Elastic wavefield tomography with probabilistic ......9 Seismic data interpretation is a crucial...
Transcript of Elastic wavefield tomography with probabilistic ......9 Seismic data interpretation is a crucial...
Elastic wavefield tomography with probabilistic petrophysical model
constraints1
Odette Aragao & Paul Sava2
Center for Wave Phenomena3
Colorado School of Mines4
5
(January 15, 2019)6
GEO-2000-00007
Running head: Tomography with probabilistic model constraints8
ABSTRACT
Seismic data interpretation is a crucial step for understanding subsurface geology. Without a rea-9
sonably accurate subsurface model, image interpretation is erroneous, which can be costly. Seismic10
exploration and characterization often use elastic wavefield tomography to derive models of subsur-11
face properties. Unconstrained multiparameter inversion may lead to model component updates that12
do not represent real lithology for independently derived parameters. Incorporating petrophysical13
information, such as that provided by well logs, during inversion constrains the inverted models14
and avoids implausible models. To facilitate constrained inversion, we develop a method for elastic15
wavefield tomography that explicitly imposes petrophysical restrictions in order to update models to16
have feasible lithology, i.e., have model components that are consistent with the underlying petro-17
physics. We enforce a feasible region with a model probability density function calculated without18
user-defined parameters. Inside this feasible region, the inverted models do not need to obey a19
specific trend, i.e., we do not link the parameters with explicit and potentially inaccurate petrophys-20
1
ical relations. We demonstrate through elastic models that incorporating probabilistic petrophysical21
constraints into the inversion objective function leads to reliable models that are superior to models22
obtained either without constraints or with approximate analytic constraints.23
2
INTRODUCTION
One of the greatest challenges in seismic exploration consists of estimating accurate subsurface24
elastic models. Seismic tomography is becoming a standard technique for subsurface imaging at25
various scales, and aims to build high-resolution models of the physical parameters underlying26
wave propagation. The main approaches to seismic tomography use either traveltimes (Woodward,27
1992; Schuster and Quintus-Bosz, 1993; Taillandier et al., 2009) or wavefields (Tarantola, 1984;28
Mora, 1989; Biondi, 2006; Duan and Sava, 2016; Diaz and Sava, 2017). Wavefield tomography29
has advantages over traveltime tomography because it recovers parameters with higher resolution30
(Tarantola, 1986; Pratt, 1999; Operto et al., 2013) by exploiting both the kinematics and dynamics31
of the observed waveforms.32
Full waveform inversion (FWI) is among the most promising techniques in the wavefield tomog-33
raphy category as it delivers highly accurate subsurface models from seismic data (Tarantola, 1984;34
Pratt, 1999; Sirgue et al., 2004; Virieux and Operto, 2009). FWI operates by iteratively updating35
model parameters based on the mismatch between observed and simulated data. This optimization36
exploits seismic wavefield modeling (the forward problem) and solves the associated inverse prob-37
lem by minimizing an objective function. The gradient of the objective function estimates model38
perturbations that progressively decrease the data residual. Conventional FWI implementations39
use the adjoint state method (Plessix, 2006) to compute the objective function gradient, since this40
method is among the most efficient and requires the least amount of storage.41
Early implementations of FWI used the acoustic wave equation (Tarantola, 1984; Pratt, 1990;42
Pratt et al., 1996). However, acoustic FWI does not take into account elastic effects and the presence43
of S waves in the data. Several parameters are necessary for describing elastic models (e.g., density44
ρ and Lame parameters λ and µ, or density and P- and S-wave velocities). Therefore, elastic FWI45
3
(EFWI) better describes the subsurface properties (Tarantola, 1988; Pratt, 1990; Plessix, 2006).46
Despite its popularity, FWI is hampered by several challenges undermining its practical appli-47
cation. FWI is a highly non-linear and ill-posed problem, so its convergence is subjected to local48
minima (Symes, 2008; Virieux and Operto, 2009). Therefore, one needs to include prior informa-49
tion, i.e., model regularization, into inversion to find a plausible solution (Tikhonov and Arsenin,50
1977; Tarantola, 2005). Also, the wave extrapolator can become unstable when model updates51
fluctuate, thus impending further updates.52
Elastic FWI suffers from a number of additional practical difficulties. First, using multiple pa-53
rameters increases the degrees of freedom in the model space, thereby increasing the non-linearity of54
the inversion (Operto et al., 2013). In addition, as model parameters are updated simultaneously but55
independently, different model components can be physically contradictory to one-another, creating56
combinations that are lithologically implausible or impossible. This situation may create inaccurate57
and unstable forward solutions and lead to poor convergence (Baumstein, 2013). Moreover, similar58
radiation patterns among various elastic parameters create crosstalk, i.e. different models are viewed59
similarly in data (Operto et al., 2013; Kamath and Tsvankin, 2016) resulting in ambiguous model60
inversion. Radiation pattern analysis can partially correct for crosstalk, but is largely ineffective in61
poorly illuminated areas.62
In order to recover models that are self-consistent and properly characterize the subsurface, we63
need to explicitly impose model constraints during the inversion (Baumstein, 2013; Peters et al.,64
2015; Duan and Sava, 2016; Manukyan et al., 2018; Zhang et al., 2018; Rocha and Sava, 2018).65
These constraints can use petrophysical information, such as that contained in well logs or from66
other sources, as core or basin analysis. Incorporating constraints into the objective function de-67
fines model space bounds, forcing the inverted models to be consistent with known petrophysical68
4
relationships and thus represent realistic lithological units.69
Baumstein (2013) and Peters et al. (2015) use the projection onto convex sets (POCS method)70
to incorporate constraints into acoustic FWI. These constraints impose velocity bounds (box con-71
straints) and specify a minimum smoothness for the model parameters. The updated models must72
be at the intersection of all convex sets, which leads to more stable simulations and geologically73
plausible structure. Manukyan et al. (2018) constrain the inversion by assuming that the elastic pa-74
rameters have structural similarity and thus mitigate parameter trade-off problems and the different75
spatial resolution of elastic parameters. They show that structurally constrained FWI can deliver76
higher-quality models when compared to conventional FWI. However, the assumption of structural77
similarity between different elastic parameters can potentially compromise the final outcome. Zhang78
et al. (2018) use seismic facies as a priori information to constrain the EFWI to derive more reliable79
and higher-resolution isotropic and anisotropic models. They achieve this goal by adding a model80
regularization term to the objective function, related to the facies. As the spatial distribution of the81
facies is unknown, they propose to update the facies at each iteration by using a Bayesian inversion82
workflow.83
Duan and Sava (2016) use a logarithmic penalty function to constrain the elastic wavefield to-84
mography assuming a general linear relationship between the model parameters. Rocha and Sava85
(2018) also use this type of constraint, but in the context of elastic reflection waveform inversion.86
Both examples demonstrate that an explicit petrophysical constraint can yield consistent and plausi-87
ble models even if conventional spatial regularization is not explicitly applied during the inversion.88
However, complex models do have non-linear petrophysical relationships between the elastic pa-89
rameters. Morever, the analytic logarithmic barrier constraint works with user-defined parameters,90
and their selection may be challenging, especially when different lithofacies are present in the same91
area. Thus, it is necessary to define more general and automated petrophysical constraints that92
5
provide flexibility and adapt to the specific geologic situation in the imaged area.93
We propose elastic wavefield tomography with an objective function that constrains the relation-94
ship between elastic parameters using model probability density functions (PDFs). In the following95
sections, we first discuss general aspects of elastic FWI, emphasizing the mechanisms used to in-96
corporate constraints into the inversion. We demonstrate that PDFs can be used in EFWI without97
knowing or defining an analytic relationship among the different elastic parameters. Finally we98
illustrate the features and performance of the proposed method with two synthetic examples.99
ELASTIC WAVEFIELD TOMOGRAPHY
We consider the isotropic elastic wave equation100
ρu− λ[∇(∇ · u)]− µ[∇ ·(∇u +∇uT
)]= f , (1)
where u(e,x, t) is the elastic wavefield, f(e,x, t) is the source function, λ(x) and µ(x) are the101
Lame parameters, ρ(x) is the density and e, x and t are, respectively, the experiment index, space102
coordinates and time. Equation 1 assumes that Lame parameters vary slowly, such that their spatial103
gradient can be neglected.104
Considering us as the source wavefield, the waveform inversion problem is solved by mini-105
mizing an objective function J (us, λ, µ) consisting of three parts: a term that evaluates the misfit106
between the simulated and observed data JD(us, λ, µ); a model regularization term JM (λ, µ); that107
enforces spatial correlation of the model parameters, and a term that enforces petrophysical model108
constraints, JC(λ, µ):109
6
J (us, λ, µ) = JD(us, λ, µ) + JM (λ, µ) + JC(λ, µ). (2)
We can define the data residual110
rD(e,x, t) = Wu(e,x, t)us(e,x, t)− dobs(e,x, t), (3)
and then the data misfit term111
JD(us, λ, µ) =∑e
1
2‖rD(e,x, t)‖2 , (4)
where Wu(e,x, t) are weights that restrict the source wavefield us(e,x, t) to the known receiver112
locations, and dobs(e,x, t) are the observed data.113
To update the model iteratively using a gradient-based method (Tarantola, 1988), one can com-114
pute the gradient of JD with respect to the model parameters λ and µ using the adjoint-state method115
(Plessix, 2006). This method consists of four steps:116
1. Compute the seismic wavefield us(e,x, t) from a source function fs, such that Lus− fs = 0,117
where L is the linear elastic wave operator derived from equation 1,118
2. Compute the adjoint source gs = ∂J /∂us, which exploits the difference between the ob-119
served and simulated data (equation 3),120
3. Compute the adjoint wavefield as(e,x, t) = LTgs, which exploits the adjoint elastic wave121
operator LT ,122
4. Compute the gradient of JD with respect to the parameters λ and µ by123
7
∂JD∂λ∂JD∂µ
=∑e
−∇(∇ · u) ? as
−∇(∇u +∇uT
)? as
, (5)
where the symbol ? represents zero-lag time crosscorrelation.124
Incorporating a model regularization term into the objective function reduces the inversion ill-125
posedness, and accelerates convergence towards the global minimum (Asnaashari et al., 2013; Jun126
et al., 2017; Zhang et al., 2018). Prior information can be estimated from non-seismic data, e.g. well127
logs, or by incorporating knowledge about the model geometry, e.g., smoothness or consistency with128
a migrated image. Considering the model residuals:129
rλ(x) = Wλ(x)(λ(x)− λ(x)), (6)
rµ(x) = Wµ(x)(µ(x)− µ(x)), (7)
the term JM (λ, µ) is130
JM (λ, µ) =1
2‖rλ(x)‖2 +
1
2‖rµ(x)‖2 , (8)
where λ(x) and µ(x) are prior (reference) models. The weighting operators Wλ(x) and Wµ(x)131
are related to the model covariance matrices such that Wλ = C− 1
2λ and Wµ = C
− 12
µ .132
The gradient of JM (λ, µ) with respect to the model parameters λ and µ is133
8
∂JM∂λ∂JM∂µ
=
WTλWλ(λ− λ)
WTµWµ(µ− µ)
(9)
It is apparent from equations 5 and 9 that the model components λ and µ for multiparameter134
inversion are updated independently and thus can become physically inconsistent with each other,135
producing inaccurate models, images, and inaccurate geologic interpretation. Explicity incorporat-136
ing petrophysical constraints improves the quality of the inversion by restricting the updated models137
to a feasible region, resulting in more accurate subsurface characterization.138
Duan and Sava (2016) propose a logarithmic penalty function JL to constrain the inversion to a139
feasible region by enforcing physical relationships between the updated model parameters:140
JL(λ, µ) = −η∑x
[log (hu) + log (hl)] . (10)
The scalar parameter η determines the strength of the constraint term in the objective function, and141
hu and hl are linear functions that define, respectively, upper and lower boundaries of the feasible142
region:143
hu = −λ+ cuβ + bu = 0, (11)
hl = λ− clβ − bl = 0. (12)
The coefficients cu and cl are the slopes of the boundary lines, and bu and bl are their intercepts.144
The distance of a given model parameter in the parameter space {λ, µ} to the boundary lines hu and145
9
hl determines the value of JL, such that this term dominates the objective function when models146
are close to either one of the barriers. This constraint forces the inverted models to move away from147
those boundaries, which act like a barrier that favors model parameters in a given feasible region.148
The gradients of JL with respect to the model parameters λ and µ are149
∂JL∂λ∂JL∂µ
= η
−1
λ− cuµ− bu− 1
λ− clµ− blcu
λ− cuµ− bu+
clλ− clµ− bl
. (13)
The term JL assumes that the relationship between the model parameters λ and µ is linear. There-150
fore, the inversion may not be able to recover complex, nonlinear true physical relationships between151
the model parameters. Alternatively, if we have access to petrophysical information obtained, for152
example, from well logs, we can impose a more general kind of petrophysical constraint. In this153
situation, we can construct a model probability density function (PDF) of the reference well log154
parameters to characterize the interdependence between the elastic model parameters.155
Consider, for example, the crossplot in Figure 1a illustrating a possible petrophysical relation-156
ship between the Lame parameters, λ and µ. The linear barrier constraint JL would not be appro-157
priate for this example, as λ and µ do not share a linear relationship. Instead, we can create a model158
PDF (Figure 1b) based on the available values of λ and µ. For any given model with components159
(λx, µx), represented by the red dot in Figure 1b, we would like to evaluate a model space distance160
to the entire distribution (Figure 2). If this model is close to regions of high probability, we would161
like the value of the petrophysical constraint JP to be small. Otherwise, the value of JP must be162
large and increasing when model parameters separate more from the distribution. To accomplish163
this, we construct a weighted sum of the distances between the model with (λx, µx) components to164
all the models of the PDF. The weight is proportional to the value of the PDF at a specific model165
10
cell and inverse proportional to the distance from the analyzed model to that cell.166
The model space distance from (λx, µx) to a cell of the PDF with indices (i, j) is167
dx(i, j) =
√[λ(i)− λx]2 + [µ(j)− µx]2, (14)
for i = 1 · · ·Nλ and j = 1 · · ·Nµ where Nλ and Nµ are the numbers of discrete intervals defined168
for parameters λ and µ, respectively. Then, the weight from a specific model (λx, µx) to the cell at169
coordinates (i, j) with the probability P (i, j) is170
wx(i, j) =P (i, j)
dx(i, j). (15)
The weighted distance from the point (λx, µx) to the entire distribution is171
Dx =∑i,j
P (i, j)
dx(i, j). (16)
Therefore, each point in the updated model space is connected to all cells of the model PDF, i.e., we172
have constructed the distance from this model to the distribution. Since Dx is small for points that173
are far from regions of high probability, we can define the petrophysical constraint JP as174
JP (λ, µ) = η∑x
1
Dx. (17)
This definition imposes the condition that all points representing inverted models are as close as175
possible to the original petrophysical distribution without being drawn especially to any value of176
the model, but rather close in a statistical sense to the entire distribution. Figure 1c shows the177
11
distribution of the constraint term calculated using equation 17 and the PDF presented in Figure 1b.178
The gradient of JP with respect to the model parameters λ and µ is179
∂JP∂λ∂JP∂µ
= −η
(∑
i,j
P (i, j) [λ(i)− λx]
[dx(i, j)]3
)∑i,j
1
D2x(∑
i,j
P (i, j) [µ(j)− µx]
[dx(i, j)]3
)∑i,j
1
D2x
. (18)
The distance to high probabilities in the model parameter space defines the value of JP , such that180
the gradient of this term dominates in the total gradient of the objective function J if the updated181
models are far from high probability. Otherwise, this term smoothly forces the models away from182
regions of low probability.183
The probabilistic petrophysical model constraint JP addresses many issues related to elastic184
FWI. First, it reduces the non-linearity and ill-posed nature of FWI as it incorporates prior infor-185
mation into the inversion, which reduces the number of model solutions and improves the inversion186
convergence, preventing instabilities. In addition, it avoids geologically implausible earth models187
and reduces the artifacts created by the interparameter crosstalk, as it links the elastic parameters188
during the inversion and restricts them to a feasible region.189
EXAMPLES
We illustrate our EFWI method with two synthetic examples and compare inversions using only the190
data misfit term JD, the data misfit with the logarithmic penalty function JD + JL, and the data191
misfit with the probabilistic petrophysical constraints JD + JP .192
12
Gaussian anomalies model193
The first synthetic example uses a model with negative and positive Gaussian anomalies centered194
at (1.25, 0.75) and (1.25, 1.75) km, respectively (Figure 3). We simulate 20 vertical displacement195
sources in a well at x = 0.1 km and a line of geophones at x = 2.4 km. Figure 3 shows the true196
λ and µ models along with a crossplot of the models. We choose this simple example to show197
how the inversion using data misfit with the logarithmic penalty function fails when the model is198
characterized by two clusters, which represent lithology with different linear relationships between199
parameters λ and µ. We initiate the inversion with constant backgrounds and run all inversions for200
10 iterations.201
Figure 4 shows the recovered λ and µ models using the objective function JD along with a202
crossplot in the λ− µ space. Without imposing petrophysical constraints, the λ and µ values devi-203
ate from the true model as the similar radiation patterns of different model parameter pertubations204
create crosstalk among physical properties. This problem is enlarged due to the limited acquisi-205
tion coverage and different illumination. The inversion does not recover the correct magnitude or206
size of the λ model. Additionally, the µ model has many artifacts due to the crosstalk between207
the elastic parameters. Although a linear relationship between the parameters characterizes the true208
model (Figure 3), the crossplot shown in Figure 4 indicates that the parameters λ and µ are uncor-209
related, which agrees with the fact the data misfit objective function updates them independently,210
thus leading to unphysical models.211
Adding the logarithmic penalty function JL to JD in the objective function, we obtain the λ and212
µ models shown in Figure 5. The lines in the crossplot correspond to the linear boundaries used to213
define the logarithmic penalty functionJL. JL is not flat in the interval between the upper and lower214
boundaries and forces the recovered models towards the middle region, as seen in Figure 5. The215
13
point (10.4, 5.2) GPa, which represents the background models, is not mid-way between the two216
boundaries and, consequently, inversion using JL changes the background parameters (Figure 5).217
Figure 6 shows the recovered λ and µ models using the objective function JD + JP . For218
simplicity, we define in this example the PDF as a Gaussian distribution covering the true model.219
This is not necessary for more complex examples, where the relationship between the parameters is220
non-linear and derived from actual well logs. The lines in the crossplot correspond to the feasible221
region used to define the probabilistic constraint JP . Notice that when we include the probabilistic222
constraint term both anomalies have amplitude and shape closer to the true λ and µ models (Fig-223
ure 3). Imposing the probabilistic constraint confines the model to the feasible region defined by224
the PDF, but within this area the models do not follow any predefined trend, which does not happen225
when we use the logarithmic boundaries. Also, the crossplot shown in Figure 6 has two main linear226
relationships between the inverted models, what conforms to the true models.227
Marmousi 2 model228
The second synthetic example uses a portion of the Marmousi 2 model (Martin et al., 2002). For229
this example, we consider that prior petrophysical data from three well logs are available and we230
use this information to build the petrophysical model contraints. Figure 7 shows the correct λ and231
µ models, the well locations for our experiment and the crossplot of the model parameters. We232
simulate a multicomponent ocean bottom seismic survey (OBS) with a line of receiver at the water233
bottom (z = 0.17 km) and 20 evenly spaced pressure sources located at z = 0.05 km. Figure 8234
shows the crossplot of λ and µ models extrected at the wells located at x = 1.1, 2.0 and 3.0 km235
(Figure 7). We use a 7.5 Hz peak Ricker source wavelet. Figure 9 shows the initial λ and µ models236
and the corresponding crossplot in the model parameter space.237
14
Figure 10 shows the recovered λ and µ models using the objective function JD. The inverted µ238
model has more structural detail than the λ model, as it only depends on density and S-velocities,239
while λ depends on density as well as the P- and S-velocities. The wavelenght of S waves is shorter240
than that of P waves, and therefore the inverted µ model has higher spatial resolution. However, as241
in this example we do not impose petrophysical constraints, the µ model has spurious artifacts due242
to parameter crosstalk. Note that the values of the main structure presented in the initial λ model243
(Figure 9) that conforms to the true λ model (Figure 7) are not preserved during the unconstrained244
inversion (Figure 10). Additionally, the crossplot shown in Figure 10 does not resamble the true245
model (Figure 7) because the updates are not restricted to a feasible petrophysical region.246
Figure 11 shows the recovered λ and µ models using the objective function JD + JL. We247
estimate the upper and lower boundaries of the logarithmic penalty function from the well logs as248
hl = λ − 1.3µ − 0.2 and hu = −λ + 1.3µ + 3.15. We ensure that initial models for the inversion249
(Figure 9) are within the feasible region. As for the case of inversion using the objective function250
JD (Figure 10), the inverted µ model has spurious artifacts. The main structure presented in the251
initial λ model (Figure 9) is not preserved during this kind of constrained inversion (Figure 11). In252
addition, the crossplot shown in Figure 11 indicates that the model samples follow the linear trend253
imposed by the barriers. However this trend does not correspond to the true models (Figure 7).254
Figure 12 shows the recovered λ and µ models using the objective function JD + JP . We255
estimate the PDF presented in the crossplot as the contour lines from the well log data. The PDF256
is estimated from the well log data without user-defined parameters. We also ensure that the initial257
model for inversion (Figure 9) is within the feasible region of this experiment. Note that both λ and258
µ models in Figure 12 present similar resolution and they are better recovered compared with the259
cases when we use the objective functions JD (Figure 10) or JD + JL (Figure 11). Additionally,260
the µ model artifacts are attenuated when we incorporate the probabilistic petrophysical constraints261
15
in the inversion. Also, the crossplot built with the inverted models (Figure 12) is closer to the262
similar crossplot built with the correct model parameters relative to the crossplots obtained for the263
less robust objective functions JD (Figure 10) or JD + JL (Figure 11).264
CONCLUSIONS
We incorporate petrophysical information into the elastic full-waveform inversion, using model265
probability density functions derived from information provided by well logs or from other forms266
of rock physics analysis. We demonstrate that imposing petrophysical constraints in elastic wave-267
field tomography improves the quality of the recovered models, by restricting them to a feasible268
region, and therefore guiding the inversion towards geologically plausible solutions. Two synthetic269
examples show that the models recovered with the proposed method are closer to the true models,270
while maintaining robust and realistic petrophysical relationships between the model parameters.271
This constraint term also helps mitigate the interparameter crosstalk ccreated by similar radiation272
patterns among the elastic parameters. One can also impose model regularization in addition to273
petrophysical model constraints to further enhance the geologic realism of models provided by274
wavefield tomography.275
ACKNOWLEDGMENTS
We acknowledge CAPES (Brazilian Federal Agency for Support and Evaluation of Graduate Ed-276
ucation) for financial support. We thank the sponsors of the Center for Wave Phenomena (CWP)277
for additional support of this research. The synthetic examples in this paper uses the Madagascar278
open-source software package (Fomel et al., 2013) freely available from http://www.ahay.org.279
16
REFERENCES
Asnaashari, A., R. Brossier, S. Garambois, F. Audebert, P. Thore, and J. Virieux, 2013, Regularized280
seismic full waveform inversion with prior model information: Geophysics, 78, R25–R36.281
Baumstein, A., 2013, POCS-based geophysical constraints in multiparameter full wavefield inver-282
sion: Presented at the 75th European Association of Geoscientists and Engineers Conference and283
Exhibition.284
Biondi, B., 2006, 3D seismic imaging: Society of Exploration Geophysics.285
Diaz, E., and P. Sava, 2017, Cascaded wavefield tomography and inversion using extended common-286
image-point gathers: A case study: Geophysics, 82, no. 5, S391–S401.287
Duan, Y., and P. Sava, 2016, Elastic wavefield tomography with physical model constraints: Geo-288
physics, 81, no. 6, R447–R456.289
Fomel, S., P. Sava, I. Vlad, Y. Liu, and V. Bashkardin, 2013, Madagascar: open-source software290
project for multidimensional data analysis and reproducible computational experiments: Journal291
of Open Research Software, 8th ed., DOI: http://dx.doi.org/10.5334/jors.ag.292
Jun, H., J. Kwon, C. Shin, H. Zhou, and M. Cogan, 2017, Regularized LaplaceFourier-Domain Full293
Waveform Inversion Using a Weighted l2 Objective Function: Pure and Applied Geophysics,294
174, no. 3, 955–980.295
Kamath, N., and I. Tsvankin, 2016, Elastic full-waveform inversion for VTI media: methodology296
and sensitivity analysis: Geophysics, 81, no. 2, C53–C68.297
Manukyan, E., H. Maurer, and A. Nuber, 2018, Improvements to elastic full-waveform inversion298
using cross-gradient constraints: Geophysics, 83, no. 2, R105–R115.299
Martin, G., K. Marfut, and S. Larsen, 2002, Marmousi-2, an updated model for the investigation of300
avo in structurally complex areas: Presented at the SEG Technical Program Expanded Abstracts.301
Mora, P., 1989, Inversion = migration + tomography: Geophysics, 54, 1575–1586.302
17
Operto, S., Y. Gholami, V. Prieux, A. Ribodetti, R. Brossier, L. Metivier, and J. Virieux, 2013, A303
guided tour of multiparameter full-waveform inversion with multicomponent data: from theory304
to practice: The Leading Edge, 32, no. 9, 1040–1054.305
Peters, B., Z. Fang, B. R. Smithyman, and F. J. Herrmann, 2015, Regularizing waveform inversion306
by projections onto intersections of convex sets: Presented at the 77th European Association of307
Geoscientists and Engineers Conference and Exhibition.308
Plessix, R. E., 2006, A review of the adjoint-state method for computing the gradient of a functional309
with geophysical applications: Geophysical Journal International, 167, no. 2, 495–503.310
Pratt, R. G., 1990, Inverse theory applied to multi-source cross-hole tomography: Geophysical311
Prospecting, 38, 311–329.312
——–, 1999, Seismic waveform inversion in the frequency domain. Part 1: Theory and verification313
in a physical scale model: Geophysics, 64, 888–901.314
Pratt, R. G., Z. M. Williamson, and M. Warner, 1996, Two-dimensional velocity model from wide-315
angle seismic data by wavefield inversion: Geophysical Journal International, 124, 323–340.316
Rocha, D., and P. Sava, 2018, Elastic reflection waveform inversion with petrophysical model con-317
strains: Presented at the SEG Technical Program Expanded Abstracts.318
Schuster, G. T., and A. Quintus-Bosz, 1993, Wavepath eikonal traveltime inversion: Theory: Geo-319
physics, 58, 1314–1323.320
Sirgue, L., , and R. Pratt, 2004, Efficient waveform inversion and imaging: a strategy for selecting321
temporal frequencies: Geophysics, 69, 231–248.322
Symes, W. W., 2008, Migration velocity analysis and waveform inversion: Geophysical Prospect-323
ing, 56, no. 6, 765–790.324
Taillandier, C. M., M. Noble, H. Chauris, and H. Calandra, 2009, First-arrival traveltime tomogra-325
phy based on the adjoint-state method: Geophysics, 74, WCB1–WCB10.326
18
Tarantola, A., 1984, Inversion of seismic reflection data in the acoustic approximation: Geophysics,327
49, 1259–1266.328
——–, 1986, A strategy for nonlinear elastic inversion of seismic reflection data: Geophysics, 51,329
1893–1903.330
——–, 1988, Theoretical background for the inversion of seismic waveforms including elasticity331
and attenuation: Pure and Applied Geophysics, 128, 365–399.332
——–, 2005, Inverse Problem Theory and Methods for Model Parameter Estimation: Society for333
Industrial and Applied Mathematics.334
Tikhonov, A., and V. Arsenin, 1977, Solution of ill-posed problems: Winston and Sons.335
Virieux, J., and S. Operto, 2009, An overview of full-waveform inversion in exploration geophysics:336
Geophysics, 74, no. 6, WCC1–WCC26.337
Woodward, M. J., 1992, Wave-equation tomography: Geophysics, 57, 15–26.338
Zhang, Z., T. Alkhalifah, E. Z. Naeini, and B. Sun, 2018, multiparameter elastic full waveform339
inversion with facies-based constraints: Geophysical Journal International, 213, 2112–2127.340
19
LIST OF FIGURES
1 (a) Example of a λ and µ crossplot. (b) Model PDF associated to the crossplot in panel (a),341
characterizing the dependence of λ and µ. The red dot represent a specific model with components342
(λx, µx). (c) The distribution of the constraint term calculated with equation 17 associated to the343
PDF in (b). Note that high probability models are associated with low values of the constraint.344
2 Sketch indicating how each point in the updated model space (red dot) communicates with345
all cells of the model PDF. Each cell of the model PDF contributes proportionally with its probabil-346
ity value and inverse proportionally to the distance to the analyzed model.347
3 True λ and µ models and their crossplot in the λ − µ space. The top panels show the348
sources (dots) located at x = 0.1 km and the vertical line corresponds to receivers at x = 2.4 km.349
The negative and positive Gaussian anomalies for the λ and µmodels are centered at (1.25, 0.75) km350
and (1.25, 1.75) km, respectively. The crossplot shows two linear relationships between λ and µ:351
corresponding to each of the Gaussian anomalies.352
4 Updated λ and µ models using the objective function JD. The Gaussian anomalies are353
not recovered well and the inverted models present artifacts from inter-parameter crosstalk. The354
crossplot shows that the parameters do not follow the same trend of the true models (Figure 3).355
5 Updated λ and µ models using the objective function JD +JL. The lines define the upper356
and lower boundaries used in the logarithmic penalty function JL. The Gaussian anomalies are not357
recovered well and the backgrounds are modified during the inversion.358
6 Updated λ and µ models using the objective function JD + JP . The contours in the359
crossplot represent the feasible area provided by the probabilistic petrophysical constraint JP . Both360
Gaussian anomalies are better recovered for the λ and µ models.361
7 Marmousi 2 model: True λ and µmodels and their crossplot. The vertical lines correspond362
to the location of the wells, providing information for the petrophysical constraint.363
20
8 Crossplot of λ and µ provided by the well logs indicated in Figure 7. The relationship364
between the model parameters is not linear and does not follow a simple trend.365
9 Marmousi 2 model: Initial λ and µ models and their crossplot in the λ− µ space.366
10 Marmousi 2 model: Recovered λ and µ models using objective function JD. The inverted367
µ model has more structural details than the λ model.368
11 Marmousi 2 model: Recovered λ and µ models using objective function JD + JL. The369
lines in the crossplot define the upper and lower boundaries used in the logarithmic penalty function370
JL. The inverted model is forced to be in the middle area of the feasible region, which does not371
reflect correctly the trend shown in Figure 7 for the true model.372
12 Marmousi 2 model: Recovered λ and µ models using objective function JD + JP . The373
contours in the crossplot represent the feasible area for the probabilistic constraint JP . The proba-374
bilistic petrophysical constraint leads to better recovered models than unconstrained inversion (Fig-375
ure 10) and inversion with approximate analytic constraints (Figure 11).376
377
21
(a)
(b)
(c)
Figure 1: (a) Example of a λ and µ crossplot. (b) Model PDF associated to the crossplot in panel (a),characterizing the dependence of λ and µ. The red dot represent a specific model with components(λx, µx). (c) The distribution of the constraint term calculated with equation 17 associated to thePDF in (b). Note that high probability models are associated with low values of the constraint.Aragao & Sava – GEO-2000-0000
22
Figure 2: Sketch indicating how each point in the updated model space (red dot) communicateswith all cells of the model PDF. Each cell of the model PDF contributes proportionally with itsprobability value and inverse proportionally to the distance to the analyzed model.Aragao & Sava – GEO-2000-0000
23
Figure 3: True λ and µ models and their crossplot in the λ − µ space. The top panels show thesources (dots) located at x = 0.1 km and the vertical line corresponds to receivers at x = 2.4 km.The negative and positive Gaussian anomalies for the λ and µmodels are centered at (1.25, 0.75) kmand (1.25, 1.75) km, respectively. The crossplot shows two linear relationships between λ and µ:corresponding to each of the Gaussian anomalies.Aragao & Sava – GEO-2000-0000
24
Figure 4: Updated λ and µ models using the objective function JD. The Gaussian anomalies arenot recovered well and the inverted models present artifacts from inter-parameter crosstalk. Thecrossplot shows that the parameters do not follow the same trend of the true models (Figure 3).Aragao & Sava – GEO-2000-0000
25
Figure 5: Updated λ and µmodels using the objective function JD+JL. The lines define the upperand lower boundaries used in the logarithmic penalty function JL. The Gaussian anomalies are notrecovered well and the backgrounds are modified during the inversion.Aragao & Sava – GEO-2000-0000
26
Figure 6: Updated λ and µ models using the objective function JD + JP . The contours in thecrossplot represent the feasible area provided by the probabilistic petrophysical constraint JP . BothGaussian anomalies are better recovered for the λ and µ models.Aragao & Sava – GEO-2000-0000
27
Figure 7: Marmousi 2 model: True λ and µmodels and their crossplot. The vertical lines correspondto the location of the wells, providing information for the petrophysical constraint.Aragao & Sava – GEO-2000-0000
28
Figure 8: Crossplot of λ and µ provided by the well logs indicated in Figure 7. The relationshipbetween the model parameters is not linear and does not follow a simple trend.Aragao & Sava – GEO-2000-0000
29
Figure 9: Marmousi 2 model: Initial λ and µ models and their crossplot in the λ− µ space.Aragao & Sava – GEO-2000-0000
30
Figure 10: Marmousi 2 model: Recovered λ and µ models using objective function JD. Theinverted µ model has more structural details than the λ model.Aragao & Sava – GEO-2000-0000
31
Figure 11: Marmousi 2 model: Recovered λ and µ models using objective function JD + JL. Thelines in the crossplot define the upper and lower boundaries used in the logarithmic penalty functionJL. The inverted model is forced to be in the middle area of the feasible region, which does notreflect correctly the trend shown in Figure 7 for the true model.Aragao & Sava – GEO-2000-0000
32
Figure 12: Marmousi 2 model: Recovered λ and µ models using objective function JD + JP .The contours in the crossplot represent the feasible area for the probabilistic constraint JP . Theprobabilistic petrophysical constraint leads to better recovered models than unconstrained inversion(Figure 10) and inversion with approximate analytic constraints (Figure 11).Aragao & Sava – GEO-2000-0000
33