Showing papers on "Wavelet published in 1988"

Orthonormal bases of compactly supported wavelets

Abstract: We construct orthonormal bases of compactly supported wavelets, with arbitrarily high regularity. The order of regularity increases linearly with the support width. We start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction. The construction then follows from a synthesis of these different approaches.

Complete discrete 2-D Gabor transforms by neural networks for image analysis and compression

Abstract: A three-layered neural network is described for transforming two-dimensional discrete signals into generalized nonorthogonal 2-D Gabor representations for image analysis, segmentation, and compression. These transforms are conjoint spatial/spectral representations, which provide a complete image description in terms of locally windowed 2-D spectral coordinates embedded within global 2-D spatial coordinates. In the present neural network approach, based on interlaminar interactions involving two layers with fixed weights and one layer with adjustable weights, the network finds coefficients for complete conjoint 2-D Gabor transforms without restrictive conditions. In wavelet expansions based on a biologically inspired log-polar ensemble of dilations, rotations, and translations of a single underlying 2-D Gabor wavelet template, image compression is illustrated with ratios up to 20:1. Also demonstrated is image segmentation based on the clustering of coefficients in the complete 2-D Gabor transform. >

Computer Processing of Remotely-Sensed Images: An Introduction

Abstract: Preface to the First EditionPreface to the Second Edition Preface to the Third EditionList of Examples1 Remote Sensing: Basic Principles11 Introduction12 Electromagnetic radiation and its properties121 Terminology122 Nature of electromagnetic radiation123 The electromagnetic spectrum124 Sources of electromagnetic radiation125 Interactions with the Earth's atmosphere13 Interaction with Earth-surface materials131 Introduction132 Spectral reflectance of Earth surface materials1321 Vegetation1322 Geology1323 Water bodies1324 Soils14 Summary2 Remote Sensing Platforms and Sensors21 Introduction22 Characteristics of imaging remote sensing instruments221 Spatial resolution222 Spectral resolution223 Radiometric resolution23 Optical, near-infrared and thermal imaging sensors231 Along-Track Scanning Radiometer (ATSR)232 Advanced Very High Resolution Radiometer (AVHRR)233 MODIS (MODerate Resolution Imaging Spectrometer)234 Ocean observing instruments235 IRS-1 LISS236 Landsat Instruments2361 Landsat Multi-spectral Scanner (MSS)2362 Landsat Thematic Mapper (TM)2363 Enhanced Thematic Mapper Plus (ETM+)2364 Landsat follow-on programme237 SPOT sensors2371 SPOT High Resolution Visible (HRV)2372 Vegetation (VGT)2373 SPOT follow-on programme238 Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER)239 High-resolution commercial and micro-satellite systems2391 High-resolution commercial satellites - IKONOS2392 High-resolution commercial satellites - QuickBird24 Microwave imaging sensors241 ERS SAR242 RADARSAT25 Summary3 Hardware and Software Aspects of Digital Image Processing31 Introduction32 Properties of digital remote sensing data321 Digital data322 Data formats323 System processing33 MIPS software331 Installing MIPS332 Using MIPS333 Summary of MIPS functions34 Summary4 Pre-processing of Remotely Sensed Data41 Introduction42 Cosmetic operations421 Missing scan lines422 De-striping methods4221 Linear method4222 Histogram matching4223 Other destriping methods43 Geometric correction and registration431 Orbital geometry model432 Transformation based on ground control points433 Resampling procedures434 Image registration435 Other geometric correction methods44 Atmospheric correction441 Background442 Image-based methods443 Radiative transfer models444 Empirical line method45 Illumination and view angle effects46 Sensor calibration47 Terrain effects48 Summary5 Image Enhancement Techniques51 Introduction52 Human visual system53 Contrast enhancement531 Linear contrast stretch532 Histogram equalisation533 Gaussian Stretch54 Pseudocolour enghancement541 Density slicing542 Pseudocolour transform55 Summary6 Image Transforms61 Introduction62 Arithmetic operations621 Image addition622 Image subtraction623 Image multiplication624 Image division and vegetation ratios63 Empirically based image transforms631 Perpendicular Vegetation Index632 Tasselled Cap (Kauth-Thomas) transformation64 Principal Components Analysis641 Standard Principal Components Analysis642 Noise-adjusted Principal Components Analysis643 Decorrelation stretch65 Hue, Saturation and Intensity (HIS) transform66 The Discrete Fourier Transform661 Introduction662 Two-dimensional DFT663 Applications67 The Discrete Wavelet Transform671 Introduction672 The one-dimensional Discrete Wavelet Transform673 The two-dimensional Discrete Wavelet Transform68 Summary7 Filtering Techniques71 Introduction72 Spatial domain low-pass (smoothing) filters721 Moving average filter722 Median filter723 Adaptive filters73 Spatial domain high-pass (sharpening) filters731 Image subtraction method732 Derivative-based methods74 Spatial domain edge detectors75 Frequency domain filters76 Summary8 Classification81 Introduction82 Geometrical basis of classification83 Unsupervised classification831 The k-means algorithm832 ISODATA833 A modified k-means algorithm84 Supervised classification841 Training samples842 Statistical classifiers8421 Parallelepiped classifier8422 Centroid (k-means) classifier8423 Maximum likelihood method843 Neural classifiers85 Fuzzy classification and linear spectral unmixing851 The linear mixture model852 Fuzzy classifiers86 Other approaches to image classification87 Incorporation of non-spectral features871 Texture872 Use of external data88 Contextual information89 Feature selection810 Classification accuracy811 Summary9 Advanced Topics91 Introduction92 SAR Interferometry921 Basic principles923 Interferometric processing923 Problems in SAR interferometry924 Applications of SAR interferometry93 Imaging spectrometry931 Introduction932 Processing imaging spectrometer data9321 Derivative analysis9322 Smoothing and denoising the reflectance spectrum Savitzky-Golay polynomial smoothing Denoising using the Discrete Wavelet Transform9323 Determinationof 'red edge' characteristics of vegetation9324 Continuum removal94 Lidar941 Introduction942 Lidar details943 Lidar applicationsAppendix A: Description of Sample Image Data SetsReferencesIndex

On the wavelet transformation of fractal objects

Abstract: The wavelet transformation is briefly presented. It is shown how the analysis of the local scaling behavior of fractals can be transformed into the investigation of the scaling behavior of analytic functions over the half-plane near the boundary of its domain of analyticity. As an example, a “Weierstrass-like” fractal function is considered, for which the wavelet transform is related to a Jacobi theta function. Some of the scalings of this theta function are analyzed, and give some information about the scaling behavior of this fractal.

The Wavelet Transform for Analysis, Synthesis, and Processing of Speech and Music Sounds

Abstract: The wavelet transform is a recent method of signal analysis and synthesis (Grossmann and Morlet 1984; Grossmann et al. 1987). It analyzes signals in terms of wavelets-functions limited both in the time and the frequency domain. In comparison, the classical Fourier analysis method analyzes signals in terms of sine and cosine wave components that are not limited in time. The wavelet transform is related to granular analysis/synthesis, first suggested by Gabor (1946). Granular synthesis has been implemented by Roads (1978) and Truax (1988). Rodet (1985) and Li6nard (1984) use adapted grains for speech signals; however, these implementations do not attempt to reconstruct an arbitrary given signal. The Gabor method uses an expansion of a function into a two-parameter family of elementary wavelets that are obtained from one basic wavelet

Wavelet Transforms and Edge Detection

Abstract: A wavelet transform of a function is, roughly speaking, a description of this function across a range of scales. We use the technique of wavelet transforms to detect discontinuities in the n-th derivative of a function of one variable.

Multiresolution representations and wavelets

Abstract: Multiresolution representations are very effective for analyzing the information in images. In this dissertation we develop such a representation for general purpose low-level processing in computer vision. We first study the properties of the operator which approximates a signal at a finite resolution. We show that the difference of information between the approximation of a signal at the resolutions 2$\sp{j+1}$ and 2$\sp{j}$ can be extracted by decomposing this signal on a wavelet orthonormal basis of ${\bf L}({\bf R}\sp{n}$). In ${\bf L}\sp2({\bf R})$, a wavelet orthonormal basis is a family of functions $\left\lbrack\sqrt{2\sp{j}}\ \psi(2\sp{j}x+n)\right\rbrack\sb{(j,n)\in{\rm Z}\sp2}$, which is built by dilating and translating a unique function $\psi(x)$, called a wavelet. This decomposition defines an orthogonal multiresolution representation called a wavelet representation. It is computed with a pyramidal algorithm of complexity n log(n). We study the application of this signal representation to data compression in image coding, texture discrimination and fractal analysis. The multiresolution approach to wavelets enables us to characterize the functions $\psi(x) \in {\bf L}\sp2({\bf R})$ which generate an orthonormal basis. The inconvenience of a linear multiresolution decomposition is that it does not provide a signal representation which translates when the signal translates. It is therefore difficult to develop pattern recognition algorithms from such representations. In the second part of the dissertation we introduce a nonlinear multiscale transform which translates when the signal is translated. This representation is based upon the zero-crossings and local energies of a multiscale transform called the dyadic wavelet transform. We experimentally show that this representation is complete and that we can reconstruct the original signal with an iterative algorithm. We study the mathematical properties of this decomposition and show that it is well adapted to computer vision. To illustrate the efficiency of this Energy Zero-Crossings representation, we have developed a coarse to find matching algorithm on stereo epipolar scan lines. While we stress the applications towards computer vision, wavelets are useful to analyze other types of signal such as speech and seismic-waves.

A method of ground-roll elimination

Abstract: Amplitude‐ and frequency‐modulated wave motion constitute the ground‐roll noise in seismic reflection prospecting. Hence, it is possible to eliminate ground roll by applying one‐dimensional, linear frequency‐modulated matched filters. These filters effectively attenuate the ground‐roll energy without damaging the signal wavelet inside or outside the ground roll’s frequency interval. When the frequency bands of seismic reflections and ground roll overlap, the new filters eliminate the ground roll more effectively than conventional frequency and multichannel filters without affecting the vertical resolution of the seismic data.

Wavelet transformations in signal detection

Abstract: It is pointed out that in the analysis of transient signals such as those encounters in speech, or in certain kinds of image processing, standard Fourier analysis is often non satisfactory because the basic functions of the Fourier analysis (sines, cosines, complex exponentials) extend over infinite time, whereas the signals to be analyzed are short-time transients. Reference is made to a method for dealing with transient signals which has recently appeared in the literature. The basis functions are referred to as wavelets, and they utilize time compression (or dilation) rather than a variation of frequency of the modulated sinusoid. Hence, all the wavelets have the same number of cycles. The analyzing wavelets must satisfy a few simple conditions, but are not otherwise specified. There is a wide latitude in the choice of these functions and they can be tailored to specific applications. The wavelets are founded on rigorous mathematical theory, and the expansions are robust. They are applied to detect ventricular delayed potentials (VLP) in the electrocardiogram. >

Ricker wavelets in their various guises

Abstract: The so-called Ricker wavelets remain in vogue among many synthetic seismogram producers, though the physical basis for the theory of such wavelets was proved to be invalid many years ago. For most synthetic seismogram studies, however, the important thing is to be able to model the wavelet remaining after processing. The wavelets used are long-range approximations to the true Ricker wavelet and the advantages claimed for them are that they are easily computed and unambiguously specifiabie by one parameter only (a frequency or period). Unfortunately they not only differ from processed wavelets in many cases, but a variety of conventions regarding their specification has crept in over the years and now there is confusion over what any practitioner means by his particular 'Ricker wavelet'. This article reviews all the known forms (and some unknown ones) that Ricker wavelets can take and points out the possible confusions that can occur as well as their lack of accord with real or processed wavelets. A recently arrived form is the minimum-phase Ricker wavelet which is examined and shown to possess a time delay which depends on the sample interval used in computing it. A two-parameter Ricker-type wavelet is described that allows greater versatility. The main conclusions are that Ricker wavelets employed by others are to be treated with extreme suspicion, and that Ricker wavelets should never be used at all if one has any choice.

Dispersive noise removal in t-x space; application to Arctic data

Abstract: Strongly dispersive noise from surface waves can be attenuated on seismic records by Flexfil, a new prestack process which uses wavelet spreading rather than velocity as the criterion for noise discrimination. The process comprises three steps: trace‐by‐trace compression to collapse the noise to a narrow fan in time‐offset (t-x) space; muting of the noise in this narrow fan; and inverse compression to recompress the reflection signals. The process will work on spatially undersampled data. The compression is accomplished by a frequency‐domain, linear operator which is independent of trace offset. This operator is the basis of a robust method of dispersion estimation. A flexural ice wave occurs on data recorded on floating ice in the near offshore of the North Slope of Alaska. It is both highly dispersed and of broad frequency bandwidth. Application of Flexfil to these data can increase the signal‐to‐noise ratio up to 20 dB. A noise analysis obtained from a microspread record is ideal to use for dispersion ...

Time-frequency and time-scale (signal processing)

Abstract: General results are presented for time-frequency and time-scale methods. Attention is given to both linear (short-time Fourier transform, wavelet transform) and bilinear (Wigner-Ville distribution, affine Wigner distribution) approaches, with emphasis put on their relationships. Also considered is the relationship of the methods examined to such approaches as constant-Q analysis and ambiguity functions. >

Correcting for coloured primary reflectivity in deconvolution1

Abstract: Statistical deconvolution, as it is usually applied on a routine basis, designs an operator from the trace autocorrelation to compress the wavelet which is convolved with the reflectivity sequence. Under the assumption of a white reflectivity sequence (and a minimum-delay wavelet) this simple approach is valid. However, if the reflectivity is distinctly non-white, then the deconvolution will confuse the contributions to the trace spectral shape of the wavelet and reflectivity. Given logs from a nearby well, a simple two-parameter model may be used to describe the power spectral shape of the reflection coefficients derived from the broadband synthetic. This modelling is attractive in that structure in the smoothed spectrum which is consistent with random effects is not built into the model. The two parameters are used to compute simple inverse- and forward-correcting filters, which can be applied before and after the design and implementation of the standard predictive deconvolution operators. For whitening deconvolution, application of the inverse filter prior to deconvolution is unnecessary, provided the minimum-delay version of the forward filter is used. Application of the technique to seismic data shows the correction procedure to be fast and cheap and case histories display subtle, but important, differences between the conventionally deconvolved sections and those produced by incorporating the correction procedure into the processing sequence. It is concluded that, even with a moderate amount of non-whiteness, the corrected section can show appreciably better resolution than the conventionally processed section.

Lithology discrimination for thin layers using wavelet signal parameters

Abstract: A forward modeling technique using Ricker wavelets demonstrates the need for a multiparameter approach in lithology determination using reflections from thin layers. The combination of time‐ and frequency‐domain analyses leads to a set of algorithms which define pore fluid and lithology from wavelet characteristics. The dispersive behavior of the thin layer varies considerably with the environment surrounding the layer, resulting in characteristic frequency‐domain behavior. With a limited prior knowledge of the formation environment, the pore fluid type can be determined using mode‐converted waves in the frequency domain.

A comparison of wavelet deconvolution techniques for ultrasonic NDT

Abstract: Wavelet deconvolution is a problem of fundamental importance in many signal processing applications. The authors are concerned with deconvolution to extract impulse responses of materials tested in ultrasonic nondestructive testing (NDT). The impulse responses can provide vital information about the flaws and material properties. Several deconvolution techniques including Wiener filtering in three versions, the spiking filter, time domain deconvolution, L1 norm deconvolution, and others are examined and critical comparisons are made for the ultrasonic NDT data. >

Measuring wavelet phase from seismic data

Abstract: Geophysicists are well into their second decade of serious interest in the seismic wavelet. Several requirements — to invert seismic complexes into their causative reflection series, to do stratigraphic modeling, to correct section displays to zero phase form, to standardize data of different vintages for composite structural mapping — put a knowledge of the basic wavelet shape, which can provide the way to satisfy these needs, near the top of every wish list. Ever since the convolutional model, with its basic wavelet and reflectivity series concepts, was proposed more than 30 years ago, there has been some kind of continuous effort to estimate the wavelet in a reliable and convenient manner. In spite of this constant attention, wavelets today are usually determined by directly measuring them in the field, or by estimating them in the processing environment through assuming their phase properties. Phase assumptions always leave the serious wavelet estimator a little queasy. He is always looking for some d...

Wavelet estimation: An interpretive approach

Abstract: There appears to be no uniform agreement on how to determine the polarity and shape of the wavelet in seismic data. In addition, when the determination is performed, the interpreter is usually the last to be informed of the results. Although the wavelet estimation problem can become buried in mathematics, the interpreter is best suited to judge the quality of the estimated wavelet and the applicability of the wavelet to the final interpretation. An approach to getting the interpreter involved in the wavelet estimation process is described in this article, and we believe that it offers some conceptual advantages over many of the popular methods currently in use.

Wavelet Analysis Using Well Log Information

Abstract: An important problem in detailed lithologic studies from seismic data is the estimation of the signal wavelet A methodology is proposed for estiiation of the wavelet using borehole information In the processing sequence the wavelet amplitude and phase spectra re gradually investigated The first stage is a multitrace coherence (MC) analysis to estimate wavelet and noise power spectra of real seismic traces Then a linear phase (LP) wavelet is estimated using well-control The final stage consists in a detailed analysis in which the amplitude and phase spectra of the wavelet are simultaneously investigated In the final stage, the impedance model in the vicinity of the available well can be modified through the two-dimensional (2D) inversion method described in Brat et al (1988) This processing scheme provides an improved matching between well-synthetic and seismic information while ensuring wavelet reliability and robustness It is presented and illustrated by numerical experiments on real data

Source wavelet estimation by upward extrapolation

Abstract: A new deterministic technique for wavelet estimation and deconvolution of seismic traces was recently introduced. This impedance‐type technique was developed for a marine environment where both the source and the receivers are located inside a homogeneous layer of water. In this work, an extension of the theory of source wavelet estimation is proposed. As in previous publications, this method is based on extrapolation of the wave field measured at depth, upward to the free surface. The extrapolation is performed by using the finite‐difference approximation to the full inhomogeneous wave equation. The extrapolation results in a wavelet which generally includes ghosts and can be used for source signature deconvolution and deghosting. The method needs two closely spaced receivers and is applicable for arbitrary locations of the source and the receivers in one‐dimensional multilayered models, provided the source is above the receivers; furthermore, it can be applied to both marine and land data. Application o...

Dyadic Wavelets Energy Zero-Crossings

Abstract: An important problem in signal analysis is to define a general purpose signal representation which is well adapted for developing pattern recognition algorithms. In this paper we will show that such a representation can be defined from the position of the zero-crossings and the local energy values of a dyadic wavelet decomposition. This representation is experimentally complete and admits a simple distance for pattern matching applications. It provides a multiscale decomposition of the signal and at each scale characterizes the locations of abrupt changes in the signal. We have developed a stereo matching algorithm to illustrate the application of this representation to pattern matching. Comments University of Pennsylvania Department of Computer and Information Science Technical Report No. MSCIS-88-30. This technical report is available at ScholarlyCommons: http://repository.upenn.edu/cis_reports/612 Dyadic Wavelets Energy Zero Crossings MS-CIS-88-30 GRASP LAB 140

Antialiasing methods for two-dimensional spatial wavelets in seismic modeling

Abstract: Two methods of circumventing spatial aliasing in seismic reflection modeling of zero‐offset data were examined. The first approach used asymptotic approximations of the near‐ and far‐field variants of the space‐frequency domain wave‐field continuation operator. This technique controls spatial aliasing but puts a lower limit on the extrapolation step size and an upper limit on frequencies that can be used in the modeling. The second method used Fourier (wavenumber) analysis of the same operator to identify the aliased components and wavenumber windowing to remove the unwanted portions of the wavenumber spectrum. In contrast to the asymptotic analysis, the Fourier analysis approach was simple and flexible to use and did not restrict other variables.

Wavelet Estimation For a Multidimensional Acoustic Or Elastic Earth

Abstract: A new and general wave theoretical wavelet estimation method is derived. Knowing the seismic wavelet is important both for processing seismic data and for modeling the seismic response. To obtain the wavelet, both statistical (e.g., Wiener-Levinson) and deterministic (matching surface seismic to well-log data) methods are generally used. In the marine case, a far-field signature is often obtained with a deep-towed hydrophone. The statistical methods do not allow obtaining the phase of the wavelet, whereas the deterministic method obviously requires data from a well. The deep-towed hydrophone requires that the water be deep enough for the hydrophone to be in the far field and in addition that the reflections from the water bottom and structure do not corrupt the measured wavelet. None of the methods address the source array pattern, which is important for amplitude-versus-offset (AVO) studies.This paper presents a method of calculating the total wavelet, including the phase and source-array pattern. When the source locations are specified, the method predicts the source spectrum. When the source is completely unknown (discrete and/or continuously distributed) the method predicts the wavefield due to this source. The method is in principle exact and yet no information about the properties of the earth is required. In addition, the theory allows either an acoustic wavelet (marine) or an elastic wavelet (land), so the wavelet is consistent with the earth model to be used in processing the data. To accomplish this, the method requires a new data collection procedure. It requires that the field and its normal derivative be measured on a surface. The procedure allows the multidimensional earth properties to be arbitrary and acts like a filter to eliminate the scattered energy from the wavelet calculation. The elastic wavelet estimation theory applied in this method may allow a true land wavelet to be obtained. Along with the derivation of the procedure, we present analytic and synthetic examples.

Post-Critical Wavelet Estimation and DECONVOLUTION1

Abstract: Using synthetic data, it is demonstrated that the amplitude spectra of post-critical plane-wave components are stable and equal to the amplitude spectrum of the input wavelet (critical reflection theorem). This analysis and physical explanation of the theorem are based only on amplitude versus offset arguments. The stability of the spectra in the post-critical region is directly related to a high amplitude post-critical reflection that dominates the trace in the slant-stack domain. The validity of the theorem for both the acoustic and elastic cases, its assumptions and limitations, are also examined with emphasis on applications for processing seismic reflection data. Based on the theorem, a deterministic procedure is developed (assuming minimum-phase properties) for wavelet estimation and subsequent deconvolution. The authors call this method Post-critical Deconvolution, which emphasizes reliance on post-critical reflection data. The performance of the method is shown with real data and the results are compared to those obtained with conventional deconvolution techniques.

Analysis of the characteristics-integration method for one-dimensional wave inversion

Abstract: Basing our work on the one‐dimensional (1-D) wave equation, we present an inverse method which we call the characteristics‐integration method. The method is derived from integration along characteristic families of straight lines of the wave equation in the time domain. With the source function known and reflection data recorded on the surface, the characteristics‐integration method can efficiently and economically recover the subsurface impedance profile, provided that the structure is inhomogeneous only in the depth direction. In general, when seismic data are contaminated by noise, the characteristics‐integration method, like any other 1-D inverse method, suffers from instability. We find that, for a smoothly varying impedance profile, the instability of inversions using characteristic methods depends heavily on the bandwidth of the source wavelet. We devised a resampling technique to stabilize the inverse scheme and to suppress the growth of errors. Numerical examples, including data contaminated by n...

Detection and Resolution of Thin Layers: a Model Seismic STUDY1

Abstract: The detection and resolution of a thin layer closely situated above a high-impedance basement are predominantly determined by both the frequency content of the incident seismic wavelet and the existence of the nearby high-impedance bedrock. The separation of the thin layer and the basement arrivals is investigated depending on the low-frequency content of the wavelet. The high-frequency content of the wavelet is kept constant. The initial wavelet spectrum with low frequencies has a rectangular shape. All wavelets used have zero-phase characteristics. Numerical and analogue seismic modelling techniques are used. The study is based on the geology of the Pachangchi Sandstone in West Taiwan. Firstly the resolution of a thin layer between two half-spaces is examined by applying the Ricker and De Voogd-Den Rooijen criteria. The lack of low-frequency components of the incident seismic wavelet reduces the shortest true two-way traveltime by about 20%. In addition, low-frequency components of the wavelet diminish the deviation between true and apparent two-way traveltime by about 65% for layer thicknesses in the transition from a thick to a thin layer. The second step deals with the influence of a high-impedance basement just below a thin layer on the detection and resolution of that thin layer. Reflected signal energies and apparent two-way traveltimes are considered. The reflected signal energy depends on the low-frequency content of the incident wavelet, the layer's thickness and the distance between the basement and the layer. This applies only to layers with thicknesses less than or equal to one-third of the mean wavelength in the layer, and a distance to basement in the range of one to one-half of the mean wavelength in the rock material between layer and basement. The minimum thin-layer thickness resolvable decreases with increasing distance to the basement; i.e. for a layer thickness of one-third of the mean wavelength in the layer the relative error of the two-way traveltime increases from 5% to 30%, if the distance is reduced from one to one-half of the mean wavelength in the material between the basement and the thin layer. Finally, a combination of vertical seismic profiling and downward-continuation techniques is presented as a preprocessing procedure to prepare realistic data for the detection and resolution investigation.

Wavelet Estimation From Noisy Data

Abstract: To reduce some of this uncertainty, most wavelet estimation techniques assume that the reflectivity is a white noise sequence. This assumption allows the estimation of the wavelet spectrum from that of the seismic trace when the additive noise is negligible. This assumption has been recently disputed in [l] and 121 who showed that the spectra of reflection coefficient sequences derived from several well logs are markedly different from white spectra. Even if r(t) is white, the noise will positively bias the spectrum of the trace with respect to the spectrum of the signal. Thus, these problems have to be addressed in order to achieve reliable estimates of the wavelet amplitude spectrum.