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Applications of Wavelet Methods to the Analysis ofMeteorological Radar Data An Overview

Stephan DahlkePhilippsUniversitat Marburg

FB Mathematik und InformatikHans Meerwein Str., Lahnberge

35032 MarburgGermany

Gerd TeschkeyUniversitat Bremen

Fachbereich 3Postfach 33 04 40

28334 BremenGermany

Volker LehmannDeutscher Wetterdienst

Met. Observatorium Lindenberg15864 Lindenberg

Germany

September 25, 2002

Abstract

The aim of this paper is to give an overview on the current applications of wavelet meth-ods to the analysis of radar data. There are two major topics where wavelet algorithms havealready been successfully applied, namely the reconstruction of continuous reflectivity densi-ties and the analysis of discrete radar wind profiler data. The first problem can be treated byusing the specific reconstruction and approximation properties of wavelet frames whereas thesecond one is treated by a suitable variant of the classical wavelet thresholding method. Bothtopics are discussed in detail and several numerical results are presented.

Key Words: Radar, reflectivity density, wideband regime, narrowband regime, meteoro-logical signal processing, radar wind profiler, frames, wavelets, continuouswavelet transforms, uncertainty principles.

AMS(MOS) Subject classification: 42C15, 42C40, 65J22The work of this author has been supported by Deutsche Forschungsgemeinschaft (DA 360/41)yThe work of this author has been supported by Deutsche Forschungsgemeinschaft (MA 1657/61)1

Contents

1 Introduction 3

2 Wavelet Analysis 4

2.1 The Discrete Wavelet Transform . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Biorthogonal Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Continuous Wavelet Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4 Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Statistical Curve Estimation Using Wavelets 13

3.1 Threshold Wavelet Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 Error Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4 Basic Radar Setting 15

4.1 Wideband Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.2 Narrowband Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.3 Localization and Radar Uncertainty Principles . . . . . . . . . . . . . . . . . . . 20

4.4 Discrete Narrowband Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5 Analysis of Continuous Reflectivity Densities 23

5.1 Basic Reconstruction Formulas in the Wideband Regime . . . . . . . . . . . . . 24

5.2 Error Bounds in the Wideband Regime . . . . . . . . . . . . . . . . . . . . . . . 26

5.3 Numerical Experiments in the Wideband Regime . . . . . . . . . . . . . . . . . 28

5.4 Basic Reconstruction Formulas in the Narrowband Regime . . . . . . . . . . . . 31

6 Analysis of Discrete Radar Wind Profiler Data 33

6.1 Radar Wind Profilers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

6.2 Problems in RWP Signal Processing . . . . . . . . . . . . . . . . . . . . . . . . 38

6.3 Improved RWP Signal Processing by Wavelet Techniques . . . . . . . . . . . . . 39

6.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

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1 Introduction

In recent years, wavelet analysis has become a very powerful tool in applied mathematics. Thefirst applications of wavelets were concerned with problems in image/signal analysis/compression.Furthermore, quite recently, wavelet algorithms have also been applied very successfully in nu-merical analysis, geophysics, meteorology, astrophysics and in many other fields. Especially, ithas turned out that the specific features of wavelets can also be efficiently used for certain prob-lems in the context of radar signal analysis. The basic radar problem asks to gain informationabout an object by analyzing waves reflected from it. However, although this fundamental prob-lem is always the same for every radar application, the algorithms that are used clearly depend onthe concrete setting and may differ dramatically. The choice of the appropriate method obviouslyrests on the properties of the object under consideration (point object, dense target environmentetc.) and on the parameters one wants to reconstruct (reflectivity densities, wind velocities etc.)For an overview on radar problems, we refer to one of the textbooks [9, 17, 22, 45, 55]. In recentstudies, it has turned out that wavelet methods are at least helpful for the following two topics: the reconstruction of continuous reflectivity densities; the analysis of discrete radar wind profiler data (RWP).The aim of this paper is to give an overview on both problems and to explain how they may betreated by wavelet methods.

The problem of reconstructing reflectivity densities can be described as follows. Let us first assumethat the object under consideration can be modeled as a single point, moving with constant velocitytowards or away from a given source. Then, the aim is to reconstruct the distance and the velocityof the object by analyzing the reflected waves. Indeed, for a simple point object, this goal can beachieved by simply computing the maxima of the continuous wavelet transform of the receivedecho, see, e.g., [30] and Section 4.1 for details. However, in many applications, one is faced witha reflecting continuum with varying reflectivities. Such a target environment is then modeled bya certain reflectivity density. In this case, this reflectivity density can not be reconstructed bysimply transmitting one single signal, see again [30] and Section 4.1. Nevertheless, a completereconstruction is possible if a family of signals is used. This approach was first suggested byNarpast [40, 41] who studied the case that the transmitted signals form an orthonormal waveletbasis. We refer to Section 2 for the definition and the basic properties of wavelets. However, therequirement of full orthonormality is quite restrictive. Therefore, quite recently, RebolloNeira,Plastino, and FernandezRubio generalized Naparsts approach to the case that the transmittedfamily forms a frame. We refer to Subsection 2.4 for a short introduction to frames. Moreover,some further generalizations concerning, e.g., rigorous error estimates, the multivariate case andnumerical examples have also been presented in [10, 11]. It has turned out that especially waveletframes perform quite successfully. The main results of these approaches are presented in Section5.

The setting of wind profiler data is quite different. In this case, one wants to gain informationconcerning the three dimensional atmospheric wind vector. This is done be sampling the reflectedradar beams at certain rates corresponding to different heights, followed by the application of wily

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radar signal processing devices. The whole analysis consists of the following steps: coherent in-tegration, Fourier analysis, spectral parameter estimation, and, finally, the wind estimation. Thewhole procedure is explained in Section 6.1. However, one of the basic problems in wind profileranalysis is that the data may be heavily contaminated by echoes returned from ground surround-ings, from targets like airplanes or birds or from external radio-frequency transmissions. Due tothe local nature of these disturbing signal components, de-noising algorithms based on Fouriertransforms are not very appropriate. This is exactly the topic where wavelet analysis suggestsitself since wavelets are by construction very localized functions. Indeed, the following variantof wavelet de-noising algorithms has already been successfully applied [32, 49]. The classicalwavelet de-noising methods consist of three steps. First of all, the signal is decomposed into awavelet series by means of the fast wavelet transform, then the small wavelet coefficients are ne-glected by applying some thresholding operator (hard/soft thresholding), and, finally, the signalis reconstructed by applying the inverse wavelet transform. We refer to the Sections 2 and 3 fordetails. However, in the setting of RWP, the disturbing components are usually much larger thanthe signal one wants to analyze. Therefore the thresholding operator is applied the opposite way:the small wavelet coefficients are kept and the large ones are neglected. The whole algorithm isexplained in Section 6.3.

This paper is organized as follows. In Section 2, we briefly recall the basic facts on wavelet analy-sis as far as they are needed for our purposes. The discussion covers orthogonal and biorthogonalwavelets, the continuous wavelet transform including associated uncertainty relations, and theconcept of frames. Section 3 is devoted to statistical estimations by means of wavelet methods.Then, in Section 4, we discuss the basic radar setting. We present the wideband as well as thenarrowband approach. Moreover, we introduce a new discrete narrow band model which is de-signed to serve as some kind of bridge between the two basic problems introduced above. Then,in Section 5, we explain how a continuous reflectivity density can be reconstructed by means of aframe approach. We also present some error estimates in weighted L2spaces and discuss severalnumerical examples. Section 6 is devoted to the radar wind profiler problems. After explainingthe basic setting in Subsection 6.1, we discuss the problems in RWP signal processing in Sub-section 6.2. In Subsection 6.3, we explain how wavelet methods can help to improve the currentalgorithms and, finally, in Subsection 6.4, we present some numerical examples.

2 Wavelet Analysis

In this section,