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

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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

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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

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(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

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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

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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

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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

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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

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Figure 9: Marmousi 2 model: Initial λ and µ models and their crossplot in the λ− µ space.Aragao & Sava – GEO-2000-0000

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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

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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