,and A . Note that the correction term in (8a) is 𝑎 𝑏 𝑐 15 𝐶 (𝑥 −𝑥 ) modulated by 𝜎 ∗(𝑥 )𝜎 ∗(𝑥 )−𝜎 .Thismodulation 𝑎 𝑖 𝑗 𝑎 𝑖 𝑎 𝑗 𝑒 has a chessboard structure, while the desired modulation revealed by the to-be-corrected deviation of A in Figure 4(a) has a banded structure (along the direction of 𝑥 + 𝑥 = 𝑖 𝑗 constant, perpendicular to the diagonal line). This explains why the correction term in (8a) offsets only a part of the deviation as revealed by the deviation of A in Figure 4(b). On the other hand, the correction term in (8b) is modulated 2 2 (𝑥 /2 + 𝑥 /2) − 𝜎 . This modulation not only retains the by 𝜎 ∗ 𝑎 𝑖 𝑗 𝑒 self-adjointness butalsohasthedesired banded structure, so thecorrectiontermin(8b)isanimprovement over thatin 0.5 1 1.5 2 2.5 3 x /L (8a), as shown by the deviation of A in Figure 4(c) versus that ＝Ｉ 𝑏 of A in Figure 4(b). However, as revealed by Figure 4(c), the Figure 2: Amplitude of spatial variation of benchmark 𝜎 (𝑥), deviation of A still has two significant maxima (or minima) 2 2 defined by max 𝜎 (𝑥) − min 𝜎 (𝑥), plotted by red solid curve as 𝑎 𝑎 along each band on the two sides of the diagonal line of 𝑥 = a function of Δ𝑥 /𝐿 . Amplitude of spatial variation of estimated co 2 2 2 𝑥 , while the to-be-corrected deviation of A in Figure 4(a) 𝑗 𝑒 𝜎 (𝑥), defined by max 𝜎 (𝑥) − min 𝜎 (𝑥) and plotted by blue ∗ ∗ ∗ 𝑎 𝑎 𝑎 has a single maximum (or minimum) along each band. This dotted curve as a function of Δ𝑥 /𝐿 . co implies that the function form of 𝐶 (𝑥 −𝑥 ) is not sufficiently 𝑎 𝑖 𝑗 wide for the correction. As a further improvement, this function form is widened to 𝐶 (𝑥 −𝑥 ) for the correction 𝑏 𝑖 𝑗 term in (8c), so the deviation of A in Figure 4(d) is further reduced from that of A in Figure 4(c). 150 𝑏 When an estimated A is used to update the background 100 error covariance in the second step for analyzing the high- resolution observations in the nested domain, the accuracy of the second-step analysis depends not only, to a certain extent, on the number of iterations performed by the minimization algorithm but also on the accuracy of the estimated A over the nested domain plus its extended vicinities within the −50 distance of 2𝐿 outside the nested domain. Here, 𝐿 is 𝑎 𝑎 −100 the decorrelation length scale of 𝐶 (𝑥) defined by 𝐿 ≡ 𝑎 𝑎 [−𝐶 (𝑥)/𝑑 𝐶 (𝑥)]| according to (4.3.10) of Daley [12], and 𝑎 𝑥 𝑎 𝑥=0 −150 𝐿 (=4.45kmfor thecaseinFigures 1and 3)canbeeasily computed as a by-product from the spectral formulation. −200 Over this extended nested domain, the relative error (RE) of the estimated A with respect to the benchmark A can be −200 −150 −100 −50 0 50 100 150 200 𝑒 measured by Figure 3: Structure of benchmark A plotted by color contours every 󵄩 󵄩 󵄩 󵄩 2 −2 I (A − A) I 󵄩 󵄩 𝑠 𝑒 𝑠 1m s for the case in Figure 1. 󵄩 󵄩 𝐹 RE (A )≡ , (9) 𝑒 󵄩 󵄩 󵄩 󵄩 I AI 󵄩 󵄩 󵄩 𝑠 𝑠 󵄩 The formulation in (8a) is conventional, as in (2.1) of Purser where I denotes the unit matrix in the subspace associated et al. [2] or originally (11) of Rutherford [11], in which the with the grid points in the extended nested domain and covariance is modified by applying 𝜎 ∗(𝑥) separately to each thus I (A − A)I (or I AI )isthe submatrixof A − A 𝑎 𝑠 𝑒 𝑠 𝑠 𝑠 𝑒 entry (indexed by 𝑖 and 𝑗)of 𝐶 (𝑥 −𝑥 ) to retain the (or A) associated only with the grid points in the extended 𝑎 𝑖 𝑗 self-adjointness. eTh second equation in (8a) shows that the nested domain and ‖()‖ denotes the Frobenius norm of conventional approach can be viewed alternatively as 𝐴 () den fi ed by the square root of the sum of the squared 𝑗𝑒𝑖 plus a correction term, the last term in (8a). Ideally, the absolute values of the elements of the matrix in () according correction term should completely offset the deviation of 𝐴 to (2.2–4) of Golub and Van Loan [13]. eTh REs of A , A , 𝑗𝑒𝑖 𝑎 𝑏 from the true covariance, but the correction term in (8a) and A can be measured by the same form of Frobenius norm offsets only a part of the deviation. ratioasthatdenfi edfor A in (9). The REs of A , A , A , 𝑒 𝑒 𝑎 𝑏 For the case in Figure 1, the benchmark analysis error and A are computed for the case in Figure 1 and listed in covariance matrix, denoted by A, is computed precisely from the first column of Table 1. As shown by the listed values, (1b) and is plotted in Figure 3, while the deviations of A , A , the RE becomes increasingly small when A is modified 𝑒 𝑎 𝑒 A ,and A from the benchmark A are shown in Figures 4(a), successively to A , A ,and A , and this is consistent with and 𝑏 𝑐 𝑎 𝑏 𝑐 2 2 (Ｇ /Ｍ ) Advances in Meteorology 5 200 200 150 150 100 100 50 50 j 0 j 0 −50 −50 −100 −100 −150 −150 −200 −200 −200 −150 −100 −50 0 50 100 150 200 −200 −150 −100 −50 0 50 100 150 200 i i (a) (b) 200 200 150 150 100 100 50 50 j 0 j 0 −50 −50 −100 −100 −150 −150 −200 −200 −200 −150 −100 −50 0 50 100 150 200 −200 −150 −100 −50 0 50 100 150 200 i i (c) (d) 2 −2 Figure 4: (a) Deviation of A from benchmark A in Figure 3 plotted by color contours every 0.5 m s .Deviationsof A , A ,and A from 𝑒 𝑎 𝑏 𝑐 2 −2 benchmark A are plotted by color contours every 0.2 m s in panels (b), (c), and (d), respectively. Here, A is the previously estimated analysis error covariance matrix with its th element 𝐴 ≡𝜎 𝐶 (𝑥 −𝑥 ) obtained from the spectral formulation, while A , A ,and A are the newly 𝑗𝑒𝑖 𝑒 𝑎 𝑖 𝑗 𝑎 𝑏 𝑐 modified estimates of A as shown in (8a), (8b), and (8c), respectively. also quantifies the successively reduced deviation shown in First, we consider the case of Δ𝑥 →0 with Δ𝑥 = co𝑚+ co𝑚− Figures 4(a)–4(d). Δ𝑥 (or Δ𝑥 →0 with Δ𝑥 =Δ𝑥 ). In this co co𝑚− co𝑚+ co case, the concerned 𝑚th observation collapses onto the same point with its right (or left) adjacent observation, that is, the 2.3. Nonuniform Coarse-Resolution Observations with Peri- (𝑚 + 1)th [or (𝑚 − 1)th] observation. eTh two collapsed odic Extension. Consider that the 𝑀 coarse-resolution obser- observations should be combined into one superobservation vations are now nonuniformly distributed in the analysis 2 2 with a reduced error variance from 𝜎 to 𝜎 /2.Theerror domain of length 𝐷 with periodic extension, so their averaged 𝑜 𝑜 variance reduction produced by this superobservation still resolution canbedenfi edby Δ𝑥 ≡𝐷/𝑀 .Thespacing co can be estimated by (3) but with of a concerned coarse-resolution observation, say the 𝑚th observation, from its right (or left) adjacent observation can be denoted by Δ𝑥 (or Δ𝑥 ). Now we can consider the 𝛾 = . (10a) co𝑚+ co𝑚− 2 2 (𝜎 +𝜎 /2) following two limiting cases. 𝑏 𝑖𝑗 6 Advances in Meteorology −1 Table 1: Entire-domain averaged RMS errors (in ms ) for the analysis increments obtained from SE, TEe, TEa, TEb, and TEc applied to the first set of innovations with periodic extension and consecutively increased 𝑛,where 𝑛 is the number of iterations. All the RMS errors are evaluated with respect to the benchmark analysis increment. The relative error (RE) of the estimated analysis error covariance for updating the background error covariance in the second step of the two-step analysis is listed with the experiment name in the first column for each two-step experiment. Experiment 𝑛 = 20 𝑛 = 50 𝑛 = 100 Final SE 0.671 0.365 0.187 0.013 at 𝑛 = 481 TEe with RE(A ) = 0.229 0.171 0.150 0.142 0.135 at 𝑛 = 210 TEa with RE(A ) = 0.156 0.169 0.142 0.144 0.144 at 𝑛 = 116 TEb with RE(A ) = 0.101 0.147 0.098 0.090 at 𝑛=67 TEc with RE(A ) = 0.042 0.145 0.063 0.062 0.032 at 𝑛 = 176 On the other hand, without super-Obbing, the error variance the result derived from (10c)-(10d). Note also that 𝛽 =1 reduction produced by the two collapsed observations will be for Δ𝑥 =0 and Δ𝑥 =Δ𝑥 (or Δ𝑥 =0 and co𝑚+ co𝑚− co co𝑚− overestimated by (3) with Δ𝑥 =Δ𝑥 ), so the adjusted error variance is 𝜎 = co𝑚+ co 𝑜𝑚 2 2 𝜎 +𝜎 which recovers the result derived from (10a)-(10b). 2 2 𝑜 𝑏 2𝜎 𝜎 𝑏 𝑏 Clearly, for Δ𝑥 = Δ𝑥 = Δ𝑥 , 𝛽 =0,so 𝜎 is not 𝛾 = = . (10b) 𝑏 co𝑚− co𝑚+ co 𝑚 𝑜 2 2 2 2 (𝜎 +𝜎 ) (𝜎 /2 + 𝜎 /2) 𝑜 𝑜 𝑏 𝑏 adjusted which recovers the result for uniformly distributed coarse-resolution observations. By comparing (10b) with (10a), it is easy to see that this 2 2 2 The above results suggest that 𝛾 =𝜎 /(𝜎 +𝜎 ) should 𝑏 𝑏 𝑏 𝑜 overestimation can be corrected if the error variance is 2 2 2 be modiefi d into inflated from 𝜎 to 𝜎 +𝜎 for each of the two collapsed 𝑜 𝑜 𝑏 observations. 𝛾 = Then, we consider the case of Δ𝑥 →0 and Δ𝑥 → co𝑚+ co𝑚− 2 2 2 (𝜎 +𝛽 𝜎 +𝜎 ) 𝑏 𝑏 𝑜 (11b) 0.Inthiscase,theconcerned 𝑚th observation collapses with its two adjacent observations, that is, the (𝑚 + 1)th and (𝑚 − 2 for the den fi ition of Δ𝜎 (𝑥 ) in (3) . 1)thobservations.Thethreecollapsedobservationsshould be combined into one superobservation with a reduced error This modification can improve the similarity of the spatial 2 2 variance from 𝜎 to 𝜎 /3. eTh error variance reduction 𝑜 𝑜 variation of ∑ Δ𝜎 (𝑥) to that of the true error variance produced by this superobservation still can be estimated by 2 2 2 reduction, denoted by Δ𝜎 (𝑥) ≡ 𝜎 −𝜎 (𝑥),butthe 𝑏 𝑎 (3) but with 2 2 maximum (or minimum) of ∑ Δ𝜎 (𝑥), denoted by Δ𝜎 𝑚 𝑚 emx 2 2 2 𝜎 (or Δ𝜎 ), is usually not very close to that of Δ𝜎 (𝑥). 𝑏 emn 𝛾 = . (10c) 2 eTh maximum (or minimum) of Δ𝜎 (𝑥) can be closely (𝜎 +𝜎 /3) 𝑏 𝑜 2 2 estimated by Δ𝜎 (or Δ𝜎 ), the maximum (or minimum) mx mn On the other hand, without super-Obbing, the error variance 2 of Δ𝜎 (𝑥) computed by (6) for uniform coarse-resolution reduction produced by the three collapsed observations will observations but with Δ𝑥 decreased to Δ𝑥 (or increased co omn be overestimated by (3) with to Δ𝑥 ), where Δ𝑥 (or Δ𝑥 )istheminimum(ormax- omx omn omx 2 2 imum) spacing between two adjacent observations among all 3𝜎 𝜎 𝑏 𝑏 𝛾 = = . (10d) nonuniformly distributed coarse-resolution observations in 2 2 2 2 (𝜎 +𝜎 ) (𝜎 /3 + 𝜎 /3) 𝑜 𝑜 𝑏 𝑏 the one-dimension analysis domain. By adjusting Δ𝜎 to emx 2 2 2 Δ𝜎 and Δ𝜎 to Δ𝜎 ,theerrorvariancereduction can By comparing (10d) with (10c), it is easy to see that this mx emn mn be estimated by overestimation can be corrected if the error variance is 2 2 2 inflated from 𝜎 to 𝜎 +2𝜎 for each of the three collapsed 𝑜 𝑜 𝑏 2 2 2 2 observations. Δ𝜎 𝑥 =𝐹 𝑥 ≡[ ∑Δ𝜎 𝑥 −Δ𝜎 ]𝜌 + Δ𝜎 , (12a) ( ) ( ) ( ) 𝑀 𝑚 emn mn Basedonthe aboveanalyses,when theerror variance 𝑚 reduction produced by the 𝑚th observation is estimated by 2 2 2 2 where 𝜌=[Δ𝜎 −Δ𝜎 ]/[Δ𝜎 −Δ𝜎 ]. (3), the error variance should be adjusted for this observation mx mn emx emn eTh analysis error variance is then estimated by 𝜎 (𝑥) ≈ unless Δ𝑥 =Δ𝑥 =Δ𝑥 .Inparticular, itserror co𝑚+ co𝑚− co 𝑎 2 2 2 2 2 2 2 2 𝜎 (𝑥) ≡ 𝜎 −Δ𝜎 (𝑥) as in (7), except that Δ𝜎 (𝑥) is variance can be adjusted from 𝜎 to 𝜎 =𝜎 +𝛽 𝜎 with ∗ 𝑎 𝑏 𝑀 𝑀 𝑜 𝑜𝑚 𝑜 𝑚 𝑏 computed by (12a) instead of (6). As shown by the example 𝛽 given empirically by in Figure 5, the estimated 𝜎 (𝑥) captures closely not only 2 2 2 [𝐶 (Δ𝑥 )+𝐶 (Δ𝑥 )−2𝐶 (Δ𝑥 )] 𝑏 co𝑚+ 𝑏 co𝑚− 𝑏 co the maximum and minimum but also the spatial variation (11a) 𝛽 = . 2 of the benchmark 𝜎 (𝑥) computed from (1b). Using this [1 − 𝐶 (Δ𝑥 )] co estimated 𝜎 ∗(𝑥),the previously estimated A from the 𝑎 𝑒 Note that 𝛽 =2 for Δ𝑥 =Δ𝑥 =0, so the spectral formulation can be modiefi d into A , A ,or A with 𝑚 co𝑚+ co𝑚− 𝑎 𝑏 𝑐 2 2 2 adjusted error variance is 𝜎 =𝜎 +2𝜎 which recovers its th element given by the same formulation as shown in 𝑜𝑚 𝑜 𝑏 𝑖𝑗 𝑏𝑎 𝑏𝑎 𝑏𝑎 Advances in Meteorology 7 small when A is modified successively to A , A ,and A , 𝑒 𝑎 𝑏 𝑐 which quantifies the successively reduced deviation shown in Figures 7(a)–7(d). 2.4. Nonuniform Coarse-Resolution Observations without Periodic Extension. Consider that the 𝑀 coarse-resolution observations are still nonuniformly distributed in the one- dimensional analysis domain of length 𝐷 but without peri- odic extension. In this case, their produced error variance reduction Δ𝜎 (𝑥) still can be estimated by (12a) except for 6 𝑀 the following three modica fi tions. 4 2 (i) eTh maximum (or minimum) of ∑ Δ𝜎 (𝑥),thatis, 2 2 Δ𝜎 (or Δ𝜎 ), should be found in the interior domain emx emn −60 −40 −20 04 20 0 60 between the leftist and rightist observation points. x (km) (ii) For the leftist (or rightist) observation that has only one adjacent observation to its right (or le)ft in the  (x)  2 2 one-dimensional analysis domain, its error variance is still  (x) ∗  a o 2 2 2 2 adjusted from 𝜎 to 𝜎 =𝜎 +𝛽 𝜎 but 𝛽 is calculated 𝑜 𝑜𝑚 𝑜 𝑚 𝑏 𝑚 2 2 Figure 5: As in Figure 1 but for 𝑀 (=10) nonuniformly distributed by setting 𝐶 (Δ𝑥 )=0 [or 𝐶 (Δ𝑥 )=0]in(11a)for co𝑚− co𝑚+ 𝑏 𝑏 coarse-resolution observations. calculating 𝛾 in (11b). (iii) Note from (12a) that ∑ Δ𝜎 (𝑥)→0 and thus 2 2 𝐹(𝑥)→Δ𝜎 −𝜌Δ𝜎 as 𝑥 moves outward far away from the mn emn leftist (or rightist) measurement point and thus far away from 2 2 all the observations points. In this case, if Δ𝜎 −𝜌Δ𝜎 <0 mn emn (as for the case in this section), then Δ𝜎 (𝑥) estimated by 𝐹(𝑥) in (12a) may become unrealistically negative as 𝑥 moves outward beyond the leftist (or rightist) measurement point, denoted by 𝑥 . To avoid this problem, (12a) is modified into 𝑚𝑏 j 0 Δ𝜎 𝑥 =𝐹(𝑥 )−[𝐹 (𝑥 )−𝐹 𝑥 ]𝑅 ( ) ( ) 𝑀 𝑚𝑏 𝑚𝑏 1 (12b) −50 for 𝑥 beyond 𝑥 , 𝑚𝑏 −100 where 𝑅 is a factor defined by −150 𝐹(𝑥 ) 𝑚𝑏 −200 𝑅 = min {1, }. (13) 2 2 [𝐹 (𝑥 )+𝜌Δ𝜎 −Δ𝜎 ] 𝑚𝑏 emn mn −200 −150 −100 −50 0 50 100 150 200 2 2 It is easy to see from (12b) that for Δ𝜎 −𝜌Δ𝜎 <0 and mn emn thus 𝑅 =1, Δ𝜎 (𝑥) = 𝐹(𝑥 ) − [𝐹(𝑥 ) − 𝐹(𝑥)]𝑅 →0 as 1 𝑀 𝑚𝑏 𝑚𝑏 1 Figure 6: As in Figure 3 but for the case in Figure 5. |𝑥| → ∞,sothe estimated Δ𝜎 (𝑥) in (12b) can never become unrealistically negative. eTh analysis error variance is estimated by 𝜎 (𝑥) ≈ (8a), (8b), or (8c). For the case in Figure 5, the benchmark 2 2 2 2 𝜎 ∗(𝑥) ≡ 𝜎 −Δ𝜎 (𝑥) as in (7), except that Δ𝜎 (𝑥) is A isplottedinFigure6,whilethedeviationsof A , A , A , 𝑎 𝑏 𝑀 𝑀 𝑒 𝑎 𝑏 computed by (12a) [or (12b)] for 𝑥 within (or beyond) the and A from the benchmark A are shown in Figures 7(a), interior domain. As shown by the example in Figure 8, the 7(b), 7(c), and 7(d), respectively. As shown, the deviation estimated 𝜎 ∗(𝑥) captures closely the spatial variation of the becomes increasingly small when the estimated analysis error covariance matrix is modified successively to A , A ,and A . benchmark 𝜎 (𝑥) not only within but also beyond the interior 𝑎 𝑏 𝑐 As explained in Section 2.2, the accuracy of the second- domain. Using this estimated 𝜎 ∗(𝑥), A can be modified 𝑎 𝑒 step analysis depends on the accuracy of the estimated A over into A , A ,or A with its th element given by the same 𝑎 𝑏 𝑐 the extended nested domain (i.e., the nested domain plus its formulation as shown in (8a), (8b), or (8c). For the case in extended vicinities within the distance of 2𝐿 on each side Figure 8, the benchmark A (not shown) has the same interior outside the nested domain), while the latter can be measured structure (for interior grid points 𝑖 and 𝑗)asthatforthecase by the smallness of the RE of the estimated A with respect to with periodic extension in Figure 6, but significant differences the benchmark A, as defined for A in (9). The REs of A , A , are seen in the following two aspects around the four corners 𝑒 𝑒 𝑎 A ,and A computed for the case in Figure 5 are listed in the (similar to those seen from Figures 7(a) and 11(a) of Xu et 𝑏 𝑐 first column of Table 2. As listed, the RE becomes increasingly al. [8]). (i) eTh element value becomes large toward the two 2 2 (Ｇ /Ｍ ) 𝑖𝑗 8 Advances in Meteorology 200 200 150 150 100 100 50 50 j 0 j 0 −50 −50 −100 −100 −150 −150 −200 −200 −200 −150 −100 −50 0 50 100 150 200 −200 −150 −100 −50 0 50 100 150 200 i i (a) (b) 200 200 150 150 100 100 50 50 j 0 j 0 −50 −50 −100 −100 −150 −150 −200 −200 −200 −150 −100 −50 0 50 100 150 200 −200 −150 −100 −50 0 50 100 150 200 i i (c) (d) Figure 7: As in Figure 4 but for the case in Figure 5. cornersalong thediagonalline(whichisconsistentwiththe 3. Analysis Error Variance Formulations for increased analysis error variance toward the two ends of the Two-Dimensional Cases analysis domain as shown in Figure 8 in comparison with 3.1. Error Variance Reduction Produced by a Single Obser- that in Figure 5). (ii) The element value becomes virtually vation. For a single observation, say, at x ≡(𝑥 ,𝑦 ) in 𝑚 𝑚 𝑚 zero toward the two off-diagonal corners (because there is the two-dimensional space of x =(𝑥,𝑦) ,the inversematrix no periodic extension). The deviations of A , A , A ,and A 𝑒 𝑎 𝑏 𝑐 −1 −1 𝑇 2 2 from the benchmark A are shown in Figures 9(a), 9(b), 9(c), (HBH + R) in (1b) also reduces to (𝜎 +𝜎 ) ,sothe 𝑖th 𝑏 𝑜 diagonal element of A is given by the same formulation as and 9(d), respectively, for the case in Figure 8. As shown, in (2) except that 𝑥 (or 𝑥 )isreplacedby x (or x ). Here, the deviation becomes increasingly small when the estimated 𝑖 𝑚 𝑖 𝑚 analysis error covariance matrix is modiefi d successively to x denotes the 𝑖th point in the discretized analysis space 𝑅 A , A ,and A .TheREs of A , A , A ,and A are listed in the with 𝑁=𝑁 𝑁 , 𝑁 (or 𝑁 )isthenumberof analysis grid 𝑎 𝑏 𝑐 𝑒 𝑎 𝑏 𝑐 𝑥 𝑦 𝑥 𝑦 first column of Table 3. As listed, the RE becomes increasingly points along the 𝑥 (or 𝑦) direction in the two-dimensional small when A is modified successively to A , A ,and A , analysis domain. eTh length (or width) of the analysis domain 𝑒 𝑎 𝑏 𝑐 which quantifies the successively reduced deviation shown in is 𝐷 =𝑁 Δ𝑥 (or 𝐷 =𝑁 Δ𝑦) and is assumed to be much 𝑥 𝑥 𝑦 𝑦 Figures 9(a)–9(d). larger than the background error decorrelation length scale Advances in Meteorology 9 spectral formulation for two-dimensional cases in Section 2.3 of Xu et al. [8]. The domain-averaged value of ∑ Δ𝜎 (x) can be com- 𝑚 𝑚 puted by 12 2 ∬ 𝑑x ∑ Δ𝜎 (x) 𝑚 𝑚 2 𝐷 Δ𝜎 ≡ (𝐷 𝐷 ) 𝑥 𝑦 2 2 𝛾 𝜎 ∑ ∬ 𝑑x 𝐶 (x − x ) 𝑏 𝑚 𝑚 𝑏 𝑏 (15a) (𝐷 𝐷 ) 𝑥 𝑦 2 2 𝛾 𝜎 ∑ ∑ 𝐶 (x − x ) 𝑏 𝑏 𝑚 𝑖 𝑏 𝑖 𝑚 ≈ , −60 −40 −20 020 40 60 x (km) where ∬ 𝑑x denotes the integration over the two- 2 2  (x)  a e dimensional analysis domain, ∑ denotes the summation 2 2  (x)  a o over 𝑖 for the 𝑁 grid points, and 𝐷 𝐷 =𝑁 Δ𝑥𝑁 Δ𝑦 = 𝑥 𝑦 𝑥 𝑦 𝑁Δ𝑥Δ𝑦 is used in the last step. By extending 𝐶 (x − x ) Figure 8: As in Figure 5 but without periodic extension. 𝑚 with the analysis domain periodically in both the 𝑥 and 𝑦 directions, Δ𝜎 canbeestimated analytically as follows: 𝐿 in x,where Δ𝑥 (or Δ𝑦) is the grid spacing in the 𝑥 (or 𝑦) ∬ 𝑑x ∑ Δ𝜎 (x) 2 𝑚 Δ𝜎 ≡ direction and Δ𝑥 = Δ𝑦 is assumed for simplicity. (𝐷 𝐷 ) 𝑥 𝑦 Since 𝐶 (x) is a continuous function of x, the aforemen- 2 2 tioned formulation for the 𝑖th diagonal element of A can 𝛾 𝜎 ∑ ∑ ∑ ∬ 𝑦𝐶𝑑 (𝑥 − 𝑥 −𝑝𝐷 ,𝑦 − 𝑦 −𝑞𝐷 ) 𝑏 𝑏 𝑚 𝑝 𝑞 𝑏 𝑚 𝑥 𝑚 𝑦 2 2 2 = (15b) be written into 𝜎 (x)≡𝜎 −Δ𝜎 (x) also as a continuous 𝑚 𝑏 𝑚 (𝐷 𝐷 ) 𝑥 𝑦 function of x,where 2 2 2 2 𝛾 𝜎 𝑀∬𝑑 x𝐶 (x) 𝛾 𝜎 𝐼 𝐿 𝑏 𝑏 𝑏 𝑏 2 2 = = , (14) Δ𝜎 (x) ≡𝛾 [𝜎 𝐶 (x − x )] Δ𝑥 (𝐷 𝐷 ) 𝑏 𝑏 𝑏 𝑚 co 𝑚 𝑥 𝑦 is the error variance reduction produced by analyzing a single where ∬𝑑 x =∬𝑥𝑑𝑦𝑑 denotes the integration over the observation at x = x . This reduction decreases rapidly and 2 entire space of x, ∑ ∑ ∑ ∬ 𝐶 (𝑥 − 𝑥 −𝑝𝐷 ,𝑦 − 𝑚 𝑥 𝑚 𝑝 𝑞 𝑏 2 2 𝐷 becomes much smaller than it peak value of 𝛾 𝜎 𝐶 at x = x 2 2 𝑏 𝑏 𝑏 𝑚 𝑦 −𝑞𝐷 )=∑ ∬𝑑 x𝐶 (x − x )=∑ ∬𝑑 x𝐶 (x)= 𝑚 𝑦 𝑚 𝑏 𝑚 𝑚 𝑏 as |x − x | increases to 𝐿 and beyond. 𝑀∬𝑑 x𝐶 (x) is used in the second to last step, and 2 2 𝐼 ≡∬𝑑 x𝐶 (x)/𝐿 is used with Δ𝑥 ≡𝐷 𝐷 /𝑀 in 2 𝑏 co 𝑥 𝑦 3.2. Uniform Coarse-Resolution Observations with Periodic the last step. For the double-Gaussian form of 𝐶 (x)= Extension. Consider that there are 𝑀 coarse-resolution 2 2 2 2 0.6 exp(−|x| /2𝐿 ) + 0.4 exp(−2|x| /𝐿 ) used in Section 4 of observations uniformly distributed in the above analysis Xuet al.[8],wehave 𝐼 = 2𝜋(0.2 + 0.48/5) . eTh derived value domain of length 𝐷 and width 𝐷 with periodic extension 𝑥 𝑦 in (15b) is very close to the numerically computed value from 1/2 1/2 along 𝑥 and 𝑦,sotheir resolution is Δ𝑥 ≡(𝐷 𝐷 ) /𝑀 , co 𝑥 𝑦 (15a). where 𝑀=𝑀 𝑀 , 𝑀 (or 𝑀 ) denotes the number of 2 𝑥 𝑦 𝑥 𝑦 With the domain-averaged value adjusted from Δ𝜎 to observations uniformly distributed along the 𝑥 (or 𝑦)direc- 2 2 Δ𝜎 , Δ𝜎 (x) can be estimated by the same formulation as in tion in the two-dimensional analysis domain, and 𝐷 /𝑀 = 𝑥 𝑥 (6) except that 𝑥 is replaced by x.Theanalysiserror variance 𝐷 /𝑀 is assumed (so Δ𝑥 =𝐷 /𝑀 =𝐷 /𝑀 ). In this 𝑦 𝑦 co 𝑥 𝑥 𝑦 𝑦 is then estimated by case, as explained for the one-dimensional case in Section 2.2, the error variance reduction produced by each observation 2 2 2 2 𝜎 (x) ≈𝜎 (x) ≡𝜎 −Δ𝜎 (x) . (16) 𝑎 𝑎 𝑏 𝑀 can be considered as an additional reduction to the reduction produced by its neighboring observations. This additional As shown by the example in Figure 10, the estimated 𝜎 ∗(x) in reduction is smaller than the reduction produced by a single (16) is very close to the benchmark 𝜎 (x) computed precisely observation, so the error variance reduction produced by from (1b), and the deviation of 𝜎 (x) from the benchmark analyzing the 𝑀 coarse-resolution observations is bounded 𝑎 2 2 −2 𝜎 (x) is within (−0.21, 0.35) m s . On the other hand, the above by ∑ Δ𝜎 (x),which is similartothatfortheone- 𝑎 2 2 −2 constant analysis error variance (𝜎 =6.7m s )estimated by dimensionalcasein(4).Forthesamereasonasexplained for the one-dimensional case in (4), this implies that the the spectral formulation deviates from the benchmark 𝜎 (x) 2 −2 domain-averaged value of ∑ Δ𝜎 (x) is larger than the true widely from −1.91 to 2.22 m s . 2 2 2 2 averaged reduction estimated by Δ𝜎 ≡𝜎 −𝜎 ,where 𝜎 is Using the estimated 𝜎 ∗(x) in (16), the previously esti- 𝑏 𝑒 𝑒 the domain-averaged analysis error variance estimated by the mated analysis error covariance matrix, denoted by A with 2 2 (Ｇ /Ｍ ) 𝑏𝑒 𝑏𝑒 𝑏𝑠 𝑑𝑥 𝑏𝑠 𝑏𝑠 𝑏𝑠 10 Advances in Meteorology 200 200 150 150 100 100 50 50 j 0 j 0 −50 −50 −100 −100 −150 −150 −200 −200 −200 −150 −100 −50 0 50 100 150 200 −200 −150 −100 −50 0 50 100 150 200 i i (a) (b) 200 200 150 150 100 100 50 50 j 0 j 0 −50 −50 −100 −100 −150 −150 −200 −200 −200 −150 −100 −50 0 50 100 150 200 −200 −150 −100 −50 0 50 100 150 200 i i (c) (d) Figure 9: As in Figure 7 but for the case in Figure 8. its th element 𝐴 ≡𝜎 𝐶 (x − x ) obtained from the benchmark A computed precisely from (1b) can be den fi ed 𝑗𝑒𝑖 𝑒 𝑎 𝑖 𝑗 spectral formulation, can be modified into A , A ,or A in the same way as that for A in (9), except that the extended 𝑎 𝑏 𝑐 𝑒 with its th element given by the same formulation as in nested domain is two-dimensional here. The REs of A , A , 𝑒 𝑎 (8a), (8b), or (8c) except that 𝑥 (or 𝑥 )isreplacedby A ,and A computed forthecase inFigure 10 arelistedinthe 𝑖 𝑗 𝑏 𝑐 x (or x ). Again, as explained in Section 2.2 but for the first column of Table 4. As listed, the RE becomes increasingly 𝑖 𝑗 two-dimensional case here, the accuracy of the second-step small when A is modified successively to A , A ,and 𝑒 𝑎 𝑏 analysis depends on the accuracy of the estimated A over A . the extended nested domain, that is, the nested domain plus its extended vicinities within the distance of 2𝐿 outside the 3.3. Nonuniform Coarse-Resolution Observations with Peri- odic Extension. Consider that the 𝑀 coarse-resolution obser- nested domain. Here, 𝐿 is the decorrelation length scale 2 2 vations are now nonuniformly distributed in the analy- of 𝐶 (x) defined by 𝐿 ≡[−2𝐶 (x)/∇ 𝐶 (x)]| according 𝑎 𝑎 𝑎 x=0 sis domain of length 𝐷 and width 𝐷 with periodic to (4.3.12) of Daley [12], and 𝐿 (=4.52 km for the case in 𝑥 𝑦 extension, so their averaged resolution can be defined by Figure 10) can be easily computed as a by-product from the 1/2 1/2 spectral formulation. Over this extended nested domain, the Δ𝑥 ≡(𝐷 𝐷 ) /𝑀 . eTh spacing of a concerned coarse- co 𝑥 𝑦 relative error (RE) of each estimated A with respect to the resolution observation, say the 𝑚th observation, from its 𝑖𝑗 𝑖𝑗 Advances in Meteorology 11 30 30 20 20 10 10 0 0 −10 −10 −20 −20 −30 −30 −60 −40 −20 020 40 60 −60 −40 −20 020 40 60 x (km) x (km) (a) (b) 2 2 −2 2 Figure 10: (a) Benchmark analysis error variance 𝜎 (x) plotted by red contours of every 2 m s and estimated analysis error variance 𝜎 ∗ (x) 𝑎 𝑎 2 2 2 −2 in (16) plotted by blue contours. (b) Deviation of 𝜎 (x) from 𝜎 (x) plotted by colored contours of every 0.2 m s . eTh black + signs show the 𝑎 𝑎 locations of 𝑀 (= M M =12 × 6) uniformly distributed coarse-resolution observations with periodic extension along 𝑥 and 𝑦.Thecoarse- x y 2 2 2 −2 resolution observation resolution is Δ𝑥 =𝐷 /𝑀 = 𝐷 /𝑀 = 10 km, and the observation error variance is 𝜎 = 2.5 m s . eTh background co 𝑥 𝑥 𝑦 𝑦 2 2 2 2 2 2 2 2 −2 error covariance 𝜎 𝐶 (x) has the double-Gaussian form with 𝐶 (x) = 0.6 exp(−|x| /2𝐿 ) + 0.4 exp(−2|x| /𝐿 ), 𝜎 =5 m s ,and 𝐿=10 km. 𝑏 𝑏 𝑏 𝑏 The analysis domain length (or width) is 𝐷 =𝑁 Δ𝑥 = 120 km (or 𝐷 =𝑁 Δ𝑦=60 km), and the number of coarse-resolution observations 𝑥 𝑥 𝑦 𝑦 is M = M M =12 × 6. x y 𝑘th adjacent observation (among the total 4 adjacent obser- where ∑ denotes the summation over 𝑘 for the four adjacent vations), can be denoted by Δ𝑥 . Now we can consider observations nearest to the concerned 𝑚th observation. With co𝑚𝑘 2 2 2 the limiting case of Δ𝑥 →0 for 𝐾 (≤4) adjacent 𝛽 givenby(18a),theadjusted 𝜎 =𝜎 +𝛽 𝜎 recovers co𝑚𝑘 𝑚 𝑜𝑚 𝑜 𝑚 𝑏 observations with Δ𝑥 =Δ𝑥 for the remaining 4−𝐾 not only the inflated observation error variance derived above co𝑚𝑘 co (≥0) adjacent observations. In this case, the concerned 𝑚th for each limiting case [with Δ𝑥 →0 for 𝑘 = 1,2,...,𝐾 co𝑘 observationcollapses ontothesamepointwithits 𝐾 adjacent (≤4) and Δ𝑥 =Δ𝑥 for the remaining 4−𝐾 (≥0) co𝑘 co observations. The 𝐾+1 collapsed observations should be observations] but also the original observation error variance combined into one superobservation with a reduced error 𝜎 for uniformly distributed coarse-resolution observations. 2 2 2 2 2 variance from 𝜎 to 𝜎 /(𝐾+1) . eTh error variance reduction 𝑜 𝑜 The above results suggest that 𝛾 =𝜎 /(𝜎 +𝜎 ) should 𝑏 𝑏 𝑜 produced by this superobservation still can be estimated by be modiefi d into (14) but with 𝛾 = 2 𝑚 2 2 2 (𝜎 +𝛽 𝜎 +𝜎 ) 𝑚 𝑜 𝑏 𝑏 (18b) 𝛾 = . (17a) 2 2 [𝜎 +𝜎 / (𝐾+1 )] 𝑏 𝑜 2 for the den fi ition of Δ𝜎 (x) in (14) . On the other hand, without super-Obbing, the error variance This modification can improve the similarity of the spatial reduction produced by the 𝐾+1 collapsed observations will variation of ∑ Δ𝜎 (x) to that of the true error variance 𝑚 𝑚 be overestimated by (14) with 2 2 2 reduction, denoted by Δ𝜎 (x)≡𝜎 −𝜎 (x),butthe 𝑏 𝑎 2 2 2 2 maximum (or minimum) of ∑ Δ𝜎 (x), denoted by Δ𝜎 (𝐾+1 ) 𝜎 𝜎 𝑚 emx 𝑏 𝑏 2 2 𝛾 = = . (17b) 𝑏 (or Δ𝜎 ), is usually not very close to that of Δ𝜎 (x).The 2 2 2 2 emn (𝜎 +𝜎 ) [𝜎 / (𝐾+1 ) +𝜎 / (𝐾+1 )] 𝑏 𝑜 𝑏 𝑜 2 maximum (or minimum) of Δ𝜎 (x) canbeestimated by 2 2 2 Δ𝜎 (or Δ𝜎 ), the maximum (or minimum) of Δ𝜎 (x) By comparing (17b) with (17a), it is easy to see that this mx mn 𝑀 computed for uniformly distributed observations but with overestimation can be corrected if the error variance is 2 2 2 2 Δ𝑥 decreased to Δ𝑥 (or increased to Δ𝑥 ), where inflated from 𝜎 to 𝜎 =𝜎 +𝐾𝜎 for each of the (𝐾 + 1) co omn omx 𝑜 𝑜𝑚 𝑜 𝑏 Δ𝑥 (or Δ𝑥 ) is the minimum (or maximum) spacing collapsed observations. omn omx of adjacent observations among all nonuniformly distributed Basedonthe aboveanalyses,when theerror variance coarse-resolution observations in the two-dimension analysis reduction produced by the concerned 𝑚th observation is domain. Specifically, Δ𝑥 (or Δ𝑥 )isestimatedby estimatedby(14),theerror variance should be adjusted for omn omx min (∑ |x − x |)/𝐾 with 𝐾=2 and Δ𝑥 is estimated this observation unless Δ𝑥 =Δ𝑥 for 𝑘 = 1,2,3 ,and 4. 𝑚 𝑘 𝑚 𝑚𝑘 omx co𝑚𝑘 co 2 2 2 2 by max (∑ |x − x |)/𝐾 with 𝐾=4 ,where x denotes In particular, 𝜎 canbeadjustedto 𝜎 =𝜎 +𝛽 𝜎 with 𝛽 𝑚 𝑘 𝑚 𝑚𝑘 𝑚 𝑜 𝑜𝑚 𝑜 𝑚 𝑏 𝑚 the 𝑚th observation point, x denotes the observation point given empirically by 𝑚𝑘 that is 𝑘th nearest to x ,min (or max ) denotes the 𝑚 m m 2 2 minimum (or maximum) over index 𝑚 for all the coarse- [∑ 𝐶 (Δ𝑥 )−4𝐶 (Δ𝑥 )] 𝑘 𝑏 co𝑚𝑘 𝑏 co (18a) 𝛽 = , resolution observation points in the two-dimension analysis [1 − 𝐶 (Δ𝑥 )] co 𝑏 domain, ∑ denotes the summation over 𝑘 from 1 to 𝐾,and y (km) y (km) 𝑏𝑎 𝑏𝑎 𝑏𝑎 12 Advances in Meteorology 30 30 20 20 10 10 0 0 −10 −10 −20 −20 −30 −30 −60 −40 −20 0 20 40 60 −60 −40 −20 020 40 60 x (km) x (km) (a) (b) Figure 11: As in Figure 10 but for the second set of innovations with nonuniformly distributed coarse-resolution observations, and the colored 2 −2 2 2 contours are plotted every 1 m s for the deviation of 𝜎 (x) from 𝜎 (x) in panel (b). 𝑎 𝑎 𝐾 is the total number of adjacent observation points (nearest where 𝑀 (or 𝑀 ) is estimated by the nearest integer to 𝑥 𝑦 to x ) used for estimating Δ𝑥 (with 𝐾=2 )or Δ𝑥 (or 𝐷 /Δ𝑥 (or 𝐷 /Δ𝑥 ). eTh total number of near-boundary 𝑚 omn omx 𝑥 co 𝑦 co 2 2 2 2 observations is thus given by 2(𝑀 +𝑀 )−8. To identify 𝐾=4 ). By adjusting Δ𝜎 to Δ𝜎 and Δ𝜎 to Δ𝜎 ,the 𝑥 𝑦 emx mx emn mn these near-boundary observations, we need to divide the 2D error variance reduction can be estimated by domain uniformly along the 𝑥-direction and 𝑦-direction into 𝑀 𝑀 boxes, so there are 2(𝑀 +𝑀 )−8 boundary boxes 2 2 2 2 𝑥 𝑦 𝑥 𝑦 Δ𝜎 (x) =𝐹 (x) ≡[ ∑Δ𝜎 (x) −Δ𝜎 ]𝜌 + Δ𝜎 , (19a) 𝑀 𝑚 emn mn (not including the four corner boxes). If a boundary box containsnocoarse-resolutionobservation,thenitisanempty 2 2 2 2 box and should be substituted by its adjacent interior box (as where 𝜌=[Δ𝜎 −Δ𝜎 ]/[Δ𝜎 −Δ𝜎 ]. mx mn emx emn 2 a substituted boundary box). From each nonempty boundary eTh analysis error variance is then estimated by 𝜎 (x)≈ box (including substituted boundary box), we can n fi d one 2 2 2 2 𝜎 (x)≡𝜎 −Δ𝜎 (x) as in (16), except that Δ𝜎 (x) is 𝑎 𝑏 𝑀 𝑀 near-boundary observation that is nearest to the associated computed by (19a). As shown by the example in Figure 11, boundary. A closed loop of observation boundary can be 2 2 the estimated 𝜎 ∗(x) is fairly close to the benchmark 𝜎 (x), 𝑎 𝑎 constructed by piece-wise linear segments with every two 2 2 and the deviation of 𝜎 ∗(x) from the benchmark 𝜎 (x)is 𝑎 𝑎 neighboring near-boundary observation points connected by 2 −2 within (−2.40, 4.20) m s . On the other hand, the constant alinearsegmentandwitheachnear-cornerobservationpoint 2 2 −2 analysis error variance (𝜎 = 6.7 m s )estimated bythe connectedbyalinear segmenttoeachofits twoneighboring spectral formulation deviates from the benchmark 𝜎 (x) near-boundary observation points. 2 −2 After the above preparations, the error variance reduction widely from −9.98 to 3.83 m s .Using this estimated 𝜎 ∗(x), Δ𝜎 (x) can be estimated by (19a) with the following three the previously estimated A from the spectral formulation can modifications: be modiefi d into A , A ,or A with its th element given 𝑎 𝑏 𝑐 (i) eTh maximum (or minimum) of ∑ Δ𝜎 (x),thatis, by thesametwo-dimensional versionof(8a),(8b),or(8c) 𝑚 𝑚 2 2 as explained in Section 3.2. eTh REs of A , A , A ,and A Δ𝜎 (or Δ𝜎 ) should be found in the interior domain of 𝑒 𝑎 𝑏 𝑐 emx emn computed for the case in Figure 11 are listed in the first column |𝑥| < 𝐷 /2 − Δ𝑥 and |𝑦| < 𝐷 /2 − Δ𝑥 . 𝑥 co 𝑦 co of Table 5. As listed, the RE becomes increasingly small when (ii) For each above defined near-boundary (or near- A is modified successively to A , A ,and A . corner) observation that has only three (or two) adjacent 𝑒 𝑎 𝑏 𝑐 observations, its error variance is still adjusted from 𝜎 to 2 2 2 3.4. Nonuniform Coarse-Resolution Observations without 𝜎 +𝛽 𝜎 but 𝛽 is calculated by setting 𝐶 (Δ𝑥 )=0 in 𝑜 𝑚 𝑏 𝑚 𝑏 co𝑘 Periodic Extension. Consider that the 𝑀 coarse-resolution (18a) for 𝑘=4 (or 𝑘=3 and 4). observations are still nonuniformly distributed in the analysis (iii) Note from (19a) that ∑ Δ𝜎 (x)→0 and thus 𝑚 𝑚 2 2 2 domain of length 𝐷 and width 𝐷 but without periodic Δ𝜎 (x)→Δ𝜎 −𝜌Δ𝜎 <0 as x moves outward far 𝑥 𝑦 𝑀 mn emn extension. In this case, their averaged resolution is still away from all the observations points. In this case, if Δ𝜎 − mn 1/2 2 2 defined by Δ𝑥 ≡(𝐷 𝐷 /𝑀) . To estimate their produced co 𝑥 𝑦 𝜌Δ𝜎 <0 (as for the case in this section), then Δ𝜎 (x) emn 𝑀 error variance reduction, we need to modify the formulations estimated by (19a) may become unrealistically negative as x constructed in the previous subsection with the following moves outward beyond the above constructed observation preparations. First, we need to identify four near-corner boundary loop. To avoid this problem, (19a) is modified into observations among all the coarse-resolution observations. Each near-corner observation is defined as the one that near- est to one of the four corners of the analysis domain. eTh n, Δ𝜎 (x) =𝐹( x )−[𝐹 ( x )−𝐹 (x)]𝑅 𝑀 𝑚𝑏 𝑚𝑏 2 we need to identify 𝑀 −2 (or 𝑀 −2) near-boundary obser- 𝑥 𝑦 (19b) vations associated with each 𝑥-boundary (or 𝑦-boundary), for x outside the observation boundary loop, y (km) y (km) 𝑖𝑗 Advances in Meteorology 13 30 30 20 20 10 10 0 0 −10 −10 −20 −20 −30 −30 −60 −40 −20 0 20 40 60 −60 −40 −20 0 20 40 60 x (km) x (km) (a) (b) Figure 12: As in Figure 11 but without periodic extension. where 𝑅 is a factor defined by analyzing the high-resolution innovations in the second step with the background error covariance updated by A , A , A , 𝑒 𝑎 𝑏 𝐹( x ) 𝑚𝑏 and A , respectively, aer ft the coarse-resolution innovations 𝑅 = min {1, }, (20) 2 2 [𝐹 (x )+𝜌Δ𝜎 −Δ𝜎 ] 𝑚𝑏 emn mn areanalyzedinthefirststep.TheTEeissimilartothefirsttype of two-step experiment (named TEA) in Xu et al. [8], but the x is the projection of x on the observation boundary loop 𝑚𝑏 TEa, TEb, and TEc are new here. As in Xu et al. [8], a single- and the projection from x is along the direction normal to step experiment, named SE, is also designed for analyzing the x-associated domain boundary (nearest to x). However, if all the innovations in a single step. In each of the above vfi e x is closer to a corner observation point than the remaining types of experiments, the analysis increment is obtained by part of the observation boundary loop, then x is simply that 𝑚𝑏 using the standard conjugate gradient descent algorithm to near-corner observation point. It is easy to see from (19b) 2 2 2 minimize the cost-function (formulated as in (7) of Xu et al. that, for Δ𝜎 −𝜌Δ𝜎 <0 and thus 𝑅 =1, Δ𝜎 (x)= mn emn 2 𝑀 [8]) with the number of iterations limited to𝑛=20 ,50, or 100 𝐹(x )−[𝐹(x )−𝐹( x)]𝑅 =𝐹( x )→0 as |x|→−∞,sothe 𝑚𝑏 𝑚𝑏 2 𝑚𝑏 2 before the final convergence to mimic the computationally estimated Δ𝜎 (x) in (19b) can never become unrealistically constrained situations in operational data assimilation. Three negative. sets of simulated innovations are generated for the above eTh analysis error variance is estimated by 𝜎 (x)≈ vfi e types of experiments. eTh first set consists of 𝑀 (=10) 2 2 2 2 𝜎 ∗(x)≡𝜎 −Δ𝜎 (x) as in (16), except that Δ𝜎 (x) is com- 𝑎 𝑏 𝑀 𝑀 uniformly distributed coarse-resolution innovations over the puted by (19a) [or (19b)] for x inside (or outside) the closed analysis domain (see Figure 1) with periodic extension and observation boundary loop. As shown by the example in 󸀠 𝑀 (=74) high-resolution innovations in the nested domain Figure 12, the estimated 𝜎 ∗(x) is fairly close to the benchmark of length 𝐷/6 (similar to those shown by the purple × signs 2 2 2 𝜎 (x), and the deviation of 𝜎 ∗(x) from the benchmark 𝜎 (x) in Figure 1 of Xu et al. [8] but generated at the grid points 𝑎 𝑎 𝑎 2 −2 is within (−4.08, 5.54) m s . On the other hand, the constant not covered by the coarse-resolution innovations within the 2 2 −2 analysis error variance (𝜎 = 6.7 m s )estimated bythe nested domain). The second (or third) set is the same as the rfi st set except that the coarse-resolution innovations spectral formulation deviates from the benchmark 𝜎 (x) very 2 −2 are nonuniformly distributed with (or without) periodic widely from −16.1 to 3.82 m s .Using theestimated 𝜎 ∗(x), extension as shown in Figure 5 (or Figure 8). All the innova- the previously estimated A from the spectral formulation tions are generated by simulated observation errors subtract- can be modified into A , A ,or A with its th element 𝑎 𝑏 𝑐 ing simulated background errors at observation locations. given by the same two-dimensional version of (8a), (8b), or Observation errors are sampled from computer-generated (8c) as explained in Section 3.2. eTh REs of A , A , A ,and 𝑒 𝑎 𝑏 −1 uncorrelated Gaussian random numbers with 𝜎 =2.5 ms A computed for the case in Figure 12 are listed in the first for both coarse-resolution and high-resolution observations. column of Table 6. As listed, the RE becomes increasingly Background errors are sampled from computer-generated small when A is modified successively to A , A ,and A . 𝑒 𝑎 𝑏 𝑐 −1 spatially correlated Gaussian random efi lds with 𝜎 =5 ms and 𝐶 (𝑥) modeled by the double-Gaussian form given in 4. Numerical Experiments for Section2.2 (alsosee thecaption of Figure 1).Thecoarse- One-Dimensional Cases resolution innovations in the rfi st, second, and third sets are thus generated in consistency with the three cases in Figures 4.1. Experiment Design and Innovation Data. In this sec- 1, 5, and 8, respectively. tion, idealized one-dimensional experiments are designed and performed to examine to what extent the successively improved estimate of A in (8a), (8b), and (8c) can improve the 4.2. Results from the First Set of Innovations. The first set of two-step analysis. In particular, four types of two-step exper- innovations is used here to perform each of the vfi e types iments, named TEe, TEa, TEb, and TEc, are designed for of experiments with the number of iterations limited to y (km) y (km) 𝑖𝑗 14 Advances in Meteorology Table 2: As in Table 1 but for the second set of innovations with periodic extension. Experiment 𝑛 = 20 𝑛 = 50 𝑛 = 100 Final SE 0.711 0.334 0.276 0.018 at 𝑛 = 404 TEe with RE(A ) = 0.355 0.482 0.439 0.442 at 𝑛=76 TEa with RE(A ) = 0.238 0.418 0.388 0.348 0.353 at 𝑛 = 108 TEb with RE(A ) = 0.197 0.318 0.288 0.257 0.243 at 𝑛 = 179 TEc with RE(A ) = 0.148 0.213 0.151 0.155 at 𝑛=52 Table 3: As in Table 2 but for the third set of innovations without periodic extension. Experiment 𝑛 = 20 𝑛 = 50 𝑛 = 100 Final SE 0.499 0.328 0.194 0.012 at 𝑛 = 451 TEe with RE(A ) = 0.355 0.463 0.424 0.399 at 𝑛=73 TEa with RE(A ) = 0.238 0.394 0.358 0.385 at 𝑛=54 TEb with RE(A ) = 0.196 0.281 0.273 0.248 at 𝑛=77 TEc with RE(A ) = 0.147 0.215 0.149 0.123 at 𝑛=77 𝑛=20 , 50, or 100 before the n fi al convergence. eTh accuracy butmuchlesssignicfi antthanthatofTEAoverSE in Table of the analysis increment obtained from each experiment 2 of Xu et al. [8]. This reduced improvement can be largely with each limited 𝑛 is measured by its domain-averaged RMS explained by the fact that the coarse-resolution innovations error (called RMS error for short hereaer) ft with respect to are generated here not only more sparsely but also more the benchmark analysis increment computed precisely from nonuniformly than those in Section 3.3 of Xu et al. [8] and (1a). Table 1 lists the RMS errors of the analysis increments the deviation of A from the benchmark A becomes much obtained from the SE, TEe, TEa, TEb, and TEc with the larger in Figure 7(a) here than that in Figure 7(b) of Xu et number of iterations increased from 𝑛=20 to 50, 100, and/or al.[8].TheTEa outperformsTEe for 𝑛=20 and 50 but the n fi al convergence. still underperforms SE for 𝑛 increased to 50 and beyond. The As showninTable 1, theTEeoutperformsSEfor 𝑛=20 , improvement of TEa over TEe is consistent with the improved 50, and 100 but not for 𝑛 increased to the final convergence. accuracy of A [RE(A ) = 0.238] over A [RE(A ) = 0.355]. 𝑎 𝑎 𝑒 𝑒 The improved performance of TEe over SE is similar to but The TEb outperforms TEa for each listed value of 𝑛 and less signica fi nt than that of TEA over SE in Table 1 of Xu et al. also outperforms SE for 𝑛 up to 100. The improvement of [8]. The reduced improvement can be largely explained by the TEb over TEa is consistent with the improved accuracy of fact that the coarse-resolution innovations are generated here A [RE(A ) = 0.197] over A .TheTEc outperforms TEb 𝑏 𝑏 𝑎 more sparsely and the deviation of A from the benchmark for each listed value of 𝑛, and the improvement is consistent A is thus increased (as seen from Figure 4(a) in comparison with the improved accuracy of A [RE(A ) = 0.148] 𝑐 𝑐 with Figure 5(b) of Xu et al. [8]). The TEa outperforms TEe over A . for 𝑛=20 and 50 before 𝑛 increased to 100 (which is very closetothe nfi alconvergenceat 𝑛 = 116 for TEa). eTh 4.4. Results from the iTh rd Set of Innovations. The third set of improvement of TEa over TEe is consistent with and can be innovations is used here to perform each of the vfi e types of largely explained by the improved accuracy of A [RE(A )= experiments with the number of iterations limited to 𝑛=20 , 𝑎 𝑎 0.156] over A [RE(A ) = 0.229].TheTEb outperforms 50,or100 before thefinalconvergence.Thedomain-averaged 𝑒 𝑒 TEa for 𝑛=20 and50(before thefinalconvergence at RMS errors of the analysis increments obtained from the four 𝑛=67 ). The improvement of TEb over TEa is consistent two-step experiments are shown in Table 3 versus those from with the improved accuracy of A [RE(A ) = 0.101] over the SE. As shown, the TEe outperforms SE for 𝑛=20 but not 𝑏 𝑏 A . eTh TEc outperforms TEb for each listed value of 𝑛,and so for 𝑛=50 . eTh improvement of TEe over SE is much less the improvement is consistent with the improved accuracy of signica fi nt than that of TEA over SE in Table 3 of Xu et al. [8], A [with RE(A ) = 0.042] over A . and this reduced improvement can be explained by the same 𝑐 𝑐 𝑏 fact as stated for the previous case in Section 4.3. eTh TEa 4.3. Results from the Second Set of Innovations. The second set outperforms TEe for 𝑛=20 and 50, and the improvement of innovations is used here to perform each of the ve fi types of is consistent with the improved accuracy of A [RE(A )= 𝑎 𝑎 experiments with the number of iterations limited to 𝑛=20 , 0.238] over A [RE(A ) = 0.355].TheTEb outperforms 𝑒 𝑒 50,or100 beforethefinalconvergence.Thedomain-averaged TEa for each listed value of 𝑛, which is consistent with the RMS errors of the analysis increments obtained from the four improved accuracy of A [RE(A ) = 0.196] over A .TheTEc 𝑏 𝑏 𝑎 two-step experiments are shown in Table 2 versus those from outperforms TEb for each listed value of 𝑛, which is consistent the SE. As shown, the TEe outperforms SE for 𝑛=20 but not with the improved accuracy of A [RE(A ) = 0.147] 𝑐 𝑐 so for 𝑛=50 .Theimprovement ofTEeoverSEissimilarto over A . 𝑏 Advances in Meteorology 15 Table 4: As in Table 1 but for the two-dimensional case in Figure 10 in which the first set of two-dimensional innovations is used with periodic extension. Experiment 𝑛 = 20 𝑛 = 50 𝑛 = 100 Final SE 0.742 0.364 0.154 0.071 at 𝑛 = 201 TEe with RE(A ) = 0.233 0.394 0.185 0.108 0.104 at 𝑛 = 140 TEa with RE(A ) = 0.181 0.397 0.186 0.102 0.097 at 𝑛 = 145 TEb with RE(A ) = 0.130 0.403 0.183 0.089 0.085 at 𝑛 = 133 TEc with RE(A ) = 0.038 0.397 0.160 0.064 0.059 at 𝑛 = 183 Table 5: As in Table 4 but for the two-dimensional case in Figure 11 where the second set of innovations is used with periodic extension. Experiment 𝑛 = 20 𝑛 = 50 𝑛 = 100 Final SE 0.757 0.364 0.175 0.069 at 𝑛 = 241 TEe with RE(A ) = 0.462 0.416 0.202 0.149 0.144 at 𝑛 = 140 TEa with RE(A ) = 0.274 0.411 0.202 0.143 0.144 at 𝑛 = 215 TEb with RE(A ) = 0.244 0.402 0.195 0.136 0.132 at 𝑛 = 191 TEc with RE(A ) = 0.165 0.403 0.179 0.106 0.103 at 𝑛 = 287 5. Numerical Experiments for value of 𝑛 before the final convergence, which is similar to the improved performance of TEA over SE shown in Table 4 Two-Dimensional Cases ofXu et al.[8].TheTEa outperformsTEeas 𝑛 increases to 100 5.1. Experiment Design and Innovation Data. In this section, and beyond, which is consistent with the improved accuracy idealized two-dimensional experiments are designed and of A [RE(A ) = 0.181] over A [RE(A ) = 0.233].The 𝑎 𝑎 𝑒 𝑒 namedsimilarly to thoseinSection 4exceptthatsimu- TEb outperforms TEa as 𝑛 increases to 50 and beyond, which lated innovations are generated in three sets for the two- is consistent with the improved accuracy of A [RE(A )= 𝑏 𝑏 dimensional cases in Figures 10, 11, and 12, respectively. 0.130] over A . eTh TEc outperforms TEb for each listed In particular, the first set consists of 𝑀 (= M M =12 value of 𝑛, which is consistent with the improved accuracy x y × 6) uniformly distributed coarse-resolution innovations of A [RE(A ) = 0.038] over A . 𝑐 𝑐 𝑏 over the periodic analysis domain (as shown in Figure 10) and 𝑀 (=66) high-resolution innovations generated at the 5.3. Results from the Second Set of Innovations. The second grid points not covered by the coarse-resolution innovations set of innovations is used here to perform each of the five within the nested domain. eTh nested domain ( 𝐷 /6 = 20 km types of experiments with the number of iterations limited to long and 𝐷 /6 = 10 km wide) is the same as that shown 𝑛=20 , 50, or 100 before the final convergence. The domain- in Figure 16 of Xu et al. [8]. Again, all the innovations averaged RMS errors of the analysis increments obtained are generated by simulated observation errors subtract- from the four two-step experiments are shown in Table 5 ing simulated background errors at observation locations. versus those from the SE. As shown, the TEe outperforms SE Observation errors are sampled from computer-generated for each listed value of 𝑛 before the n fi al convergence. eTh −1 uncorrelated Gaussian random numbers with 𝜎 =2.5ms TEa outperforms TEe slightly, and the improved performance for both coarse-resolution and high-resolution observations. is consistent with the improved accuracy of A [RE(A )= 𝑎 𝑎 Background errors are sampled from computer-generated 0.274] over A [RE(A ) = 0.462].TheTEb outperforms 𝑒 𝑒 −1 TEA for each listed value of 𝑛, which is consistent with the spatially correlated Gaussian random efi lds with 𝜎 =5ms and 𝐶 (x) modeled by the double-Gaussian form given in improved accuracy of A [RE(A ) = 0.244] over A .TheTEc 𝑏 𝑏 𝑎 outperforms TEb for 𝑛>20 ,andtheimprovedperformance Section 3.2 (also see the caption of Figure 10). The second is consistent with the improved accuracy of A [RE(A )= (or third) set is the same as the first set except that the 𝑐 𝑐 coarse-resolution innovations are nonuniformly distributed 0.165] over A . with (or without) periodic extension as shown in Figure 11 (or Figure 12). 5.4. Results from the Third Set of Innovations. The third set of innovations is used here to perform each of the vfi e types of 5.2. Results from the First Set of Innovations. The first set of experiments with the number of iterations limited to 𝑛=20 , innovations is used here to perform each of the vfi e types of 50,or100 before thefinalconvergence.Thedomain-averaged experiments with the number of iterations limited to 𝑛=20 , RMS errors of the analysis increments obtained from the 50,or100 beforethefinalconvergence.Thedomain-averaged four two-step experiments are shown in Table 6 versus those RMS errors of the analysis increments obtained from the four from the SE. As shown, the TEe outperforms SE for each two-step experiments are shown in Table 4 versus those from listed value of 𝑛 before the n fi al convergence. eTh improved the SE. As shown, the TEe outperforms SE for each listed performance of TEe over SE is similar to but less significant 16 Advances in Meteorology Table 6: As in Table 5 but for the two-dimensional case in Figure 12 where the third set of innovations is used without periodic extension. Experiment 𝑛 = 20 𝑛 = 50 𝑛 = 100 Final SE 0.808 0.453 0.176 0.078 at 𝑛 = 235 TEe with RE(A ) = 0.462 0.509 0.179 0.136 0.134 at 𝑛 = 161 TEa with RE(A ) = 0.305 0.497 0.184 0.146 0.140 at 𝑛 = 213 TEb with RE(A ) = 0.258 0.495 0.179 0.135 0.127 at 𝑛 = 170 TEc with RE(A ) = 0.240 0.473 0.157 0.103 0.101 at 𝑛 = 201 than that ofTEAoverSEinTable 5ofXuetal.[8],andthe simplest type, the total error variance reduction is estimated reason is mainly because the coarse-resolution innovations in two steps. First, the error variance reduction produced are generated more sparsely and nonuniformly than those by analyzing each coarse-resolution observation as a single in Section 4.3 of Xu et al. [8]. eTh TEa outperforms TEe observation is equally weighted and combined into the total. for 𝑛=20 but not so as 𝑛 increases to 50 and beyond, Then, the combined total error variance reduction is adjusted although A has an improved accuracy [RE(A ) = 0.305] by a constant to match to the domain-averaged total error 𝑎 𝑎 over A [RE(A ) = 0.462]. eTh TEb outperforms TEa for variance reduction estimated by the spectral formulation 𝑒 𝑒 each listed value of 𝑛,andtheimprovedperformance is [see (5a), (5b), (15a), and (15b)]. eTh estimated analysis consistent with the improved accuracy of A [RE(A )= error variance (i.e., the background error variance minus the 𝑏 𝑏 0.258] over A . eTh TEc outperforms TEb for each listed adjusted total error variance reduction) captures not only the value of 𝑛, which is consistent with the improved accuracy domain-averaged value but also the spatial variation of the of A [RE(A ) = 0.240] over A . benchmark truth (see Figures 1, 2, and 10). 𝑐 𝑐 𝑏 (ii) The second type consists of nonuniformly distributed coarse-resolution observations with periodic extension. For 6. Conclusions this more general type, the total error variance reduction is also estimated in two steps: The rfi st step is similar to In this paper, the two-step variational method developed that for the first type but the combination into the total is in Xu et al. [8] for analyzing observations of different weighted based on the averaged spacing of each concerned spatial resolutions is improved by considering and efficiently estimating the spatial variation of analysis error variance observation from its neighboring observations [see (11a), (11b), (18a), and (18b)]. In the second step, the combined produced by analyzing coarse-resolution observations in the total error variance reduction is adjusted and scaled to first step. eTh constant analysis error variance computed from the spectral formulations in Xu et al. [8] can represent the match the maximum and minimum of the true total error variance reduction estimated from the spectral formulation spatial averaged value of the true analysis error variance but for uniformly distributed coarse-resolution observations but itcannotcapturethespatialvariation inthetrueanalysis with the observation resolutions set, respectively, to the error variance. As revealed by the examples presented in minimum spacing and maximum spacing of the nonuni- this paper (see Figures 1, 2, 5, and 8 for one-dimensional formly distributed coarse-resolution observations [see (12a) cases and Figures 10–12 for two-dimensional cases), the true and (19a)]. The estimated analysis error variance captures analysis error variance tends to have increasingly large spatial notonlythe maximumandminimumbutalsothespatial variations when the coarse-resolution observations become increasingly nonuniform and/or sparse, and this is especially variation of the benchmark truth (see Figures 5 and 11). (iii) The third type consists of nonuniformly distributed true and serious when the separation distances between coarse-resolution observations without periodic extension. neighboring coarse-resolution observations become close to or even locally larger than the background error decorrelation For this most general type, the total error variance reduction is estimated with the same two steps as for the second type, length scale. In this case, the spatial variation of analysis error except that three modica fi tions are made to improve the variance and associated spatial variation in analysis error estimation near and at the domain boundaries [see (i)–(iii) covariance need to be considered and estimated efficiently in in Sections 2.4 and 3.4]. The analysis error variance na fi lly order to further improve the two-step analysis. eTh analysis error variance can be viewed equivalently estimated captures the spatial variation of the benchmark truth not only in the interior domain but also near and at the and conveniently as the background error variance minus domain boundaries (see Figures 8 and 12). the total error variance reduction produced by analyzing all the coarse-resolution observations. To ecffi iently estimate the The above estimated spatially varying analysis error latter, analytic formulations are constructed for three types of variance is used to modify the analysis error covariance coarse-resolution observations in one- and two-dimensional computed from the spectral formulations of Xu et al. [8] spaces with successively increased complexity and generality. in three different forms [see (8a), (8b), and (8c)]. eTh The main results and major ndings fi are summarized below rfi st is a conventional formulation in which the covariance for each type of coarse-resolution observations: is modulated by the spatially varying standard deviation (i) eTh rst fi type consists of uniformly distributed coarse- separately via each entry of the covariance to retain the resolution observations with periodic extension. For this self-adjointness. This modulation has a chessboard structure Advances in Meteorology 17 but the desired modulation has a banded structure (along References the direction perpendicular to the diagonal line) as revealed [1] W.-S. Wu, R. J. Purser, and D. F. Parrish, “Three-dimensional by the to-be-corrected deviation from the benchmark truth variational analysis with spatially inhomogeneous covariances,” (see Figure 4(a)), so the deviation is only partially reduced Monthly Weather Review,vol.130,no. 12,pp.2905–2916, 2002. (see Figure 4(b)). The second formulation is new, in which [2] R.J.Purser, W.-S.Wu, D.F.Parrish,andN.M.Roberts, the modulation is realigned to capture the desired banded “Numerical aspects of the application of recursive filters to structure and yet still retain the self-adjointness. The devia- variational statistical analysis. 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Liu,“Multistep constructing covariance matrices with nonconstant variances variational data assimilation: Important issues and a spectral for general applications beyond this paper. approach,” Tellus Series A: Dynamic Meteorology and Oceanog- eTh formulations constructed in this paper for estimating raphy, vol. 68, no. 1, Article ID 31110, 2016. the spatial variation of analysis error variance and associated [9] Q. Xu and L. Wei, “Multistep and multi-scale variational data spatial variation in analysis error covariance are effective for assimilation: Spatial variations of analysis error variance,” in further improving the two-step variational method devel- Proceedings of the 17th Conference on Mesoscale Processes, opedin Xuetal.[8],especiallywhenthe coarse-resolution American Meteorological Society,SanDiego,CA,USA,2017. observations become increasingly nonuniform and/or sparse. [10] A. H. Jazwinski, Stochastic Processes and Filtering eo Th ry , These formulations will be extended together with the spec- Academic Press, New York, NY, USA, 1970. tral formulations of Xu et al. [8] for real-data applications [11] I. D. Rutherford, “Data assimilation by statistical interpolation in three-dimensional space with the variational data assim- of forecast error fields,” Journal of the Atmospheric Sciences,vol. ilation system of Gao et al. [5], in which the analyses are 29,no.5, pp.809–815,1972. univariate and performed in two steps. Such an extension is [12] R. Daley, Atmospheric Data Analysis, Cambridge University currently being developed. Press, Cambridge, UK, 1991. [13] G.H.GolubandC.F.Van Loan, Matrix computations,vol.3of Conflicts of Interest Johns Hopkins University Press,Baltimore,MD,USA, 1983. eTh authors declare that there are no conflicts of interest regarding the publication of this paper. Acknowledgments eTh authors are thankful to Dr. Jindong Gao for their constructive comments and suggestions that improved the presentation of the paper. eTh research work was supported by the ONR Grants N000141410281 and N000141712375 to the University of Oklahoma (OU). Funding was also provided to CIMMS by NOAA/Office of Oceanic and Atmospheric Research under NOAA-OU Cooperative Agreement no. NA11OAR4320072, US Department of Commerce. 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