Rain Pattern Detection by means of Packet Wavelets Andrea Marazzi , Paolo Gamba and Roberto Ranzi y Dipartimento di Elettronica, Universita di Pavia, Via Ferrata, 1, I-27100 Pavia Tel: +39-382-505923 Fax: +39-382-422583 E-mail:
[email protected] y Dipartimento di Ingegneria Civile, Universit a di Brescia, Via Branze, 38, I-25123, Brescia ( )
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Abstract { The recent advances in microwave telecommunications and the need for more precise weather forecasting systems are two of the many elds where is extremely important to be able to analyze quickly, precisely and, possibly, in a fully or partially automatic way, the data obtained by systems like metereological radars. In this work it is presented a wavelet packet based algorithm, combined with a C-means classier, for rain patterns detection and tracking from this data. The use of this kind of classication chain is motivated by the high eciency and low computational load of the wavelet transform algorithm and by the observation that a large class of natural textures can be modeled as quasi-periodic signal, whose dominant frequencies are located in the middle frequency channels, easily provided by this transform. The chain was applied to a radar data sequence of a rain event on Northern Italy, interesting interpretation of the dynamics of storm structures at dierent meso-scales.
1. INTRODUCTION Communication systems working in the microwave frequency domain above 10 GHz are now widely spread, and there is therefore an increasing need for very detailed investigations on the propagation characteristics of the atmosphere 1]. In particular, great interest is shown for the information on the spatial and temporal organization of precipitation patterns. A very useful method to obtain these information is by means of weather-radars, that supply an estimation of the rain intensity for suciently wide geographic regions to consider large scale rain structures, but with a ne space resolution 2]- 3]. These data allow the study, within a coarse-to-ne strategy, of the spatial structure of precipitation. However, the raw records of radar reectivities need a complex processing procedure to provide to the nal user the information on the spatial/temporal characteristics of rain events. To achieve this goal, a system able to track the spatial evolution of rain patterns has to be determined, and this system must be able to tackle a number of problems. The tracking process is not, in fact, an easy task, even if performed by a human observer in real time: the temporal evolution of the rain events results from complex processes, organized at dierent space-time
scales. Each scale, ranging from the synoptic external scale of frontal systems to the meso-gamma scale of convective cells, displays its own dynamic, which is sometimes coupled with the others, sometimes not. Thus the patterns to be tracked change continuously their shape while moving 5]. We propose a chain based on a wavelet representation of the rain data, on which a classication is performed in order to extract the textured rain patterns at dierent scales. A single image, representing the rain intensity in the environment around the radar, is preprocessed by a wavelet packet algorithm and divided in subimages, different representations of the same scene: the use of this transform is motivated by the observation that a large class of natural textures can be modeled as quasi-periodic signal, whose dominant frequencies are located in the middle frequency channels. The subband images are then processed by an envelope signal estimationto provide a method for features extraction: dierent textures have dierent \energy" in the detail subband related to the magnitude of oscillation of wavelet coecients in each subband. The image can be nally seen as a multiband representation of the same scene and thus resolved as a multidimensional data clustering problem, and a C-means algorithm is applied to the image to achieve an ecient segmentation of the dierent rain patterns.
2. THE PACKET WAVELET TRANSFORM The general structure and computational framework of the discrete wavelet transform (DWT) are similar to those found in subband coding system, but wavelet lters are required to be regular. In this study, we consider the discrete wavelet packet transform (DWPT), which corresponds to a general tree-structured lter bank. Wavelet transforms are entirely specied in terms of a prototype lter h that satises the standard quadrature mirror lter condition: ( ) ( ; 1) + H (z )H (;z ; 1) = 1
H z H z
(1)
where H (z ) denotes the z -transform of h. The lter h is also required to satisfy the lowpass constraint: H (z )jz=1 = 1. A complementary high pass lter is obtained by shift
Figure 1: Three frames from the recorded radar data of a rain event in Northern Italy.
Figure 2: The two dierent meso-scale rain patterns detected from the images of g. 1. and modulation:
h
( ) = zH (;z ; 1) (2) These prototypes are then used to generate, in an iterative fashion (initial condition H0 (z ) = 1), a sequence of lters of increasing width (indexed by i): Hi+1 (z ) = H (z 2i )Hi (z ) = G(z 2i )Hi (z ) (i = 0 : : : I ; 1) (3) Gi+1 (z ) The pyramidal structure of the DWF decomposes a signal into a set of dierent frequency channels that have narrower bandwidths in the lower frequency region. The application of a simple DWF to a texture is completely useful for segmentation because a large class of natural textures can be modeled as quasi-periodic signal, whose dominant frequencies are located in the middle frequencies channels. Therefore, the concept of DWF must be generalized using a library of modulated waveform orthonormal basis called wavelet packets. The ltering is recursively applied to both the low frequency and high frequency components creating a binary tree. The 2-D wavelet can be seen by a tensor product of two 1-D wavelet basis function along the horizontal and vertical directions. the corresponding coecients can be expressed as hLL (k l ) = h(k )h(l ) hLH (k l ) = h(k )g (l ) G z
HL (k l) = g(k)h(l)
h
HH (k l) = g(k)g(l)
(4)
where the rst and second subscript denote the lowpass and highpass ltering in the x- and y- directions.
3. THE WAVELET CLASSIFICATION CHAIN Exploiting the concept shown in the preceding section, a wavelet classication chain has been implemented, followed by a standard procedure able to measure, according to dierent methods, the velocities of the dierent rain patterns. The aim is to extract automatically how dierent rain patterns evolve during the recorded rain event. This wavelet-based method can be summarized in the following steps (for a more detailed description, refer to 7]): 1. First of all, a 2-D Battle-Lemarie packet wavelet transform is applied to each original radar image. By means of this operation, the components of the rain patterns at the dierent spatial and frequency scales are subdivided. This result allows to discard information related to high spatial frequencies, that individuate very short-term phenomena of little or no interest for our investigation. In particular, we discard all the data about very little rain cells, usually with life too short to be trackable within the frame
temporal frequency (an image each 15 min.) of our radar data.
of the 0-isoallobaric contour line. A small scale motion of cell clusters was recognized to be consistent with the 700 hPa wind speed vector, which is directed toward N2. Each of the chosen packet wavelet subimages is mod- NW, a dierent direction respect to the large-scale one. ied by an envelope estimation algorithm (EEA). It This de-coupled dynamic was recognized in the eld after can be used a simple zero-crossing algorithm, where its wavelet decomposition into the small- and large-scale the maximum value between two adjacent zero-crossings structures (see also 8]). is found and assigned to all points within the interIn fact, after classication, further operations are perval. The EEA is applied row-wise or column-wise formed on the data to extract information on the bedepending to the wavelet-bank lter that originated haviours of the rain patterns at the dierent meso-scales. each subimage. In particular, since we are interested in their movement and life, a rst operation is to detect their velocities by 3. The gray value of the subimages is normalized. This lag-correlation or by means of an advettive model with a is done because the variance of the gray level can vary single velocity or with more components. The velocities of a lot from subimage to subimage. Remember that the two patterns along the sequence of g. 2 are obtained the result of the wavelet transform are coecient and as de-coupled, as can be easily veried in the frames. not gray level value, so they can have a range quite dierent from the usual 8-bit image coding. 4. The subimages can be considered as a multiband image of the same region and a clustering algorithm (the classical C-means or the more recent Fuzzy-C-means
6]) is applied to the wavelet coecients obtained in the preceding steps. The output of such a classication algorithm are, for each frame, the rain patterns associated to dierent values of the \energy" represented by the wavelet coecients. The intensity of the rainfall eld is derived from the reectivity of a C-band Doppler Radar operating at the 5.5 cm wavelength. Reectivity was converted into rainfall rate using the Marshall-Palmer formula, which was veried to be accurate in the event investigated. In g. 1 three frames of a radar sequence are shown, that refer to the rain event that occurred on 4 October 1992 and that was examined in this study. It represents an episode of cyclogenesis in the lee of the Alps, whose occurrence is rather frequent in the Gulf of Genova. A deep and stationary low developed for more than ve days, reaching a minimum of 989 hPa at 6.00 GMT of 4 October 1992. The convergence over the Italian Peninsula of southerly winds, carrying warm and moist air, with cold and dry air masses owing over the Rodano Valley, in France, caused persistent precipitation over the Po Valley. Vertical soundings recorded over the meteorological stations of Milano Linate, Udine and S. Pietro Capoume, close to Bologna, showed that warm advection occurred in the lower levels. The vertical wind prole recorded at S. Pietro Capoume at 11.00 GMT of 4 October reveals the 0 C isotherms to be located at the 720 hPa level, at a height of about 2700 m a.s.l., where a bright band is detected by the radar. Several rainbands were observed in the structure of the cyclone, during its occluded stage of development. A large scale synoptic motion was detected from the geostationary METEOSAT satellite pictures and from the movement
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