125
THE FULL WAVEFORM ACOUSTIC LOG INVERSION PROBLEM
by
G.H. Garcia and C.H. Cheng
Earth Resources Laboratory
Department of Earth, Atmospheric, and Planetary Sciences
Massachusetts Institute of Technology
Cambridge, MA 02139
ABSTRACT
This paper presents the full waveform acoustic log inversion problem for a
multisource - multireceiver tool logging a cylindrical fluid-filled borehole. The probiem
is formulated in the frequency-depth domain. A priori knowledge about the source
array spectra and the borehole formation parameters (velocity, attenuation, density)
is taken Into account. It is shown that the inversion problem can be formulated as a
Separable Nonlinear Least Squares (SNLS) problem with separable nonlinear equality
constraints. The case where the source Is known and parameterized is studied with
synthetic data and an inversion for
Vi' and Pb is implemented and studied.
v..,v.,
INTRODUCTION
Full waveform acoustic logging has become a widely used tool in formation
evaluation. However, up to now the techniques for determination of formation
compressional and shear wave velocities and formation attenuation have been done
separately. In "slow" formations shear wave velocities cannot be determined directly
and have to be inverted from Stoneley wave velocities. In some tools, the sourcereceiver characteristics prevent one from identifying the Stoneley wave in "slow"
formations. Measurements of formation attenuation are complicated by the effects
of modal dispersion. The obvious answer to all these problems is the simultaneous
determination of the formation velocity, attenuation, and density from inversion of the
fUll waveform data using a model of elastic wave propagation in a borehole. In this
paper, we address the full waveform inversion problem from a theoretical point of
view of a constrained least squares inversion. We introduce the concept of
separable least squares to decouple the effects of the source from the responses of
the formation. We then present some results from the inversion of synthetic full
waveform acoutic logging data and study the sensitivity of the waveform to the
variations in each individual parameter.
INVERSION THEORY
Statisticians usually distinguish two broad kinds of data analysis problems
(Parzen, 1978):
(1)
"Statistical Inference-type 1" or model analysis:
Garcia and Cheng
126
Problem: What is the answer to a mathematically well-posed question?
Approach: Theorems on the properties of algorithms derived from estimation and
optimization theory.
(2)
"Statistical Inference-type 2" or model synthesis:
Problem: What is the question, or equivalently what is the model?
Approach: Data analysis procedures to be learned by doing (observations,
experimentation, modelling•.•)
This paper deals with a "statistical inference-type 1" problem, the full waveform
acoustic inversion in a cylindrical borehole geometry. This is a parametric inversion
method as opposed to non-parametric inversion methods such as semblance or
correlation.
Previous work in parametric and non-parametric inversion
Parametric modelling is common practice in many applications, the very general
Autoregressive Moving Average (ARMA) formulation for time series being one of the
better known examples. In ARMA modelling, the spectrum of the process Is
parameterized by a rational fraction In the frequency domain.
For the cylindrical full waveform inversion problem, one considers a parametric
form of the frequency-wavenumber spectrum with a finite number of parameters.
Much Is lost in the inference process if the parametric form is ignored, because when
the number of parameters is small compared with the total number of observations,
the results of the inference will be much stronger than when using a non-parametric
approach. Parametric methods involve statements about the probability distribution
of the observed data. Parametric Inference presupposes a known form of the
statistical distribution of the data being inverted for. Because of the finite amount of
data, the class of probability distribution is relatively restricted, usually to the
exponential family including Poisson, Gaussian, Binomial, etc., via the probability
distribution. The model is above all a mathematical formulation of the knowledge
about the tool-borehole-formation system under study. Equation (1), for exampie,
reflects an hypothetical link between the observed signal and the unknown signal. In
other words, there exists an expected value of the borehole wavefield signai in the
form of a model-hypothesis built from physical laws (the wave equation in cylindrical
geometry).
By comparison, the non-parametric approaches, suoh as semblance or oorrelation
methods, do not make any assumption about the statistical distribution of the data.
For the determination of P- and S-wave velocities for instance, most of the work has
concentrated so far on non-parametric inversion methods (Willis, 1983; Willis and
Toksoz, 1982). Non-parametric methods are usually more expedient and robust (i.e.,
results are practically independent of any statistical distribution of the data) than
their parametric counterpart. Recent asymptotic optimality results, however, show
that these methods cannot be substantially improved without the more stringent
modelling assumptions (Yakowitz, 1985). They are used in situations where other
estimators are difficult to implement because of mathematical manipulative
difficulties. They often rely, however, on heuristic techniques such as windowing,
threshold detection, etc.
6-2
Full Waveform Inversion
127
Moreover, in slow formations, where the shear' wave velocity is lower than the
borehole fluid velocity, non-parametric semblance/correlation methods cannot
extract the shear wave information -- although present -- directly from the data. In
these cases, one is justified in trying the more "fragile" and compute-intensive
parametric inversion methods by using models of the tool-borehole system with
additional a priori knowledge in the form of constraints.
In this paper, the forward model structure assumed is set forth and the method
of Maximum Likelihood (ML), one of the most versatile and powerful tools of
estimation, will be used for inference on the parameters and the associated
estimation error covariance.
When there is a priori knowledge on the parameters (for example, data from a
density tool, a Caliper Log, or an initial inverse from a non-parametric method), the
Maximum A Posteriori (MAP) method will provide (via the Bayes theorem) a way to
systematically incorporate these a priori in the inversion.
Since the source function is in general not known, this will then lead to the
Constrained Separable inversion method which eliminates the scurce function
inversion from the formation parameters inversion proper and allows a priori about the
source and the formation parameters to be taken into account.
THE CYLINDRICAL FORWARD MODEL ASSUMPTIONS
The assumptions are illustrated in Figure 1. The pressure frequency response
P(r,z,G)) in a fluid filled borehole at an axial distance z and radial distance r from a
point isotropic source is well known (Tsang and Rader, 1979; Cheng et al., 1982). it
Is given by
(1)
where X(G)) is the complex Fourier spectrum of the source. The first·term within the
brackets represents the direct source term. A(@ll.)' the modal coefficient, is given
by
A(all.)
=
g(a,,)K,(f R) -Ko(f R)
g(all.)I, (f R) + IoU R)
(2)
and
(3)
where all. is the parameter vector:
6-3
Garcia and Cheng
128
(4)
and
=
w is the angular frequency; G is the phase velocity; k wi G is the axial wave
number; V.
and V are the P and S wave velocity of the formation and the
f
borehole li'uid veiocity; respectively; R is the borehole radius; Pb and Pf are the
formation and fluid density; and 1; and Ki are the modified Bessel functions of the ith.
order. The term A(0a,)Io(/r) represents the response of the borehole.
,v.
Formation and fluid attenuations appear as the imaginary parts in the complex
velocities
Vf (Cheng et aI., 1982).
.
v;, ,v.,
THE -DISCRETE FORWARD CYLINDRICAL MODEL
The one source, one receiver (1S-1R) tool
For one source and one receiver separated by a distance z, equation (1) above
can be written:
P(w,z ,a)
= G(w,z ,0)
X(w)
(5)
The observed waveform W at frequency w is modelled as:
W(w)
= G(w,z)
X(w)
+N,,(w)
(6)
where W(r.:J) is the Fourier transform of the observed output field, P(w,z ,a) is the
modelled output pressure field, X(w) is the input source and N" (w) is the corrupting
observation noise.
G(w, z, 0) is the complex frequency response of the tool-borehole formation
system at frequency w. It represents the term within brackets in Equation (1).
is
a parameter vector, a subset of aa, shown in Equation (4). One or more of the
parameters in aa, may be fixed or known a priori with some uncertainty. The source
function may also be parameterized by a vector ~
a
X(w)
=X(w,~)
6-4
(
Full Waveform Inversion
and
~
129
may be included in the inversion.
The Fourier spectrum of the pressure can be discretized at N" frequencies
!(;Ji l, i = 1 ,N", as follows: (This is a valid approximation by the Stone-Weirstrass
theorem if the power spectral density matrix of the modelled process is positive
definite and continuous)
In matrix form the N" equations above become:
P(z,B)
=G(z,S)
(7.a)
X
where:
p(z,e) = !P«;Ji'z,S), i=1, . , .
N"J
G(z,S) = DiagHG«;Ji'z,S», i=1,···
XW
= !X«;Ji'~)' i=1,...,N"l
(7.b)
N"J
(7.c)
(7.d)
Note that P(z,S), X(~) are complex valued vectors and that G(z,e) is the complex
transmittance matrix from the source to the receiver at position z.
Multi-source/multi-receiver discrete model
In a similar fashion one can derive (see Appendix) the multi-receiverjmultisource discrete model. The equation has the same form as above. For a tooi with N~
receivers, Ns sources and N" sampling points in frequency domain, Equation (7a) can
be generalized into the form
P(S) =G(e) . X
(8)
where P is an (N x 1) complex vector, with N = N,,'Ns ·NT ; S is a vector of
borehole/formation parameters; G is a N xl.! matrix where M = N,,'Ns with the
structure shown in Figure 4; and X is a M x 1 source array vector. The details are
given in the Appendix.
NUMERICAL EXAMPLES
The case where the source characteristics are perfectly known will now be
studied on synthetic data. The purpose of this exercise is to find out if the problem as defined above is properly formulated and study its conditioning. In addition it brings information about the sensitivity of the functional being minimized
with respect to the parameter being inverted in a controllable manner.
The synthetic waveforms have been generated with the following formation
parameters: Tj,
4 km/s, V.
2 km/s, VI
1.5 kmjs, Ph (bulk density)
2.3
=
=
=
6-5
=
Garcia and Cheng
130
=
=
=
=
gm/cm 3 , P
1.2 gm/cm3 , Qp
60, Q. 60, Qf 20, and the radius is 0.10 m. We
used a Kely source (Toksoz et al., 1 983) with a center frequency c.Jo = 10kHz.
Figures 6 to 12 show the shape of the residual energy as a function of various
formation and borehole parameters. In each of the figures, we have kept all but one
parameter constant. We vary that parameter about the actual model and compute
the square of the difference in the pressure function (in the frequency domain)
between the varied and actual models.
Figures 6 and 7 point out the fact that the determination of >j, is not as well
conditioned as the determination of Ys. The shape of the residual energy function
(the "incoherence") is much flatter for >j, than for Ys. This is due in part to the small
P-wave energy in the signai. Furthermore, as pointed out by Cheng et al. (1 982),
there is very little P-wave energy in the guided wave packet, which is usually the
dominant arrival in a microseismogram. The P-wave velocity is nevertheless
determined by the whole waveform, rather than the mere arrival time. It can be
shown that this increases the achievable signai-to-noise ratio compared with any
other inversion method which does not use the complete waveform. The formation
shear wave veiocity, on the other hand, is very well resolved. A small error in the
estimation of the shear wave velocity results in a large residual energy. Since a
least squares inversion minimizes the residual energy, the formation shear wave
velocity is very well constrained.
In Figure 8 one sees that a good determination of the borehole fluid velocity is
necessary since the shape of the residual energy-function is drastically affected by
the value of the fluid velocity. Note the same kind of behavior is observed for the
variation of the energy norm as a function of the borehole radius (Figure 10). It
appears therefore. useful to include the caliper and independent measurements of the
borehole fluid P.-wave velocity as a priori in the inversion with an appropriate errcr
covariance to take into account this inherent sensitivity.
Figure 11 illustrates the poor resolution of the density parameter Ph' The scale
has to be magnified two orders of magnitUde to see the variations as a function c r
Ph' This suggests that formation density is not well resolved in the inversion as it is
set up at the present time. However, by using an external source of density
measurement as a constraint to the problem, we may improve. the resolution of the
formation density..
In Figure g variations in the residual energy as a function of the P-wave quality
factor Qp are plotted. Qp influences the function in an asymmetrical manner, and the
minimum is not as sharp as for the other parameters. This suggests a
reparameterization of the inversion. Instead of using Qp as the variable, 1/ Qp would
give the problem a better conditioning and avoid the sharp angular features in the
residual energy. Physically this makes much more sense since 1/ Q is a direct
measure of attenuation (amplitude decay). The variations as a function of the pwave attenuation are representative of the variations for all the 3 attenuation
parameters.
In Figure 12 the sensitivity to the source center frequency is plotted. One can
see that the source parameters do affect the waveform quite drastically and that it
is a potential problem since there is plenty of experimental and theoretical evidence
of the variability of the source as a function of environmental conditions. The best
6-6
,"
Full Waveform Inversion
solution would be to measure the source and constrain the problem with that
measurement, or to use the Separable Inversion method to circumvent the source
inversion problem.
Rgure 13 and following illustrate the inversion of synthetic data for 4
parameters: formation p- and S-wave velocity, P-wave velocity in the borehole fluid,
and formation density. Figure 13 shows the residual energy function or incoherence
as a function of the number of iterations. We can see that the incoherence
decreases monotonically as a function of the number of iterations. Note that Vs and
Vi are much better behaved during the inversion than Y;, and Ph (Figure 14). This
i1fustrates the fact that the introduction of the density and the P-wave velocity
creates large excursions in the parameter space. These non-physical excursions do
slow down the inversion problem B.nd can cause divergence in the case of large
residuals. Again the behavior for the synthetic case suggests the limitations of the
tool in real situations, and the use of additional sources of informatior. to "robustify"
the problem. One can expect therefore, without any additional a priori knowledge,
that there will be comparatively large uncertainties on some of the ML answers. The
MAP criterion will have to be used in the real world to obtain more reliable information
about the shear veiocity and attenuation.
The evolution of the synthetic full waveform that would be generated from the
successive values of the 4-parameter vector and their spectra is shown in the last
two figures (Figures 15 and 16). One can see the significant changes in the
microseismograms and their associated spectra as the parameters converge to the
model solution. Note that even between the last two iterations (7 and 11) there is a
substantial change in the waveform character despite the small changes in the
parameters (Figure 14).
SUMMARY
In this paper we have formulated the full waveform inversion problem for a simple
cylindrical borehole. The probiem is formulated in the frequency-depth demain. We
demonstrated our algorithm using synthetic data in a simple one source - one receiver
model. The sensitivity and the shape of the residual energy function are studied.
The problem is well behaved. It is shown that formation shear wave velocity has the
most influence on the residual energy function, while the formatlcn compressional
wave velocity and formation density have less influence. This information is vaiuable
for the applying of constraints in the inversion of actual field data.
ACKNOWLEDGEMENTS
This research is supported by the Full Waveform Acoustic Logging Consortium at M.I.T.
6-7
. i '
131
132
Garcia and Cheng
REFERENCES
Cheng, C.H., M.N. Toksoz, and MoE. Willis, 1982, Determination of in situ attenuation
from full waveform acoustic logs: J. Geophys. Res., 87, 5477-5484.
Golub, G.H. and V. Pereyra, 1973, The differentiation of pseudo-inverses and
nonlinear least squares problems whose variables separate: SIAM J. Numer. Anal.,
40,413-432.
Parzen, E., 1978, Discussion on histogram splines: J. R. Stat. Soc., B, 40, 1-25.
Toksoz, M.N., C.H. Cheng and M.E. Willis, 1983, Wave types in a borehole -- a review:
M.I.T. Full Waveform Acoustic Logging Consortium Annual Report, 19S3.
Tsang L. and D. Rader, 1979, Numerical evaluation of the transient acoustic waveform
due to a point source in a fluid·filled borehole: Geophysics, 44, 1706-1720.
Willis, M.E., 19S3, Seismic velocity and attenuation from full waveform acou.stic iogs:
Ph.D. Thesis, Massachusetts Institute of Technoiogy, Cambridge, Massachusetts.
Willis, M.E. and M.N. Toksoz, 1983, Automatic P and S velocity determination from full
waveform digital acoustic logs: Geophysics, 48, 1631·1644.
Yakowitz, S.J., 1985, Nonparametric estimation, prediction and regression for Markow
sequences: J. Am. Stat. Assoc., 80, 215-221.
6-8
Full Waveform Inversion
133
APPENDIX
The multi-receiver forward model
For a tool with NT receivers. the observed set of waveforms at each receiver
location z,.zZ.....zN Is modelled as
T
(A-1 )
In matrix form the model of the pressure field for an NT receiver tool is
pea)
= G(e)
(A-2)
X(O
where:
pea)
= nP(CVi,Zj.a).
G(a)
=[(G(z,.a)•.. ,
i=1 .....N,,; j =1 •...,NT
nT
G(Zj.a)•... ,G(zN ,e)]T
T
with
G(Zj.a)
=DiagnG(cvi,Zj.a),
i=1, ....
N"n
N"n T
X(~") = nX(cvi'~)' i=1 •... •
As shown in Figure 3. the matrix approximation for NT receivers is obtained by
algebraically "stacking" NT models. one for each source-receiver separation, Zj' The
transfer matrix G( a) is obtained by also stacking the individual transfer matrices.
The vector pea) has length (Nw . NT) • Matrix G has dimensions (NT xN) and the
source vector has length N ,,'
The multi-source!multi-receiver forward model
When the tool has an array of Ns diffecent sources represented by complex
vectors X, • Xz • ",X N (the Fourier spectra of each source function). a new matrix
s
approximation is obtained by stacking Ns models as follows (see also Figure 4):
For each source indexed by u the model of the pressure field for Nr receivers is
(A-3)
where
with
6-9
Garcia and Cheng
134
and z}u) is the distance between source (u) and receiver (j).
The ."I. models of equation (A-3) "see" the same
one equation:
p(e,O
e and can be aggregated intc
= G(e) X(O
(A-4)
We can re-organize the equations such that at each receiver (u) the ."I.
waveforms are stacked. One now has the ."I. equations:
(A-5)
with the structure shown in Figure 4. This can be written in one matrix form:
(A-6)
(
including all the receivers amounts to stacking the matrix equations above Nr times
into one equation:
pee)
=G(e)
X(O
(A-71
where G now has the structure shown in Figure 4. G depends on a vector parameter
e, and on the unknown source vector X.
The separable inversion method
The function to minimize, the residual energy, is a sum of squares of functions.
The frequency domain formulation reveals that the problem has more structure than
the classical nonlinear least squares because the source vector X appears linearly in
the functional E:
E(6,X)
= I/w-G(e) XI/ 2 ,
(A-8)
The approach to the solution of
is to modify the residual energy functional E( a,X) such that consideration of the
unknown source parameter X is deferred. The variables
and X are separated so
that in each iteration the solution for 6 is obtained first and X is obtained in ONE
STEP after that.
e
6-10
Full Waveform Inversion
135
For any fixed @ , the problem of finding the optimal source vector X
corresponding to that 19 is a linear least squares problem. The solution is found by:
~
(A-9)
where G+(e)is the pseudo-inverse of Gat
e =e
Define now a new functional E 2 (0) by
(A-10)
One sees now that E 2 depends on 19 only and can be minimized with respect to 0 by
means of a Gauss-Newton algorithm to give an optimal
So, if there is no a priori
knowledge, the separable method is
1) mini I (I-G(0) G+(e))WI1 2 gives e
e.
9
Golub and Pereyra (1973) had shown that this two-step method is equivalent to
the direct inversion, provided G(e) has a constant rank around
By taking
advantage of this structure, one does not have to minimize the residual energy E
with respect to the source parameters X and so the size of the nonlinear problem has
been reduced.
e.
6-11
Garcia and Cheng
136
sources
__ "s (~)
Receivers
(
Model (6)
Source ( f)
Figure 1: The Cylindrical Fluid-Filled Borehole Model.
6-12
Full Waveform Inversion
137
one receiver,one source
S
Vector-Matrix form:
.Borehole + Formation
FFT Data
Source
Figure 2: The matrix approximation structure for a one source, one receiver tool.
6-13
138
Garcia and Cheng
Multiple Receivers (Nrol them)
p = G(e)·S(~) + N
r
N = (~1'~2 .... ~N)
Figure 3: The multireceiver model.
6-14
Full Waveform Inversion
139
Multiple sources (Nrof them)
Multiple Receivers
eNs of them)
Sources
vector
Data
p=G(e).s(~)
Figure 4: The multisource, multireceiver model.
6-15
+N
Garcia and Cheng
140
MEASUREMENT SPACES
(kz,t)
!)~/'
..
",
(z,t)
/"f/
.... ,
'Unitary
'"
.
Transforms
,,
,,
,,
,
~
\l
c.,-<"
(z,c.J)
THE INVERSE PROBLEM
Measurement Space
Parameter Space
,
Estimation Theory
1-"----1
OptU:zation
--;;:t'
I.. . ----:~~:
Figure 5: The unitary transform between the 4 measurement spaces.
6-16
~.~
Full Waveform Inversion
141
~
~-.-------,..---,..-----,..----.,
~
~
~ -'
~ - :
-: : -,-:- """! -:- -;-
-; -
-; -
- - --
-
II p _.- -
- -
w~ J~
S . G( V p)
- -
-
-; -
-
... :- ... -';_
~
-;
---
- -
... -
- -, -
;..
..
;-
-----;-~~-~--
;-
-~-
- .
....:, - , -:_:_ .... _ ..... _:_1..._:_
'-
~
- : - -0 -
-, - - -
-,-
~
~
-,---~~-r-;-
__ :_ ~ _L.
j_~
_:_
- -: - - -: - .... -:- - -:- - - :- ..... - t-
-i-
;
,.
,.:;,;;,
--;----~--------~-~_.-
- - - - - - -' - - - - - - - _:.... ....: - ;... -.- - - - ~ -, - - -,- ~ -:- ~ - ~ ~ - ~ -;..;
a..
.- -
- - - - -. - - - - - :- -: - _.- _:.._'- - - -'-": -'-....:- ~ -:- - -- - - -,- - - : - - -:-
-:
;-
.~
~
-:-
~
~
~
--~-~~------'---~~-~-'-
gl-~.oo
4.20
3.80
5.00
4.60
VP
~.,..-----------.,.---.,.-.,.-------,
-
- ,- -; -
:
-: -
.... -
-
~
-:- :
-:
~
-
r
-,-
--------------~f----c--
II
IIp-s.G(Vp )
-=====:=
- :... - -,- - :- - - - _.--------------------_.'-
~
~
-:
,
"
-- - - - -- - - - - :...
..;.
-,
-
- -
---
.:..
-'-
;-. ~
' , , '
-
-:-'
- - '-
..;
-
'-
- '-
-'
- :.. -' -
....: -
- - -
t.:l
X
(flO
o
'0
a.. '"
o
o
+-----,----,-.,:",.-""'-,....---.,.-----1
~.oo
J.40
4.20
3.80
4.60
5.00
VP
Figure 6: The one source - one receiver (15-1 R) model residual energy as a function
of
y;,.
6-17
Garcia and Cheng
142
0
0
0
LI>
(\J
-j
,
'"0 0
-~
XO
--'
0
-,
(\J
-
-
~
P
-
-.
-;-
~
-
-'
-0
(JJLI>
-;
>-
--'
r.=:
-
~
-
L
- ,
,
- ,- -+ -
-
-,-,-
-i
Xo
G( Vs )
,-
~
""
-<
.,. - ;- ..,
;- -;
- '- --'
-
-
- ; - , - - , - ;-, - ;- -,- +
!
- , , '1"
;.. -I
-"
,- ""1'
Ir 1
...;..--
~
S
'0
_0
'-
--
-
-
~
- , - t-
-,-
-!' _.
--;
-
!
!
-r-
+-. -f,
1
'- -! - i-
-1-
,.
-~-
,
i-
,
-t-
~
-
--;
~O
""
Do
--'
0: 0
-;
ZLI>
- ,.. -,- .,.
c
- - - c
~
-'
""
'-
-'
-
~
-i
-
-,
-
'- --' -
-'
-,-
-,..
!
,
,
!..
_i_
~
- ;-
~
-,
0
0
0
("-!-
l.._1._..
~'
'- .J oj
r- -j j .., - '- --' ,
, ,- - -;
, -; ~
0
I 0
ll.
r- 4 -
L,
!
;- -;
T
'- - -
(JJ~
;
.
1 00
.
1 40
.
1 80
2. 20
2. 60
VS
Figure 7: 18-1 R model residual energy as a function of
6-18
10;,.
3 • 00
,
f
Full Waveform Inversion
o
o
•
o
U"l.,..-....,.........,
l'\J
"'0
0
143
....,........,...-,.._......,......,......,....,..._......,......,....,.......,..._......,....,...--,
! ! !
!- "1 "r" '-j" ··t···-·j·- ·t"
. . J . _.. L.-1.- . 1.. _.1__ J.. _.L . J L.. _L_ . 1 . _.1._.. J..
. . _["- 'I ··-t- ·,t" ---1- '"t
-c:
1
;
!:
,
,
ii
,
j
,
"r- "-r- "T '-'r- .._;- -"r "-r "; "-1
:1:0
. .L.._1._ . 1...
i 1 j
··-t·_··+··
! i
o
N
"T"
,
s
p
.+.. _.I-. .+
.J
i
.-;
!
!,
,
"T"'-T-"';'
Iii
I
i
i
i
G( Vf
2
)
.
o
(.f)0
o
0...
i
!
i
... _'..
;
i
i
. ._+_.+.i
1
1
i
i
!
. .-.f.. _..t-" ._~.- .
1
! !
· ·..T-··!"·-r-.,
. . -!.:
.-j'
-l-.
o
o
t
;
,
:
~
!
...;.
i
! ! i
1
;
,
'_
,
i
-.I.
.L. '_~"_" i . _.!,_ . 1. _, !__,.J..
!
I! 1 i
·f-··_(--t -1- -t- -'i---t·
i
I
.1-, _ ' . _
-toi .-. i1__
j
.:...
j
...l.
i
!
i i
f- ~
+_. -[ -". +- ---1 _.-+ -"i-··
!
1 l
! , !
i
!- I" -'r.. _! '-"T -T- "1' '-"i-"
1-~. - . i- -!
+- . _.j _. -1,
i
;
!
I
;
!
;
!
1
T'-T-
r'l
:-~
-1-.-1
t,
i-.. J
J.
I
i
t- . -t.
J
.;.. _;i
l
l
~
j, 50
VF
1. 60
Figure 8: 1S-1 R model residual energy as a function of V .
f
6-19
..
!
;
+-,..
, _!._...
,
l
!
.t- ·-t·i
~
..
!
. _+._. +-..
_+_..
. ---"r"
r
1
. --f.. _.. t-. --f--"
L- . --l ,L..-+._.
, ,
,;"" :
;-.--;
I
j
-i-
I
j,1.J0
;
,~,
I
!
!' -'r"J'
i
:- -r- T
1
,
,
~
-+,
-~,-
.J...
,
·T··-1'-"·
,
;
f- .-: .+- . _,[._ .
i ~ . L._l._ .
j_.~
i __ L .. J.
- t - "t
!
"'~"-"T""'
i
,
;
...--;..- ..t- ··-t·_···
~."!
:
-r- i --t-"-i
;
..1- ..
,
t-···-l.. _·· t· --·1·-- .-t. --··1- -f
1
i
l ........i
:
1 ··-t-.. t· -"i-' l
"!"-r-!,
!
! -·r..··!
I
. . --t..- .'t-'
-;-,..--;..- .+- ·-r-···
L.
. J . _; i
! i "r-- ·-t·_··,
~
._,~-,
r.=J
:1:0
'-1 ·_·.. t '-1"'-'"
L .. J . _...L ._1._..
~
! 1 !
."~
1. 70
t
-1-.. _1._..
:
y-·-f·_..
! !
1.80
144
Garcia and Cheng
o
o
:;;,---.-----~--O:___:_7""'"'7'""_:_~"'"'7"'"""':"_;___:__:__::__r
N
!
j
.4,
_:_
-l
I- _:_
. . _1
p-s G(
'_
..... -
~""""t
Qp)
,, ,, ,
.-f. _., j-... -f-'
~
r-
,
,
,
..-
;-
,
~
Il.
- i- -+- - 'f- -+- -1- +
,
i
- , , - ! , -ii - I - i-- I
i. - !- i.
I - i.- -I - -i- ., I, .j. - ~, -r-1-' -1, , _.L-J.._.'
, _!.l.
,
J
! !
.i.
~
-
~
,
- ,
,
i-
!
,
~"1'
,
.-
-r _. -
-'
-,;-
,
~
i
, ,
'i
-
_. t-tI
I
~' -t·_·
!
-'---: : : -r-'
L _L_
_.
j-1- t _.[- .... j-"
, ,
,
1 -'- - ~-'
, , i
~
~
i !
i i
~-:- ... -i-- - ,- ..;
',
,
,
,
;
..;
-f-i--~-+-;- , I
,
, ! 1 ! !
!- -i !-. -!
1- -i- +- -! - -+ - [- oj - '!.
f , - r , , - I -; - ,:
!
,
, , .l. ,
- - - j
'I
,
_.:_.~.
-;- i
,
--1··
r-'~'
iii
;
,
~. -t -
I
_. r- -r- t
-t _
. . . _1._ ...
- ""
-. - - - .-.- -_.-.+- !
-,-1 -
_. t' -t-·
..J_:"._1._
,
,
,
~
90.00
QP
-t
- l,
-'- ..l.,
i
130.00
Figure 9: 1 8-1 R model residual energy as a function of Qp'
6-20
;.. -
-
t- -!.l.
r
-,
-+
,..
, - '-
-
-.
-i= ~ =~-~
- r
\
_ l... _;_
170.00
210.00
Full Waveform Inversion
145
o
o
o
0....-....,.......,..._.,..-....,.......,...__.,..-....,.......,...---,_.,.-.,.-__....,.......,.......,.......,..._....,......,.......,
i
!
. .-t..- . t"" .t· ! i -:-"-; 't-,··_!,·_..'t···-t-···t
i···_..·t- '-i" -t ·-t-_..·
. . J . _.. L.. _1._.. 1. _ 1_ 1. _..L J i
.1.. _1_.. J..
.J . L.--1.. _ . L--1.-..
!
i [
! ! i
!
!
iii i i
.._+._.+ ··-f·- . -,-- -r- T ,r"
...;......_.;._...-.;..
'
"
"1
"-"r "-1"'- '0; "-r-'"
:
.L..J . _ j __1._..
. .J . _..1.. . _1._.1 _l_. J..._..L ..
! ._J._
i ! ! i i
,or i
i
j
! 'r i
i i i
.
.
.
~
. .-;.._ . +- ··-f·- .+. -;-" -+ .f-..
!
!
! ! i
r
-t-··-f·_···
! !
0
~o
"'-1"- T "-r-'
~o
",-!.,
0.
0:0
0:(0
t:l
:K
(1')0
o
10
i
~"
!
;
;
~.
!
;
... _'.
i
:::l'
Cl..
!!
!
1
~·-t·
f
!
''-''
1 [
!
T
;
..";"'..
i
-.-
!!
f..··-f·- -+.
1
i
-!-
!
1
-+
!
!
i
"r- o'-i
0
0
._.L_.
..L, .-!.-
!
.t-.
i
'-'"1- "-r . ,!
,
0
_ ..!-. ....1..
L..
!
P
--i
i,- ...;,
!
!
S
_+.
1-. ~ .. . f- .. .. .. ._.!.__.+..
;
i !
i ! j
.!
"-'j-' t·
'
'
T
'-r'-'
!
.
...;
,-L
~
1- _.l.-. -1..!
!
!
! i
! -! t ! i
L.. J L ._1 1, _.i_.. 1..
! ! ! ! !
:- i
i- -1 T -'1'- -r
_~
:
o
._~.
0
0.13
i
0.17
RADIUS
Figure 10: 15-1 R model residual energy as a function of R.
6-21
"-r-'"
o
,
!
--1'-- . L
~ "-1'-'"
!
"J.. _ . t-.. -!._..
!
! i
! ! 1
. _+_.
+ .._ ..j- . -+. +- -+.- ·+··-t·_..·
- j.._.
i
0.09
,
-i
z
;
.. --!
;
1
o
o
"'---1
~
i
-i- -,-
!
!
i
!
r,'
...j
J.
t· -to ··t -t- i
;
. 1..
!
..-i.
!
J.
.:
9J.05
.!
i
.....
!
!
..--1
i
..--+,
!
;
~
G(r)
. -f
"":
--i
i,
...l.
i
'i'
!
.. ~
.~.
. +- ..-i·_..
;
! i i
T- .....1"-. '-r-'"
.l. j . - . L . -1._..
! ! ! !
-! t"
,--.l._.
L. _1._.
L
i i Ii l
r'
0
r-'
" -r-'T"-r-'
!
o
0.21
0.25
Garcia and Cheng
146
o
~~,-.,---,-,---,-,-,--,-,,-.,-,--:--.,......,
-: - !'" -; - to - , - - - ,- ..... - .- -; -
~; t= =
~
1" - ; -
+ -
i - -t - : - -I -
f 1~
1"" - ; -
trt
= =:= ==!= ==C1= =l = =1= =!= ·· =
=,=.
- .
........
. - _._.-
~=~=:===llp-s.G(p)
'b
:::5 ____
00
:I:u>
0:-
._~_
- -.; - c -: - - - ,-
~
0
-
~~
-
;.. -: -
II
, .
, .
,
·, .
.,
~_:........;-;.._:-
~_!--i_i.._i_
;- - : - ., - : -
~
-
r- -i -
~ -:-
-,- ... -;-""'! - t- -!-.,. -;-"'!" -!- "": - r- -: - t'"
;
:
: ,
:
--..- - :- , - ' -'-.-'- : -'-~-
xo
;
010
~
o
~
~
~
;
~
~
;
;
~
~
10
___ . . . _ ... _:_ ... _:_
a...
~
_:_ ..... _;... ....
_L_~_
:
'
~'
:
-~~-~~-~-i-~-~~-~~-~-!"
:I: 0
0:0
00
-i - ----.---.--
:_ ~ _,_ .,:
Zu>
ck.oo
-; -
~
.. - : - '7 - ,- "'7 - : - --: -
-: -
;'" - : -
"- - -:- . . . - ~ -:- ~ -;- 7-:- ~ - ~,- ~ -i.
, .. , . . . " " ,
-----------"----------_.-
I- -; - o
0-l-
- -
--.
.------~~---.-.--"":-------
...;-...:.-_..:.-.,.....;.....--'-;._....;.....-l
2,10
2.20
RHO
2.30
2.40
2.50
:i\,---------------.,--....,-....,--,
-, -,.. - - r -.-'"
-:-"""'! - ... -;-:- -,- -: -;-
'1-~ ~
- : -;-
.
..
----------,-----------, . , . , ..
,
. ,.
.",
"."
.
"",
-~--~---,-~----.---~~-~-'-
--
.;
- .- -: -
;..
-:-
---'-'--'-'-
----'-:~
- _.-
.;
_:....
-
-
(
:.. _:-
~---:----.-.-~------,-
x
.-
010
o
10
a... '"
~
-'
- -._. -"~
,
,
--~--------
,.
.
-------
-.-....;
_...
~---,-
"2.00
2.10
2.20
RHO
2.30
2.40
Figure 11: 1 S-l R model residual energy as a function of Pb'
6-22
2.50
Full Waveform Inversion
147
0
0
0
,
In
N
..
r-
--;
;
0 0
-~
x0
0
N
_0
_0
<.::lo
x In
a3:
~o
..., - - - ,.. -,
_.
-! - i..
-, - ,..
_.
-' ~
--j
-;
(f)0
-;.!.
.:.
-
...j
::l:0
0: 0
Oo
Zin
-
~
-'
-
~
t· -i - 1"
- -- -
t -i-' """!
1 _.1- ;
_.i -',-! -i!
-,-- "'!
--r-
t'
2
-;
--'
C-
_. -
- i! ! ,
- --'
,
;
,~ -,- -. - ;-
r-
-1 - t
f-
-'
,
,
;- -oj
~
~
- ;-
-
-;
-
,
-i-
,
,-
~
!
,
!
_. !
-
-
,..
,-
'-;
'-
;- -~-
,
'-
+ -1- -j
-
_.'-
;
+-
,.
,
,
,
,
t
-t-
;
;
~
!
L -.-
, ;
-!' -
1
r-
i- -1 _.
i
- f-
i _.i_..
i
'- -, - '- -,- -,.
,
- - ;,
~
--'
'-
L.
! ....... _.
1-"
! !
!--+.. -
;
,
p - S . G( CJ o )
,
, - ,
- ,.. .., c- , - ,
- t -i- t
,-
0-
,
~
0
I 0
- ;- """!
,
;
-t , - ,
T
;
-;
'-
t
+-
-l-
r-
- - - - -:-
:- -,- -'.,.
- - - .,.
-: -
,.. -,
-' - '-
-'
'-
-
0
0
95. 00
7. 00
9. 00
1 1 00
CEN TER F REQUE NC Y
1 3. 00
15. 00
W0
Figure 12: 18-1 R model residual energy as a function of wo ' the source center frequency.
6-23
148
Garcia and Cheng
0
0
0
0
0
-
j-..._ ...t-..._ ... -j!
II>
X
.i
.1.
;
-t.
I
"T
'r'
. . _.. L .L
0
0
co
1
;
.. ~
;
.l..
1.
ot-
..l.
i
t-
+1
J' ·"t
!
0
0
co
.L. . .L
+-.! . t·i
W
u
zo
a:o
;
w~
~.
i;
c-
W:::I'
I
~
;
i
::I:
+!
•..i....
I
.1.
i
··t·;
'i
,
i
j
I
I
~.
'ri ..!.
zo
;
C\l
i
j-
L
. . _.. t-.i
0
0
i
1 00
0
.-+
,
!
=r=( V
!
P - S G( Vs )
1
( Vf ) i- "I! "I 1"_..1-. .L.
.J. -!._. -1··_·
,;
i
i ( Pb )
-'i- '-"1- f-"
-j
-1"-" -1"-'
i
;
~
;
i;
-;
,
;
~
~
..-
;
'-1"I
-i
..;.;
I
I
-+
~
I
I
I
..:
i;
;
I
;
;
i
-0
'-'!-
,_1 .. _ .
i
-,; -
:
;
._.1._.
!
!
!
I
I
i;
t
I
i
,-1"-,, -.j._. -,,1- ._.. j'-i"-'
;
i
I;
,
i
-1--
-1,,-- -: -
,
i
; _.. i_
7 00
0
.
1a 00
I TERA T I CJ N S TEP
Figure 13: The residual energy reduction during a 4 parameter
sion.
6-24
;...
I
;
...
;
.;..
!j-
,L
i
.,
'Y-'
;
!
13 00
0
(v."
.~
.+- .f! I
TO'
, 'I'
L
i
-·f-.. --i-
I
I
t-
-i--
;
!
TO,
,; .
i
I
;
I
'-'1'-'
r';
+-: +-;
i
;
-_.!-_.
..L.
;
;
~
;
-
,~
.1--...- .;...
._+_. -1"-- --!.- . _..!- -_.-!;
I
i
~
i
'-",-
I
.~.
i
!
1 '-r- '-j'-' -l- '-"f-' -'i- r~
-\.- ,-1.-. -.1--. -·l- -·1-. "i
i
!
-[-- -"i-1 _. -1 -
"T"
00
;,...
._.~-
r
.,...,;
't t··
;
;
..l.
I
l-,
I
I
0
;
i
._!..-
-~ ..
lj
t-
r--j-
._L_.. _.1._. _.1_.. _..J_ ;,... +,.. .1..
,,
!
I
I
!
:
i
i
._"!""-1
-[,._.
·_··f- '-"1- 'r- +_. 'r
-r'-;
;
I
i.
i
.J i"I '1"-'
!-_. L L ._.. 1.... _..
1I
i
1
i
-t,. .~ -i._
p )
+-.
t2 i !
..1
.-:i
...;.
t
1
,.;
1--1- --j"-"'-,],- '-"1-"'-"1-
_.+
1
.J.
...
0
0
~
.~
I;
...... 0
1
-1.
1;
I
+
:
..._ ...1-
0
W
0
1
i
i
i ._.. -.;.-i ....'f I
0
0
I
i;
'1
~
.L
!
°0
'-'0
!
i
t
.;...
.
,1
;
;
1 6 00
0
v., vf ' p) inver-
Full Waveform Inversion
o
"''"
N,.------------------,
~:.,-------f..\----,--------,--.
~-~-~--i--i--~-~-r-~.. ., ., ,. ., ., ., .,
-
-:-
-:- -:- -:- -:-
-;-
-:-
-
...
~
M
_
...
~-;'i--!--i-
__
r:-..~,"--
o
o
.;
-
_. __.
-';"'-
- -:- -;- -;- -!- -i- -:- - i--;"-
o
o
149
_
---------~-~&-+-------
N
_l-.-:-_i-_i-_
.
----------_._-~-------,..-
-~-~-~---~--,------~-~-~-
- ':"
___
~
-
-;
:..._~_~
- --- ~
-:
~
-
-;-
- - -.- -
__ __ .__
~
;-
-
,... -
-
;-
_
~_~_L
------------------.--:--~---,...---
-
- - - - "-" -
- -
__
.... - - - - - --
~_-_~_~_~_--_.
'"> --~-~---~--_~_--~--~--.-_~_--;---- - .,. - - - - - - - - - - - - -'- - ... - - - --
~-7---~-i-~-~--i--i--~-~-t-t--~-~_"'-~----'--'--l----~-~_~-
...
o
o
..;
----- -- --
;-
_:... _.<. -
-' -
-- - -
... -
-
.,. -. .,
-,
-
-;
...
-,-
-;-
- ,- -.- -
-..: -
-,- -:- -:- -'- -
-
-:
-'.,
... -,- - ,- - -
;-
...
:-
'-
...
In
-'
'-
- ......... -
-----_~_~_~_--_._---------
-_~_~_-'--------_:..._-Q
o:-I-__~--~--'--.;_--_;....--'-___l
4.00
7.00
!O.OO
13.00
ITERATION STEP
o
o
... ;- - -'- ... - -:- - -: -
-1.00
...
;--;--~-t
....... -
'- -
... -
....
...
7,00
10.00
13.00
ITERRTION STEP
15.00
-:.,.----,---,------,----~
_ ;.. _ ~ _ ~ _ i _ i _ -: _
o
-:--i--~--!--;--;....-:'_-;...-;-
~
11.00
o
o
o
-----------~-~-1--:--j--r-~-L-L... ,.. - ;:.' - .,. - - ... .., -
-------.
0-l---'----.----,-__---r--'_ _.;_--___l
16.00
"'"':,.----------------~
'"
N
.'.
---;--;---_._-,--;--;----
-;--:---:---
'"
'- -:.. ... : ' -
-,....
o
~.OO
._--_~_~-
_._~_
-
'"
- ..; ... -:- -:- -i- ... ;- ... !- ., i- ... ,.. -
;.. - :- -
':" -
j __
!__:__ :__ _:. . _:- _ _
~
~
; - ..; - ...; - -: - -; - - j - -j- -;- - :- - :- - ;.. -
- .... - ..: - "1 -
- - - -,..
-i - -: - -:- -:- -:- -;- - . . - . . - -, -:- "- - ,.. - .,... ~
-~-
-:-
~-i-i-~--i--i--~-r-;--T-
_-l
...
-i
_.j ... _: __ ~ __ ' __ ~ __ ;_ _ ,- ...
L_
o
o
lU~
....
a:
a::
-,"'
.............
C
:I:
a::
----_~_~-~-~-~-_._-,--~--~-~_~_--
"'
'"
- - -:- - -:- - . - - - - ... - .., ':"
~
-;- -:-
-,-
-;-
--
__
- -. - -: - -,- - ,- -
-~-:.._~_~-~--_~-~-_.
.,
-;-
- ~ - - _...: - - - -'. -:- -!-
-;-
o
o
-:--
~_~_~-:.._:..-
_~-
,-
- ,.
- ,... -
-t.OO
-1.00
10.00
ITERATION STEP
_. - - - - - -'-,.. '- - -
-;..
-
_:- -,- -:...-
o
7.00
~--"----;----------
..;
g:+:-:----"'T--,-.,..-----.:...---T~---~'----.:..---_T',__- .,..-.,__---.:..---_T.----~'------C _q.oO
..: - -!- -"- -;- -;- -;- -'- _:... -;--
""0
........................... -~~_'-"
...,;,. - - - - - -- - -i- -i- -:- - '- - - _:..-:- -:-
-~-~----;--~-~-~---:--
lU o
__ ..,. ... .;. ... ....: ... ...; ... ..: _ ......... ; ... _:_ ... i...... ;... _ :.. ... :. _
13.00
16.00
0:1-....:..---'_~-'--....:..--...,-'---~....:..-'---.;_--!j.OO
7.00
Figure 14. Residual energy reduction individual contributions
6-25
10.00
ITERRTION STEP
0:;', y", VI ,p).
13.00
16.00
150
Garcia and Cheng
-
....
1.e
1
-
0-
e.•
f\
1\
e.e
-2l!l~-
-<.S
-.. a~9-f-·-~-1 ._•. e
e.,
.,
"""-r-r-- i
e.•
e.e
e••
1.e
I
Le
l
7
e.s
I
e.e
.L.i•••
I
•••
I
e.•
I
e.•
I
e.e
I
I.'
, .e
e.s
II
'.e
-a.s
-<.s
-, .•.:j..---r--,--T.-'-'-,-~-"-~-T--,---j
a.a
if.iI
e..
t..
e.'
1.'
-I.il.+_~_ 1- --·-"'-T--'-l·-r·--,,-...,.-~
0..
a.,i
It...
e..
t.,
l..
Figure 15. The time series evoiution during inversion.
5-26
(
Full Waveform Inversion
151
. ..,....--------------,
-..,...---------------,
sa
1M
..
,
••
;
!
•
,
-r-r-rrr-Tl-T"T""f!T'-TT" TTTTTT"
•
,.
IS
e.
-
.,
--,,-,-,-j,.
I
I
'U~~TTTrPTTTTT
.
.. ~
•
,
"
eo
,.
.. .
-...
7
/
"
-
e..
...
•
J'
,.
•
,
..
rrTT
Figure 16. The spectral density evolution during inversion.
6-27
,.
II
..~
•
,.
15
eo
.
JO
,.
152