Quantifying the Location Error of Precipitation Nowcasts

Arthur Costa Tomaz de Souza; Georgy Ayzel; Maik Heistermann

doi:10.1155/2020/8841913

Quantifying the Location Error of Precipitation Nowcasts

Costa Tomaz de Souza, Arthur;Ayzel, Georgy;Heistermann, Maik 2020-12-03 00:00:00 Hindawi Advances in Meteorology Volume 2020, Article ID 8841913, 12 pages https://doi.org/10.1155/2020/8841913 Research Article Arthur Costa Tomaz de Souza , Georgy Ayzel , and Maik Heistermann University of Potsdam, Institute of Environmental Science and Geography, Potsdam 14476, Germany Correspondence should be addressed to Arthur Costa Tomaz de Souza; costatomazde@uni-potsdam.de Received 11 September 2020; Accepted 21 October 2020; Published 3 December 2020 Academic Editor: Francesco Viola Copyright © 2020 Arthur Costa Tomaz de Souza et al. 'is 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. In precipitation nowcasting, it is common to track the motion of precipitation in a sequence of weather radar images and to extrapolate this motion into the future. 'e total error of such a prediction consists of an error in the predicted location of a precipitation feature and an error in the change of precipitation intensity over lead time. So far, veriﬁcation measures did not allow isolating the extent of location errors, making it diﬃcult to speciﬁcally improve nowcast models with regard to location prediction. In this paper, we introduce a framework to directly quantify the location error. To that end, we detect and track scale-invariant precipitation features (corners) in radar images. We then consider these observed tracks as the true reference in order to evaluate the performance (or, inversely, the error) of any model that aims to predict the future location of a precipitation feature. Hence, the location error of a forecast at any lead timeΔt ahead of the forecast time t corresponds to the Euclidean distance between the observed and the predicted feature locations at t +Δt. Based on this framework, we carried out a benchmarking case study using one year worth of weather radar composites of the German Weather Service. We evaluated the performance of four extrapolation models, two of which are based on the linear extrapolation of corner motion from t − 1 to t (LK-Lin1) and t − 4 to t (LK-Lin4) and the other two are based on the Dense Inverse Search (DIS) method: motion vectors obtained from DIS are used to predict feature locations by linear (DIS-Lin1) and Semi-Lagrangian extrapolation (DIS-Rot1). Of those four models, DIS-Lin1 and LK-Lin4 turned out to be the most skillful with regard to the prediction of feature location, while we also found that the model skill dramatically depends on the sinuosity of the observed tracks. 'e dataset of 376,125 detected feature tracks in 2016 is openly available to foster the improvement of location prediction in extrapolation-based nowcasting models. 'e present study focuses on nowcasts that are based on 1.Introduction ﬁeld tracking. 'e performance (or skill) of ﬁeld tracking Forecasting precipitation for the imminent future (i.e., techniques is mostly veriﬁed by comparing the forecast minutes to hours) is typically referred to as precipitation precipitation ﬁeld F for time t +Δt against the observed t+Δt nowcasting. A common nowcasting technique is to track the precipitation ﬁeld O at time t +Δt, where t is the forecast t+Δt motion of precipitation from a sequence of weather radar time and Δt is the lead time. A large variety of veriﬁcation images and to extrapolate that motion into the future [1]. For measures have been suggested in the literature (see, e.g., that purpose, we often assume that the intensity of pre- [4, 5]). Most of them, however, struggle with disentangling cipitation features in the most recent image remains con- diﬀerent sources of error: when we compare F to O , t+Δt t+Δt stant over the lead time period—an assumption commonly how can we know the cause of the disagreement? Was it our referred to as “Lagrangian persistence” [2]. In Lagrangian prediction of the future location of a precipitation feature, or ﬁeld tracking, a velocity vector is obtained for each pixel of a was it how precipitation intensity changed over time? Some precipitation ﬁeld, and that vector ﬁeld is used to extrapolate veriﬁcation scores, such as the Fractions Skill Score [6], the motion of the entire precipitation ﬁeld—as opposed to apply a metric over spatial windows of increasing size in cell tracking in which contiguous high-intensity objects are order to examine how the forecast performance depends on tracked (see [3] for a discussion of both methods). the spatial scale. Yet, we still lack the ability to explicitly 2 Advances in Meteorology isolate and quantify the location error. 'is makes it diﬃcult True feature track to benchmark and optimize the corresponding components of nowcast models. Forecast feature track Location In this study, we introduce an approach to directly error t + 3 quantify the location error of precipitation nowcasts which is based on the extrapolation of ﬁeld motion. With location t + 2 error, we refer to the spatial oﬀset (or Euclidean distance) t + 1 between the true and the forecast locations of a precipitation Forecast time feature (Figure 1). In this context, the term “feature” does not refer to a contiguous object but to a distinct point in the t – 1 precipitation ﬁeld, and we make use of the ability of the t – 2 OpenCV library to detect and track the true motion of such distinct points. In a veriﬁcation case study, we will dem- Figure 1: Illustration of the location error for a prediction at forecast time t that is based on the linear extrapolation of feature onstrate the ability to quantify the location error by motion from t − 1 to t. benchmarking a set of routine extrapolation techniques for one year of quality-checked radar data in Germany. Section 2 highlights the approach to quantify the loca- To collect all feature tracks T in any given time period tion error and describes a set of tracking and extrapolation with a length of n time steps, we detect “goodFeaturesTo- techniques based on optical ﬂow, as well as the radar data for Track” (Shi and Tomasi, 1994) at each time step k∈ [1, . . ., n], our case study. Section 3 presents the results of our case and track these features over as many subsequent time steps study, and Section 4 concludes the paper. as possible. Accordingly, each track T could be identiﬁed i,j,k by a unique tuple (i, j, k) that carries its starting point (by the grid’s row and column indices, i and j) and its starting time 2.Methods and Data index k. In this study, we use an analysis period of one year 2.1. Feature Detection and Tracking. We suggest quantifying (2016, a leap year) and a time step length of 5 minutes, so the location error of a forecast by comparing the observed that k∈ [1, . . ., 105408]. location (or displacement) of a precipitation feature against In summary, the tracking process consists of six steps: its predicted location. In visual computing, a feature is new deﬁned as a point that stands out in a local neighborhood (1) Identify the features p using goodFeaturesToTrack and is invariant in terms of scale, rotation, and brightness [11] at any time k. [7]. For a radar image, a feature (or corner) represents a old (2) If there are already features being tracked, p , from point with a sharp gradient of rainfall intensity [2]. new k − 1 to k, we consider only those features p for In this study, features are detected using the approach old which the distance to any feature p is greater than of Shi and Tomasi [8] If a feature is detected at one time 7 km (with this threshold, we enforce consistency step, we attempt to track that feature in any subsequent with the minDistance parameter of the Shi-Tomasi time step until it is no longer trackable. 'e feature tracking corner detection; see Table 1). 'e trackable features follows the approach of Kanade [9], as implemented by old new p are hence the union of p and p . k k Bouguet [10]. 'e tracking error (or, inversely put, the (3) Track p from k to k + 1 using calcOpticalFlowPyrLK robustness of tracking a feature from one radar image to the k [11]; next) is quantiﬁed in terms of the minimum eigenvalue of a 2 × 2 normal matrix of optical ﬂow equations (this matrix is (4) Backwards track (from k + 1 to k) those features p k+1 called a spatial gradient matrix in Bouguet [10]), divided by that were obtained in step 3. the number of pixels in a neighborhood window. In the (5) Calculate the distance, d , from the features p to tracking step, that minimum eigenvalue has to exceed a back the backward-tracked locations resulting from step threshold in order for a feature to be considered as suc- cessfully tracked. Table 1 provides an overview of pa- rameters used for both feature detection and tracking. (6) Keep only those features p where the distance k+1 old 'ese values are based on the ones presented by Ayzel et al. d is less than 1 km. 'ese features are now p . back k+1 [2]. 'e underlying equations are well documented in For statistical analysis, each track T is characterized by OpenCV [11]. i,j,k its duration τ (the number of time steps over which the track In order to increase the robustness of track detection, the persists), the overall displacement distance d of the feature tracking was also performed backwards at each time step along its track, the average feature velocity υ � d/τ, and the (Figure 2): let p signify a feature that was identiﬁed at frame t straightness of the feature’s displacement in terms of the and tracked to the next frame at time t + 1 at the position p . t+1 sinuosity index (SI) (which is calculated by dividing d by the 'e same tracking process was then applied backwards from back Euclidean distance between the feature origin and end lo- the point p to time t, yielding the point p . Only the t+1 cations). 'e concept of sinuosity is widely used to char- trajectories where the distance d between the source point back back acterize river curvatures as introduced by Mueller [12] and p and the backwards tracked point p was less than one t K was also applied to atmospheric science by Terry and Feng kilometer (the grid resolution) were considered in our analysis. Linear extrapolation Advances in Meteorology 3 Table 1: OpenCV function parameters used for feature detection and tracking. Parameter name Value Meaning maxCorners 200 Maximum number of features qualityLevel 0.2 Minimum accepted quality of features minDistance 7 Minimal Euclidean distance between features blockSize 21 Size of pixel neighborhood for covariance calculation winSize (20, 20) Size of the search window maxLevel 2 Maximal number of pyramid levels Table 2: Overview of extrapolation models. p t + 1 Forward # Time steps Name Main approach looking back Persist Eulerian persistence 0 back Backward LK- Linear extrapolation based on Lukas Lin1 Kanade back LK- Linear extrapolation based on Lukas Lin4 Kanade DIS- Linear extrapolation from DIS Lin1 motion ﬁeld Semi-Lagrangian extrapolation based DIS- Figure 2: Illustration of the backward tracking test performed at on motion ﬁeld obtained by dense 1 Rot1 each time step for all features. optical ﬂow [13] to quantify the sinuosity of typhoon tracks. In our 2.3.1. Eulerian Persistence. As a trivial benchmark, we use the assumption of Eulerian persistence, meaning that the analysis, we will also use the sinuosity index in order to precipitation feature will simply remain at its position at understand the error of predicted feature locations. forecast time; that is, P � p . t+Δt t 2.2. Error of Predicted Locations. Let p be the true location and let P be the predicted location of a point feature in a 2.3.2. Linear Extrapolation. Linear extrapolation of feature motion assumes that a feature moves, over any lead time, at Cartesian coordinate system. At forecast time t, p will be equal to P . Consider P � f (p , Δt, S ) any function or constant velocity and in the same direction. 'e displace- t t t t+Δt ment vector representing this motion can be obtained in algorithm that predicts the future location P of point p t+Δt from any set S of predictors that is available at time t or diﬀerent ways. 'ese ways constitute three diﬀerent models exempliﬁed in the present study: LK-Lin1, LK-Lin4, and DIS- before. In the context of our study, that set of predictors could be, for example, the previous locations p , p , . . . of Lin1. In the case of LK-Lin1 and LK-Lin4, the displacement t−1 t−2 vector is obtained from “looking back” m time steps from p . We then deﬁne the error of our prediction, henceforth forecast time t to previous feature locations at t − m (tracked referred to as location error ε, as the Euclidean distance between P and p . by using the Lucas–Kanade method, hence the LK label). For t+Δt t+Δt LK-Lin1, m equals 1, so the vector v(t, p ) to displace feature p is the connection from p to p ; for LK-Lin4, m equals 4, t t−1 t 2.3. Extrapolation Techniques. In a veriﬁcation experiment, so that the displacement vector results from the connection we can use our collection of tracks T in order to retrieve between p and p , where the length of the vector is divided t−4 t points p for which the location P at t +Δt should be by 4 in order to obtain the displacement velocity. Hence, a t+Δt predicted, points that could be used as predictors (S ), as well forecast at lead time Δt extends the vector v(t, p ) corre- as the true location p of the point at t +Δt. Assuming that spondingly. Please see Figure 3 for an illustration of both the t+Δt an extrapolation of motion uses feature locations from m LK-Lin1 and the LK-Lin4 method. Of course, any other time steps before t, the minimum feature track length to look-back time m could be used to obtain a displacement produce a forecast would be m + 1. In order to retrieve the vector. In this study, we arbitrarily used m∈ {1, 4} in order to location error of such a prediction at time t +Δt, we would examine the eﬀect of m on the forecast performance. need a minimum track length of m +Δt + 1. For the DIS-Lin1 model, a complete ﬁeld of motion Based on the above terminology, we present in the vectors V is obtained from the Dense Inverse Search DIS following the extrapolation models analyzed in the present (DIS) method [14]; the underlying concept and equations of study. 'ese models are based on the models that were also the DIS method have been elaborated by Kroeger et al. [15] evaluated in a recent benchmarking study on optical-ﬂow- and then used for the extrapolation. A point p is linearly based precipitation nowcasting [2]. Table 2 gives an overview extrapolated from t to t + n by n times the velocity vector of model acronyms and their main properties. v (t, p ), where v (t, p ) is the vector closest to p in the DIS t DIS t 4 Advances in Meteorology ε (t ) t + n n t + n ε (t ) v (t) t + 3 v (t) t + 2 t – 1 v (t) v (t) t + 1 v (t) v (t) t – 2 v (t) v (t) t – 1 v (t) t – 3 v (t) t – 4 Figure 3: Illustration of the linear extrapolation schemes for the LK group: on the left LK-Lin1 and on the right LK-Lin4. 'e location error is displayed by ε(t ). V (t) ﬁeld (Figure 4). V (t) is calculated by OpenCV’s DIS DIS t + n cv2.DISOpticalFlow_create function, which returns velocity V (t) DIS vectors for each grid pixel based on the radar frames from t + 3 t − 1 to t. In a recent benchmarking study about optical-ﬂow- based precipitation nowcasting, Ayzel et al. [2] showed that t + 2 the DIS-based model (referred to as the “Dense” model in that paper) is an eﬀective method for radar-based precipi- t + 1 tation nowcasting. t – 1 2.3.3. Semi-Lagrangian Approach Based on Dense Optical Flow. In a Semi-Lagrangian approach, the motion ﬁeld is typically assumed as constant over the forecast period and Figure 4: In the DIS-Lin1 model, the vector v (t, p ) (light red DIS t the feature trajectory is determined by following the arrow) obtained from V (t) is transferred to the p location and DIS streamlines [16]. Following this concept, the DIS-Rot1 linearly extended to t + n. model (corresponding to “Dense rotation” in [2]) uses the two most recent radar images, t − 1 and t, to estimate V (t) DIS by cv2.DISOpticalFlow_create function. Similar to the DIS- t + n Lin1 model, the displacement vector v (t, p ) which is DIS t V (t) DIS closest to p is used to extrapolate the motion of p from its t t t + 3 position at t to t + 1, providing the location of P . 'is t+1 t + 2 process is repeated at all lead time steps until the maximum lead time is achieved. Hence, at each lead time step n, we t + 1 retrieve the vector v (t, P ) which is closest to P in DI S t+n t+n order to extrapolate the feature location, P . Accord- t t+n+1 ingly, the velocity vector is updated at each lead time step t – 1 from V (t), allowing for rotational or curved motion DIS patterns (Figure 5). Figure 5: Schematic of the DIS-Rot1 model (orange path), where 2.4. Weather Radar Data and Experimental Setup. Our the velocity is updated every time step by transferring the velocity benchmarking case study is based on weather radar data vector v (t, p) (light orange arrow) closest to p (black circles, for DIS from the German Weather Service, namely, the RY product t � 0) or P (orange circles, for t> 0) in V , to the P +Δt t+1 DIS(t) t location to advect. generated as part of the RADKLIM radar reanalysis of the German Weather Service DWD [17]. 'e RY product represents a quality-controlled national precipitation in- tensity composite from 18 C-Band radars covering Germany mountains that would interfere with the beam propagation. at 5-minute intervals and a spatial resolution of 1 km at an Quality control includes a wide range of correction methods extent of 1100 × 900 km. 'e basis of the composite product for, e.g., clutter or partial beam blockage (see [17] for is the so-called “precipitation scans” from each of the 18 details). radar locations. 'e precipitation scan is designed to follow 'e year 2016, selected for this experiment, was char- the horizon as closely as possible at an azimuth resolution of acterized by an annual precipitation close to the climato- 1 and a radial resolution of 1 km, adjusting the elevation logical mean for most regions in Germany, as can be seen in angle for each azimuth depending on the presence of the German Climate Atlas [18]. However, the precipitation Advances in Meteorology 5 mean during autumn was below the normal average and general, expect the length of a track to increase with its during the winter months slightly above the climatological duration. Yet, there are also months—most notably the summer months from May to August—where this expec- mean. As 2016 was a leap year, this experiment was carried out tation is not met; and, of course, the length of a track de- on 105408 radar composite images. Since none of the pends not only on its duration but also on a feature’s methods under evaluation required any kind of training, velocity. 'e average feature velocity in 2016 amounted to a there was no need to split the data into sets for calibration value of 42 km/h; and, in fact, not only does velocity show a and validation. Instead, we used all tracks for veriﬁcation. clear seasonal pattern (with minimum velocities in the For each track, we always use, as forecast time t, a time of 20 summer months; see Figure 6(d)), but also the seasonal minutes after the feature was detected for the ﬁrst time. 'at pattern helps us to understand where the patterns of track is because our model LK-Lin4 needs to look back four time length and duration appear to be “inconsistent.” For ex- steps (i.e., 20 minutes) in order to make a forecast, and we ample, the track velocity is at a minimum in May and June, need to make sure, for a fair comparison, to compare all which decreases the length of track despite the rather high duration values for these two months. models for the same forecast times. 'e clearest seasonal pattern can be observed for rainfall intensity (Figure 6(e)). 'at pattern is very much in line with 2.5. Computational Details. 'e analysis was carried out in a our expectation as rainfall in the summer months is gov- Python 3.6 environment using the following main open- erned by convective events that tend to be more intense than source libraries: NumPy (https://numpy.org), NumExpr stratiform event types. However, if we assumed that a higher (https://github.com/pydata/numexpr), and SciPy (https:// rainfall intensity along a track is caused by the convective www.scipy.org) for general computations; OpenCV nature of the underlying event, the track duration in the (https://opencv.org) for feature tracking; and Pandas corresponding months (e.g., May and June) is at least (https://pandas.pydata.org) and h5py (https://www.h5py. surprising: we would expect a convective event not only to be org). more intense but also to be rather short (in comparison to widespread stratiform rainfall). 'e apparent inconsistency between the patterns of rainfall intensity and track duration 3.Results and Discussion points us to one of the key issues with the presented track 3.1. Properties of Collected Tracks. 'e identiﬁcation and inventory: we must not misinterpret a “track” as an “event” tracking process detected 376,125 features above the rainfall in a hydrometeorological sense. 'e corner detection al- rate threshold of 0.2 mm/h and lasted over 20 minutes, gorithm (see Section 2.1) searches for pronounced features which resulted in 337,776 eligible tracks after applying the in the sense of strong local gradients and tracks a feature for extrapolation step. A track was considered as “eligible” in as long as it stands out. While we deﬁne a rainfall event as case all models had a predicted location at all lead times, some coherent process in space and time, the tracking al- from t to t + n. 'e loss of 10.2% that is implied by the above gorithm could “lose” a feature right in the course of an numbers was caused by the DIS group of models which did ongoing event and maybe, at the same time, ﬁnd another feature to track somewhere else in the ﬁeld. Obviously, the not generate a valid velocity vector v (t, p ) near every p DIS t point, in the V (t) ﬁeld, within a 3.5 km threshold. tracking algorithm was able to track features over a longer DIS duration in May, June, and September of 2016. However, as Figure 6 gives an overview of the properties of the valid tracks. 'e ﬁgure also shows the seasonal dependency of of now, we do not know which properties of the corre- sponding rainfall events caused that eﬀect. We should just these track properties by summarizing their distribution on a per month basis. We would like to emphasize that this emphasize that the duration of a track does not necessarily analysis must not be interpreted as a “climatology” of track correspond to the duration of an event. In the same way, we properties as it only contains data from a single year. Still, we cannot expect the tracking algorithm to ﬁnd features at consider it as illustrative to investigate which properties tend “representative” locations of a convective cell. It will detect such features anywhere in a rainfall ﬁeld where local gra- to exhibit a seasonal pattern and also to discuss whether the observed properties can be considered as representative for dients meet the tracking criteria. 'at could be right not only in the middle of heavy rainfall but also at the edges. Hence, the governing rainfall processes in Germany. In an average month of 2016, we identiﬁed and tracked the reported precipitation intensities along the tracks will not be representative of the mean precipitation intensities of 28,146 features (Figure 6(a)). 'e largest number of tracks is found from April to August (all above the average). Yet, the corresponding precipitation ﬁelds. there is no continuous seasonal pattern in the number of Altogether, we have to emphasize at this point that the detected tracks because, e.g., January and October also show seasonal track statistics are indeed plausible. But it must be rather large counts. clear that track statistics are not necessarily representative No pattern at all can be found for the track length for “event” statistics. 'at notion might be irritating for (Figure 6(b)). With an average track length of 128 km, those who have been deﬁning and tracking features in terms of coherent rainfall objects over their lifetime from initiation monthly maximum mean and median track lengths occur in January, April, and September. A partly similar pattern can to dissipation. A new feature track as we understand it in our analysis could be found right in the middle of an ongoing be found for the track duration that amounts to 207 minutes on average (Figure 6(c)). 'is is plausible as we would, in event, and it can be lost long before the actual rainfall 6 Advances in Meteorology Ave = 28146 Ave = 128 Jan 875.8 Jan Mean Median Feb 793.06 25% 95% Feb Mar 774.18 Min Max Mar 5% Apr 869.54 75% Apr May 882.9 May Jun 825.71 Jun Jul 969.44 Jul Aug 963.59 Aug 908.94 Sep Sep Oct 1233.41 Oct Nov 812.08 Nov Dec 766.41 Dec 10000 20000 30000 40000 50000 0 100 200 300 400 500 Count Track length (km) (a) (b) Ave = 207 Ave = 42 Jan 1870 Jan 133.77 Feb 135.21 Feb 1685 Mar 1315 Mar 117.67 Apr 1350 Apr 115.16 May 1690 May 105.78 Jun 2235 Jun 108.56 Jul 1635 Jul 114.58 Aug 1040 Aug 111.83 Sep 1825 Sep 122.65 Oct 1955 Oct 130.8 Nov 2170 Nov 133.6 Dec 2605 Dec 137.57 0 200 400 600 800 025 50 75 100 125 Duration (min) Velocity (km/h) (c) (d) Ave = 1.6 Ave = 1.09 Jan 14.85 Jan 101.09 Feb 18.48 Feb 108.42 Mar 16.99 Mar 46.27 Apr 23.99 Apr 166.94 May 68.16 May 73.83 Jun 90.7 Jun 88.6 Jul 61.64 241.42 Jul Aug 82.5 Aug 78.15 Sep 35.95 Sep 162.16 Oct 39.19 Oct 110.55 Nov 14.44 Nov 107.92 Dec 12.61 53.9 Dec 0 2468 1.0 1.1 1.2 1.3 1.4 1.5 Rainfall intensity (mm/h) Sinuosity index (e) (f) Figure 6: Statistical properties of detected tracks, organized by month: (a) number of detected tracks, (b) track length, (c) track duration (time elapsed from detection and loss of a feature), (d) feature velocity, (e) rainfall intensity of a detected feature, and (f) sinuosity index of a track. Advances in Meteorology 7 In order to convey a better idea about the rainfall pat- “object” dissolves. However, that does not at all lessen the value of these tracks for the purpose of our analysis, which is terns in the examples, the observed rainfall intensity at forecast time t is plotted as a background in grey scale. to quantify the forecast location error based on well-deﬁned and scale-invariant features. Furthermore, the sinuosity index and the track duration are Having said that, one ﬁnal track property shown in printed in the corresponding subplots. Figure 6(f) has not been discussed yet: the sinuosity index. Please note that the duration of the observed tracks in As pointed out above (Section 2.1), the sinuosity index il- Figure 7 can extend over many hours; very long tracks were lustrates how much the shape of a track deviates from a capped at a duration of 300 minutes for the purpose of straight line (which would correspond to a sinuosity index of plotting. Furthermore, the lead time of the predictions in the examples was set to the (capped) track duration minus 20 1). Figure 6(f) shows rather large sinuosity values for the summer months, May to September, but there is no obvious minutes (which corresponds to the period t − 4 until forest time t). As a consequence, the lead times illustrated in seasonal pattern. More strikingly, the distribution of the sinuosity index is very heavily tailed. 'e average value Figure 7 are mostly longer than the maximum lead time of 120 minutes, which is used in our veriﬁcation experiment amounts to approximately 1.10 in the year 2016, which is, at the same time, the 90th percentile of the sinuosity values. (see the next section). Hence, the ﬁrst visual impression of 'at means, in turn, that the vast majority of tracks are Figure 7 is dominated by the considerable errors that can rather straight, while the remaining tracks show all kinds of occur for such long lead times. But, of course, we should curved, meandering, twisted, or just erratic behavior. rather be aware of the behavior for shorter lead times up to Hence, before we systematically show the results of our 120 minutes. For that reason, the 120-minute lead time is veriﬁcation experiment with regard to the location error (see highlighted by a larger dot. Not surprisingly, most of the competing methods appear to Section 3.3), we would like to illustrate, in the following paragraph, the behavior of observed tracks in comparison to remain rather close to the observed track for short lead times of up to 30 minutes (except, e.g., in subplot Figure 7(j) in which the forecast tracks under diﬀerent sinuosity conditions. the DIS-based methods entirely fail to capture the direction of feature movement). After that, the lead time over which the 3.2. Visual Examples of Observed and Predicted Tracks. extrapolation models adequately predict the observed feature Before we systematically evaluate the performance of dif- track varies, depending on the persistence of the motion be- ferent extrapolation techniques, we would like to provide havior and the validity of the underlying model assumption. some illustrative examples of observed versus predicted For example, all models perform quite well for very long times tracks. 'e selection of tracks for this illustration is arbitrary in subplot (f). In subplot (i), the Semi-Lagrangian approach and does not intend to be representative of the performance (DIS-Rot1) shows a clear advantage, while in subplots of of any of the extrapolation methods. Instead, we aim to Figures 7(c) and 7(k), DIS-Rot1 is outperformed by all other exemplify shapes of observed and predicted tracks under models. Surely, there are several examples (Figures 7(b), 7(d), diﬀerent sinuosity conditions in order to convey a better 7(e), and 7(g)) in which all models entirely fail to anticipate the understanding of the various constellations that will ﬁnally motion for lead times beyond 120 minutes. be condensed into one single location error value. As this compilation of examples is deliberately arbitrary, Figure 7 shows a “gallery” of 11 observed tracks in it does not provide a basis to infer the general superiority or diﬀerent subplots (From Figure 7(a) to 7(k)). Each subplot inferiority of one or the other method. All models appear to also contains the tracks that were predicted by the diﬀerent struggle with predicting very sinuous tracks (subplots in extrapolation models. Each dot represents one feature Figures 7(b), 7(d), 7(e), and 7(g)), which is what we would location in a 30-minute time step, except the ﬁrst one that expect. However, while the ﬁgure makes it diﬃcult to represents the ﬁrst prediction step at ﬁve-minute lead time. compare the absolute location error between the examples LK-Lin1 and LK-Lin4 infer the displacement vector di- (due to the diﬀerent scales), it still appears that the absolute rectly from the feature positions at t and t − 1 or t and t − 4, location error does not necessarily depend on the sinuosity. respectively. As a reminder, DIS-Lin1 and DIS-Rot1 obtain For example, the location error of LK-Lin1 after the max- the displacement vector of a feature from the DIS algo- imum lead time (280 minutes) is higher in subplot 7(i) rithm, a dense optical ﬂow technique that produces motion (almost straight, SI � 1.01) than it is in subplot 7(d) ﬁelds based on the radar images at t and t − 1; DIS-Lin1 (SI � 1.36). In fact, straight tracks can imply a large error if extrapolates the closest vector linearly over the entire lead the initial motion vector of a forecast method fails to rep- time, while DIS-Rot1 uses a Semi-Lagrangian scheme in resent the average long-term direction (see subplot 7(j) for a which the displacement vector is updated as the feature very impressive example). 'en again, large errors can occur moves through the velocity ﬁeld obtained from the DIS if a strong sinuosity of the track coincides with a large technique. Further details have been provided in Section overestimation of the absolute velocity (e.g., subplots 7(b) 2.3. As in all forecasts of our veriﬁcation experiment, the and 7(g)). In that case, the linear extrapolation quickly forecast time t corresponds to the 5th feature of the ob- departs from the track origin, while the actual feature track served track. 'at is because the LK-Lin4 method needs to meanders slowly and remains in the close vicinity of the look four steps back in time (t − 4) in order to produce a origin. For such a scenario, the trivial persistence model (the forecast, while the other methods only look back one step in feature just remains at the origin) will be superior even for time (t − 1). short lead times. 8 Advances in Meteorology Sl = 1.01 τ = 300 Sl = 1.12 τ = 175 Sl = 1.36 τ = 300 Sl = 1.01 τ = 165 (a) (d) (f ) (h) 10 km 10 km Sl = 1.01 τ = 300 (i) Sl = 1.33 τ = 300 (b) 10 km 10 km Sl = 2.22 τ = 300 Sl = 1.36 τ = 300 10 km (e) 10 km (j) Sl = 1.0 τ = 170 (c) Sl = 1.01 τ = 245 10 km (k) Sl = 1.0 τ = 300 (g) 10 km 10 km 10 km 10 km 05 10 15 20 25 30 mm/h Observation DIS-Lin1 LK-Lin1 DIS-Rot1 LK-Lin4 Figure 7: Compilation of forecast versus observed tracks under diﬀerent sinuosity conditions. Due to the diﬀerent spatial extents of the windows, the scale of each subplot is diﬀerent. Hence, a 10 km scale bar is provided for orientation. For each example, the observed track duration τ (in hours) and its sinuosity index SI are shown. 'e lead time of 120 minutes is highlighted by a larger dot. Some very long tracks have been capped at a maximum of 300 minutes for illustrative purposes. Altogether, these diﬀerent examples give us a better idea all models, dramatically lower than that for the persistence of how location errors can develop from both inadequate model; the mean error of persistence is higher than the mean error of any model at any lead time, which means that all model assumptions (e.g., linear approximation versus curved or sinuous conditions) and a failure to approximate models, on average, have positive skill at all lead times. For all models, the error distribution is obviously positively skewed, the average motion from the initial feature locations. It is impossible, though, to diagnose the superiority of one or the with the mean error being much higher than the median, and other model from these examples. Hence, we will now thus there is a heavy tail towards high location errors. systematically examine the results of our model veriﬁcation For very short lead times of up to 10 minutes, the mean experiment. We will not only analyze how the location error error is about one kilometer for all competing models except depends on lead time, but we will also investigate how the for persistence which is already up at more than seven ki- model performance relative to the persistence model de- lometers after ten minutes. After 60 minutes, the mean pends on the sinuosity of the underlying tracks. location error of all models exceeds a distance of 5 kilo- meters, as well as 10 kilometers after 110 minutes. For all models, at least 25% of all forecasts exceed an error of 5 3.3. Systematic Quantiﬁcation of the Location Error. After kilometers after 50 minutes and an error of 10 kilometers having exempliﬁed diﬀerent observed and predicted tracks in after 90 minutes. After 75 minutes, at least 5% of all forecasts the previous section, we now present the results of our exceed an error of 15 kilometers. benchmarking experiment. Figure 8 shows the distribution of Altogether, the location error can be substantial for a locations errors for diﬀerent models and lead times up to 120 signiﬁcant proportion of forecasts, while the median loca- minutes. For each lead time, the box plots specify mean, tion error grows at a more moderate rate. median, interquartile range, and the 5th and 95th percentiles While this general pattern governs the behavior of all of the location error. For all models, the error quantiles in- models, there are clear diﬀerences between the performances crease slightly exponentially but almost linearly with lead time. of the competing models. 'ese diﬀerences, however, are not 'e rate at which the location error grows with lead time is, for always coherent across all error quantiles and lead times, Advances in Meteorology 9 Median Mean 25% 95% 5% 75% 5 101520253035404550556065707580859095 100 105 110 115 120 Lead time (min) Lk-Lin1 DIS-Rot1 Lk-Lin4 Persist DIS-Lin1 Figure 8: 'e distribution of location errors for diﬀerent extrapolation models and lead times. except for the DIS-Rot1 model, which has the weakest per- examine the skill of our models more closely. Skill scores rate formance of all models at virtually all lead times and for all the score of a forecast in relation to the score of a reference quantiles, and the LK-Lin1 model, which performs better than forecast, in our case persistence. 'ey are particularly useful in benchmark studies such as the present one. Equation (1) DIS-Rot1 but ranks second last. As for the best forecast performance, the LK-Lin4 and the DIS-Lin1 models take turns shows the general deﬁnition of skill as derived from any forecast score, as well as the speciﬁc formula if we use the depending on error quantile and lead time: For the 5th and the 25th percentiles, the LK-Lin4 model performs best for lead location error ε as the “score” (which becomes zero for a times up to 100 minutes, for the median up to 80 minutes, and perfect forecast) and persistence as the “reference”: for the mean up to 55 minutes. 'e DIS-Lin1 model shows the ε − ε Score − Score forecast persistence forecast reference strongest changes of relative performance over lead time: as for Skill � � . Score − Score −ε the mean error, DIS-Lin1 starts to outperform LK-Lin4 at a perfect reference persistence lead time of 60 minutes and continues this way until the (1) maximum lead time of 120 minutes. As for the median error, We examine the forecast skill under diﬀerent sinuosity DIS-Lin1 only catches up with LK-Lin4 after 90 minutes. For conditions. As already pointed out in Section 3.1, the dis- the 75th percentile, DIS-Lin1 outperforms LK-Lin4 after 50 tribution of sinuosity is highly skewed and 90% of observed minutes and for the 95th percentile already after 20 minutes. tracks would pass as at least “rather straight” with a sinuosity In summary, LK-Lin4 tends to outperform DIS-Lin1 in the index equal to or lower than 1.1. Hence, we split the forecasts ﬁrst hour, while DIS-Lin1 becomes superior in the second into three unequal groups, depending on quantiles of the hour, apparently because it tends to avoid very high errors sinuosity index: 'e ﬁrst group contains the “straight” 90% more eﬃciently than LK-Lin4 does. of the forecasts with a sinuosity index below 1.1. We consider In the following, we would like to better understand how the value of 1.1 as an—admittedly—arbitrary threshold model skill is aﬀected by sinuosity. In Section 3.2, we have between “rather straight” and “rather winding” tracks. 'e already indicated that the absolute values of location errors remaining 10% of tracks are split into two equally sized do not clearly depend on sinuosity. 'at was conﬁrmed by groups, again based on sinuosity: the 5% with the highest the systematic veriﬁcation experiment (results not shown). sinuosity, exceeding an SI value of 1.2, could be labelled as Yet, the diﬀerence between an extrapolation model and the “twisted,” and the remaining 5% with intermediate SI values (trivial) persistence model might very well depend on sin- between 1.1 and 1.2 could be labelled as “winding.” Figure 9 uosity. In order to formally evaluate that hypothesis, we now Distance (km) 10 Advances in Meteorology Straight : Sl < 1.1 Winding : 1.1 ≤ Sl < 1.2 Twisted : Sl ≥ 1.2 0.88 0.66 0.5 0.64 0.4 0.86 0.62 0.3 0.60 0.84 0.2 0.58 0.82 0.56 0.1 0.54 0.80 0.0 0.52 0.78 0.50 –0.1 5 30 60 90 120 5 30 60 90 120 5 30 60 90 120 Lead time (min) Lead time (min) Lead time (min) Lk-Lin1 DIS-Lin1 Lk-Lin1 DIS-Lin1 Lk-Lin1 DIS-Lin1 Lk-Lin4 DIS-Rot1 Lk-Lin4 DIS-Rot1 Lk-Lin4 DIS-Rot1 Figure 9: 'e mean model skill over each lead time with regard to location prediction for diﬀerent extrapolation models and sinuosity conditions. Please note that the very low skill values of the DIS-based models at 5-minute lead time (in the winding and twisted groups) are hidden by the scaling of the y-axis. At ﬁve-minute lead time, both models only have a skill of about 0.35 (winding) and −0.55 (twisted). shows the average model’s skill over every lead time for these images. In our study, we detected features by using the three sinuosity classes. Clearly, the model skill dramatically approach of Shi and Tomasi (1994) and tracked these fea- varies between these three groups: it ranges between 0.79 tures following the approach of Lucas and Kanade [9], using and 0.87 for the “straight” category, mostly between 0.5 and both algorithms as implemented in the OpenCV library. We 0.65 for the “winding” category, and mostly between 0 and increased the robustness of extracted feature tracks by 0.5 for the “twisted” category. 'is decrease of skill with making sure that the features can be successfully tracked forwards and backwards. 'at approach, together with a increasing sinuosity is well in line with our expectation. Furthermore, the ranking of all models based on skill is quite rather strict deﬁnition of parameter values for feature de- coherent across all categories and also consistent with our tection and tracking, increases our conﬁdence in the reli- previous analysis of location errors. DIS-Lin1 becomes ability of the detected tracks. Still, we have to assume that the superior within the second forecast hour, while LK-Lin1 feature locations themselves are, as any measurement, un- performs better in the ﬁrst forecast hour. Only in the certain. We expect the main sources of uncertainty to be the “twisted” category do LK-Lin1 and, even more, LK-Lin4 grid resolution (which does not allow resolving errors below outperform DIS-Lin1 across all lead times. It should be 1 km), and complex small-scale intensity dynamics that can noted, though, that the overall skill in the twisted category is interfere with motion patterns. For future studies, we suggest very low for all competing models. In the “winding” cate- a comprehensive sensitivity analysis with regard to the gory, LK-Lin1 slightly outperforms LK-Lin4 in the ﬁrst 20 parameters of the feature detection and tracking algorithms minutes. Finally, DIS-Rot1 performs worst at all lead times in order to better understand the eﬀects on both the number in all categories. and the robustness of detected tracks in the context of 'e change of model skill with lead time should be rainfall motion analysis. Still, we assume that the error of interpreted with care, as it depends on both the performance extrapolating feature motion is substantially larger than the of the extrapolation model itself and the location error of the error of feature tracking itself. In summary, we consider it persistence model. For most models and SI categories, the warranted to use the observed tracks as a reference in order skill appears to reach an optimum at some lead time, which to evaluate the performance (or, inversely, the error) of any implies that the superiority of the model over persistence model that aims to predict the future locations of such reaches a maximum. precipitation features. For that purpose, we deﬁned the location error of a forecast at any lead time Δt ahead of the forecast time t as the Euclidean distance between the ob- 4.Conclusions served and the predicted feature locations at t +Δt. One might want to use this approach to comprehensively In this paper, we have introduced a framework to isolate and quantify the location error of any forecast model for the full quantify the location error in precipitation nowcasts that are based on ﬁeld-tracking techniques. While it is often assumed spatial domain of a forecast grid, for example, a national radar composite. In such a case, we would need to assume that errors in precipitation nowcasts are dominated by the temporal dynamics of precipitation intensity, the location that the average of forecast errors that we have quantiﬁed from observed feature locations in a forecast domain is error of predicted precipitation features has so far not been explicitly and formally quantiﬁed. representative for the average error of all location predic- tions in that domain. We have not yet investigated the 'e main idea of our framework is to detect and track scale-invariant precipitation features (corners) in radar validity of that assumption. One might argue that the Skill Skill Skill Advances in Meteorology 11 precipitation nowcasting, for example, in the context of early behavior of locations identiﬁed as “corners” or “good fea- tures to track” might not be representative for the motion warning systems for pluvial ﬂoods in urban environments (see [19]), it becomes obvious that location errors matter: the order behavior of the entire precipitation ﬁeld; however, it will be diﬃcult to ﬁnd evidence to either verify or falsify such a of magnitude of these errors is about the same as the typical hypothesis, as it would require another independent way to extent of a convective cell or of a medium-sized city. Hence, the quantify the location error. Still, we are convinced that the uncertainty of precipitation nowcasts at such length sca- proposed framework is useful: even without the need of les—just as a result of locational errors—can be substantial strong assumptions on representativeness, the framework already at lead times of less than an hour. allows us to compare and benchmark the ability of diﬀerent While similar conclusions have already been drawn by using spatially sensitive veriﬁcation measures such as the models to forecast future locations of precipitation features and thus to speciﬁcally focus on improving that ability by Fractions Skill Score (see, e.g., [6]), our framework allows us to isolate the location error for speciﬁc models and situations, to future model development. 'e hypothesis that such further model developments better understand the factors that govern these errors, and hence to use that knowledge in order to speciﬁcally improve the are urgently required is supported by the results of our benchmarking study. It should be clariﬁed again that this extrapolation of motion patterns in existing nowcasting models. benchmark study does not intend to suggest better ex- As an example, we have demonstrated how the use of the trapolation models but to demonstrate the ability of our sinuosity index can help us to better understand the predictive framework to unravel the location errors that are produced skill and hence the uncertainty of our models in speciﬁc sit- by state-of-the-art extrapolation methods. For that purpose, uations. We hope that the large number of extracted tracks will we compared four models: two models use the feature lo- help to foster the development of new techniques that use data- driven machine learning models for the extrapolation of feature cations before and at forecast time t in order to derive displacement vectors which are then used to linearly ex- location. For that purpose, we have made openly available the full set of extracted feature tracks for the year 2016 (https://doi. trapolate feature movement over the lead time. Model LK- Lin1 uses the feature locations at t and t − 1, and LK-Lin4 org/10.5281/zenodo.4024272 [20]) to serve as input to future studies. However, such future studies should also use radar data uses the feature locations at t and t − 4. 'e other two models are based on the dense optical ﬂow algorithm DIS that from a longer time period in order to learn more about the generates a full motion vector ﬁeld under various seasonal eﬀects related to the properties of feature tracks. smoothness constraints. 'e model DIS-Lin1 obtains the Data Availability displacement vector for a feature at t from the nearest motion vector in the ﬁeld based on the radar images at times 'e radar data are provided by DWD at https://opendata. t and t − 1 and uses that vector over the entire lead time. DIS- dwd.de/weather/radar/radolan/ry (last access: Sept. 2020). Rot1, in contrast, uses a Semi-Lagrangian scheme in which 'e code of this analysis is available in the Github repository the displacement vector is updated as the feature moves under https://github.com/arthurcts/loc_error (last access: through the motion ﬁeld obtained from the DIS technique. Sept. 2020). 'e dataset of extracted feature tracks has been 'e motivation behind the DIS-Rot1 model is to better deposited in the Zenodo repository (https://doi.org/10.5281/ represent rotational or curved motion patterns. From these zenodo.4024272). four competing models, LK-Lin4 appears to be the best model in the ﬁrst forecast hour and DIS-Lin1 the best in the Conflicts of Interest second. DIS-Rot1 performs consistently the worst. 'at is not quite in line with our naive expectation in which we 'e authors declare that there are no conﬂicts of interest would hope that a Semi-Lagrangian approach should be able regarding the publication of this paper. to better capture at least curved motion patterns. But not even in the winding category does the complexity of the DIS- Acknowledgments Rot1 approach pay oﬀ. Whether that is due to the imple- 'e authors acknowledge the German Weather Service, mentation of the Semi-Lagrangian approach or due to the namely, Dr. Tanja Winterrath, for making the RY data lack of validity of the approach should be the subject of available from the latest RADKLIM reanalysis. Arthur Costa future research. Comparing LK-Lin1 to LK-Lin4, we see a Tomaz de Souza has been funded by a Ph.D. scholarship of clear advantage in looking back in time more than one step. the German Academic Exchange Service (DAAD). Georgy It appears that, this way, we can retrieve more reliable, more Ayzel was partly funded by the ClimXtreme project (BMBF, representative, and less noisy displacement vectors, which FKZ 01LP1903B). shows in the superiority of LK-Lin4 over LK-Lin1. For all competing models, the mean location error exceeds References a distance of 5 kilometers after 60 minutes and 10 kilometers after 110 minutes. At least 25% of all forecasts exceed an error [1] M. 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Publisher: Hindawi Publishing Corporation
Copyright: Copyright © 2020 Arthur Costa Tomaz de Souza et al. 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.
ISSN: 1687-9309
eISSN: 1687-9317
DOI: 10.1155/2020/8841913
Publisher site: See Article on Publisher Site

Abstract

Hindawi Advances in Meteorology Volume 2020, Article ID 8841913, 12 pages https://doi.org/10.1155/2020/8841913 Research Article Arthur Costa Tomaz de Souza , Georgy Ayzel , and Maik Heistermann University of Potsdam, Institute of Environmental Science and Geography, Potsdam 14476, Germany Correspondence should be addressed to Arthur Costa Tomaz de Souza; costatomazde@uni-potsdam.de Received 11 September 2020; Accepted 21 October 2020; Published 3 December 2020 Academic Editor: Francesco Viola Copyright © 2020 Arthur Costa Tomaz de Souza et al. 'is 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. In precipitation nowcasting, it is common to track the motion of precipitation in a sequence of weather radar images and to extrapolate this motion into the future. 'e total error of such a prediction consists of an error in the predicted location of a precipitation feature and an error in the change of precipitation intensity over lead time. So far, veriﬁcation measures did not allow isolating the extent of location errors, making it diﬃcult to speciﬁcally improve nowcast models with regard to location prediction. In this paper, we introduce a framework to directly quantify the location error. To that end, we detect and track scale-invariant precipitation features (corners) in radar images. We then consider these observed tracks as the true reference in order to evaluate the performance (or, inversely, the error) of any model that aims to predict the future location of a precipitation feature. Hence, the location error of a forecast at any lead timeΔt ahead of the forecast time t corresponds to the Euclidean distance between the observed and the predicted feature locations at t +Δt. Based on this framework, we carried out a benchmarking case study using one year worth of weather radar composites of the German Weather Service. We evaluated the performance of four extrapolation models, two of which are based on the linear extrapolation of corner motion from t − 1 to t (LK-Lin1) and t − 4 to t (LK-Lin4) and the other two are based on the Dense Inverse Search (DIS) method: motion vectors obtained from DIS are used to predict feature locations by linear (DIS-Lin1) and Semi-Lagrangian extrapolation (DIS-Rot1). Of those four models, DIS-Lin1 and LK-Lin4 turned out to be the most skillful with regard to the prediction of feature location, while we also found that the model skill dramatically depends on the sinuosity of the observed tracks. 'e dataset of 376,125 detected feature tracks in 2016 is openly available to foster the improvement of location prediction in extrapolation-based nowcasting models. 'e present study focuses on nowcasts that are based on 1.Introduction ﬁeld tracking. 'e performance (or skill) of ﬁeld tracking Forecasting precipitation for the imminent future (i.e., techniques is mostly veriﬁed by comparing the forecast minutes to hours) is typically referred to as precipitation precipitation ﬁeld F for time t +Δt against the observed t+Δt nowcasting. A common nowcasting technique is to track the precipitation ﬁeld O at time t +Δt, where t is the forecast t+Δt motion of precipitation from a sequence of weather radar time and Δt is the lead time. A large variety of veriﬁcation images and to extrapolate that motion into the future [1]. For measures have been suggested in the literature (see, e.g., that purpose, we often assume that the intensity of pre- [4, 5]). Most of them, however, struggle with disentangling cipitation features in the most recent image remains con- diﬀerent sources of error: when we compare F to O , t+Δt t+Δt stant over the lead time period—an assumption commonly how can we know the cause of the disagreement? Was it our referred to as “Lagrangian persistence” [2]. In Lagrangian prediction of the future location of a precipitation feature, or ﬁeld tracking, a velocity vector is obtained for each pixel of a was it how precipitation intensity changed over time? Some precipitation ﬁeld, and that vector ﬁeld is used to extrapolate veriﬁcation scores, such as the Fractions Skill Score [6], the motion of the entire precipitation ﬁeld—as opposed to apply a metric over spatial windows of increasing size in cell tracking in which contiguous high-intensity objects are order to examine how the forecast performance depends on tracked (see [3] for a discussion of both methods). the spatial scale. Yet, we still lack the ability to explicitly 2 Advances in Meteorology isolate and quantify the location error. 'is makes it diﬃcult True feature track to benchmark and optimize the corresponding components of nowcast models. Forecast feature track Location In this study, we introduce an approach to directly error t + 3 quantify the location error of precipitation nowcasts which is based on the extrapolation of ﬁeld motion. With location t + 2 error, we refer to the spatial oﬀset (or Euclidean distance) t + 1 between the true and the forecast locations of a precipitation Forecast time feature (Figure 1). In this context, the term “feature” does not refer to a contiguous object but to a distinct point in the t – 1 precipitation ﬁeld, and we make use of the ability of the t – 2 OpenCV library to detect and track the true motion of such distinct points. In a veriﬁcation case study, we will dem- Figure 1: Illustration of the location error for a prediction at forecast time t that is based on the linear extrapolation of feature onstrate the ability to quantify the location error by motion from t − 1 to t. benchmarking a set of routine extrapolation techniques for one year of quality-checked radar data in Germany. Section 2 highlights the approach to quantify the loca- To collect all feature tracks T in any given time period tion error and describes a set of tracking and extrapolation with a length of n time steps, we detect “goodFeaturesTo- techniques based on optical ﬂow, as well as the radar data for Track” (Shi and Tomasi, 1994) at each time step k∈ [1, . . ., n], our case study. Section 3 presents the results of our case and track these features over as many subsequent time steps study, and Section 4 concludes the paper. as possible. Accordingly, each track T could be identiﬁed i,j,k by a unique tuple (i, j, k) that carries its starting point (by the grid’s row and column indices, i and j) and its starting time 2.Methods and Data index k. In this study, we use an analysis period of one year 2.1. Feature Detection and Tracking. We suggest quantifying (2016, a leap year) and a time step length of 5 minutes, so the location error of a forecast by comparing the observed that k∈ [1, . . ., 105408]. location (or displacement) of a precipitation feature against In summary, the tracking process consists of six steps: its predicted location. In visual computing, a feature is new deﬁned as a point that stands out in a local neighborhood (1) Identify the features p using goodFeaturesToTrack and is invariant in terms of scale, rotation, and brightness [11] at any time k. [7]. For a radar image, a feature (or corner) represents a old (2) If there are already features being tracked, p , from point with a sharp gradient of rainfall intensity [2]. new k − 1 to k, we consider only those features p for In this study, features are detected using the approach old which the distance to any feature p is greater than of Shi and Tomasi [8] If a feature is detected at one time 7 km (with this threshold, we enforce consistency step, we attempt to track that feature in any subsequent with the minDistance parameter of the Shi-Tomasi time step until it is no longer trackable. 'e feature tracking corner detection; see Table 1). 'e trackable features follows the approach of Kanade [9], as implemented by old new p are hence the union of p and p . k k Bouguet [10]. 'e tracking error (or, inversely put, the (3) Track p from k to k + 1 using calcOpticalFlowPyrLK robustness of tracking a feature from one radar image to the k [11]; next) is quantiﬁed in terms of the minimum eigenvalue of a 2 × 2 normal matrix of optical ﬂow equations (this matrix is (4) Backwards track (from k + 1 to k) those features p k+1 called a spatial gradient matrix in Bouguet [10]), divided by that were obtained in step 3. the number of pixels in a neighborhood window. In the (5) Calculate the distance, d , from the features p to tracking step, that minimum eigenvalue has to exceed a back the backward-tracked locations resulting from step threshold in order for a feature to be considered as suc- cessfully tracked. Table 1 provides an overview of pa- rameters used for both feature detection and tracking. (6) Keep only those features p where the distance k+1 old 'ese values are based on the ones presented by Ayzel et al. d is less than 1 km. 'ese features are now p . back k+1 [2]. 'e underlying equations are well documented in For statistical analysis, each track T is characterized by OpenCV [11]. i,j,k its duration τ (the number of time steps over which the track In order to increase the robustness of track detection, the persists), the overall displacement distance d of the feature tracking was also performed backwards at each time step along its track, the average feature velocity υ � d/τ, and the (Figure 2): let p signify a feature that was identiﬁed at frame t straightness of the feature’s displacement in terms of the and tracked to the next frame at time t + 1 at the position p . t+1 sinuosity index (SI) (which is calculated by dividing d by the 'e same tracking process was then applied backwards from back Euclidean distance between the feature origin and end lo- the point p to time t, yielding the point p . Only the t+1 cations). 'e concept of sinuosity is widely used to char- trajectories where the distance d between the source point back back acterize river curvatures as introduced by Mueller [12] and p and the backwards tracked point p was less than one t K was also applied to atmospheric science by Terry and Feng kilometer (the grid resolution) were considered in our analysis. Linear extrapolation Advances in Meteorology 3 Table 1: OpenCV function parameters used for feature detection and tracking. Parameter name Value Meaning maxCorners 200 Maximum number of features qualityLevel 0.2 Minimum accepted quality of features minDistance 7 Minimal Euclidean distance between features blockSize 21 Size of pixel neighborhood for covariance calculation winSize (20, 20) Size of the search window maxLevel 2 Maximal number of pyramid levels Table 2: Overview of extrapolation models. p t + 1 Forward # Time steps Name Main approach looking back Persist Eulerian persistence 0 back Backward LK- Linear extrapolation based on Lukas Lin1 Kanade back LK- Linear extrapolation based on Lukas Lin4 Kanade DIS- Linear extrapolation from DIS Lin1 motion ﬁeld Semi-Lagrangian extrapolation based DIS- Figure 2: Illustration of the backward tracking test performed at on motion ﬁeld obtained by dense 1 Rot1 each time step for all features. optical ﬂow [13] to quantify the sinuosity of typhoon tracks. In our 2.3.1. Eulerian Persistence. As a trivial benchmark, we use the assumption of Eulerian persistence, meaning that the analysis, we will also use the sinuosity index in order to precipitation feature will simply remain at its position at understand the error of predicted feature locations. forecast time; that is, P � p . t+Δt t 2.2. Error of Predicted Locations. Let p be the true location and let P be the predicted location of a point feature in a 2.3.2. Linear Extrapolation. Linear extrapolation of feature motion assumes that a feature moves, over any lead time, at Cartesian coordinate system. At forecast time t, p will be equal to P . Consider P � f (p , Δt, S ) any function or constant velocity and in the same direction. 'e displace- t t t t+Δt ment vector representing this motion can be obtained in algorithm that predicts the future location P of point p t+Δt from any set S of predictors that is available at time t or diﬀerent ways. 'ese ways constitute three diﬀerent models exempliﬁed in the present study: LK-Lin1, LK-Lin4, and DIS- before. In the context of our study, that set of predictors could be, for example, the previous locations p , p , . . . of Lin1. In the case of LK-Lin1 and LK-Lin4, the displacement t−1 t−2 vector is obtained from “looking back” m time steps from p . We then deﬁne the error of our prediction, henceforth forecast time t to previous feature locations at t − m (tracked referred to as location error ε, as the Euclidean distance between P and p . by using the Lucas–Kanade method, hence the LK label). For t+Δt t+Δt LK-Lin1, m equals 1, so the vector v(t, p ) to displace feature p is the connection from p to p ; for LK-Lin4, m equals 4, t t−1 t 2.3. Extrapolation Techniques. In a veriﬁcation experiment, so that the displacement vector results from the connection we can use our collection of tracks T in order to retrieve between p and p , where the length of the vector is divided t−4 t points p for which the location P at t +Δt should be by 4 in order to obtain the displacement velocity. Hence, a t+Δt predicted, points that could be used as predictors (S ), as well forecast at lead time Δt extends the vector v(t, p ) corre- as the true location p of the point at t +Δt. Assuming that spondingly. Please see Figure 3 for an illustration of both the t+Δt an extrapolation of motion uses feature locations from m LK-Lin1 and the LK-Lin4 method. Of course, any other time steps before t, the minimum feature track length to look-back time m could be used to obtain a displacement produce a forecast would be m + 1. In order to retrieve the vector. In this study, we arbitrarily used m∈ {1, 4} in order to location error of such a prediction at time t +Δt, we would examine the eﬀect of m on the forecast performance. need a minimum track length of m +Δt + 1. For the DIS-Lin1 model, a complete ﬁeld of motion Based on the above terminology, we present in the vectors V is obtained from the Dense Inverse Search DIS following the extrapolation models analyzed in the present (DIS) method [14]; the underlying concept and equations of study. 'ese models are based on the models that were also the DIS method have been elaborated by Kroeger et al. [15] evaluated in a recent benchmarking study on optical-ﬂow- and then used for the extrapolation. A point p is linearly based precipitation nowcasting [2]. Table 2 gives an overview extrapolated from t to t + n by n times the velocity vector of model acronyms and their main properties. v (t, p ), where v (t, p ) is the vector closest to p in the DIS t DIS t 4 Advances in Meteorology ε (t ) t + n n t + n ε (t ) v (t) t + 3 v (t) t + 2 t – 1 v (t) v (t) t + 1 v (t) v (t) t – 2 v (t) v (t) t – 1 v (t) t – 3 v (t) t – 4 Figure 3: Illustration of the linear extrapolation schemes for the LK group: on the left LK-Lin1 and on the right LK-Lin4. 'e location error is displayed by ε(t ). V (t) ﬁeld (Figure 4). V (t) is calculated by OpenCV’s DIS DIS t + n cv2.DISOpticalFlow_create function, which returns velocity V (t) DIS vectors for each grid pixel based on the radar frames from t + 3 t − 1 to t. In a recent benchmarking study about optical-ﬂow- based precipitation nowcasting, Ayzel et al. [2] showed that t + 2 the DIS-based model (referred to as the “Dense” model in that paper) is an eﬀective method for radar-based precipi- t + 1 tation nowcasting. t – 1 2.3.3. Semi-Lagrangian Approach Based on Dense Optical Flow. In a Semi-Lagrangian approach, the motion ﬁeld is typically assumed as constant over the forecast period and Figure 4: In the DIS-Lin1 model, the vector v (t, p ) (light red DIS t the feature trajectory is determined by following the arrow) obtained from V (t) is transferred to the p location and DIS streamlines [16]. Following this concept, the DIS-Rot1 linearly extended to t + n. model (corresponding to “Dense rotation” in [2]) uses the two most recent radar images, t − 1 and t, to estimate V (t) DIS by cv2.DISOpticalFlow_create function. Similar to the DIS- t + n Lin1 model, the displacement vector v (t, p ) which is DIS t V (t) DIS closest to p is used to extrapolate the motion of p from its t t t + 3 position at t to t + 1, providing the location of P . 'is t+1 t + 2 process is repeated at all lead time steps until the maximum lead time is achieved. Hence, at each lead time step n, we t + 1 retrieve the vector v (t, P ) which is closest to P in DI S t+n t+n order to extrapolate the feature location, P . Accord- t t+n+1 ingly, the velocity vector is updated at each lead time step t – 1 from V (t), allowing for rotational or curved motion DIS patterns (Figure 5). Figure 5: Schematic of the DIS-Rot1 model (orange path), where 2.4. Weather Radar Data and Experimental Setup. Our the velocity is updated every time step by transferring the velocity benchmarking case study is based on weather radar data vector v (t, p) (light orange arrow) closest to p (black circles, for DIS from the German Weather Service, namely, the RY product t � 0) or P (orange circles, for t> 0) in V , to the P +Δt t+1 DIS(t) t location to advect. generated as part of the RADKLIM radar reanalysis of the German Weather Service DWD [17]. 'e RY product represents a quality-controlled national precipitation in- tensity composite from 18 C-Band radars covering Germany mountains that would interfere with the beam propagation. at 5-minute intervals and a spatial resolution of 1 km at an Quality control includes a wide range of correction methods extent of 1100 × 900 km. 'e basis of the composite product for, e.g., clutter or partial beam blockage (see [17] for is the so-called “precipitation scans” from each of the 18 details). radar locations. 'e precipitation scan is designed to follow 'e year 2016, selected for this experiment, was char- the horizon as closely as possible at an azimuth resolution of acterized by an annual precipitation close to the climato- 1 and a radial resolution of 1 km, adjusting the elevation logical mean for most regions in Germany, as can be seen in angle for each azimuth depending on the presence of the German Climate Atlas [18]. However, the precipitation Advances in Meteorology 5 mean during autumn was below the normal average and general, expect the length of a track to increase with its during the winter months slightly above the climatological duration. Yet, there are also months—most notably the summer months from May to August—where this expec- mean. As 2016 was a leap year, this experiment was carried out tation is not met; and, of course, the length of a track de- on 105408 radar composite images. Since none of the pends not only on its duration but also on a feature’s methods under evaluation required any kind of training, velocity. 'e average feature velocity in 2016 amounted to a there was no need to split the data into sets for calibration value of 42 km/h; and, in fact, not only does velocity show a and validation. Instead, we used all tracks for veriﬁcation. clear seasonal pattern (with minimum velocities in the For each track, we always use, as forecast time t, a time of 20 summer months; see Figure 6(d)), but also the seasonal minutes after the feature was detected for the ﬁrst time. 'at pattern helps us to understand where the patterns of track is because our model LK-Lin4 needs to look back four time length and duration appear to be “inconsistent.” For ex- steps (i.e., 20 minutes) in order to make a forecast, and we ample, the track velocity is at a minimum in May and June, need to make sure, for a fair comparison, to compare all which decreases the length of track despite the rather high duration values for these two months. models for the same forecast times. 'e clearest seasonal pattern can be observed for rainfall intensity (Figure 6(e)). 'at pattern is very much in line with 2.5. Computational Details. 'e analysis was carried out in a our expectation as rainfall in the summer months is gov- Python 3.6 environment using the following main open- erned by convective events that tend to be more intense than source libraries: NumPy (https://numpy.org), NumExpr stratiform event types. However, if we assumed that a higher (https://github.com/pydata/numexpr), and SciPy (https:// rainfall intensity along a track is caused by the convective www.scipy.org) for general computations; OpenCV nature of the underlying event, the track duration in the (https://opencv.org) for feature tracking; and Pandas corresponding months (e.g., May and June) is at least (https://pandas.pydata.org) and h5py (https://www.h5py. surprising: we would expect a convective event not only to be org). more intense but also to be rather short (in comparison to widespread stratiform rainfall). 'e apparent inconsistency between the patterns of rainfall intensity and track duration 3.Results and Discussion points us to one of the key issues with the presented track 3.1. Properties of Collected Tracks. 'e identiﬁcation and inventory: we must not misinterpret a “track” as an “event” tracking process detected 376,125 features above the rainfall in a hydrometeorological sense. 'e corner detection al- rate threshold of 0.2 mm/h and lasted over 20 minutes, gorithm (see Section 2.1) searches for pronounced features which resulted in 337,776 eligible tracks after applying the in the sense of strong local gradients and tracks a feature for extrapolation step. A track was considered as “eligible” in as long as it stands out. While we deﬁne a rainfall event as case all models had a predicted location at all lead times, some coherent process in space and time, the tracking al- from t to t + n. 'e loss of 10.2% that is implied by the above gorithm could “lose” a feature right in the course of an numbers was caused by the DIS group of models which did ongoing event and maybe, at the same time, ﬁnd another feature to track somewhere else in the ﬁeld. Obviously, the not generate a valid velocity vector v (t, p ) near every p DIS t point, in the V (t) ﬁeld, within a 3.5 km threshold. tracking algorithm was able to track features over a longer DIS duration in May, June, and September of 2016. However, as Figure 6 gives an overview of the properties of the valid tracks. 'e ﬁgure also shows the seasonal dependency of of now, we do not know which properties of the corre- sponding rainfall events caused that eﬀect. We should just these track properties by summarizing their distribution on a per month basis. We would like to emphasize that this emphasize that the duration of a track does not necessarily analysis must not be interpreted as a “climatology” of track correspond to the duration of an event. In the same way, we properties as it only contains data from a single year. Still, we cannot expect the tracking algorithm to ﬁnd features at consider it as illustrative to investigate which properties tend “representative” locations of a convective cell. It will detect such features anywhere in a rainfall ﬁeld where local gra- to exhibit a seasonal pattern and also to discuss whether the observed properties can be considered as representative for dients meet the tracking criteria. 'at could be right not only in the middle of heavy rainfall but also at the edges. Hence, the governing rainfall processes in Germany. In an average month of 2016, we identiﬁed and tracked the reported precipitation intensities along the tracks will not be representative of the mean precipitation intensities of 28,146 features (Figure 6(a)). 'e largest number of tracks is found from April to August (all above the average). Yet, the corresponding precipitation ﬁelds. there is no continuous seasonal pattern in the number of Altogether, we have to emphasize at this point that the detected tracks because, e.g., January and October also show seasonal track statistics are indeed plausible. But it must be rather large counts. clear that track statistics are not necessarily representative No pattern at all can be found for the track length for “event” statistics. 'at notion might be irritating for (Figure 6(b)). With an average track length of 128 km, those who have been deﬁning and tracking features in terms of coherent rainfall objects over their lifetime from initiation monthly maximum mean and median track lengths occur in January, April, and September. A partly similar pattern can to dissipation. A new feature track as we understand it in our analysis could be found right in the middle of an ongoing be found for the track duration that amounts to 207 minutes on average (Figure 6(c)). 'is is plausible as we would, in event, and it can be lost long before the actual rainfall 6 Advances in Meteorology Ave = 28146 Ave = 128 Jan 875.8 Jan Mean Median Feb 793.06 25% 95% Feb Mar 774.18 Min Max Mar 5% Apr 869.54 75% Apr May 882.9 May Jun 825.71 Jun Jul 969.44 Jul Aug 963.59 Aug 908.94 Sep Sep Oct 1233.41 Oct Nov 812.08 Nov Dec 766.41 Dec 10000 20000 30000 40000 50000 0 100 200 300 400 500 Count Track length (km) (a) (b) Ave = 207 Ave = 42 Jan 1870 Jan 133.77 Feb 135.21 Feb 1685 Mar 1315 Mar 117.67 Apr 1350 Apr 115.16 May 1690 May 105.78 Jun 2235 Jun 108.56 Jul 1635 Jul 114.58 Aug 1040 Aug 111.83 Sep 1825 Sep 122.65 Oct 1955 Oct 130.8 Nov 2170 Nov 133.6 Dec 2605 Dec 137.57 0 200 400 600 800 025 50 75 100 125 Duration (min) Velocity (km/h) (c) (d) Ave = 1.6 Ave = 1.09 Jan 14.85 Jan 101.09 Feb 18.48 Feb 108.42 Mar 16.99 Mar 46.27 Apr 23.99 Apr 166.94 May 68.16 May 73.83 Jun 90.7 Jun 88.6 Jul 61.64 241.42 Jul Aug 82.5 Aug 78.15 Sep 35.95 Sep 162.16 Oct 39.19 Oct 110.55 Nov 14.44 Nov 107.92 Dec 12.61 53.9 Dec 0 2468 1.0 1.1 1.2 1.3 1.4 1.5 Rainfall intensity (mm/h) Sinuosity index (e) (f) Figure 6: Statistical properties of detected tracks, organized by month: (a) number of detected tracks, (b) track length, (c) track duration (time elapsed from detection and loss of a feature), (d) feature velocity, (e) rainfall intensity of a detected feature, and (f) sinuosity index of a track. Advances in Meteorology 7 In order to convey a better idea about the rainfall pat- “object” dissolves. However, that does not at all lessen the value of these tracks for the purpose of our analysis, which is terns in the examples, the observed rainfall intensity at forecast time t is plotted as a background in grey scale. to quantify the forecast location error based on well-deﬁned and scale-invariant features. Furthermore, the sinuosity index and the track duration are Having said that, one ﬁnal track property shown in printed in the corresponding subplots. Figure 6(f) has not been discussed yet: the sinuosity index. Please note that the duration of the observed tracks in As pointed out above (Section 2.1), the sinuosity index il- Figure 7 can extend over many hours; very long tracks were lustrates how much the shape of a track deviates from a capped at a duration of 300 minutes for the purpose of straight line (which would correspond to a sinuosity index of plotting. Furthermore, the lead time of the predictions in the examples was set to the (capped) track duration minus 20 1). Figure 6(f) shows rather large sinuosity values for the summer months, May to September, but there is no obvious minutes (which corresponds to the period t − 4 until forest time t). As a consequence, the lead times illustrated in seasonal pattern. More strikingly, the distribution of the sinuosity index is very heavily tailed. 'e average value Figure 7 are mostly longer than the maximum lead time of 120 minutes, which is used in our veriﬁcation experiment amounts to approximately 1.10 in the year 2016, which is, at the same time, the 90th percentile of the sinuosity values. (see the next section). Hence, the ﬁrst visual impression of 'at means, in turn, that the vast majority of tracks are Figure 7 is dominated by the considerable errors that can rather straight, while the remaining tracks show all kinds of occur for such long lead times. But, of course, we should curved, meandering, twisted, or just erratic behavior. rather be aware of the behavior for shorter lead times up to Hence, before we systematically show the results of our 120 minutes. For that reason, the 120-minute lead time is veriﬁcation experiment with regard to the location error (see highlighted by a larger dot. Not surprisingly, most of the competing methods appear to Section 3.3), we would like to illustrate, in the following paragraph, the behavior of observed tracks in comparison to remain rather close to the observed track for short lead times of up to 30 minutes (except, e.g., in subplot Figure 7(j) in which the forecast tracks under diﬀerent sinuosity conditions. the DIS-based methods entirely fail to capture the direction of feature movement). After that, the lead time over which the 3.2. Visual Examples of Observed and Predicted Tracks. extrapolation models adequately predict the observed feature Before we systematically evaluate the performance of dif- track varies, depending on the persistence of the motion be- ferent extrapolation techniques, we would like to provide havior and the validity of the underlying model assumption. some illustrative examples of observed versus predicted For example, all models perform quite well for very long times tracks. 'e selection of tracks for this illustration is arbitrary in subplot (f). In subplot (i), the Semi-Lagrangian approach and does not intend to be representative of the performance (DIS-Rot1) shows a clear advantage, while in subplots of of any of the extrapolation methods. Instead, we aim to Figures 7(c) and 7(k), DIS-Rot1 is outperformed by all other exemplify shapes of observed and predicted tracks under models. Surely, there are several examples (Figures 7(b), 7(d), diﬀerent sinuosity conditions in order to convey a better 7(e), and 7(g)) in which all models entirely fail to anticipate the understanding of the various constellations that will ﬁnally motion for lead times beyond 120 minutes. be condensed into one single location error value. As this compilation of examples is deliberately arbitrary, Figure 7 shows a “gallery” of 11 observed tracks in it does not provide a basis to infer the general superiority or diﬀerent subplots (From Figure 7(a) to 7(k)). Each subplot inferiority of one or the other method. All models appear to also contains the tracks that were predicted by the diﬀerent struggle with predicting very sinuous tracks (subplots in extrapolation models. Each dot represents one feature Figures 7(b), 7(d), 7(e), and 7(g)), which is what we would location in a 30-minute time step, except the ﬁrst one that expect. However, while the ﬁgure makes it diﬃcult to represents the ﬁrst prediction step at ﬁve-minute lead time. compare the absolute location error between the examples LK-Lin1 and LK-Lin4 infer the displacement vector di- (due to the diﬀerent scales), it still appears that the absolute rectly from the feature positions at t and t − 1 or t and t − 4, location error does not necessarily depend on the sinuosity. respectively. As a reminder, DIS-Lin1 and DIS-Rot1 obtain For example, the location error of LK-Lin1 after the max- the displacement vector of a feature from the DIS algo- imum lead time (280 minutes) is higher in subplot 7(i) rithm, a dense optical ﬂow technique that produces motion (almost straight, SI � 1.01) than it is in subplot 7(d) ﬁelds based on the radar images at t and t − 1; DIS-Lin1 (SI � 1.36). In fact, straight tracks can imply a large error if extrapolates the closest vector linearly over the entire lead the initial motion vector of a forecast method fails to rep- time, while DIS-Rot1 uses a Semi-Lagrangian scheme in resent the average long-term direction (see subplot 7(j) for a which the displacement vector is updated as the feature very impressive example). 'en again, large errors can occur moves through the velocity ﬁeld obtained from the DIS if a strong sinuosity of the track coincides with a large technique. Further details have been provided in Section overestimation of the absolute velocity (e.g., subplots 7(b) 2.3. As in all forecasts of our veriﬁcation experiment, the and 7(g)). In that case, the linear extrapolation quickly forecast time t corresponds to the 5th feature of the ob- departs from the track origin, while the actual feature track served track. 'at is because the LK-Lin4 method needs to meanders slowly and remains in the close vicinity of the look four steps back in time (t − 4) in order to produce a origin. For such a scenario, the trivial persistence model (the forecast, while the other methods only look back one step in feature just remains at the origin) will be superior even for time (t − 1). short lead times. 8 Advances in Meteorology Sl = 1.01 τ = 300 Sl = 1.12 τ = 175 Sl = 1.36 τ = 300 Sl = 1.01 τ = 165 (a) (d) (f ) (h) 10 km 10 km Sl = 1.01 τ = 300 (i) Sl = 1.33 τ = 300 (b) 10 km 10 km Sl = 2.22 τ = 300 Sl = 1.36 τ = 300 10 km (e) 10 km (j) Sl = 1.0 τ = 170 (c) Sl = 1.01 τ = 245 10 km (k) Sl = 1.0 τ = 300 (g) 10 km 10 km 10 km 10 km 05 10 15 20 25 30 mm/h Observation DIS-Lin1 LK-Lin1 DIS-Rot1 LK-Lin4 Figure 7: Compilation of forecast versus observed tracks under diﬀerent sinuosity conditions. Due to the diﬀerent spatial extents of the windows, the scale of each subplot is diﬀerent. Hence, a 10 km scale bar is provided for orientation. For each example, the observed track duration τ (in hours) and its sinuosity index SI are shown. 'e lead time of 120 minutes is highlighted by a larger dot. Some very long tracks have been capped at a maximum of 300 minutes for illustrative purposes. Altogether, these diﬀerent examples give us a better idea all models, dramatically lower than that for the persistence of how location errors can develop from both inadequate model; the mean error of persistence is higher than the mean error of any model at any lead time, which means that all model assumptions (e.g., linear approximation versus curved or sinuous conditions) and a failure to approximate models, on average, have positive skill at all lead times. For all models, the error distribution is obviously positively skewed, the average motion from the initial feature locations. It is impossible, though, to diagnose the superiority of one or the with the mean error being much higher than the median, and other model from these examples. Hence, we will now thus there is a heavy tail towards high location errors. systematically examine the results of our model veriﬁcation For very short lead times of up to 10 minutes, the mean experiment. We will not only analyze how the location error error is about one kilometer for all competing models except depends on lead time, but we will also investigate how the for persistence which is already up at more than seven ki- model performance relative to the persistence model de- lometers after ten minutes. After 60 minutes, the mean pends on the sinuosity of the underlying tracks. location error of all models exceeds a distance of 5 kilo- meters, as well as 10 kilometers after 110 minutes. For all models, at least 25% of all forecasts exceed an error of 5 3.3. Systematic Quantiﬁcation of the Location Error. After kilometers after 50 minutes and an error of 10 kilometers having exempliﬁed diﬀerent observed and predicted tracks in after 90 minutes. After 75 minutes, at least 5% of all forecasts the previous section, we now present the results of our exceed an error of 15 kilometers. benchmarking experiment. Figure 8 shows the distribution of Altogether, the location error can be substantial for a locations errors for diﬀerent models and lead times up to 120 signiﬁcant proportion of forecasts, while the median loca- minutes. For each lead time, the box plots specify mean, tion error grows at a more moderate rate. median, interquartile range, and the 5th and 95th percentiles While this general pattern governs the behavior of all of the location error. For all models, the error quantiles in- models, there are clear diﬀerences between the performances crease slightly exponentially but almost linearly with lead time. of the competing models. 'ese diﬀerences, however, are not 'e rate at which the location error grows with lead time is, for always coherent across all error quantiles and lead times, Advances in Meteorology 9 Median Mean 25% 95% 5% 75% 5 101520253035404550556065707580859095 100 105 110 115 120 Lead time (min) Lk-Lin1 DIS-Rot1 Lk-Lin4 Persist DIS-Lin1 Figure 8: 'e distribution of location errors for diﬀerent extrapolation models and lead times. except for the DIS-Rot1 model, which has the weakest per- examine the skill of our models more closely. Skill scores rate formance of all models at virtually all lead times and for all the score of a forecast in relation to the score of a reference quantiles, and the LK-Lin1 model, which performs better than forecast, in our case persistence. 'ey are particularly useful in benchmark studies such as the present one. Equation (1) DIS-Rot1 but ranks second last. As for the best forecast performance, the LK-Lin4 and the DIS-Lin1 models take turns shows the general deﬁnition of skill as derived from any forecast score, as well as the speciﬁc formula if we use the depending on error quantile and lead time: For the 5th and the 25th percentiles, the LK-Lin4 model performs best for lead location error ε as the “score” (which becomes zero for a times up to 100 minutes, for the median up to 80 minutes, and perfect forecast) and persistence as the “reference”: for the mean up to 55 minutes. 'e DIS-Lin1 model shows the ε − ε Score − Score forecast persistence forecast reference strongest changes of relative performance over lead time: as for Skill � � . Score − Score −ε the mean error, DIS-Lin1 starts to outperform LK-Lin4 at a perfect reference persistence lead time of 60 minutes and continues this way until the (1) maximum lead time of 120 minutes. As for the median error, We examine the forecast skill under diﬀerent sinuosity DIS-Lin1 only catches up with LK-Lin4 after 90 minutes. For conditions. As already pointed out in Section 3.1, the dis- the 75th percentile, DIS-Lin1 outperforms LK-Lin4 after 50 tribution of sinuosity is highly skewed and 90% of observed minutes and for the 95th percentile already after 20 minutes. tracks would pass as at least “rather straight” with a sinuosity In summary, LK-Lin4 tends to outperform DIS-Lin1 in the index equal to or lower than 1.1. Hence, we split the forecasts ﬁrst hour, while DIS-Lin1 becomes superior in the second into three unequal groups, depending on quantiles of the hour, apparently because it tends to avoid very high errors sinuosity index: 'e ﬁrst group contains the “straight” 90% more eﬃciently than LK-Lin4 does. of the forecasts with a sinuosity index below 1.1. We consider In the following, we would like to better understand how the value of 1.1 as an—admittedly—arbitrary threshold model skill is aﬀected by sinuosity. In Section 3.2, we have between “rather straight” and “rather winding” tracks. 'e already indicated that the absolute values of location errors remaining 10% of tracks are split into two equally sized do not clearly depend on sinuosity. 'at was conﬁrmed by groups, again based on sinuosity: the 5% with the highest the systematic veriﬁcation experiment (results not shown). sinuosity, exceeding an SI value of 1.2, could be labelled as Yet, the diﬀerence between an extrapolation model and the “twisted,” and the remaining 5% with intermediate SI values (trivial) persistence model might very well depend on sin- between 1.1 and 1.2 could be labelled as “winding.” Figure 9 uosity. In order to formally evaluate that hypothesis, we now Distance (km) 10 Advances in Meteorology Straight : Sl < 1.1 Winding : 1.1 ≤ Sl < 1.2 Twisted : Sl ≥ 1.2 0.88 0.66 0.5 0.64 0.4 0.86 0.62 0.3 0.60 0.84 0.2 0.58 0.82 0.56 0.1 0.54 0.80 0.0 0.52 0.78 0.50 –0.1 5 30 60 90 120 5 30 60 90 120 5 30 60 90 120 Lead time (min) Lead time (min) Lead time (min) Lk-Lin1 DIS-Lin1 Lk-Lin1 DIS-Lin1 Lk-Lin1 DIS-Lin1 Lk-Lin4 DIS-Rot1 Lk-Lin4 DIS-Rot1 Lk-Lin4 DIS-Rot1 Figure 9: 'e mean model skill over each lead time with regard to location prediction for diﬀerent extrapolation models and sinuosity conditions. Please note that the very low skill values of the DIS-based models at 5-minute lead time (in the winding and twisted groups) are hidden by the scaling of the y-axis. At ﬁve-minute lead time, both models only have a skill of about 0.35 (winding) and −0.55 (twisted). shows the average model’s skill over every lead time for these images. In our study, we detected features by using the three sinuosity classes. Clearly, the model skill dramatically approach of Shi and Tomasi (1994) and tracked these fea- varies between these three groups: it ranges between 0.79 tures following the approach of Lucas and Kanade [9], using and 0.87 for the “straight” category, mostly between 0.5 and both algorithms as implemented in the OpenCV library. We 0.65 for the “winding” category, and mostly between 0 and increased the robustness of extracted feature tracks by 0.5 for the “twisted” category. 'is decrease of skill with making sure that the features can be successfully tracked forwards and backwards. 'at approach, together with a increasing sinuosity is well in line with our expectation. Furthermore, the ranking of all models based on skill is quite rather strict deﬁnition of parameter values for feature de- coherent across all categories and also consistent with our tection and tracking, increases our conﬁdence in the reli- previous analysis of location errors. DIS-Lin1 becomes ability of the detected tracks. Still, we have to assume that the superior within the second forecast hour, while LK-Lin1 feature locations themselves are, as any measurement, un- performs better in the ﬁrst forecast hour. Only in the certain. We expect the main sources of uncertainty to be the “twisted” category do LK-Lin1 and, even more, LK-Lin4 grid resolution (which does not allow resolving errors below outperform DIS-Lin1 across all lead times. It should be 1 km), and complex small-scale intensity dynamics that can noted, though, that the overall skill in the twisted category is interfere with motion patterns. For future studies, we suggest very low for all competing models. In the “winding” cate- a comprehensive sensitivity analysis with regard to the gory, LK-Lin1 slightly outperforms LK-Lin4 in the ﬁrst 20 parameters of the feature detection and tracking algorithms minutes. Finally, DIS-Rot1 performs worst at all lead times in order to better understand the eﬀects on both the number in all categories. and the robustness of detected tracks in the context of 'e change of model skill with lead time should be rainfall motion analysis. Still, we assume that the error of interpreted with care, as it depends on both the performance extrapolating feature motion is substantially larger than the of the extrapolation model itself and the location error of the error of feature tracking itself. In summary, we consider it persistence model. For most models and SI categories, the warranted to use the observed tracks as a reference in order skill appears to reach an optimum at some lead time, which to evaluate the performance (or, inversely, the error) of any implies that the superiority of the model over persistence model that aims to predict the future locations of such reaches a maximum. precipitation features. For that purpose, we deﬁned the location error of a forecast at any lead time Δt ahead of the forecast time t as the Euclidean distance between the ob- 4.Conclusions served and the predicted feature locations at t +Δt. One might want to use this approach to comprehensively In this paper, we have introduced a framework to isolate and quantify the location error of any forecast model for the full quantify the location error in precipitation nowcasts that are based on ﬁeld-tracking techniques. While it is often assumed spatial domain of a forecast grid, for example, a national radar composite. In such a case, we would need to assume that errors in precipitation nowcasts are dominated by the temporal dynamics of precipitation intensity, the location that the average of forecast errors that we have quantiﬁed from observed feature locations in a forecast domain is error of predicted precipitation features has so far not been explicitly and formally quantiﬁed. representative for the average error of all location predic- tions in that domain. We have not yet investigated the 'e main idea of our framework is to detect and track scale-invariant precipitation features (corners) in radar validity of that assumption. One might argue that the Skill Skill Skill Advances in Meteorology 11 precipitation nowcasting, for example, in the context of early behavior of locations identiﬁed as “corners” or “good fea- tures to track” might not be representative for the motion warning systems for pluvial ﬂoods in urban environments (see [19]), it becomes obvious that location errors matter: the order behavior of the entire precipitation ﬁeld; however, it will be diﬃcult to ﬁnd evidence to either verify or falsify such a of magnitude of these errors is about the same as the typical hypothesis, as it would require another independent way to extent of a convective cell or of a medium-sized city. Hence, the quantify the location error. Still, we are convinced that the uncertainty of precipitation nowcasts at such length sca- proposed framework is useful: even without the need of les—just as a result of locational errors—can be substantial strong assumptions on representativeness, the framework already at lead times of less than an hour. allows us to compare and benchmark the ability of diﬀerent While similar conclusions have already been drawn by using spatially sensitive veriﬁcation measures such as the models to forecast future locations of precipitation features and thus to speciﬁcally focus on improving that ability by Fractions Skill Score (see, e.g., [6]), our framework allows us to isolate the location error for speciﬁc models and situations, to future model development. 'e hypothesis that such further model developments better understand the factors that govern these errors, and hence to use that knowledge in order to speciﬁcally improve the are urgently required is supported by the results of our benchmarking study. It should be clariﬁed again that this extrapolation of motion patterns in existing nowcasting models. benchmark study does not intend to suggest better ex- As an example, we have demonstrated how the use of the trapolation models but to demonstrate the ability of our sinuosity index can help us to better understand the predictive framework to unravel the location errors that are produced skill and hence the uncertainty of our models in speciﬁc sit- by state-of-the-art extrapolation methods. For that purpose, uations. We hope that the large number of extracted tracks will we compared four models: two models use the feature lo- help to foster the development of new techniques that use data- driven machine learning models for the extrapolation of feature cations before and at forecast time t in order to derive displacement vectors which are then used to linearly ex- location. For that purpose, we have made openly available the full set of extracted feature tracks for the year 2016 (https://doi. trapolate feature movement over the lead time. Model LK- Lin1 uses the feature locations at t and t − 1, and LK-Lin4 org/10.5281/zenodo.4024272 [20]) to serve as input to future studies. However, such future studies should also use radar data uses the feature locations at t and t − 4. 'e other two models are based on the dense optical ﬂow algorithm DIS that from a longer time period in order to learn more about the generates a full motion vector ﬁeld under various seasonal eﬀects related to the properties of feature tracks. smoothness constraints. 'e model DIS-Lin1 obtains the Data Availability displacement vector for a feature at t from the nearest motion vector in the ﬁeld based on the radar images at times 'e radar data are provided by DWD at https://opendata. t and t − 1 and uses that vector over the entire lead time. DIS- dwd.de/weather/radar/radolan/ry (last access: Sept. 2020). Rot1, in contrast, uses a Semi-Lagrangian scheme in which 'e code of this analysis is available in the Github repository the displacement vector is updated as the feature moves under https://github.com/arthurcts/loc_error (last access: through the motion ﬁeld obtained from the DIS technique. Sept. 2020). 'e dataset of extracted feature tracks has been 'e motivation behind the DIS-Rot1 model is to better deposited in the Zenodo repository (https://doi.org/10.5281/ represent rotational or curved motion patterns. From these zenodo.4024272). four competing models, LK-Lin4 appears to be the best model in the ﬁrst forecast hour and DIS-Lin1 the best in the Conflicts of Interest second. DIS-Rot1 performs consistently the worst. 'at is not quite in line with our naive expectation in which we 'e authors declare that there are no conﬂicts of interest would hope that a Semi-Lagrangian approach should be able regarding the publication of this paper. to better capture at least curved motion patterns. But not even in the winding category does the complexity of the DIS- Acknowledgments Rot1 approach pay oﬀ. Whether that is due to the imple- 'e authors acknowledge the German Weather Service, mentation of the Semi-Lagrangian approach or due to the namely, Dr. Tanja Winterrath, for making the RY data lack of validity of the approach should be the subject of available from the latest RADKLIM reanalysis. Arthur Costa future research. Comparing LK-Lin1 to LK-Lin4, we see a Tomaz de Souza has been funded by a Ph.D. scholarship of clear advantage in looking back in time more than one step. the German Academic Exchange Service (DAAD). Georgy It appears that, this way, we can retrieve more reliable, more Ayzel was partly funded by the ClimXtreme project (BMBF, representative, and less noisy displacement vectors, which FKZ 01LP1903B). shows in the superiority of LK-Lin4 over LK-Lin1. For all competing models, the mean location error exceeds References a distance of 5 kilometers after 60 minutes and 10 kilometers after 110 minutes. At least 25% of all forecasts exceed an error [1] M. 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Quantifying the Location Error of Precipitation Nowcasts

Quantifying the Location Error of Precipitation Nowcasts

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Quantifying the Location Error of Precipitation Nowcasts

Quantifying the Location Error of Precipitation Nowcasts

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