Minimum-Variance and Maximum-Likelihood Deconvolution for Noncausal Channel Models

This paper extends the previous works of Mendel and his students on the subject of deconvolution from causal channel (wavelet) models to noncausal channel models. Noncausal wavelets occur, for example, in seismic data processing when a land vibrator is used to excite the Earth. Minimum-variance and maximum-likelihood deconvolution algorithms are developed herein for symmetrical and/or nonsymmetrical time-invariant wavelets that are excited by stationary and/or nonstationary white noise inputs. Minimum-variance deconvolution algorithms for a noncausal wavelet turn out to be quite different than those for a causal wavelet; however, maximum-likelihood deconvolution algorithms for a noncausal wavelet, which involve event detection and amplitude restoration, are essentially the same as those for a causal wavelet. Examples are provided that illustrate the performance of the different deconvolution algorithms.     

One-dimensional normal-incidence inversion: A solution procedure for band-limited and noisy data

In this paper we present a one-dimensional normal-incidence inversion procedure for reflection seismic data. A lossless layered system is considered which is characterized by reflection coefficients and traveltimes. A priori knowledge for the unknown parameters, in the form of statistics, is incorporated into a nonuniform layered system, and a maximum a posteriori estimation procedure is used for the estimation of the system's unknown parameters (i.e., we assume a random reflector model) from noisy and band-limited data. Our solution to the inverse problem includes a downward continuation procedure for estimation of the states of the system. The state sequences are composed of overlapping wavelets. We show that estimation of the unknown parameters of a layer is equivalent to estimation of the amplitude and detection of the time delay of the first wavelet in the upgoing state sequence of the layer. A suboptimal maximum-likelihood deconvolution procedure is employed to perform estimation and detection. The most desirable features of the proposed algorithm are its layer-recursive structure and its ability to process noisy and band-limited data.     

Statistical pulse compression

The seismic method in petroleum exploration is an echo-location technique to detect interfaces between the subsurface sedimentary layers of the earth. The received seismic reflection record (field trace), in general, may be modeled as a linear time-varying (LTV) system. However, in order to make the problem tractable, we do not deal with the entire field trace as a single unit, but instead subdivide it into time gates. For any time gate on the trace, there is a corresponding vertical section of rock layers within the earth, such that the primary (direct) reflections from these layers all arrive within the gate. Each interface between layers is characterized by a local (or Fresnel) reflection coefficient, which physically must be less than unity in magnitude. Under the hypothesis that the vertical earth section has small reflection coefficients, then within the corresponding time gate the LTV model of the seismic field trace reduces to a linear time-invariant (LTI) system. This LTI system, known as the convolutional model of the seismic trace, says that the field trace is the convolution of a seismic wavelet with the reflection coefficient series. If, in addition, the reflection coefficient series is white, then all the spectral shape of the trace within the gate can be attributed to the seismic wavelet. Thus the inverse wavelet can be computed as the prediction error operator (for unit prediction distance) by the method of least squares. The convolution of this inverse wavelet with the field trace yields the desired reflection coefficients. This statistical pulse compression method, known as predictive deconvolution with unit prediction distance, is also called spike deconvolution. Alternatively, predictive deconvolution with greater prediction distance can be used, and it is known as gapped deconvolution. Other pulse compression methods used in seismic processing are signature deconvolution, wavelet processing, and minimum entropy deconvolution.     

Simultaneous wavelet estimation and deconvolution of reflectionseismic signals

The problem of simultaneous wavelet estimation and deconvolution is investigated with a Bayesian approach under the assumption that the reflectivity obeys a Bernoulli-Gaussian distribution. Unknown quantities, including the seismic wavelet, the reflection sequence, and the statistical parameters of reflection sequence and noise are all treated as realizations of random variables endowed with suitable prior distributions. Instead of deterministic procedures that can be quite computationally burdensome, a simple Monte Carlo method, called Gibbs sampler, is employed to produce random samples iteratively from the joint posterior distribution of the unknowns. Modifications are made in the Gibbs sampler to overcome the ambiguity problems inherent in seismic deconvolution. Simple averages of the random samples are used to approximate the minimum mean-squared error (MMSE) estimates of the unknowns. Numerical examples are given to demonstrate the performance of the method     

On homomorphic deconvolution of bandpass signals

A convolution may be represented as x(.)=r(.)* w(.). The goal of deconvolution is to extract r(.) and w(.) from a knowledge of x(.) and it finds numerous applications in digital signal processing. Of practical interest in oil exploration is the case where w(.) is a seismic pressure wavelet, x(.) is the observed seismic response, and r(.) is the reflectivity of the Earth. A number of procedures have been proposed, including predictive, deterministic, and homomorphic deconvolution. Homomorphic deconvolution has been found to be particularly efficient for those cases where x(.) is known to be fullband. This paper presents a robust constructive procedure for efficient homomorphic deconvolution for those cases where x(.) is a bandpass signal. Extensive comparisons with other methods for deconvolving bandpass signals on measured seismic data traces (including the Novaya Zemlya event) illustrate the improvement in the deconvolution     

Performance of the SMLR deconvolution algorithm

A performance analysis is presented for false alarms, correct detections, and the resolution of the suboptimal maximum-likelihood deconvolution (MLD) algorithm, called the single most likely replacement (SMLR) algorithm. It is assumed that the source wavelet and statistical parameters are given a priori. It is shown analytically that the performance improves as the signal-to-noise ratio (SNR) increases and as the mainlobe width of the normalized autocorrelation function of the source wavelet decreases. For the same performance, a higher SNR is required as the mainlobe width of the normalized autocorrelation function increases. Some simulation results which are consistent with the analytic results are presented     

Unsupervised deconvolution of sparse spike trains using stochasticapproximation

This paper presents an unsupervised method for restoration of sparse spike trains. These signals are modeled as random Bernoulli-Gaussian processes, and their unsupervised restoration requires (i) estimation of the hyperparameters that control the stochastic models of the input and noise signals and (ii) deconvolution of the pulse process. Classically, the problem is solved iteratively using a maximum generalized likelihood approach despite questionable statistical properties. The contribution of the article is threefold. First, we present a new “core algorithm” for supervised deconvolution of spike trains, which exhibits enhanced numerical efficiency and reduced memory requirements. Second, we propose an original implementation of a hyperparameter estimation procedure that is based upon a stochastic version of the expectation-maximization (EM) algorithm. This procedure utilizes the same core algorithm as the supervised deconvolution method. Third, Monte Carlo simulations show that the proposed unsupervised restoration method exhibits satisfactory theoretical and practical behavior and that, in addition, good global numerical efficiency is achieved     

Restoration of turbulence degraded images using improved multi-frame blind deconvolution algorithm

For the restoration of astronomical objects which are degraded mainly by turbulence, an improved multi-frame blind deconvolution method using the generalised expectation-maximisation (GEM) on the basis of the penalised maximum-likelihood estimation method is presented. Experimental results indicate that this method has better performance and costs less time.     

Semi-blind identification of ARMA systems using a dynamic-based approach

A novel dynamic-based semi-blind approach is proposed to identify an autoregressive and moving average (ARMA) system in this paper. By using a chaotic driving signal, an ARMA system can be identified accurately by a dynamic-based estimation method called the ergodic-based minimum phase space volume (EMPSV). A maximum-likelihood formulation of EMPSV is provided to certify its unbiasedness and asymptotical efficiency. Monte Carlo simulations show that the EMPSV approach has a smaller mean-square error performance than the minimum phase space volume method and the conventional identification approach based on least-squares estimation with white Gaussian probing signals. The proposed approach is then applied to blind deconvolution of real audio signals and semi-blind channel equalization for chaos communications. It is shown that the EMPSV approach has improved deconvolution and equalization performances compared to conventional techniques in both applications.     

Multiscale deconvolution of sensor array signals viasum-of-cumulants

This correspondence presents a solution to a multiscale deconvolution problem using higher order spectra where the data to be deconvolved consist of noise-corrupted sensor array measurements. We assume that the data are generated as a convolution of an unknown wavelet with reflectivity sequences that are linearly time-scaled versions of an unknown reference reflectivity sequence. This type of data occurs in many signal processing applications, including sonar and seismic processing. Our approach relies on exploiting the redundancy in the measurements due to time scaling and does not require knowledge of the wavelet or the reflectivity sequences. We formulate and solve the deconvolution problem as a quadratic minimization subject to a quadratic constraint in the sum-of-cumulants (SOC) domain. The formulation using the SOC approach reduces the effect of additive Gaussian noise on the accuracy of the results when compared with the standard time-domain formulation. We demonstrate this improvement using a simulation example     

一种新的地震子波估计方法

地震子波估计问题是地震勘探信号处理和分析中的关键一环。本文基于遗传算法提出了一个新的地震子波估计方法。该方法用ARMA模型描述地震子波，用遗传算法交替迭代地估计AR和MA参数。与其它方法相比，本文提出的方法具有高度稳定性和很好的精度，并适应于估计非最小相位地震子波。     

Wavelet estimation

An important problem in seismic exploration is the estimation of and correction for the seismic wavelet. A seismic signal may be modeled as a convolutional model with the wavelet as one component. The wavelet propagated by the seismic energy source is complicated by transmission and recording filters. Some filters in the system can be deterministically defined while others are more conjectural. The estimation of the wavelet is useful in two major ways. Borehole measurements are used to model the surface seismograms. The wavelet used in the model needs to match that of the seismogram to correlate the two measurements. Conversely, the estimated wavelet can be used to design inverse filters which make the seismogram approach the borehole measures. Some well-known methods for estimation of the wavelet are based on assumptions about the wavelet or the earth reflectivity. Examples of the methods indicate success on some data even though each makes different assumptions. The methods serve to point out basic problems in reliably estimating the wavelet from the seismogram. Basic problems include noise, band-limiting, nonstationarity, uncertain theoretical models, assumption failure, and widely diverse geological sequences of the earth. Quality control or evaluation of the performance of an estimation algorithm is a nontrivial problem. The estimation of the wavelet from a seismic recording remains an area of challenging research and importance in exploration for hydrocarbons.     

A comparison of deconvolution techniques for the ultrasonicnondestructive evaluation of materials

Several major deconvolution techniques commonly used for seismic applications are studied and adapted for ultrasonic NDE (nondestructive evaluation) applications. Comparisons of the relative merits of these techniques are presented based on a complete set of simulations on some real ultrasonic pulse echoes. Methods that rely largely on a reflection seismic model, such as one-at-a-time L(1) spike extraction and MVD (minimum variance deconvolution), are not suitable for the NDE applications discussed here because they are limited by their underlying model. L(2) and Wiener filtering, on the other hand, do not assume such a model and are, therefore, more flexible and suitable for these applications. The L(2) solutions, however, are often noisy due to numerical ill conditions. This problem is partially solved in Wiener filtering, simply by adding a constant desensitizing factor q. The computational complexities of these Wiener filtering-based techniques are relatively moderate and are, therefore, more suitable for potential real-time implementations.     

Maximum-Likelihood Header Estimation: A Cross-Layer Methodology for Wireless Multimedia

We propose a novel cross-layer header estimation methodology that can be used by UDP-based wireless multimedia applications to estimate corrupted packet headers, thereby realizing significant throughput improvements. The proposed methodology requires only minor modifications to the protocol stack at the receiver while no modifications are needed to senders or intermediate nodes. We formulate header estimation as a problem of maximum-likelihood estimation of known parameters in noise. We derive likelihood functions for two wireless channel models, namely Markov and multifractal wavelet models. Our trace-driven video simulations at 2, 5.5 and 11 Mbps data rates of an 802.11b LAN demonstrate that significant improvements over normal UDP and UDP Lite can be achieved by employing header estimation with UDP.     

一种改进的大气模糊图像恢复算法

针对天文图像受大气影响而变模糊的问题,利用多帧盲去卷积方法对变质图像的恢复及点扩展函数的辨识进行了研究,并采用基于受惩极大似然估计的GEM改进算法从模糊图像序列中得到目标以及点扩展函数的迭代解。实验结果表明该算法能有效地恢复出原物体并估计出点扩散函数。     

Applications of detection and estimation theory to large array seismology

The statistical theory of signal detection and estimation has been applied to problems in large array seismology. Using this theory the structure of the optimum detector for a known signal and long observation time in additive Gaussian noise is derived. The array processing filter employed by the optimum detector is known as the maximum-likelihood filter. This filter also has the property that it provides a minimum-variance unbiased estimate for the input signal when it is not known, which is the same as the maximum-likelihood estimate of the signal if the noise is a multidimensional Gaussian process. A series of experiments was performed using data from the large aperture seismic array to determine the effectiveness of the maximum-likelihood method relative to simpler methods such as beam-forming. These results provide significant conclusions regarding the design and processing of data from large seismic arrays. The conventional and high-resolution estimation of the frequency-wavenumber spectrum of the background microseismic noise is also presented. The diffuse structure of this spectrum is shown to aid in explaining the relative performance of array processing methods.     

A robustness test for the MVD filter and MLD algorithm

J.P. Todoeschuck and O.G. Jensen (Geophysics, vol.53, no.11, p.1410-14, 1988) recently reported that some reflectivity sequences denoted μ(k), calculated from sonic logs, are not white and have a power spectral density approximately proportional to frequency, which is called a Joseph spectrum. A robustness test is now presented for the case of μ(k) having a Joseph spectrum for the minimum-variance deconvolution (MVD) filter and the maximum-likelihood deconvolution (MLD) algorithm, which were developed based on the whiteness assumption about μ(k). From the simulations performed, it is concluded that the possible Joseph spectrum of μ(k) is not a concern when applying the MVD filter and MLD algorithm towards estimating μ(k) from seismic data     

On the Application of the Fast Kalman Algorithm to Adaptive Deconvolution of Seismic Data

The application of a recently proposed fast implementation of the recursive least squares algorithm, called the Fast Kalman Algorithm (FKA) to adaptive deconvolution of seismic data is discussed. The newly proposed algorithm does not require the storage and updating of a matrix to calculate the filter gain, and hence is computationally very efficient. Furthermore, it has an interesting structure yielding both the forward and backward prediction residuals of the seismic trace and thus permits the estimation of a ?smoothed residual? without any additional computations. The paper also compares the new algorithm with the conventional Kalman algorithm (CKA) proposed earlier [3] for seismic deconvolution. Results of experiments on simulated as well as real data show that while the FKA is a little more sensitive to the choice of some initial parameters which need to be selected carefully, it can yield comparable performance with greatly reduced computational effort.     

Radio-astronomical imaging in the presence of strong radiointerference

Radio-astronomical observations are increasingly contaminated by interference, and suppression techniques become essential. A powerful candidate for interference mitigation is adaptive spatial filtering. We study the effect of spatial filtering techniques on radio-astronomical imaging. Current deconvolution procedures, such as CLEAN, are shown to be unsuitable for spatially filtered data, and the necessary corrections are derived. To that end, we reformulate the imaging (deconvolution/calibration) process as a sequential estimation of the locations of astronomical sources. This not only leads to an extended CLEAN algorithm, but also the formulation allows the insertion of other array signal processing techniques for direction finding and gives estimates of the expected image quality and the amount of interference suppression that can be achieved. Finally, a maximum-likelihood (ML) procedure for the imaging is derived, and an approximate ML image formation technique is proposed to overcome the computational burden involved. Some of the effects of the new algorithms are shown in simulated images     

An Instrumental Variable Approach to Minimum-Variance Seismic Deconvolution

A new method is proposed in this paper for sequential parameter estimation and deconvolution of seismic signals. An on-line algorithm of the instrumental variables method is appropriately combined for this purpose with a fixed-lag version of minimum-variance seismic deconvolution. A consistency and an accuracy analysis of the proposed algorithm are presented, using a special type of instrumental variables. The performance of the method is examined in the cases of nonminimum phase or slowly time-varying seismic wavelets. The results are illustrated by simulation studies using examples already studied in the literature.