2.41 RADAR STORM MOTION ESTIMATION AND BEYOND: A SPECTRAL ALGORITHM AND RADAR OBSERVATION BASED DYNAMIC MODEL

Gang Xu *

V. Chandrasekar Colorado State University,

Fort Collins, Colorado, USA 1. INTRODUCTION Storm motion tracking using a temporal sequence of radar images is an important step in computer-aided operational nowcasting (Browning and Collier, 1989; Chornoboy et al, 1994; Wilson et al, 1998). There exist three commonly used approaches for radar storm tracking. The first approach is based on motion field that is identified by employing cross-correlation technique over two local blocks in two successive radar images (Rinehart and Garvey, 1978; Chornoboy et al, 1994). The second approach is referred to as “centroid tracking” (Austin and Bellon, 1982), such as the storm cell identification and tracking (SCIT) algorithm developed by Johnson et al (1998). The third approach is based on identification of storm’s position, size, mergers and splits that was implemented in the TITAN algorithm, referring to thunderstorm identification, tracking, analysis and nowcasting, developed by Dixon and Wiener (1993). Various improved methods have been developed based on local pattern matching and cross-correlation techniques. For example, Wolfson et al (1999) recently have developed a technique commonly referred as “growth-decay storm tracker”. The “growth-decay storm tracker” employs an elliptically shaped spatial filter such as to enable tracking systematic growth-decay propagations of the larger scale component in storms. In this paper we present the development of a new algorithm developed in spectral domain for estimating the motion field of * Corresponding author address: Gang Xu, Colorado State University, Electrical & Computer Engineering Department, Fort Collins, CO 80523; e-mail: xgang@lamar.colostate.edu.

storms. It is a global algorithm in the sense that it does not employ local block windows in radar images. The estimated motion field can be globally constructed over the whole spatial region where radar images are rendered. The smoothness of estimated motion field is controlled by the choice of the Fourier coefficients for each dimension. The motion-flow equation for radar images has been formulated and solved in the spectral domain. A global optimal solution in the least-square sense is guaranteed and the numerical computation for solving linear inversion problem is also fast and efficient. The performance of the new algorithm is evaluated using both simulated data and observed radar images. For observed radar data, we compared the motion-tracking based nowcasting using the spectral algorithm with the “growth-decay storm tracker”, henceforth referred as GDST. 2. GENERAL MODEL FOR RADAR STORM MOTION ESTIMATION: FORMAL DEVELOPMENT AND SPECTRAL ALGORITHM The general motion-flow model for radar observation field F(x, y, t) is written in a modified form here, as follows: ∂ ∂

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coordinates, in which the radar observation field, F(x,y,t), can be conveniently represented. The discrete version of F(x, y, t) can be written as F(i, j, k). The differential equation given by (1) can be written in frequency domain, in the discrete form as k =

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In Eq. 2 FDFT’s are the 3D Discrete Fourier Transform (DFT) coefficients of observed radar field F(i, j, k), which are computed from discrete space-time observations. UDFT’s are the 2D DFT coefficients of U(i, j), VDFT’s are the 2D DFT coefficients of V(i, j) and SDFT’s are the 3D DFT coefficients of S(i, j, k), which are unknowns that will be estimated. It should be noted that, Eq. 2 gives us a linear inversion problem when FDFT’s are known, so as to estimate UDFT’s, VDFT’s and SDFT’s. By choosing fewer leading coefficients for UDFT’s, VDFT’s and SDFT’s, Eq. 2 may form an over-determined linear system that can be solved by linear least-square-estimation (LSE) algorithm. Although Eq. 1 is a simple model, it offers several advantages when combining with the spectral algorithm as shown in Eq. 2. Firstly, the motion estimation is formulated and solved in spectral domain so that the estimated motion field is globally constructed over the whole spatial region where radar images are rendered. This way we not only avoid the issue of block window size versus the accuracy of local-point estimation, but also minimize the “aperture-effect” caused by local bock windows (Chornoboy et al, 1994). In general, motion fields vary slowly over the spatial domain, so we can select fewer leading Fourier coefficients to estimate and automatically obtain a smooth motion field. Secondly, the model in Eq. 1 has the ability to separate other dynamic mechanisms from advection terms. This is achieved via the extra term S(x, y, t). As demonstrated in the simulation of next section, this can alleviate the

influence of other mechanisms, such as local growth-decay of storms, on estimated motion fields. The implication of this property of the spectral algorithm is that, by explicitly introducing other linear mechanisms in the model, we may be able to separate the storm motion better from other dynamic mechanisms. Thirdly, to the best knowledge of authors, the cross-correlation technique is widely used is mainly for its stability performance. However, the high computational cost of cross-correlation method is also well known. This is due to that the vast searching has to be conducted to obtain a good and robust matching. To avoid occasionally unsmoothed estimation, the heuristic hierarchical procedure from coarser scales to finer scales usually has to be conducted. Our new spectral algorithm employs the linear inversion algorithm for reduced Fourier coefficients. It has a closed form of optimal solution and the computation for linear LSE is efficient. As demonstrated in the next section, the new spectral algorithm also shows good performance for both simulated data and observed radar images.

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3. PERFORMANCE EVALUATION OF SPECTRAL ALGORITHM FOR RADAR STORM MOTION ESTIMATION To validate the spectral algorithm developed above, we have applied it to both simulated radar data and observed radar images for estimating motion fields. For simulated data we know the ground truth of motion field, so we directly compare it with estimated motion field. Quantitative statistics are computed to evaluate the performance of the spectral algorithm. For observed radar data, we apply the spectral algorithm to estimate motion fields and apply them to extrapolate radar images for up to one hour in advance. The nowcasting using the spectral algorithm is compared with the “growth-decay storm tracker” (GDST). 3.1 Simulations We have simulated three data sets to test the spectral algorithm for global motion field estimation. For all three data sets, we have simulated a steady motion-flow field over the 2D region: -50 km ≤ x ≤ 50 km and -50 km ≤ y ≤ 50 km. The sampling resolution

is 1 km on both x-axis and y-axis. We use an observed radar reflectivity (dBZ) map (section 3.2) as initial radar image in these simulations (Fig. 1).

Fig. 1 Initial radar reflectivity (dBZ) image used in

simulations Simulation 1: in this simulation we generate a temporal sequence of radar images of 80 time steps span. The simple passive advection is simulated for this data set. The initial storm cell cluster (Fig. 1) is evolved by advection toward the north-east corner of map using the pre-generated steady motion field. By applying the spectral algorithm to the simulated temporal sequence of radar images, we estimate and reconstruct the motion-flow field to compare with the given flow field. Results show the good agreement between estimated flow field and true flow field. In this simulation the estimated motion field agrees fairly well with the true flow field within both data region and non-data region (Figs. 2, 3). The non-data region is where the storm cluster is not observed among the temporal sequence of radar images. The statistics for pixel-by-pixel comparison of flow velocities in both x-direction and y-direction are presented in Table 1. Simulation 2: during this simulation a localized steady source is added along with advection terms. Here the term, S(x, y, t)≡S(x,y), in Eq. 1 is interpreted as a growth mechanism that is steady (time-independent) and spatially localized. In the simulation a Gaussian-shaped source is centered at (10km, 10km), as shown in Fig. 4. We have compared two different ways applying the spectral algorithm to the

Fig. 2 Comparison of true flow-field and estimated flow-field using spectral algorithm: the zoomed-in

region where storm cell cluster is observed among the simulated sequence of radar images

Fig. 3 Comparison of true flow-field and estimated flow-field using spectral algorithm: the zoomed-in

region where storm cell cluster is not observed among the simulated sequence of radar images

simulated image sequence: 1) S-term is not added in the estimation algorithm and, 2) S-term is added in the estimation algorithm (Eq. 2). With the S-term added in the spectral algorithm, we gain significant improvement for the flow field estimation near the region where the growth mechanism presents (Figs. 6, 7). The quantitative comparison is also presented in Table 1. For the estimation with S-term added, the spectral algorithm also correctly identifies the source mechanism (Fig. 5).

Fig. 4 Steady S-term used in simulation 2

Fig. 5 Estimated S-term using the spectral algorithm in

simulation 2 Simulation 3: during this simulation a more complex S-term (slowly moving and non-steady) is added (Figs. 8a-8c). As in simulation 2 we have compared the estimated flow fields using spectral algorithm without S-term and with S-term added. With the S-term added in the spectral algorithm, we gain significant improvement for the flow field estimation (Table 1), though the reconstructed S-term has some artificial fluctuation components due to the more complex spatiotemporal pattern of growth mechanism. In this case it is again found that, the spectral algorithm can effectively separate advection component from growth-decay component so that a better estimation of flow field can be obtained. 3.2 Observations To further validate our new spectral algorithm, we have applied it to the observed radar reflectivity (dBZ) data from the WSR-88D radar located at Melbourne, Florida. The data were collected by the WSR-88D

Fig. 6 Estimated flow field near growth center, (10km,

10km), using the spectral algorithm without S-term (simulation 2)

Fig. 7 Estimated flow field near growth center, (10km, 10km), using the spectral algorithm with S-term added

(simulation 2) radar during the storm event from 21:02:09 (GMT), August 23rd to 00:57:26 (GMT), August 24th, 1998. This temporal sequence of radar images spans approximately 4 hours. The WSR-88D radar takes approximately 5 min to finish a volume scan. Each volume of PPI scan was converted to CAPPI data in Cartesian coordinates. The converted 2D radar images at the height of 1 km above the ground are used for this study. The re-sampled radar images lie in the 2D region: -50 km ≤ x ≤ 50 km and -50 km ≤ y ≤ 50 km. The spatial sampling resolution is 1 km on both x-axis and y-axis. The temporal sampling resolution is 5 min whereas each image is projected onto regular points on time axis. Therefore we obtain

(a)

(b)

(c)

Fig. 8 The non-steady source term that is slowly moving toward north-east corner of map (simulation 3) a temporal sequence of 48 radar images evenly sampled on time axis, as illustrated in Figs. 9a-9d. The spectral algorithm is applied on each 6 consecutive radar images that span 25 min. Each estimated motion field is used to extrapolate for the next 12 radar images. So this gives us the predicted radar images of up to one hour ahead. We adopt

three validation scores (e.g., Wilson et al, 1998, Wolfson et al 1999): 1) critical success index (CSI), 2) probability of detection (POD), and 3) false alarm rate (FAR) to evaluate the performance of the motion-tracking based nowcasting. These scores are computed on 4 km x 4 km grids with one level of reflectivity threshold (25 dBZ). For observed radar data we do not have the knowledge of true flow field for storm motion. Therefore we compare the nowcasting scores of the spectral algorithm with that of the “growth-decay storm tracker” (Theriault et al, 2000). Results are presented in Fig. 10, in which scores are averaged over all predictions of the same leading time. These results reveal that the spectral algorithm performs equally well or slightly better than the “growth-decay storm tracker”. 5. CONCLUSIONS In this paper we have developed a new algorithm in Fourier domain for radar observation based storm motion estimation. A linear dynamic model (Eq. 1) was developed along with the spectral algorithm (Eq. 2) that can be easily implemented for radar observation data. Our new algorithm provides several advantages compared to conventional cross-correlation method using local image blocks. It is demonstrated, by simulations, that the spectral algorithm has the ability to mitigate the influence of local growth-decay mechanisms on motion estimation. Therefore the spectral algorithm is able to improve the estimation of storm motion fields in complex situations. The implication of this capability of the spectral algorithm is that, by explicitly introducing different linear mechanisms in the model, we may better separate storm advection from other dynamic mechanisms. We also applied the spectral algorithm to observed radar data collected by the WSR-88D radar (Melbourne, FL). A 4-hr data set of the summer storm was used to validate our new algorithm. The estimated motion fields are used to nowcasting for up to one hour ahead. Results reveal that the spectral algorithm performs equally well or slightly better than the “growth-decay storm tracker”. 6. ACKNOWLEDGMENT

This research was supported by the Engineering Research Centers Program of the National Science Foundation under NSF award number 0313747. 7. REFERENCES Austin, G.L. and A. Bellon, 1982: Very-short-range forecasting of precipitation by the objective extrapolation of radar and satellite data. Nowcasting, A. K. Browning Ed., Academic Press, 177-190. Browning, K.A. and C.G. Collier, 1989: Nowcasting of Precipitation Systems. Reviews of Geophysics, 27, 345-370. Chornoboy, E.S., A.M. Marlin and J.P. Morgan, 1994: Automated storm tracking for terminal air traffic control. The Lincoln Laboratory Journal, 7, 427-448. Dixon, M. and G. Wiener, 1993: TITAN: thunderstorm identification, tracking, analysis, and nowcasting – a radar-based methodology. Journal of Atmosphric and Oceanic Technology, 10, 785-797. Johnson, J.T., P.L. Mackeen, A. Witt, E.D. Mitchell, Stumpf, G.J., M.D. Eilts and K.W. Thomas, 1998: The

storm cell identification and tracking algorithm: an enhanced WSR-88D algorithm. Weather and Forecasting, 13, 263-275. Rinehart, R.E. and E.T. Garvey, 1978: Three-dimensional storm motion detection by conventional weather radar. Nature, 273, 287-289. Theriault, K.E., M.M. Wolfson, B.E. Forman, R.G. Hallowell, M.P. Moore, R.J. Johnson, Jr., 2000: FAA terminal convective weather forecast algorithm assessment. The 9th Conference on Aviation, Range & Aerospace Meteorology, Orlando, FL. Wilson, J.W., N.A. Crook, C.K. Mueller, J. Sun and M. Dixon, 1998: Nowcasting thunderstorms: a status report. Bulletin of the American Meteorological Society, 79, 2079-2099. Wolfson, M.M., B.E. Forman, R.G. Hallowell, and M.P. Moore, 1999: The growth and decay storm tracker. The American Meteorological Society 79th Annual Conference, Dallas, TX.

Table 1: Statistics for pixel-by-pixel comparison between estimated flow fields and true flow fields in simulations (see full text in section 3.1). The unit of flow-field velocities is km/step. CORR is the correlation coefficient. NSE

is the normalized standard error in percent. SNR is the equivalent signal-to-noise ratio for estimation in dB.

(a) (b)

(c) (d)

Fig. 9 Example images in temporal sequence of observed radar reflectivity (dBZ) by the WSR-88D radar (Melbourne, FL) during the period from 21:02:09 (GMT), August 23rd to 00:57:26 (GMT), August 24th, 1998. The radar is at the origin of the map. Spatial sampling resolution is 1 km and temporal sampling resolution is 5 min.

Fig. 10 Nowcasting scores for observed radar data collected by the WSR-88D radar (Melbourne, FL) during the

period from 21:02:09 (GMT), August 23rd to 00:57:26 (GMT), August 24th, 1998. The spectral algorithm is compared with the “growth-decay storm tracker” (GDST). Time axis is the leading time for nowcasting in step of

5 min. CSI is the critical success index. POD is the detection rate. FAR is the false alarm rate.