Reply by the authors to A. Walden and R. White

Geophysics ◽  
1988 ◽  
Vol 53 (11) ◽  
pp. 1491-1492
Author(s):  
Anton Ziolkowski ◽  
Jacob Fokkema
Keyword(s):  
Impulse Response ◽  
Finite Length ◽  
Statistical Properties ◽  
Finite Segment ◽  
The Earth ◽  
Response Formula

We thank Andrew Walden and Roy White for their interest in our paper and their explanation of the practical whiteness assumption in deconvolution. As we understand it, what they are saying is this: True whiteness is not at issue when we are dealing with finite chunks of data. The only thing that matters is whether the statistical properties of a finite segment of the impulse response of the earth (what Walden and White call the reflection response [Formula: see text]) are those of a finite length sample from an uncorrelated sequence. Quite. And how are we going to find that out unless we first do the signature deconvolution with a known signature? In other words, we can only test this assumption in circumstances where we have no need of it.

Can we perform statistical deconvolution by polynomial factorization?

Geophysics ◽  
1991 ◽  
Vol 56 (9) ◽  
pp. 1423-1431 ◽  
Author(s):  
Anton Ziolkowski ◽  
Evert Slob
Keyword(s):  
Impulse Response ◽  
Seismic Data ◽  
Statistical Properties ◽  
Polynomial Factorization ◽  
The Earth ◽  
True Source ◽  
Free Case ◽  
Multichannel Seismic Data

We investigate the possibility of finding the source signature from multichannel seismic data by factorization of the Z-transforms of the seismic traces. In the convolutional model of the data, the source signature is the same from trace to trace within a shot gather, while the impulse response of the earth varies. In the noise‐free case, the roots of the Z-transform of the wavelet are the same from trace to trace, while the roots of the Z-transform of the impulse response of the earth must move from trace to trace. It follows that the roots of the wavelet can be found by the invariance of their positions. We demonstrate this using a simple wedge model. No assumptions about the length of the wavelet or the statistical properties of the impulse response of the earth are required. It is shown that this idea cannot work on real seismic data. There are two difficulties which we regard as insuperable. First, even without noise, a seismic trace cannot be regarded as a complete convolution, because the data are always truncated. This causes the factorization to be inexact: the wavelet roots move from trace to trace and are indistinguishable from the roots of the earth’s impulse response. Second, the addition of a small amount of noise alters the root pattern unpredictably from trace to trace and the roots of the wavelet are again indistinguishable from the roots of the earth’s impulse response. We conclude that it is impossible to identify and extract the true source signature from real seismic data using no assumptions about the statistical properties of the impulse response of the earth. We propose that the signature should be measured.

Statistical properties of Rayleigh waves due to scattering by topography

1967 ◽  
Vol 57 (1) ◽  
pp. 83-90
Author(s):  
J. A. Hudson ◽  
L. Knopoff
Keyword(s):  
Surface Waves ◽  
Rayleigh Waves ◽  
Seismic Noise ◽  
Forward Scattering ◽  
Statistical Properties ◽  
Body Waves ◽  
Simplified Model ◽  
Two Dimensional ◽  
Especial Reference ◽  
The Earth

abstract The two-dimensional problems of the scattering of harmonic body waves and Rayleigh waves by topographic irregularities in the surface of a simplified model of the earth are considered with especial reference to the processes of P-R, SV-R and R-R scattering. The topography is assumed to have certain statistical properties; the scattered surface waves also have describable statistical properties. The results obtained show that the maximum scattered seismic noise is in the range of wavelengths of the order of the lateral dimensions of the topography. The process SV-R is maximized over a broader band of wavelengths than the process P-R and thus the former may be more difficult to remove by selective filtering. An investigation of the process R-R shows that backscattering is much more important than forward scattering and hence topography beyond the array must be taken into account.

A NOTE ON THE RELATION OF SUDDENLY APPLIED DC EARTH TRANSIENTS TO PULSE RESPONSE TRANSIENTS

Geophysics ◽  
1939 ◽  
Vol 4 (4) ◽  
pp. 279-282 ◽  
Author(s):  
G. E. White
Keyword(s):  
Steady State ◽  
Experimental Method ◽  
Impulse Response ◽  
Pulse Response ◽  
Heaviside Function ◽  
Electrical Measurements ◽  
The Earth

From some work by Carson, the relation between the earth response to a Heaviside function voltage and the response to an impulse is pointed out. A method of obtaining all other electrical measurements from the impulse response is indicated. It is suggested that a new experimental method might yield more accurate measurements of the electrical earth responses than can be had from suddenly applied DC transients, or any of the steady state measurements.

Determination of the signature of a dynamite source using source scaling, Part 1: Theory

Geophysics ◽  
1993 ◽  
Vol 58 (8) ◽  
pp. 1174-1182 ◽  
Author(s):  
Anton Ziolkowski
Keyword(s):  
Impulse Response ◽  
Seismic Data ◽  
Scaling Law ◽  
Cube Root ◽  
The Earth ◽  
Shot Point ◽  
Source Scaling ◽  
Seismic Data Acquisition ◽  
Injection Function

It is normally impossible to measure the source signature in land seismic data acquisition with a dynamite source, because it is normally impossible to separate the incident field from the scattered field. Nevertheless, in any serious attempt to invert the seismic data, it is essential to know the source signature; for the dynamite source this is the volume injection function. The problem can be solved by using two different shots at each shot point and relating the source signatures by the source scaling law, which follows from the invariance of the medium parameters with the size of the charge. The volume injection function of the larger shot is an amplified and stretched version of that of the smaller shot, the amplification factor being equal to the ratio of the charge masses and the time stretch factor being equal to the cube‐root of this ratio. At a given receiver, the response to one shot is a convolution of the source signature with the impulse response of the earth, plus noise. The two shots and the scaling law give three independent equations relating the three unknowns: the two source signatures and the impulse response of the earth (plus noise). This theory may be put at risk in a physical experiment which requires a third shot at the same shot point, using a known mass of dynamite, different from the first two. The resulting shot record should be different from the first two and, apart from the noise, should be predictable from them.

Constant Q attenuation of subsurface radar pulses

Geophysics ◽  
1994 ◽  
Vol 59 (8) ◽  
pp. 1192-1200 ◽  
Author(s):  
Greg Turner ◽  
Anthony F. Siggins
Keyword(s):  
Linear Function ◽  
Impulse Response ◽  
Seismic Waves ◽  
Impulse Response Function ◽  
Single Parameter ◽  
Water Tank ◽  
Radio Waves ◽  
The Earth ◽  
Energy Dissipated ◽  
Laboratory Water

Q is a measure of the energy stored to the energy dissipated in a propagating wave and can be estimated from the ratio of attenuation and frequency. For seismic waves, Q has been found to be essentially independent of frequency. As a result, attenuation is an approximately linear function of frequency and the impulse response function of the earth. Hence, the distortion of a seismic pulse as it propagates can be described by a single parameter. Laboratory measurements show that the attenuation of radio waves in some geological materials can also be approximated by a linear function of frequency over the bandwidths of typical subsurface radar pulses. We define a new parameter Q* to describe the slope of this linear region. The impulse response of the transfer function for a given value of Q* differs from that of the same value of Q only in total amplitude. Thus the change of shape of a radar pulse as it travels through these materials can also be described by a single parameter. The constant Q* model successfully describes the distortion of radar pulses propagated through a laboratory water tank and through weathered granite in a borehole field study.

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