Geotomography has emerged from medical tomography but unlike medical tomography it largely does not work well. Why it happens? The answer is trivial: ...
Inversao Sismica / Seismic Imrersion Quinta-Feira (Manha) / Thursday (Morning) SIN.1
Is Geotomography Doomed? Philip Carrion, G. Boehm, A. VeSnaver, I. Pettenati, OGS ~
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where only limited
view angles ate possible to
achieve. What is the most detrimental is that limited angular coverage causes the so-called aperture instabilities which account for the majority of cases
Abstract
In seismic experiments. Thus running conventional
tomography one runs a high risk to obtain unstable
Geotomography has emerged from medical tomography but unlike medical tomography it largely does not work well. Why it happens? The
results.
Aperture instability is expressed in high
sensitivity
of
results
with
angular
coverage:
changing sHghtly aperture coverage one can come
answer is trivial: seismic experiments are different from medical and do not suit tomographk procedure. What can be done? To reject tomography as a tool for velocity reconstruction or (0 adopt It wifli modifications? GeotofTTography is probably a unique tool for velocity reconstruction. Conventional velocify analysis is too simplistic and is limited to layered media which do not bear much interest In this presentati.on we introduce a possibility to significantly improve tomography and to make use of it in real situations related to typical seismic experiments. We achieve it with the help of the so-called compensation operator which reduces the dependence of results upon angular coverage and thus adjusts it to real world.
up with completely different results. That is why conventional
tomography
usually
leads
10
misinterpretation and false structures appeared in computed tomograms. Carr/on and Came/ro (1089/ and Carrion (1981) have developed the so-called dual tomography with the compensation operalor
Z which incorporates hard constraints in the traveHime inversion. In this presentation, jf wilf be shown that this operator in facts 5ignmc~ntly reduces dependence upon aperture and completely suppresses aperture instabilities assoCIated with limited aperture coverage.Duaf or aperture compensated tomography, thus, leads to direct applications to real data obtained in typical sei:::>mic experiments It ususalty gives accurate imnges when other tomographies bog down.
Introduction In seismic tomography. the quality of reconstruction depends on different parameters such as angular coverage, SNR, etc. F=or basic definitions. the reader can be adverted to the texts of Nolet (1987) and Carrion (1987). Recently Carrion and Vesnaver (1991) have inverted 3-D Fermat's integral and found deterministic and statIstic criteria for recoverability of velocity from measured traveltimes. They found that in fact the velocity function is sought as the integration carried out over the domain of take~of( angles. In the case of medical tomography, sources and receivers can be rotated around the unknown scaHersr tt is precludea, however, in the case of geotomography ~-
Aperture Compensated Tomography. Geotomography is based on a solution of the traveltime equation (1 )
ObViously, a solution to this equation does n01 eXist in general and thus a least square approxill1atlon is sought instead. A least~squares substitute is found in the following form: (2)
...... 796
2
Is Geotomography Doomoed ?
model where
W
is
thl~
matrix
that
provides
stable
space.
This is an examlJ1e of p:::'lrtial
compensation.
inversion. This technique that works quite well in different areas 01' science does not work well in
Conclusion
geophysics. This is because (2) does not contain any information about angular coverage and there is no any assurance that if (2) works for large apertures it will Nork with the same success for rather small angular recording (typical to seismic exploration) Let us modify the traveltime expression in the following manner:
Aperture compensated tomography advocated in this presentation largely depends upon the compensation operator Z. This operator is built of constraints imposed in the model space on the velocity function.
Constraints are alway available
in geophysics. They can be taken as 0 for the lower
aT = AzaLi
bound and quite loo.se for the high bound. Even
(3)
tfiese A least-squares version of (3) reads in terms of a
hypothetical
constraints
buil
in
the
compensation operator significantly improve 'he
system of two parametric equations:
performance of tomographic inversion. later on these constraints can be updated in the course of iterations. What is the most interesting that these constraints can be taken in only portion of the model space. In this case we are talkin about
and
au- = ZA rA
p~rtial compensation.
(51
Suppose that in a window
(portion of the model space when the compensation It is possible to e>lplicitly deduce the expression for the operator Z from the constrained optimization
operator is identity) constraints are not taken into consideration. It is good enough though to identify
theory. What is the most exciting is that operalnr Z which is _called the compensation operator reduces the influence of limited aperture and leads to
the compensation operator outside the window What happens in this case that although thl"! compensation operator is identified outside the
drastic improvements in computed seismograms.
window, the resolution inside the window will 1)1~ significantly improved.
Numerical Example Let us c.onsider
i::l
References
numerical example with cross
borehole se~ting (Figure 1). The uncompensated tomography gives a blurred image shown in Figure 2. Figure 3 is the result obtained using aperture compensated tomography with loose constraints. One can notice lhat even loose constraints (the lower bound is :1:ero and the high bound 5.000 m/sec significantly improves the output of tomographic inversion. The compensation operator
1.
Carrion.
P.,
1987,
Inverse
problems
and
tomography in acoustics and seismology, Penn, Pub!. Co., Atlanta, GA. 2. Carrion, P., 1989.
Robust
constrained
tomography. SEG (slide sets series), Tulsa, OK. 3. Carrion, P., and Carneiro. D., 1989, Linear inversion using the dual transform, Geophys. Rf.'!~.
was introduced only in the portions of the model
Lett., 16, 1039·1042.
space (near the top and near the bottom of the
4. Nolet, G., (ed) 1987, Seismic wave propagation and seismic tomography with applications in global seismology and exploration geophysics. Pub!. Co., Dordrecht.
797
Reidel
Is Geotomography Doomoed ?
3
receivers well
sources well
Cl=1k~ec
Figure 1 - Model. Cross-well experimenL
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Figure 3" - Aperture compensatiort effect using dual tomography With the same angular coverage.
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