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Statistics of precipitation reflectivity images and cascade of Gaussian-scale mixtures in the wavelet domain: A formalism for reproducing extremes and coherent multiscale structures

2011; American Geophysical Union; Volume: 116; Issue: D14 Linguagem: Inglês

10.1029/2010jd015177

2156-2202

Mohammad Ebtehaj, Efi Foufoula‐Georgiou,

Climate variability and models

Journal of Geophysical Research: AtmospheresVolume 116, Issue D14 Climate and DynamicsFree Access Statistics of precipitation reflectivity images and cascade of Gaussian-scale mixtures in the wavelet domain: A formalism for reproducing extremes and coherent multiscale structures Mohammad Ebtehaj, Mohammad Ebtehaj Department of Civil Engineering, Saint Anthony Falls Laboratory and National Center for Earth-Surface Dynamics, University of Minnesota, Twin Cities, Minneapolis, Minnesota, USA School of Mathematics, University of Minnesota, Twin Cities, Minneapolis, Minnesota, USASearch for more papers by this authorEfi Foufoula-Georgiou, Efi Foufoula-Georgiou [email protected] Department of Civil Engineering, Saint Anthony Falls Laboratory and National Center for Earth-Surface Dynamics, University of Minnesota, Twin Cities, Minneapolis, Minnesota, USASearch for more papers by this author Mohammad Ebtehaj, Mohammad Ebtehaj Department of Civil Engineering, Saint Anthony Falls Laboratory and National Center for Earth-Surface Dynamics, University of Minnesota, Twin Cities, Minneapolis, Minnesota, USA School of Mathematics, University of Minnesota, Twin Cities, Minneapolis, Minnesota, USASearch for more papers by this authorEfi Foufoula-Georgiou, Efi Foufoula-Georgiou [email protected] Department of Civil Engineering, Saint Anthony Falls Laboratory and National Center for Earth-Surface Dynamics, University of Minnesota, Twin Cities, Minneapolis, Minnesota, USASearch for more papers by this author First published: 23 July 2011 https://doi.org/10.1029/2010JD015177Citations: 11AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Abstract [1] To estimate precipitation intensity in a Bayesian framework, given multiple sources of noisy measurements, a priori information about the multiscale statistics of precipitation is essential. In this paper, statistics of remotely sensed precipitation reflectivity imageries are studied using two different data sets of randomly selected storms for which coincident ground-based and spaceborne precipitation radar data were available. Two hundred reflectivity images of independent storm events were collected over two ground validation sites of the Tropical Rainfall Measurement Mission (TRMM) in the United States. Comparing ground-based and spaceborne images, second-order statistics of the measurement error is characterized. The average spectral signature and second-order scaling properties of those images are documented at different orientations in the Fourier domain. Decomposition of images using band-pass multiscale oriented filters reveals remarkable non-Gaussian marginal statistics and scale-to-scale dependence. Our results show that despite different physical storm structures, there are some inherent statistical properties which can be robustly parametrized and exploited as a priori information for parsimonious multiscale estimation of precipitation fields. A particular mixture of Gaussian random variables in the wavelet domain was found to be a suitable probability model that can reproduce the non-Gaussian marginal distribution as well as the scale-to-scale joint statistics of precipitation reflectivity data, important for properly capturing extremes and the coherent multiscale features of rainfall fields. Key Points Multiscale statistics of precipitation Gaussian-scale mixtures in wavelet domain for precipitation image modeling Estimation and multisensor precipitation data fusion 1. Introduction [2] In the past decades, a considerable research effort has been devoted to developing parsimonious stochastic models of space-time rainfall [e.g., Lovejoy and Mandelbrot, 1985; Gupta and Waymire, 1990, 1993; Veneziano et al., 1996; Deidda, 2000; Deidda et al., 2006; Lovejoy and Schertzer, 2006; Venugopal et al., 2006; Mandapaka et al., 2010]. The related theories of multiscale process representation, e.g., in Fourier or wavelet domains, have proven to be useful for quantifying the rainfall variability at multiple scales. A large body of these developments has exploited the way that the second-order statistics of the rainfall process vary across different scales (i.e., 1/f spectra). Beyond this, observing non-Gaussian characteristics of precipitation fields and scaling in higher-order statistical moments, the theory of Multifractals and Multiplicative Random Cascades has extensively been used to capture these distinct properties of the rainfall fields [e.g., Lovejoy and Schertzer, 1990; Gupta and Waymire, 1990, 1993]. Simultaneously, it has been shown that oriented subband encoding of precipitation fields using wavelets can lead to an efficient and rich multiscale representation of spatial rainfall [e.g., Kumar and Foufoula-Georgiou, 1993a, 1993b]. Subsequently, an appreciable amount of work has been devoted to extracting the dependency of the parameters of those stochastic models to the underlying physics of the storm [e.g., Over and Gupta, 1994; Perica and Foufoula-Georgiou, 1996; Harris et al., 1996; Badas et al., 2006; Nykanen, 2008; Parodi et al., 2011]. [3] The purpose of this paper is to: (1) demonstrate that precipitation reflectivity images exhibit some remarkably regular multiscale statistical characteristics, mainly related to non-Gaussian (heavy tail) marginals and scale-to-scale dependency, and (2) introduce a new modeling framework based on Gaussian Scale Mixtures (GSM) on wavelet trees which can be explored towards non-Gaussian, multiscale/multisensor data fusion of precipitation fields. In section 2, we present basic statistics from a diverse array of precipitation reflectivity images collected coincidentally from ground-based NEXRAD and the spaceborne Precipitation Radar (PR) abroad the TRMM satellite for two TRMM Ground Validation (GV) sites in Texas and Florida. In section 3, an extensive analysis and comparison of these images in the Fourier domain is undertaken. In section 4, the marginal and joint statistics of these precipitation reflectivity images in the wavelet domain (using an advantageous Undecimated Orthogonal Discrete Wavelet transform) are presented. A novel model based on the GSM on wavelet trees is introduced in section 5, and its potential for reproducing the observed heavy tail and covariance of the rainfall wavelet coefficients at multiple scales is demonstrated. The potential application of this model is also briefly discussed. Finally, section 6 presents conclusions and directions for future research. 2. Precipitation Data and Elementary Statistics [4] A major portion of the available remotely sensed precipitation data is acquired via imaging in the microwave band of the electromagnetic spectrum. For active microwave sensors, such as ground or spaceborne radars, the precipitation fields are retrieved via physical or statistical relationship from the reflectivity images obtained as a result of the detected back-scattered energy of microwave signals emitted from the precipitation radar. On the other hand, for passive microwave sensors such as the TRMM Microwave Imager (TMI), the precipitation fields are retrieved indirectly via conditional inversion of the observed "brightness temperature" [e.g., Kummerow et al., 1996]. In this study, we use coincidental reflectivity data of the spaceborne TRMM precipitation radar (PR) and the land-based NEXRAD radar to demonstrate that despite different physical structures of the studied storms, the near-surface images of precipitation reflectivity exhibit remarkably regular and stable statistical properties, which can be explicitly characterized within a novel formalism based on GSM in the wavelet domain. [5] Specifically, the data set used in this study is populated by near-surface reflectivity images from 200 independent storms coincidentally observed by TRMM and NEXRAD precipitation radars. The TRMM-2A25 and NEXRAD (level III) long-range reflectivity products over two TRMM-GV sites: Houston, Texas (HSTN), and, Melbourne, Florida (MELB), were collected on the basis of the TRMM overpass information provided by the GV Office at the Goddard Space Flight Center, Maryland. Using orthodromic distance, the NEXRAD product provides reflectivity at an horizontal resolution of about 1 km and up to the range of 460 km with minimum reflectivity detection of 5 dBZ. The TRMM, 2A25 product provides an orbital track that spans a swath of 250 km at nadir with a resolution of about 4–4.5 km and minimum detection sensitivity of 17 dBZ. A lucid explanation of the TRMM-GV sites and the available data at each site are provided by Wolff et al. [2005]. Note that as the quantitative comparison of the two sensors is of interest in this research, the NEXRAD near-surface long-range reflectivity product was selected to maximize the coincidental coverage between the two sensors. Obviously, rainfall rate estimation from this single level reflectivity product via a Z-R relationship needs to be limited to lower ranges (e.g., <230 km) to minimize the range effect estimation errors. [6] The data set used in our study comprises reflectivity images of 95 and 105 storm events from both sensors over the HSTN and MELB sites from 1998 to 2010, respectively (see Figure 1). Concerning the sufficiency of the data for robust statistical inference, the images were carefully selected from storm events with adequate areal coverage during the TRMM overpasses. The data set spans a wide range of storms with different physical structures and geometrical shapes ranging from highly localized convective storms to frontal and synoptically induced hurricane systems (see Figures 2 and 3). It is emphasized that no attempt was made to convert these reflectivity images to precipitation intensity values, a task that would be a research topic by itself, given the diversity of storms and the ground-based radar range-dependent estimation issues. In the rest of the paper we refer to these reflectivity fields as "precipitation reflectivity images" or "precipitation images." Figure 1Open in figure viewerPowerPoint For statistical analysis, a total of 200 independent storms were selected from the TRMM-PR and the NEXRAD reflectivity data set at the TRMM ground validation (GV) sites of Houston, Texas (HSTN), and Melbourne, Florida (MELB). The distribution of these events by year is shown. Figure 2Open in figure viewerPowerPoint The collected data sets summarized in Figure 1 span a wide range of storms with different spatial structures and geometrical shapes. The NEXRAD reflectivity images for four selected storms are shown above; they are labeled according to the GV site, date (yyyymmdd), and time in UTC: (a) MELB_19980217_131700, (b) HSTN_19981113_000200, (c) HSTN_20020620_172600, and (d) MELB_20040926_045000. Figure 3Open in figure viewerPowerPoint (a) The geographic locations of the study sites (MELB and HSTN) and the orbital track 61698 of the TRMM satellite which captured a hurricane storm over Texas on 13 September 2008. (b) Reflectivity images of the storm captured by ground-based NEXRAD at 11:16:00 UTC and (c) the coincidental TRMM-2A25 overpass. [7] Focusing on characterization of the error variance, these sensors were compared over the intensity range detectable by both. Accordingly, the mean reflectivity (in dBZ) of the TRMM images was compared with the mean of the corresponding NEXRAD images, conditioned on reflectivity values exceeding 17 dBZ; see Figure 4. For this case, the standard bias was found to be −2% and −1.8% for the HSTN and MELB sites, respectively. This indicates that the TRMM-PR overestimates the reflectivity intensity in the range that both of the sensors can detect reflected echoes. This bias is not unexpected and is mainly due to the inherent differences in the way that the two sensors interrogate the vertical profile of the atmosphere. The variance of error is estimated and reported in Table 1 based on two different definitions of signal-to-noise ratio metric. This characterization has an important implication in the context of linear multisensor fusion of precipitation products [e.g., see Chou et al., 1994; Gorenburg et al., 2001; Tustison et al., 2002; Willsky, 2002]. To this end, the bias was adjusted to zero via enforcing the regression line to pass through the origin, and also the data pairs with normalized residual values (by the standard deviation) beyond the interval [−2, 2] were excluded from the estimation process (see Figure 4). The latter treatment makes the estimation more robust to probable outliers. Figure 4Open in figure viewerPowerPoint TRMM versus NEXRAD reflectivity values in (a) HSTN and (c) MELB sites. The data pairs are the spatially averaged reflectivity values of coincidental pairs of images computed over the range of intensity values which is detectable by both sensors (≥17 dBZ). The solid line is the best least squares fitting and the broken line is the 1:1 line. The (b) HSTN and (d) MELB normalized regression residuals, with the [−2, 2] lines, marked to indicate the values that fall outside the ±2 times standard deviation of residuals. Table 1. Standardized Error Variance in Terms of Two Signal-to-Noise Ratio (SNR) Metrics and Kullback-Leibler (KL) Divergence of the Marginal Histograms of the TRMM and NEXRAD Coincidental Reflectivity Observationsa HSTN MELB NEXRAD TRMM NEXRAD TRMM SNR1 11.9 (10.4–12.9) 13.0 (11.6–13.6) 12.4 (11.2–13.6) 13.6 (13.0–14.4) SNR2 8.4 (5.8–9.75) 13.0 (11.6–13.6) 9.0 (7.45–10.0) 7.9 (6.5–9.6) KLb 1.0488 (0.7959–1.5681) 1.0488 (0.7959–1.5681) 0.6749 (0.6108–0.7494) 0.6749 (0.6108–0.7494) a Values in parentheses indicate the 95% quantile range of estimation. The two different metrics of SNR are: SNR1 = 10 log10 (μs/σn) and SNR2 = 10 log10 (σs/σn), where μs and σs are the mean and standard deviation of the signal and σn is the noise standard deviation. b KL is a mutual property between NEXRAD and TRMM. Therefore the entries repeated here are actually shared between NEXRAD and TRMM. [8] The Kullback-Leibler (KL) divergence, also known as the relative entropy, was also studied to characterize the degree of proximity of the marginal densities of the observations, provided by the two sensors. The KL divergence is defined as where pj = p(x∣Hj) is the conditional marginal density of the precipitation reflectivity values under different measurement hypotheses with j = 0,1 corresponding to TRMM and NEXRAD observations, respectively. The KL divergence is a positive quantity which is equal to zero if and only if the compared densities are equal almost everywhere in their domain. The KL is not a conventional distance since it is not symmetric and does not satisfy the triangle inequality for three arbitrary densities. Yet, it has been shown to be a useful measure of density mismatch in statistical modeling [Levy, 2008]. As can be seen from Table 1, this metric demonstrates a statistically significant deviation from zero for both GV sites implying a deviation of the marginal densities. This particular observation along with the least squares analysis of the data set (see summary in Table 1) indicates that on the average the overall quality of the selected TRMM-PR overpass observations in the MELB site is superior to that of the HSTN site. 3. Spectral Signature [9] Several studies [e.g., Lovejoy and Schertzer, 1990; Harris et al., 1996, 2001; Menabde et al., 1997; Morales and Poveda, 2009; Lovejoy et al., 2010; Ebtehaj and Foufoula-Georgiou, 2010] have reported the presence of scale invariance in the form of f−β average Fourier spectrum (i.e., [∣(f)∣2]) in precipitation fields. The Fourier transformation, as an approximation to the Karhunen-Loève expansion, allows us to decouple the correlation structure of the rainfall fields into a set of almost uncorrelated Fourier coefficients with a nearly diagonal covariance matrix. Therefore, knowing that the inner product in L2() is conserved under the Fourier transformation (i.e., Parseval's Theorem), the one-dimensional representation of the average power spectrum [∣(f)∣2] = Af−β is indeed diagonalization of the covariance in the frequency domain. Besides the information content of the spectral decay rate as depicting the second-order scaling law and degree of differentiability (smoothness) of a field, this diagonal representation of the covariance yields a computationally more efficient least square optimal filtering in the Fourier domain which also might be useful for filtering of high-dimensional strongly correlated rainfall fields [see, e.g., Simoncelli and Adelson, 1996; Gonzalez and Woods, 2008]. [10] By construction, the Fourier spectrum of an image is insensitive to spatial translation, but it is not rotation invariant and can explain the anisotropy of a field. Accordingly, in addition to the energy distribution of the intensity values in the frequency domain, the 2D spectrum depicts the orientation of the edges and regions of sharp gradients in a 2D field. For instance, it has been reported that as horizontal and vertical edges are dominant in man-made scenes (e.g., cities), the spatial distribution of the spectrum of these images is more elongated along the vertical and horizontal orientations [Torralba and Oliva, 2003]. In light of this, studying the spatial distribution, orientation and total energy of the precipitation reflectivity images in the spectral domain might be useful not only for exploring scale invariance and optimal estimation but also for studying the regional organization of storm systems for retrieval applications. [11] To this end, we compute here a more general representation of the mean spectral signature of the precipitation images at different orientations θ, where (·) denotes the Fourier transformation in polar coordinates, A(θ) is a prefactor and β(θ) is the dropoff rate of the spectrum at angle θ. Using discrete Fourier transform, the square of the absolute values of the Fourier coefficients were calculated to obtain the 2D power spectrum for each individual image. This provides a set of Fourier power spectra which can be averaged over the entire data set for each site (i.e., 95 images over HSTN and 105 images over MELB) to obtain the so-called ensemble power spectrum; see Figures 5c and 5d. Using the least squares regression in a log-log scale, the power spectral model in the form of equation (2) can be fitted at different orientations to each individual or ensemble 2D spectra. The regressions were performed in the radial frequency interval of [0.03,0.50] cycle/pixel [c/p] corresponding to the pseudo spatial scale (i.e., Euclidian distance) of 2–32 km. Table 2 reports the results of directional estimation of power spectral slopes for the NEXRAD data sets. It is observed that the estimated spectral slopes vary between 2.35 and 2.75 for the HSTN site and between 2.45 and 2.85 for the MELB site. By averaging a 2D power spectrum over all angles, a one-dimensional representation can also be obtained in which the parameters in equation (2) are independent of orientation. Figures 5a and 5b show the radially averaged ensemble spectra for the NEXRAD data sets. The estimated dropoff rate of the radially averaged ensemble spectra in the two sites is about 2.70–2.75, which implies that the precipitation images are globally much smoother than many other natural images (e.g., β ≈ 2) [Ruderman, 1994]. Interestingly, despite the different geographic locations of the two sites and different physical structures of the storms, the statistics of the spectral parameters vary within a very narrow range; see Table 2. This observation implies that the spectral signature (i.e., spatial correlation structure) of the near-surface reflectivity images may not be a discriminatory measure of the physical structure of the storms. On the other hand, this universal behavior gives us a priori knowledge about the correlation structure of these type of rainfall images which can be useful for noise removal and optimal estimation of precipitation data in the Fourier domain. Figure 5Open in figure viewerPowerPoint Radially averaged spectra for the ensemble of NEXRAD reflectivity images at the (a) Houston and (b) Melbourne GV sites. The 2D ensemble spectra for (c) Houston and (d) Melbourne GV sites depicting directional anisotropy at small scales (large frequencies). Table 2. Estimated Parameters of the Spectral Model in Equation (2) for NEXRAD Data at Multiple Directionsa HSTN MELB log [A (θ)] β (θ) log [A (θ)] β (θ) 5.18(0.18) 2.68 (0.12) 5.11(0.19) 2.75(0.10) θ = 0.0° 4.86(0.22) 2.66 (0.22) 4.80(0.27) 2.72(0.26) θ = 90° 5.02(0.25) 2.71 (0.27) 4.92(0.25) 2.80(0.21) θ = 135° 4.50(0.19) 2.75 (0.22) 4.46(0.22) 2.85(0.21) θ = 45° 4.85(0.20) 2.35 (0.20) 4.75(0.19) 2.45(0.19) a See Figure 5. The values in parentheses are the standard deviations. The parameters reported for denote those obtained from the radially averaged ensemble spectra. [12] As mentioned before, the shape of the spectrum can also speak for the regional organization of the rain cells. Pronounced abrupt changes in the spatial domain intensity values (i.e., horizontal edges) cause spectral skewness (elongation) in the perpendicular direction (i.e., vertical direction) at the frequency domain [Gonzalez and Woods, 2008]. In the collected storm images, for the low frequency components of less than 0.1 [c/p], the spectral signature shows a more dense and isotropic behavior, meaning that on average the large-scale features of the storms do not have any particular spatial orientation. However, for high-frequency components (i.e., small-scale features) the ensemble power spectra are tilted and more elongated towards the northeast (NE) and southwest (SW) directions (see Figures 5c and 5d). This similar asymmetric signature in both sites may mainly arise due to a regionally governing synoptic meteorological condition that gives rise to a directionally dominant formation of the rain patches with a length scale smaller than 10 km. [13] Due to the limited swath width and flight orientation, the TRMM-PR orbital observations often cannot capture the entire spatial extent of the storm events. TRMM-PR products often provide a cropped version of the whole storm with abrupt changes of intensity values on the swath boundaries. These artificial edges contaminate the spectral signature and give rise to some spectral leakages (see Figure 6), which do not allow us to properly study the directional organization of the rain cells (i.e., edges) from this product. Although this boundary effect may be handled, for instance by padding the TRMM-PR images with mirror reflection of themselves across the boundaries, this obviously does not add any new information that can be exploited to study the spatial organization of the rain cells. However, the decay rate of the radially averaged ensemble spectrum at similar frequency bands confirms that the reflectivity images of the TRMM sensor exhibit slightly weaker correlation structure (more irregular) compared to the NEXRAD products. Figure 6Open in figure viewerPowerPoint Radially averaged spectra for the ensemble of TRMM-PR reflectivity images in the (a) Houston and (b) Melbourne GV sites. The smaller values of the spectral slopes compared to those of the corresponding NEXRAD reflectivity images of Figure 5 imply that the reflectivity images produced by the spaceborne TRMM-PR exhibit a weaker spatial correlation than those produced by the ground-based NEXRAD. The 2D ensemble spectra of the TRMM-PR data at (c) HSTN and (d) MELB sites indicate similarity to the corresponding NEXRAD spectra but also show significant spectral leakage due to the artificial edge effects introduced by the swath boundaries (see text for more explanation). 4. Statistics of Subband Components in the Wavelet Domain [14] Natural processes exhibit variability over a broad range of scales, often manifesting itself in isolated singularities in the form of edges or nested areas of intense activity. The decay of the Fourier spectrum captures the global distribution of variance without providing information about the local distribution of the process variability at different scales. Using a set of multiscale band-pass filters at different orientations has been found to be extremely useful for extracting the information content of the local jump discontinuities and abrupt fluctuations of these fields [e.g., Kumar and Foufoula-Georgiou, 1993a, 1993b; Perica and Foufoula-Georgiou, 1996; Huang and Mumford, 1999; Lee et al., 2001; Mallat, 2009]. Spatial precipitation fields are highly clustered and exhibit strong correlation along with sparseness (zeroes) in the real domain, mainly recognizable as the presence of oriented edges between rain and no-rain areas. Consequently, precipitation images often exhibit a stronger sparseness condition in the wavelet domain as the coherent cells and broad homogeneous areas would map into near-zero wavelet high-pass coefficients. This often manifests itself in the marginal histogram of the wavelet subbands having a sharp peak at the center (i.e., around zero) and extended heavy tails which cannot be modeled in a Gaussian framework. As a simple treatment to overcome this leptokurtic behavior, Perica and Foufoula-Georgiou [1996] proposed a Gaussian density model for the so-called "standardized rainfall fluctuations", defined as the high-pass orthogonal wavelet coefficients divided by their corresponding low-pass coefficients. Although that treatment can partially model the observed thick tail behavior, it cannot flexibly account for the cusp singularity or large mass of the wavelet coefficients around the center of the distribution. [15] In this study, we demonstrate that the Generalized Gaussian (GG) density which has been widely used for statistical modeling of the high-pass wavelet subbands of natural images [e.g., Huang and Mumford, 1999; Lee et al., 2001] can be employed to fully characterize the marginal statistics and heavy tail properties of the precipitation reflectivity images. As the rainfall imageries generally suffer from a considerable number of zero intensity values at the background (nonrainy areas within the field of view), a method is also presented to allow characterization of only the "relevant zeroes", i.e., the zeroes that correspond to small isolated dry areas within the storm domain and to those corresponding to the storm edges [see also Kumar and Foufoula-Georgiou, 1994]. In addition, it is shown that despite the decorrelation capacity of the wavelet transformation [e.g., see Wornell, 1990], the wavelet coefficients of the rainfall images exhibit a weak correlation structure and a considerably regular higher-order scale-to-scale dependence, which needs to be addressed for proper multiresolution modeling of precipitation imageries. Wavelet Decomposition and Marginal Statistics [16] The Orthonormal Discrete Wavelet Transformation (OWT) [Mallat, 1989] decomposes a 2D signal f(x,y) of size K × L into a pair of almost uncorrelated expansion coefficients dm,k,li (called wavelet coefficients) and orthonormal basis functions ψm,k,li (x, y) in the form of where ψm,k,li (x, y) is the wavelet basis function at subband i = {H, V, D} (i.e., Horizontal, Vertical and Diagonal directions in this study), m denotes the scale level and (k, l) are translation indices (see also detailed exposition by Kumar and Foufoula-Georgiou [1993a]). This representation uses the orthogonal wavelet bases functions ψm,k,li (x, y) in a "critically sampling rate" (meaning that the size (N) of the input signal is equal to the total size of the output subbands). Owing to the orthogonality of the bases and critical sampling rate, the inverse transformation allows a perfect reconstruction with a computational complexity of the order of O(N). However, this critical sampling rate makes the wavelet representation shift variant and imposes significant aliasing in each individual subband. Although the aliasing artifacts will cancel out in the reconstruction phase, this would be troublesome for processing and parametrization of each individual subband [e.g., Nason and Silverman, 1995; Simoncelli and Freeman; 1995]. [17] In this study, a shift-invariant Undecimated Orthogonal Discrete Wavelet Transform (UOWT) [Nason and Silverman, 1995] is used for decomposition of the precipitation reflectivity images and statistical characterization of their wavelet coefficients. This decomposition produces nearly alias-free and overcomplete subband information. The latter property is another great advantage over the conventional OWT in which the size of the signal is downsampled by a factor of 2 at each level of decomposition, giving rise to inferential problems in subband parametrization of rainfall images with small wetted area. Obviously, the advantages of this overcomplete frame expansion come at the expense of a higher computational complexity of the order of O(N log N). [18] It is noted that in the wavelet domain, background zeroes will remain zeroes at consecutive scales in both high and low-pass subbands, while a range of zero intensity values within the storm domain (i.e., those zero intensity pixels which define the boundaries of the wetted