In this paper we show that given prior information in terms of a lower and upper bound, a prior bias, and constraints in terms of measured data, minimum relative entropy (MRE) yields exact expressions for the posterior probability density function (pdf) and the expected value of the linear inverse problem. In addition, we are able to produce any desired confidence intervals. In numerical simulations, we use the MRE approach to recover the release and evolution histories of plume in a one-dimensional, constant known velocity and dispersivity system. For noise-free data, we find that the reconstructed plume evolution history is indistinguishable from the true history. An exact match to the observed data is evident. Two methods are chosen for dissociating signal from a noisy data set. The first uses a modification of MRE for uncertain data. The second method uses “presmoothing” by fast Fourier transforms and Butterworth filters to attempt to remove noise from the signal before the “noise-free” variant of MRE inversion is used. Both methods appear to work very well in recovering the true signal, and qualitatively appear superior to that of Skaggs and Kabala [1994]. We also solve for a degenerate case with a very high standard deviation in the noise. The recovered model indicates that the MRE inverse method did manage to recover the salient features of the source history. Once the plume source history has been developed, future behavior of a plume can then be cast in a probabilistic framework. For an example simulation, the MRE approach not only was able to resolve the source function from noisy data but also was able to correctly predict future behavior.

Minimum Relative Entropy Inversion: Theory and Application to Recovering the Release History of a Groundwater Contaminant

A wide‐band, frequency‐modulated, subbottom profiling system (the chirp sonar) can remotely determine the acoustic attenuation of ocean sediments and produce artifact‐free sediment profiles in real time. The chirp sonar is controlled by a minicomputer which performs analog‐to‐digital and digital‐to‐analog conversion, correlation processing, and attenuation estimation in real time. The minicomputer generates an FM pulse that is phase‐ and amplitude‐compensated to correct for the sonar system response. Such precise waveform control helps suppress correlation noise and source ringing. The chirp sonar, which has an effective bandwidth of 5 kHz, can generate chirp (Klauder) wavelets with a tuning thickness (Rayleigh’s criterion for resolution) of approximately 0.1 ms. After each return is correlated, a computationally fast algorithm estimates the attenuation of subbottom reflections by waveform matching with a theoretically attenuated waveform. This algorithm obtains an attenuation estimate by minimizing the m...

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Chirp subbottom profiler for quantitative sediment analysis

The present invention is a method of processing seismic data in which one or more seismic vibrators are activated with one or more pilot signals and vibrator motions are recorded along with seismic data. Vibrator signatures (4) are computed from measured vibrator motions (2), such as the ground force signal. A desired impulse response (5) is specified from either a measured vibrator motion (2) or from test data or field data from a location near the location from which the seismic data was acquired. A deconvolution filter (8) is computed from the impulse response (5) and from the vibrator signatures (4) from multiple vibrators and sweeps (1). The deconvolution or deconvolution and separation filter (8) is used to process the seismic data. The vibrators are then moved to a new location, and the activation is repeated.

Shaped high frequency vibratory source

A method is provided for significantly reducing the distortion and crossfeed from any selected order harmonic for any number of vibratory seismic sources operated concurrently, at the same time providing for separation of the signals from the different sources and for improving the signal-to-noise ratio. After determining the highest order harmonic likely to cause distortion, a number of sweeps of each source in each position is selected. This number depends upon the number of sources and the highest order harmonic to be suppressed. Initial phase angles for each sweep of each source are then selected to permit signal separation while suppressing harmonics up to and including that highest order harmonic.

Method of source coding and harmonic cancellation for vibrational geophysical survey sources

A method of processing seismic data, said seismic data having been obtained by: performing a plurality of sweeps, wherein each sweep comprises generating seismic signals in the earth (1) using a plurality of vibrators (4a-4d) by applying a pilot sweep waveform to each vibrator, each pilot sweep being a waveform of changing frequency; measuring the force applied to the earth by each vibrator to determine a measured force waveform; and measuring the seismic signals at one or more locations (5a-5e) remote from the vibrators; said method comprising: filtering the measured force waveform to remove harmonics of the pilot sweep and thus determining a filtered force waveform; generating an inversion operator from the filtered force waveform for each vibrator; and applying said inversion operator to the measured seismic signals to determine the contribution of each vibrator to the seismic signals.

Processing simultaneous vibratory seismic data

Due to non-linear effects, the swept frequency signals (sweeps) transmitted into the subsurface by vibrators are contaminated by harmonics. Upon correlation of the recorded seismograms, these harmonics lead to noise trains which are particularly disturbing in the case of down-sweeps. The method described in this paper—which can be regarded as a generalization of Sorkin's approach to the suppression of even order harmonics—allows elimination, from the final vibratory source seismogram, of harmonics of the sweep up to any desired order. It requires that not one single signal but rather a series of M signals is employed where each signal has an initial phase differing from that of the previous one of the series by the phase angle 2πM. Prior to stacking, the seismograms generated with the different signals have to be brought into the form they would have if they had been generated with the same signal.

The method seems also to be capable of reducing the correlation noise if sign-bit recording techniques are used.

Reduction of harmonic distortion in vibratory source records

The amplitude of the signal and the energy of the noise on each of at least three traces can be estimated provided that the signal has the same form (but not necessarily the same amplitude) on these traces and that the noise on any trace is correlated with neither the signal nor the noise on any other trace. This estimation of signal amplitude and noise energy can be achieved by a rather simple algorithm. The accuracy of the estimate depends, of course, on the degree to which the assumption that signal and noise on the different traces are mutually uncorrelated is actually met. The accuracy tends to improve with increasing number of traces.

Estimation of the signal‐to‐noise ratio of seismic data with an application to stacking*

For a swept frequency signal with constant envelope the power spectral density is approximately inversely proportional to the signal\\\'s rate of frequency change. Hence, by means of this relation, the phase function which describes the sweep\\\'s frequency variation may be derived from a predefined power spectrum. This is in contrast to the present use where the power spectrum is the result of a rather arbitrarily chosen phase function. A numerical algorithm based on the relation between power spectrum and phase function was found to lead to sweeps whose power spectra matched the prescribed ones rather closely.

Vibroseis signals with prescribed power spectrum

Computerized evaluation of Vibroseis similarity test data is the logical consequence of the increasing quality requirements for signal reproducibility and for synchronization of vibrators. The differences in start time and phase as well as an indication of local phasing problems between any two Vibroseis signals can be obtained from an analysis of the difference of their respective phase spectra. This method appears to be accurate and stable with respect to harmonics which usually plague the signals from transducers monitoring the motion of the vibrators'base plates.

Computerized analysis of vibroseis signal similarity

Determination of impedance or velocity from a stacked seismic trace generally suffers from noise and the fact that seismic data are bandlimited. These deficiencies can frequently be alleviated by ancillary information which is often expressed more naturally in terms of probabilities than in the form of equations or inequalities.



In such a situation information theory can be used to include ‘soft’information in the inversion process. The vehicle used for this purpose is the Maximum Entropy (ME) principle. The basic idea is that a prior probability distribution (pd) of the unknown parameter(s) or function(s) is converted into a posterior pd which has a larger entropy than any other pd which also accounts for the information. Since providing new information generally lowers the entropy, this means that the ME pd is as non-committal as possible with regard to information which is not (yet) available. If the information used is correct, then the ME pd cannot be contradicted by new, also correct, data and thus represents a conservative solution to the inverse problem.



In the actual implementation, the final result is, generally, not the pd itself (which may be quite broad) but rather the expectation values of the desired parameter(s) or function(s).



A general problem of the ME approach is the need for a prior pd for the parameter(s) to be estimated. The approach used here for the velocity is based on an invariance criterion, which ensures that the result is the same whether velocity or slowness is estimated. Unfortunately, this criterion does not provide a unique prior pd but rather a class of functions from which a suitable one must be selected with the help of other considerations.

E. Rietsch

Papers

Reduction of harmonic distortion in vibratory source records

Estimation of the signal‐to‐noise ratio of seismic data with an application to stacking*

Vibroseis signals with prescribed power spectrum

Computerized analysis of vibroseis signal similarity

The maximum entropy approach to the inversion of one-dimensional seismograms1