Kristopher Innanen†, †† and Arthur Weglein†† ††University of Houston,

Linear and Non-linear Inversion for Absorptive/Dispersive Medium Parameters
Kristopher Innanen†, †† and Arthur Weglein†† ††University of Houston, † University of British Columbia M-OSRP Annual Meeting, University of Houston March 31 – April 1, 2004

Acknowledgments CDSST (UBC) sponsors and members Tad Ulrych Simon Shaw
M-OSRP sponsors and members CDSST (UBC) sponsors and members Tad Ulrych Simon Shaw Bogdan Nita

Motivations Study of the forward scattering series (MOSRP02) suggests that – properly posed – the ISS will attempt to accomplish Q-estimation and Q-compensation in the absence of prior knowledge of medium parameters. This promise motivates, for ISS-based Q procedures, investigation of theoretical underpinnings, investigation of practical issues and data requirements And it motivates the (skeptical) acceptance of an unusual way to detect the presence of Q in our data…

Key references Linear inversion for multiple parameters.
Clayton & Stolt (1981) Raz (1981) Weglein (1985) Absorption/Dispersion. Aki & Richards (2002) [and refs therein] Kjartansson (1979) Innanen and Weglein (2003) Nonlinear inversion & processing. Zhang and Weglein (2002)

Background and review The quantitative inclusion of friction into the modelling of wave propagation is a mixture of physical theory and empiricism: “Keep the constitutive relations linear, impose causality, and keep the amplitude-change per cycle (Q) independent of frequency.”

Background and review Causality dictates that absorption cannot exist without dispersion…

Background and review Constant Q demands that the frequency-dependence of the absorption parameter  be linear. However linear () leads to a divergent Hilbert transform. The challenge is to depart a small amount from one of these assumptions…

Background and review A fairly well-accepted choice leads to the following form, explicitly in terms of the single parameter Q and a reference frequency r: This form relies on a linearization, i.e. it requires …in other words, Q and  can’t both be small.

Background and review Using a Q model of this kind, the forward scattering series (acoustic reference, absorptive/dispersive non-reference media) construction of a viscoacoustic wave field was investigated (M-OSRP02). Indicated: (1) generalized imaging subseries will be a de-propagation subseries, namely it will attempt to accomplish Q compensation (2) generalized inversion subseries will accomplish Q estimation The first step in considering such methodologies is to pose the linear inverse problem for the absorptive/dispersive case.

A viscous linear inverse
What should V1 look like? The answer depends on our purposes – and we have several here. How many parameters do we wish to include in the problem? What is the dimensionality of the problem? And most importantly: Are we interested in the linear inverse as a valuable product on its own, or as the input to a higher order nonlinear scheme?

Let us begin with the most general case we’ve tackled and go from there. Consider the linear data equations associated with a shot record-like measurement of the scattered wave field, and a V1 that may vary in depth only:

Alternately, consider the 1D normal incidence linear data equations:

V1 is obtained from the difference between appropriate wave operators. In our case we have wavenumbers and

V1 is obtained from the difference between appropriate wave operators. In our case we have wavenumbers set =1/Q and

V1 is obtained from the difference between appropriate wave operators. In our case we have wavenumbers and

set F(k)i/2-(1/)ln(/r)
A viscous linear inverse V1 is obtained from the difference between appropriate wave operators. In our case we have wavenumbers set F(k)i/2-(1/)ln(/r) and

V1 is obtained from the difference between appropriate wave operators. In our case we have wavenumbers set 1/c2(z)=1/c02[1-(z)] and

expand and linearize (for now!)
A viscous linear inverse V1 is obtained from the difference between appropriate wave operators. In our case we have wavenumbers and expand and linearize (for now!)

And so

write as F(k) or F(), since k/kr= /r
A viscous linear inverse And so write as F(k) or F(), since k/kr= /r

And so

Write the linear components as

and transforming to the wavenumber domain:

Switch to `=` for now…

The 1D normal incidence case cannot in general be used to solve for two parameters. For our purposes consider a case involving contrasts in Q only:

The 1D normal incidence case cannot in general be used to solve for two parameters. For our purposes consider a case involving contrasts in Q only: ( = 0)

The 1D normal incidence case cannot in general be used to solve for two parameters. For our purposes consider a case involving contrasts in Q only:

With the tools gathered below we can investigate a set of slightly differing linear inversions for c(z) and/or Q(z). The most general case is to use data with offset and the two parameter perturbation.

Assume we can deconvolve the source wavelet over the requisite frequency bandwidth.

Use the definition of the wavenumber k=/c0 …

These are the data equations with which we solve for 1 and 1 from D. We’ve heard and will continue to hear discussion of which parameter to use in our exploitation of the extra degree of freedom. We’ll choose angle for the offset case, frequency for the 1D case. The former comes about as follows…

Offset geometry The wavenumbers and incidence angle  are related as follows:

Offset geometry The absorption/dispersion factor F(k) figures importantly in this inversion procedure. We have:

Offset geometry So we can express F(k) as a function of depth wavenumber and angle also. Returning to the inverse problem, then…

Re-write this as…

where and

The problem of estimating the spatial distribution of the absorptive/dispersive parameters 1 and 1 is therefore well-posed; i.e., for every qz, variability in offset () provides an overdetermined set of linear equations for the unknowns.

Intermediately summarizing: (1) Using a well-known Q model we pose a linear two- parameter (c & Q) inverse problem using a shot- record like input data set (2) The dispersion associated with the Q model (i.e, the frequency dependence of F(k)) permits the inversion to take place (3) The implication is that we will be interrogating the absorptive/dispersive data for Q information in an AVO-like way. This is an unorthodox Q estimation…

A simplified case We can investigate some basic numerical aspects of this problem more easily by posing a slightly simpler version of the problem. (1) The 1D normal incidence two-parameter problem is well-posed IF we assume a spatial form for the desired perturbation (2) We anticipate a greatly exaggerated transmission error for deeper interfaces – absorptive/dispersive propagation effects Return to the various linear inversion quantities we generated…

A simplified case We require the 1D normal incidence data equations, and the two parameter linear scattering potential…

A simplified case At normal incidence, k = qz…

A simplified case Assume again the source wave form is known and removable over the desired bandwidth.

A simplified case Assume again the source wave form is known and removable over the desired bandwidth. Then substitute in V1…

A simplified case For each wavenumber k, this is a single equation in two unknowns, and thus can’t be solved without some kind of assumption or prior information.

A simplified case This information can take the form of an assumption regarding the spatial form of the model. For instance, we could specify a single-interface model: acoustic reference medium attenuative non-reference medium

A simplified case – 1 interface
This information can take the form of an assumption regarding the spatial form of the model. For instance, we could specify a single-interface model: acoustic reference medium attenuative non-reference medium

Substituting this spatial form for the perturbations:

We know what the data from such an experiment looks like: acoustic reference medium attenuative non-reference medium

Substituting this data form in gives:

Now we have two unknowns and as many equations as we have frequency components in the data, or the measured reflection coefficient R(k): .

With perfect data we can study this problem using two frequencies only:

A viscous reflection coefficient
Pretty clearly these inversion procedures rely on the ability to detect the frequency-dependence of the viscous reflectivity. Let’s consider the R(k) associated with an acoustic incidence medium and an absorptive/dispersive target medium:

A viscous reflection coefficient
Let’s consider the R(k) [or R(f)] associated with an acoustic incidence medium and an absorptive/dispersive target medium: c0 = 1500 m/s c1 = 1600 m/s Q1 = 1000, 500, 100, 50 (real component) R(f) is complex, and, more importantly, frequency dependent.

Let us numerically compute these quantities and display their associated c1 and Q1 counterparts over a range of k1, k2…

Let us numerically compute these quantities and display their associated c1 and Q1 counterparts over a range of f1, f2…

Let us numerically compute these quantities and display their associated c1 and Q1 counterparts over a range of k1, k2… f1, f2 = Hz

The wavespeed inversion result is m/s, which is what we would get in the acoustic wavespeed linear inverse. Inputs: c0 = 1500 m/s c1 = 1800 m/s Q1 = 100 f2 (Hz) f2 (Hz) f1 (Hz) f1 (Hz)

At large Q contrast the low-frequency pairs produce results deflected away from the standard Born approximation. Inputs: c0 = 1500 m/s c1 = 1800 m/s Q1 = 10 f2 (Hz) f2 (Hz) f1 (Hz) f1 (Hz)

Large wavespeed contrast leads to increased estimation error for Q, still within a few % Inputs: c0 = 1500 m/s c1 = 2500 m/s Q1 = 100 f2 (Hz) f2 (Hz) f1 (Hz) f1 (Hz)

At large c, Q contrast we see deflection again at low f pairs, results still quite accurate. Inputs: c0 = 1500 m/s c1 = 2500 m/s Q1 = 10 f2 (Hz) f2 (Hz) f1 (Hz) f1 (Hz)

Another intermediate summary: (1) The single-interface inversion (i.e. known overburden) produces “standard” linear wavespeed numbers and quite accurate Q numbers (2) Caveat: these results are based on perfect data, fairly subtle changes in R with f. (3) Analogy is of a linear regression problem, detecting the slope and intercept of a line through (probably) noisy data (4) The stability of the estimation across the utilized frequency band is reassuring – lots of useful data

A simplified case – single layer
The extension of absorptive/dispersive inversion to multiple interfaces – i.e. permitting an unknown overburden – is more “perilous” than for the associated acoustic extension, because of the natural exaggeration of transmission error, aka attenuation. Nevertheless, let’s explore the problem and these issues…

Once again consider this simplified case… acoustic reference medium attenuative non-reference medium

Once again consider this simplified case… …of a 1D normal incidence experiment over a two-interface medium. acoustic reference medium attenuative non-reference medium

The data associated with such a model at normal incidence is

Consider again the 1D normal incidence surrogate case. Write the linear perturbations due to each event individually: …in which case the data equations become

One way to proceed here is to assume we can separately analyze the two events. Putting in our perfect data:

One way to proceed here is to assume we can separately analyze the two events. Putting in our perfect data: and as before assume that the events are due to step-like contrasts at the pseudo-depths associated with c0

One way to proceed here is to assume we can separately analyze the two events. Putting in our perfect data: we separate effects due to like data events…

One way to proceed here is to assume we can separately analyze the two events. Putting in our perfect data: and all that remains of the exponential factors is folded into the effective lower reflection coefficient:

We have a look at the numerical behaviour of this inversion by again picking pairs of input frequencies. Then we can solve the two equations separately for 11, 12, 11, and 12. We have some expectations about error levels. Consider the inversions again over the full range of possible input frequency pairs…

Q1 = 300 Q2 = 10 c1 = 1550 m/s c2 = 1600 m/s

Q1 = 250 Q2 = 10 c1 = 1550 m/s c2 = 1600 m/s

Q1 = 200 Q2 = 10 c1 = 1550 m/s c2 = 1600 m/s

Q1 = 150 Q2 = 10 c1 = 1550 m/s c2 = 1600 m/s

Q1 = 100 Q2 = 10 c1 = 1550 m/s c2 = 1600 m/s

We note: (1) The first-interface results remain of high-accuracy (2) There is a clear decay of the accuracy of the c2, Q2 estimates with increasing Q1. (3) However, the decay depends strongly on the input frequency pairs: low frequency pairs fare better than high. This is because of the preferential decay of amplitude at high frequency. (4) A weighting scheme on such an overdetermined problem could be geared to capitalize on this

A layer-stripping correction
Although it is not our main purpose, it seems worth mentioning that there are ways of “patching up” the errors associated with propagation. We estimate c1, Q1 and have an approximate (z2-z1). This information could be incorporated into a correction of the next deepest reflected event:

…with helpful consequences. Consider the inversion for troublesome high frequencies without the correction:

…with helpful consequences. Consider the inversion for troublesome high frequencies with the correction:

Caveat: The layer stripping works well, but (1) it relies on linear estimates of c and Q, and so is subject to error that will likely accumulate with depth, and (2) is an ad hoc patch, and will rely on assumptions about the medium to be applied. This last point is not too important in an example already full of assumptions, but in any application of the general form of the c/Q linear inverse it is more of an issue.

Summary • We develop a linear inversion for a two-parameter (c/Q), depth-variable medium based on a single shot-record data set. • We derive some basic insights by investigating a normal incidence version of the problem in which we assume a step-like model format. • Accuracy is considered in the absence and presence of an unknown absorptive and dispersive overburden. • Preferentially using low frequency inputs and an ad hoc layer-stripping are suggested as means to give the linear inverse value as a standalone procedure.

Non-linear inversion: some preliminary comments
Back to the question about a viscous V1: what should it look like? Now as the input to higher order terms in the ISS… The 1D framework was useful in the previous sections, but the spatial form of the linear reconstruction is contrived. Consider a 1D normal incidence case involving perturbations in Q only…

Consider:

We had developed some tools for absorptive/dispersive inversion.

Equating these and deconvolving the wavelet,

Equating and deconvolving the wavelet,

Equating and deconvolving the wavelet, Let us substitute in the analytic data, and get an idea of what the linear inverse for a single interface “naturally” looks like…

Equating and deconvolving the wavelet, and investigate the form by substituting in our analytic data… Heaviside fn. Filter L1(-2k)

In other words the “form” of a viscous perturbation is a filtered Heaviside function (if the model is a true Heaviside function). Let’s consider some numerical examples…

Q1 = 100 c0 = 1500 m/s

Q1 = 10 c0 = 1500 m/s

(1) The closer L1(z) is to a delta function, the less the inversion process deflects the spatial form of the model estimate away from the true model. (2) At large contrast such a deflection is readily visible: a “droop” in the step. (3) Deflection is greatly reduced in comparison with a simple integration of the trace; the division by F(k) is a fairly effective deconvolution of absorptive/dispersive reflectivity.

For a two-interface case, we can show a similar result:

For a two-interface case, we can show a similar result: i.e., both steps are linearly positioned at their depths z1 and z2, and both are filtered by functions of the data and F(k). L2(z), however, contains data artifacts that F(k) has a harder time dealing with…

Q1 = 100 Q2 = 20 c0 = 1500 m/s

(1) The linear inversion has the ability to intercede to process the phase/amplitude deflections of absorptive/dispersive reflectivity, but no ability to compensate for the attenuation of propagation. (2) Hence the linear inversion results are smoothed in the presence of unknown overburden. (3) This is as expected: what we have is a sense of what a “natural” input to the higher order terms in the series is.

Summary 1. We pose the linear absorptive/dispersive problem for a depth variable c/Q medium using frequency and offset information. 2. We investigate using a simplified form of this inversion, 1D normal incidence with structural assumptions. We see encouraging results, however, attenuation in the overburden generates problems for deeper parameter estimates. 3. The error may be mitigated by weighting in the estimation (the problem is overdetermined), or with a layer-stripping like patch-up.

Summary 4. We are further interested in understanding the form of this linear inverse that would be used as input to the higher order terms in the inverse scattering series; a 1-parameter 1D case is investigated, and we see effects due to (i) viscous contrasts and (ii) viscous propagation. 5. We consider the inversion (target-identification) subseries in a basic way. The inter-event communication appears to preferentially involve terms that aggregate all shallower events at once. 6. When applied to a Q compensated linear input, the amplitudes are corrected. Suggests the I.S. might be a last step for absorptive/dispersive case, after a nonlinear Q correction.

Kristopher Innanen†, †† and Arthur Weglein†† ††University of Houston,

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