## 1. Introduction

The term “nowcasting” refers to short-term (0–6 h or less) forecasting. Nowcasts of high-impact weather events, such as flood-producing rains and hail, can be made with sufficient lead time accuracy and spatial specificity within this time frame such that appropriate actions can be taken to effectively mitigate the loss of life and property. Thus, the term nowcasting emphasizes specificity and the short time nature of a weather event forecast (Browning 1982).

In current operational nowcasting systems, extrapolation of radar echoes, satellite imagery of clouds, and/or lightning location data are the primary mechanisms that are used to generate forecasts in the 0–3-h time frame, with such forecasts gradually being combined with numerical weather prediction (NWP) model forecasts made during the 3–6-h forecast lead time period (Wolfson et al. 2008; Dupree et al. 2009; Bowler et al. 2004; Li and Lai 2004). The underlying assumption that the relatively predictable translation of precipitation patterns dominates the relatively less predictable growth and decay of precipitation intensity forms the basis for using extrapolation-based nowcasting methods over such short prediction periods. Although studies have shown that the accuracy of extrapolation-based nowcasts decreases rapidly during the 0–1-h forecast window with a rate closely related to the scale of the precipitation pattern and the associated forcing mechanism (Browning 1980; Wilson et al. 1998), extrapolation-based nowcasts of radar fields up to 30 min have led to significant improvement in warnings and advisories resulting in substantial savings of life and property (National Research Council 1995).

The Collaborative Adaptive Sensing of the Atmosphere (CASA) radar network provides atmospheric measurements covering an approximate 7000 km^{2} area using low-cost and easily deployable [relative to current operational Weather Surveillance Radar-1988 Doppler (WSR-88D) radars] X-band radars (McLaughlin et al. 2009). Nowcasting is currently running in the CASA Distributed Collaborative Adaptive Sensing (DCAS) system to provide predictions of precipitation pattern locations up to 10 min into the future for emergency manager decision-making support. Nowcasting is also used to steer the four radar nodes comprising the CASA radar network to better observe the evolution of severe storm events using 1–5-min nowcasts to benefit operational forecasters. Data collection and processing are currently performed at Norman, Oklahoma, and the data are transferred to the University of Massachusetts at Amherst where the actual nowcasting computations are performed. Thus, the distributed processing environment and relatively large high-resolution datasets require the use of a fast nowcasting algorithm and efficient operational implementation to provide useful products to the end users of the system.

The Dynamic and Adaptive Radar Tracking of Storms (DARTS) nowcasting method was developed to meet the operational nowcasting requirements of the CASA DCAS system. DARTS represents the general continuity equation describing the flux and evolution of an observed precipitation field represented by a sequence of radar reflectivity fields as a discrete spatiotemporal linear model that is formulated in the Fourier domain and solved using linear least squares estimation (Xu and Chandrasekar 2005). Advection is performed using an efficient method based on a sinc kernel representation of the reflectivity field. This formulation allows for efficient predictions of storm motion, areal evolution, and position using high-resolution CASA radar data in an operational distributed processing environment.

The purpose of this paper is twofold: (a) to describe the theory and implementation of the DARTS method for motion vector estimation and the sinc kernel–based advection method in the context of current radar-based nowcasting approaches, and (b) to show that this nowcasting methodology was successfully implemented in the CASA DCAS system. The DARTS nowcasting and sinc-based advection methods are described in section 2. An overview of the CASA DCAS system and the architecture of the operational CASA nowcasting system are described in section 3. The assessment of nowcasting performance during the 2009 Integrated Project 1 (IP1; Brotzge et al. 2005) experiment is presented in section 4. The paper concludes with a summary and discussion.

## 2. Nowcasting methodology

### a. Overview of current radar-based nowcasting approaches

Currently, strictly radar-based nowcasting methods belong to one of the following four categories (or a combination thereof): area-based, object-based, statistical, and probabilistic approaches.

Area-based nowcasting approaches estimate a motion vector field over the entire radar coverage domain and have shown effectiveness in estimating the translation of a variety of precipitation pattern types. Additionally, the motion vector fields generated by area-based nowcasting methods cannot only provide a means for tracking internal storm motions (e.g., rotation and vortices), but also vector convergence (representing areal decay) and divergence (representing areal growth). Area-based nowcasting approaches also require a separate advection algorithm to suitably translate the latest observed (or predicted) data field recursively with the estimated motion vector field to produce future predicted fields (Germann and Zawadzki 2002; Rood 1987). Thus, the estimation of a distributed motion vector field and the advection process can make such approaches computationally expensive. Examples of area-based methods include Tracking of Radar Echoes by Correlation (TREC; Rinehart and Garvey 1978; Rinehart 1981; Tuttle and Foote 1990; Chornoboy et al. 1994), where locations of maximum cross-correlation coefficients between subgrids of successive radar fields are determined to estimate motion; Continuity of TREC (COTREC) vectors (Li et al. 1995), where a variational constraint is applied to smooth the TREC-derived motion vector field; Growth and Decay Storm Tracker (GDST; Wolfson et al. 1999), where radar data fields are prefiltered with a directional elliptical filter to separate precipitation pattern envelope motion from internal motions prior to estimation using a cross-correlation-based approach; and the McGill Algorithm for Precipitation Nowcasting by Lagrangian Extrapolation (MAPLE; Germann and Zawadzki 2002), where a cost function based on atmospheric physics is minimized to estimate the motion vector field and a wavelet-based filtering method is applied to predicted fields to remove unpredictable scales (Turner et al. 2004).

Object-based nowcasting schemes attempt to identify areas of high reflectivity in radar fields and track the size, shape, and translation of coherent features in time across successive fields. Object-based nowcasting methods work very well with strong, well-defined, and well-behaved storm structures. Examples of object-based nowcasting methods include the Storm Cell Identification and Tracking (SCIT; Johnson et al. 1998) method, where reflectivity objects are tracked using a *k*-means algorithm in a 3D paradigm; the Thunderstorm Identification Tracking, Analysis, and Nowcasting (TITAN) algorithm (Dixon and Wiener 1993), where combinatorial optimization techniques with the capability to detect cell mergers and splits are used to track reflectivity objects; TRACE3D (Handwerker 2002), where a two-stage cell detection method is combined with a tracking procedure similar to that used in SCIT to account for cell splitting and merging and track crossings; and the approach used in the Weather Decision Support System 2 (WDSS2) “SegMotion” module (Lakshmanan et al. 2003), where a hierarchical *k*-means algorithm is used to identify reflectivity objects in a 3D paradigm and an optimization technique with Kalman filtering is used to estimate object motion. The Enhanced TITAN (ETITAN; Han et al. 2009) method combines the TREC and TITAN approaches.

Nowcasting methods also exist where atmospheric evolution is represented by statistical models that are able to incorporate knowledge of atmospheric dynamics. Radar reflectivity fields are modeled as random processes using radial (Cornford 2004) or parameterized elliptical basis functions (Xu et al. 2005; Fox and Wikle 2005). Future fields are then predicted within a Bayesian hierarchical model framework using Markov chain Monte Carlo (MCMC; Jones and Hobert 2002) simulation techniques where the mean field of the resulting posterior distribution is taken to be the nowcast and the standard deviation field is the measure of forecast uncertainty. Physical characteristics of precipitation patterns are modeled as parameters for which a range is preselected based on meteorological expertise. Although each level of the Bayesian hierarchical model can be parameterized, computational complexity of such models is high and a comprehensive evaluation of such methods has yet to be performed. Another statistical approach is called spectral prognosis (S-PROG; Seed 2003), where a radar reflectivity field is decomposed into a cascade of random fields using a notch filter in the frequency domain. Nowcasts are generated by extrapolating each level in the cascade of precipitation scales by a single motion vector estimated by maximizing cross correlation over similar levels in successive radar fields and by modeling the temporal development at each level in the cascade using a second-order autoregressive model.

Several probabilistic nowcasting approaches also exist where the probability of encountering a precipitation rate above a certain threshold at each point in the field of radar coverage is computed for a given lead time (Germann and Zawadzki 2004; Megenhardt et al. 2004; Schmeits et al. 2008; Dance et al. 2010).

### b. The DARTS model

*F*(

*x*,

*y*,

*t*) is the sequence of radar reflectivity fields,

*U*(

*x*,

*y*) is the east–west component of the velocity field,

*V*(

*x*,

*y*) is the north–south component of the velocity field, and

*S*(

*x*,

*y*,

*t*) can be interpreted as the sequence of additive evolution mechanisms, such as the growth and decay of intensity. DARTS estimates precipitation pattern motion in terms of a motion vector field by representing Eq. (1) as a discrete spatiotemporal linear model, given by Xu and Chandrasekar (2005),where

*F*

_{DFT}(

*k*,

_{x}*k*,

_{y}*k*) represents the 3D discrete Fourier transform (DFT) coefficients of the discrete observed radar field sequence

_{t}*F*(

*i*,

*j*,

*k*),

*U*(

*i*,

*j*),

*V*(

*i*,

*j*), and S

_{DFT}(

*k*,

_{x}*k*,

_{y}*k*) represents the 3D DFT coefficients of the sequence of estimated evolution (i.e., growth and decay) fields

_{t}*S*(

*i*,

*j*,

*k*). Respectively,

*T*and

_{x}*T*are the lengths of the east–west and north–south dimensions of the observed gridded reflectivity fields,

_{y}*T*is the number of reflectivity fields considered for motion estimation (i.e., the temporal span of the sequence of gridded reflectivity fields), and

_{t}*N*and

_{x}*N*are the maximum harmonic numbers of

_{y}*F*

_{DFT}(

*k*,

_{x}*k*,

_{y}*k*) in the horizontal and vertical dimension, respectively. The further details and the derivation of Eq. (2) can be found in appendix A.

_{t}Equation (2) can be implemented in matrix form *F*(*i*, *j*, *k*), and **x** is the vector of Fourier coefficients representing *U*(*i*, *j*) and *V*(*i*, *j*). Details of the implementation of Eq. (2) can be found in appendix B. The operational CASA nowcasting system does not include the *S* term in the prediction model [Eq. (2)] based on the fact that several previous studies have shown attempts to predict short-term growth and decay based on intensity trends in past radar observation sequences, which do not provide significant improvement in nowcasting performance (Tsonis and Austin 1981; Wilson et al. 1998; Grecu and Krajewski 2000; Germann and Zawadzki 2002).

DARTS is “dynamic” in the sense that the motion estimation model is based on a dynamic equation and is considered “adaptive” because the smoothness of the estimated motion field and the scale of motion can be controlled. By truncating the DFT coefficients representing the estimated 2D motion vector field components *U*(*x*, *y*) and *V*(*x*, *y*), smoothness of the estimated motion field can be controlled, where choosing lower values of *M _{x}* and

*M*(see appendix B) provides a smoother motion field. By truncating the model coefficients representing the band limit in each dimension of the DFT,

_{y}*N*and

_{x}*N*, Fourier low-pass filtering can be implemented on the sequence of observed reflectivity fields. In this sense, high wavenumbers correspond to small spatial scales and lower cut-off frequencies in each field dimension imply that the motion of larger-scale weather features will be estimated. Additionally, the ability to compute the DFTs using fast Fourier transform (FFT) algorithms (Frigo and Johnson 2005) allows for computationally efficient motion estimation.

_{y}### c. The sinc kernel–based advection model

*F*(

*x*,

*y*,

*t*) is approximated according to the Whittaker–Shannon–Kotelnikov sampling theorem (Whittaker 1915; Shannon 1949; Higgins 1996) as follows:where the equidistant samples of

*F*(

*x*,

*y*,

*t*),

*F*(

_{kl}*t*) =

*F*(

*k*Δ

*x*,

*l*Δ

*y*,

*t*), may be interpreted as the coefficients of the 2D product basis that are obtained by appropriate translation and rescaling of the sinc kernel function sinc(

*x*) = sin(

*πx*)/(

*πx*). The discrete approximation of Eq. (3) can be written as (appendix C)where

Equations (4)–(9) show that numerical advection is conducted by a matrix computation and the temporal integration is done by the iteration of matrix computations over small steps [*δ _{t}* in Eq. (5)]. Figure 1 shows that the sinc kernel–based advection scheme does not introduce significant numerical diffusion and thus adequately preserves power in the smallest spatial scale (1 km) relative to results of a similar study presented by Germann and Zawadzki (2002).

## 3. Nowcasting system implementation

### a. Overview of CASA DCAS

The CASA radar network consists of four low-power (10-kW peak radiated), short-range (40 km max), X-band (9.41 GHz) dual-polarization Doppler radar units mounted on telecommunication towers covering an area of approximately 7000 km^{2}. The four polarimetric radar nodes are installed along U.S. Interstate 44, southwest of Oklahoma City, Oklahoma, and are located within the coverage area of the KFDR and KTLX WSR-88D radars. Each radar node is located approximately 30 km away from the next unit providing a triangulated, overlapping coverage area. Multiple simultaneous radar observations of the same weather event by the radars in the CASA network facilitate the use of an attenuation correction procedure to improve rainfall estimates (Lim and Chandrasekar 2009; Chandrasekar and Lim 2008). Attenuation correction and clutter removal procedures are performed in real time at each of the four CASA radar nodes.

The network is designed with the system goal of mapping severe weather events in the lowest 3 km of the troposphere with high spatial and temporal resolution. Each radar node was developed to accomplish this goal through the coordinated interaction with the other radars in the network via a real-time, closed-loop software control system. This software control system consists of a cluster of computers and storage devices referred to as the System Operation and Control Center (SOCC) that runs a suite of algorithms known as the Meteorological Command and Control (MC&C; Zink et al. 2005) responsible for managing the automated operation of the network. The SOCC (or multiple SOCCs) can be physically located wherever there is an adequate network connection. Radio links provide Internet connectivity to the radar node sites allowing radar data products to be provided to local emergency managers and National Weather Service (NWS) operational forecasters in real time. Further details of the IP1 radar network infrastructure, radar node architecture, hardware, software, operation, and examples of features such as scan patterns are described by McLaughlin et al. (2009) and Junyent et al. (2010).

### b. Nowcasting system architecture

The nowcasting system development and maintenance are performed by Colorado State University located at Fort Collins, Colorado. The MC&C located at the SOCC at the University of Oklahoma at Lawton, Oklahoma, ingests data from the remote radars, identifies meteorological features in the data, and determines each radars’ future scan strategy based on detected feature contours and end-user requirements. The data are converted to Network Common Data Format (NetCDF) and is sent to the SOCC located at the University of Massachusetts at Amherst via a local data manager (LDM). The overall system architecture is depicted in Fig. 2. While the entire nowcasting process could have been performed at the SOCC located at Norman, this distributed process was established as a demonstration experiment to understand the challenges associated with distributed processing as part of the CASA network scaling (Jogalekar and Woodside 2000) operation.

After the radar data files are suitably synchronized by the ingester, the individual radar data files are gridded and merged. These gridded and merged data files serve as input to the DARTS nowcasting module, which provides predicted reflectivity fields presented to the end user via an Internet-based display. The diagram of the operational DARTS software module is shown in Fig. 3 with an example observed reflectivity image and corresponding 10-min prediction, as shown on the Internet-based display depicted in Fig. 4.

Evaluation of the performance of the DARTS nowcasting system during the 2009 IP1 experiment is described in the next section.

## 4. Assessment of nowcasting performance during the 2009 CASA IP1 experiment

The performance of the CASA nowcasting system during the 2009 IP1 experiment was evaluated. The ability of the system to provide predictions of radar reflectivity fields up to 10 min into the future was evaluated quantitatively while the effectiveness of the system to provide radar scanning adaptation was evaluated qualitatively. This section describes the dataset used for evaluation along with the methodology and results of the quantitative evaluation of nowcasting products. The benefits to nowcasting of adaptive radar scanning are also illustrated.

### a. Data

Radar reflectivity data collected from 17 events during the 2009 CASA IP1 experiment were considered for evaluation. The average duration of each event in the dataset was approximately 3.9 h for a total of approximately 66 h of data (approximately 3960 data frames). The set of events covered a wide range of precipitation pattern types, consisting of supercellular, strong and steady quasi-linear, and disorganized multicellular events (Byers and Braham 1949; Weisman and Klemp 1986). Details of the events are given in Table 1.

Summary of precipitation event data collected during the 2009 CASA IP1 experiment used for nowcasting performance evaluation.

Constant altitude plan position indicator (CAPPI) grids at an altitude of 1 km above ground level (AGL) with grid spacing of 0.5 km covering an area of dimension ±70 km in the east–west and north–south directions were generated by merging attenuation-corrected reflectivity data from each of the four network radar nodes during each 1-min volume scan. A threshold of 20 dB*Z* was applied. Figure 5 shows examples of the reflectivity and DARTS motion vector fields for 4 of the 17 events (one for each of the storm types identified: supercell, line, single cell, and multicell). This figure illustrates the differences in structure between the various case types and the corresponding estimated motion vector fields. It should also be noted that the nature of the nowcasting model imposes the continuity of motion vectors over the entire grid, even where no data were available to estimate the motion vector field. This is important because while observations may not exist in a particular region at time *t*_{0}, observations (and subsequent predictions) may be advected into these regions at times up to *t*_{0} + *τ*, where *τ* is the maximum lead time for prediction.

### b. Quantitative performance assessment methodology

While nowcasting for hydrological applications are typically assessed using continuous scalar measures of forecast accuracy such as mean absolute error (MAE) and mean square error (MSE), nowcasting applications involving warning decision support for detection of potentially severe weather typically measure nowcasting performance in terms of categorical yes/no (e.g., rain/no rain) detection relative to a predetermined measurement threshold representative of a desired threat (Doswell et al. 1990; Ebert and McBride 2000; Schaefer 1990; Stensrud and Wandishin 2000). This paradigm is preferred for nowcasting applications in the CASA network where the ability of the nowcasting method to predict a sufficiently high reflectivity value (which can be converted to rain rate) in a given region is important for end-user emergency decision support.

*A*represents the intersection of the areas over which the event was forecast and subsequently occurred,

*B*represents the area over which the event was forecast and subsequently did not occur, and

*C*is the area over which the event occurred but was not forecast to occur. The CSI, POD, and FAR range from 0 to 1, with 1 being a perfect CSI and POD score and 0 being a perfect FAR score.

*F*,

_{i}*O*) is the

_{i}*i*th of

*N*pairs of forecasts and observations.

The CSI, FAR, POD, and MAE score statistics for lead times up to 10 min averaged over all events are shown in Fig. 6. The scores are favorable, with mean CSI scores for DARTS consistently higher than those using a persistence forecast [i.e., a commonly used reference forecast that considers the latest observation as all future predictions (Germann and Zawadzki 2002; Pierce et al. 2004; Ebert et al. 2004)]. At a lead time of 10 min, mean CSI scores for the CASA nowcasting system are approximately one standard deviation higher than those for persistence and the corresponding MAE scores are approximately 1 dB*Z* lower, corresponding roughly to a factor-of-2 difference in rain-rate estimates. As expected, standard deviation increases with increasing lead time. A 1 km × 1 km scoring area size and threshold of 30 dB*Z* were used in the CSI, FAR, and POD scoring procedures. An assimilation time of 10 min (10 data frames) was used for motion field estimation.

The results presented in Fig. 6 show that most of the relative improvement in nowcasting performance is attributed to the reduction of the FAR. Given the high spatial resolution of the data considered, it is reasonable for data to advect to neighboring grid points even at a lead time of 1 min (i.e., an advection speed of about 30 km h^{−1} would move a datum to a neighboring grid point). Thus, based on the nature of the POD and FAR scores, the results presented in Fig. 6 show that advection according to estimated motion vectors likely increases the number of “hits” (Fig. 6c), while subsequently reducing the number of “false positives” (Fig. 6b) even at a lead time of 1 min.

The high computational efficiency of the DARTS algorithm is critical when running in the CASA radar network with scan update rates of 1 min. Average motion field generation and advection times were approximately 0.30 and 47 s, respectively, when running the nowcasting system on a standard Linux-based compute server.

### c. Improved radar observations via adaptive scanning

Until the 2009 IP1 experiment, the CASA system was optimized to scan based on observations from past scans. This latency was found to be detrimental in fast-moving weather systems and was significant enough that the automated scans occasionally cut off the leading edges of moving storms. To mitigate this condition, nowcasting was introduced into the DCAS closed loop to generate fields of predicted reflectivity to provide better estimates of storm location for scanning purposes. In this sense, the motion vector field estimates generated by DARTS can be used to schedule future scanning strategies based on predicted motion (and thus future location) of storms. This information can be used to generate very specific warning areas and allow for the precise deployment of spotters and first responders during severe weather events. Such capability also allows for faster sampling of important weather features because the radars are not scanning relatively unimportant regions throughout the entire 360° sweep.

The ability of nowcasting to adjust the CASA radar scan strategy to observe the leading edge of a moving storm is illustrated in Fig. 7, which depicts an image of the MC&C display of the 0246 UTC 17 May observation comparing coverage afforded by radar node steering using previous observations versus steering using 5-min nowcasts. In this case, the storm was advecting toward the northeast. It is apparent the leading edge is observed when steering is based on 5-min nowcasts and is missed when steering uses previous observations.

The usefulness of such capability was assessed by NWS forecasters, local emergency managers, and researchers during the 2009 IP1 experiment via end-user surveys (Phillips et al. 2008). End-user feedback on radar scanning strategy adaptation was unanimously positive.

## 5. Summary and conclusions

Nowcasting capability has been successfully incorporated into the operational CASA radar network. Computational efficiency is a key concern given the 0.5 km–1 min data resolution and distributed nature of the system. To mitigate this issue, the DARTS nowcasting algorithm was developed to estimate storm motion using linear least squares estimation formulated in the Fourier domain and advection performed using a sinc kernel–based algorithm.

The DARTS and sinc kernel–based advection algorithms along with the operational architecture within the CASA network have been described. Nowcasting performance was evaluated using a set of approximately 66 h of high-resolution radar reflectivity data collected during the CASA IP1 experiment from February to May 2009 representing a range of convective precipitation patterns.

Nowcasting out to 10 min was beneficial for emergency decision-making support. In this context, DARTS was evaluated quantitatively using CSI, FAR, and POD scores as well as MAE and showed favorable performance during the 2009 IP1 experiment. The efficiency of the nowcasting method allowed for successful operation given the 1-min temporal resolution of the data and the distributed nature of the distributed architecture.

Nowcasting was also used at 1–5-min time scales to set up the radar network scanning strategy where steering the radar using nowcasting allowed the radar network to better observe the atmosphere. End-user feedback survey is the most important performance metric in CASA and operational forecaster surveys and quantitative metrics were favorable.

The DARTS algorithm is neither specific to operation in the CASA system nor to the use of CASA reflectivity data. DARTS was evaluated for use in the Weather Support for Deicing Decision Making (WSDDM; Rasmussen et al. 2001, 2003) system to make 0–60-min nowcasts of ground-based aircraft icing conditions at airports with favorable results. In this context, DARTS estimated motion vector fields using WSR-88D level III reflectivity products with 1 km × 1 km spatial resolution and 5–10-min update rates to provide a significant performance increase of over the existing cross-correlation radar-based nowcasting method used in the WSDDM system.

## Acknowledgments

This work was supported by the Engineering Research Center Program of the National Science Foundation under NSF Award 0313747 and the NSF-LEAD ITR program.

## APPENDIX A

### Derivation of Dynamic and Adaptive Radar Tracking of Storms Model

*F*(

*m*

_{1},

*m*

_{2}, … ,

*m*) are defined in the domain of a

_{D}*D*-dimensional space;

*x*∈ [−

_{i}*l*, +

_{i}*l*] and

_{i}*T*≡ 2

_{i}*l*(

_{i}*i =*1, 2, … ,

*D*) is related to the discrete Fourier transform (Oppenheim and Schafer 2009)where DFT coefficients

*F*

_{DFT}(

*k*

_{1},

*k*

_{2}, … ,

*k*) are defined in a

_{D}*D*-dimensional discrete space

*n*∈′,

_{i}*N*is the grid number in dimension

_{i}*i*, where

*i =*1, 2, … ,

*D*, by the sampling theorem (Oppenheim and Schafer 2009) by

*F*(

*x*,

*y*,

*t*),

*U*(

*x*,

*y*,

*t*),

*V*(

*x*,

*y*), and

*S*(

*x*,

*y*,

*t*) in Eq. (1) (along with some index variable transformations), and noting the derivative property of the DFT, it can be shown that (Asmar 2004)

Considering only those harmonics that lie within the band-limited region

## APPENDIX B

### Implementation of a Linear Model for Nowcasting Motion Estimation

Equation (2) can be converted to linear form allowing for efficient implementation. Let the parameter sets *N =* {*N _{x}*,

*N*,

_{y}*N*},

_{t}*M =*{

*M*,

_{x}*M*}, and

_{y}*L =*{

*L*,

_{x}*L*,

_{y}*L*} be the integer sets of selected maximum harmonic numbers for the DFT of the observation sequence

_{t}*F*

_{DFT}(

*k*,

_{x}*k*,

_{y}*k*), the estimated motion vector fields

_{t}*U*

_{DFT}(

*k*) and

_{x}*V*

_{DFT}(

*k*,

_{x}*k*), and the evolution term

_{y}*S*

_{DFT}(

*k*,

_{x}*k*,

_{y}*k*), respectively. For implementation in the CASA system

_{t}*N*= {30, 30, 4},

*M*= {1, 1},

*L*= {0, 0, 0} were the selected parameters sets determined from empirical studies.

*Y*

_{0}=

*Y*(0, 0, 0),

*U*

_{0}=

*U*

_{DFT}(0, 0),

*V*

_{0}=

*V*

_{DFT}(0, 0),

*S*

_{0}=

*S*

_{DFT}(0, 0, 0),

The matrix pseudoinverse

## APPENDIX C

### Formulation of the Method for Advection of Radar Echoes Using the Sinc Kernel

*F*(

*x*,

*y*,

*t*) as follows (noting the change of indices):where the Dsinc function is defined in Eq. (9) and elements of matrices

*S*(

*x, y, t*) term] yieldswhich leads to Eq. (5).

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