Inversão com sigmoides

Seismic traveltime inversion using Metropolis-Hasting method with sigmoidal
parameterization
Lucas F. Palma1, Juarez S. Azevedo1 , Saulo P. Oliveira2,∗, and Wilson M. Figueiró1
1UFBA, 2UFPR.
Copyright 2021, SBGf - Sociedade Brasileira de Geofı́sica.
This paper was prepared for presentation at the 17th
International Congress of the
Brazilian Geophysical Society, held in Rio de Janeiro, Brazil, August 16-19, 2021.
Contents of this paper were reviewed by the Technical Committee of the 17th
International Congress of the Brazilian Geophysical Society and do not necessarily
represent any position of the SBGf, its officers or members. Electronic reproduction
or storage of any part of this paper for commercial purposes without the written consent
of The Brazilian Geophysical Society is prohibited.
Abstract
Seismic traveltime tomographic inversion is essential to
help estimate the internal structure of the solid earth. For
such, we propose to approximate the complex geological
structure of a given media based on parameterization by
sigmodal functions of discontinuous velocity fields. In
addition, we use the Metropolis-Hasting (global-scope)
algorithm for the inversion procedure. Through this
inversion method, we provide high-resolution estimates
of the model’s parameters and ensure that the results
obtained are in accordance with actual data. Synthetic
models are employed to validate the method.
Introduction
Seismic traveltime tomography inversion methods aim
to produce a better mapping of geological structures
through seismic data processing and interpretation, with
an essential contribution to estimate reservoir continuity.
However, these methods depend on the choice of
parameterization of velocity model (Kissling et al., 2001;
Tong et al., 2019) and cannot capture, with desired
efficiency, discontinuities in velocity model. Moreover,
experiments involving parameterizations with splines
and wavelets functions (Michelini and McEvilly, 1991;
Cerqueira et al., 2016) have shown that velocity fields can
be represented without violating discontinuities.
Several authors have suggested improvements in the
process of parameterization of models (Tikhotskii et al.,
2011; Evensen and Landrø, 2010; Belhadj et al., 2018).
Motivated by these works, we show that, under certain
restrictions, a rigorous formulation of the inversion
procedure from traveltimes for the discontinuous velocity
models is possible in the scope of parameterization through
a linear combination of sigmoidal functions (Costarelli and
Spigler, 2013; Oliveira et al., 2020). This methodology
allows sharp variations of the velocity model without
neglecting continuity, besides, it does not need domain
decomposition. To this end, we invert the travel times to
obtain velocity field and interface structure at the same time
using the method of ray tracing.
Realistic models can be better represented by this
sigmoidal functions and the ambiguity, commonly present
in inversion techniques results, can be reduced if sufficient
data is available. Furthermore this kind of parameterization
have a good adjustability in the discontinuous model
since its approximation strongly depends on the sigmoidal
parameter which requires no additional computational
cost, especially for complex models whose difficulty in
representing the discontinuity by a fine mesh is inherent.
In the inversion procedure we use Metropolis Hasting
method (MH) (global scope) (Azevedo et al., 2014)
as a Bayesian inversion strategy to minimize the error
between observed and computed data, adjusting the model
parameters and boundary nodes simultaneously in each
iteration. First, an initial set of samples for the MH
algorithm is assembled through a small-scale random
search generated by the Monte Carlo method, with no
prior knowledge of the minimum. We expect that random
sequences generated by the Monte Carlo method will
improve the performance of the MH method when applied
to the initial population. Then, after a certain number
of samples, MH method is started from the initial guess
in such a way that the distribution of values converges
to the desired distribution considering that the candidate
point is either accepted or rejected using the acceptance
probability.
In this paper, we start with a qualitative 1D geological
model and typical velocities values are arbitrated for each
layer, extending these results to the two-dimensional case.
From then on, a mesh of points is placed over the domain of
velocity model, having at each point a horizontal coordinate
(which symbolizes the distance on the surface), a vertical
coordinate (which symbolizes the depth of the point) and
an associated seismic velocity. The number and position
of each parameter of the model is made in general and
inserted in the data set. Then we use ray tracing to
determine traveltimes for a fixed set of ray take-off angles.
The results obtained by both experiments were compared
with Very Fast Simulated Annealing (VFSA) method.
Model Parameterization
The parameterization of the velocity field is based on the
sigmoidal approximation of piecewise-constant functions,
which we describe next.
Following the notation of Oliveira et al. (2020), we
approximate the one-dimensional velocity field v(z) as
Seventeenth International Congress of the Brazilian Geophysical Society

SEISMIC TRAVELTIME INVERSION WITH SIGMOIDAL PARAMETERIZATION 2
follows:
va(z) =
N
∑
n=−1
vnφa
z
h
−n

, (1)
where the sigmoidal function φa is defined as
φa(x) =
1
1+e−ax
−
1
1+e−a(x−1)
, (2)
and a represents the parameter that controls the transition
between plateaus, which may be smooth or sharp if a is
small or large, respectively. This is confirmed by the one-
dimensional velocity model shown in Fig. 1.
(a) (b)
Figure 1: Sigmoidal scaling function (a) and approximation
of a piecewise-constant function (b) with slope parameters
a =10 and a =100.
Approximation (1) can also be easily extended to a
rectangular domain [0,Lx]×[0,Lz] with grid points (xm,zn) =
(mhx,nhz), 0 ≤ m ≤ Nx and 0 ≤ n ≤ Nz, by the following
tensor-product expansion:
va(x,z) =
Nz
∑
n=−1
Nx
∑
m=−1
vm,nφa

x
hx
−m

φa

z
hz
−n

, (3)
which can be approximated as
va(x,z) ≈
1
∑
i=−1
1
∑
j=−1
vmi,nj φa

x
hx
−mi

φa

z
hz
−nj

, (4)
where mi = m+i, nj = n+ j, (x,z) ∈ [xm,xm+1]×[zn,zn+1] and
the summation limits may be modified to guarantee 0 ≤ m+
i ≤ Nx and 0 ≤ n+ j ≤ Nz.
Ray tracing in heterogeneous models
The traveltime from source A to receiver B on two-
dimensional media can be obtained through the velocity
model V solving the following line integral
TAB =
Z
lAB
1
v(l)
dl (5)
where V(l) is the (unknown) local velocity along the
(unknown) actual ray path lAB between A and B, which is
calculated numerically by the following equation:
TAB(xN+1,zN+1) =
N
∑
i=0
∆Ti =
N
∑
i=0
1
vi
kxi+1 −xik2 (6)
where vi represents the wave velocity at position xi =
(xi,zi) .
Using the Fermat’s Principle of minimum time to
characterize ray paths between two points, the following
system of equations needs to be solved numerically
(Červený, 2001):















dx
ds
=
1
|p|
p,
dp
ds
= ~
∇

1
v(x)

,
dT
ds
=
1
v(x)
.
(7)
where p(s) represents the slowness vector parameterized
by the arc length s along the ray. Vector p is related to the
velocity field v through the eikonal equation |p|2 = 1/v2(x).
System (7) can be written in vector form as
dy
ds
= f(s,y), (8)
where y = [x,p,T] and f = [p/|p|,−(1/v2)∇v,1/v]. We
discretize system (8) using the fourth-order Runge-Kutta
method with step length ∆s, i.e., we consider y(n∆s) ≈ yn,
where
yn+1 = yn +
1
6
(k1 +2k2 +2k3 +k4), (9)











k1 = ∆sf(sn,yn),
k2 = ∆sf(sn + 1
2 ∆s,yn + 1
2 k1),
k3 = ∆sf(sn + 1
2 ∆s,yn + 1
2 k2),
k4 = ∆sf(sn +∆s,yn +k3).
Equation (9) is the basis for an algorithm that numerically
calculates the trajectory of the seismic ray as well as the
travel time of the seismic waves along these trajectories.
The slowness vector is updated at each iteration of the
Runge-Kutta method through the eikonal equation. For
the initial condition, we assume the ray leaves a source
point located at x0 with a prescribed take-off direction that
defines the initial slowness p0, and set T0 = 0.
In next section we will present tools that will allow us
to estimate unknown parameters of the velocity field by
parameterization of sigmoid functions by inverting the
traveltimes.
Metropolis Hasting (MH) method
Let us introduce: a model vector m written as
m = (m1,m2,...,mL)T
, (10)
an observed data vector dobs, and the calculated vector
data dcalc obtained directly from a synthetic model m.
To compute the misfit between the observed and the
simulated data we use L1-norm misfit given by
φ(m) = kdobs −dcalc(m)k1. (11)
In our context, m represents the velocity field while
dobs = (Tobs(θ1),Tobs(θ2),...,Tobs(θNS))T ,
dcalc(m) = (T[m](θ1),T[m](θ2),...,T[m](θNS))T ,
(12)

AZEVEDO ET AL. 3
where Tobs(θi) and T[m](θi) are, respectively, the observed
and the calculated traveltime for the ray with take-off angle
θi, while Nr is the number of rays that reach the surface in
both cases.
In order to compare the calculated data with the observed
data we use the likelihood function
p(dobs
|m) = exp

−
φ2(m)
2Nsσ2

. (13)
where Ns is the number of samples and σ is a standard
deviation of the data error (Cho et al., 2017). The standard
deviation σ is used as a fitting criterion for accepting or
refusing in the MH method.
The MH also uses a proposal distribution q(m,m∗), which
depends on the current state m, to generate a new
proposed sample m∗ and a prior PDF uniformly distributed
p(m). Then, the values proposed are accept or rejected
with probability
ν(m,m∗
) = min
(
1,
p(m∗)p(dobs
|m∗)q(m,m∗)
p(m)p(dobs
|m)q(m∗,m)
)
. (14)
If the proposal is not accepted, then the current value of
m is retained: m∗ = m. In our experiments we assume
that the proposal distribution q is symmetric, q(m,m∗) =
q(m∗,m). In this case, the MH method reduces to a
procedure called random walk.
In this inversion method, the random walk produces Markov
chains after the first N iterations, which are discarded by
a process called burn in period. Algorithm 1 provides a
summary of this methodology.
Algorithm 1: MH algorithm for velocity field
Input: Build initial velocity field m(0) by MC method
for n ∈ {1,...,Nmh} do
ξ(n−1)
∼ N(0,I);
mc ←
√
1−δ2m(n−1) +δξ(n−1)
, 0 δ ≤ 1;
Acceptance Probability:
ν(mc
,mn−1
) = min
(
1,
p(mn)p(dobs
|mn)
p(m)p(dobs
|m)
)
;
u ∼ U(0,1);
if u ν then
Accept the velocity field: m(n) ← mc;
else
Reject the velocity field: m(n) ← m(n−1);
end
end
Results and discussion
In the following experiments, the dimensions of the
domain are fixed in Lx = 9.4 km and Lz = 3.0 km for all
examples. In this case the rays depart from x0 = [x0,z0]
with take-off angle θ with respect to the z-axis, i.e., p0 =
[sin(θ),cos(θ)]/v0. In the experiments we considered 1000
samples in the MH method after a burn in period of 100
samples, totaling 1100 realizations in the inverse process.
Horizontally layered model (M1)
Let us choose as target model an elementary one-
dimensional model of three horizontal layers of thickness
L = 0.75 km and velocities v1 = 2.0 km/s, v2 = 2.5 km/s,
and v3 = 3.0 km/s with a single surface source at the origin
(Oliveira et al., 2020). The velocity below the third layer is
v4 = 4.0 km/s. The target model and reference ray paths
are displayed Fig. 2.
This model allows travel times to be calculated analytically.
For the purpose of comparison with the numerical solution,
the arrival reflection locations x(i) and traveltimes T(i) (i =
1,2,3) for an arbitrary take-off angle θ are defined by
x(i)
= 2L
i
∑
k=1
tanθk, T(i)
= 2L
i
∑
k=1
secθk
vk
, (15)
where θ1 = θ and θi = sin−1((vi/vi−1)sinθi−1) for i = 2,3.
Figure 2: Reference solutions for the horizontally layered
model (M1): paths of 40 rays with take-off angles uniformly
distributed in [0◦,90◦[.
In this example the forward responses obtained by the
MH method were compared with the Very Fast Simulated
Annealing (VFSA) which is also a global optimization
method used for finding the global minimum of a function
(see, e.g., Oliveira et al., 2018). In this case, the initial
and final temperatures are T0 = 1 and Tf = 0.01, whereas
the parameter c is defined by average rate of change of
temperature in log scale given by ∆t = 0.01. Here, the
number of models tested by temperature is 100.
In the following, we compare ray paths obtained with MH
and VFSA (Fig. 3) with the target solution given by Fig.
2. The MH inversion with slope parameter a = 10 yields
smoothed velocity fields and ray paths (Fig. 3a). A similar
result was obtained by the VFSA method and therefore
it was omitted here. However there is a good fit with
the observed data for both methods when parameter a
= 100 (see Figs. 3b-3c). In addition, Table 1 shows
the superiority of MH over VFSA. In both parameters, the
results presented by misfit and CPU time had lower values
in MH method.
The travel times of the sigmoidal representation selected
in Fig. 3 are shown in Fig. 4. The straight lines are the
travel times obtained by the equation (15). We can infer
that the results from MH method have acceptable fitness

a Method Misfit CPU
10
MH 209.0455 1619.61
VFSA 233.5181 164331
100
MH 53.6060 1133.93
VFSA 103.2884 158661
Table 1: Misfit and CPU time for model M1 considering MH
and VFSA methods.
due to its low relative error. It is also important to note that
in a = 100, the VFSA method obtained better results than
the MH in relation to travel times, however, the MH is able
effectively generate a velocity field profile where the rays
were reflected in the third layer.
High-Velocity Intrusion Model (M2)
In the next synthetic test, we consider a representation
of velocity field with intrusion of high velocity at the
center of the domain. Let us consider a domain with
having 3 rows and 4 columns totaling 12 coefficients to
determine. The velocities vary from 1.5 km/s to 5.5 km/s
(see Fig. 5), establishing the target model. Similar study
involving interpolation Haar functions and the Metropolis
algorithm was carried out in Cerqueira et al. (2016). The
arrangement consists of one source located at (4.65,0) km
with take-off angle θ uniformly distributed in [−45◦,45◦].
Figure 6 shows the ray paths from sigmoidal
representations with inverted model assuming parameters
a = 10 and a = 100. As in Fig. 4, we employ 40 rays
to illustrate the ray trajectories through the model. It
should be noted that the solution by MH method manged
to highlight the intrusion zone. Note at a = 100 we got
better results in the velocity field. However, in both slopes
parameters, the velocity field has been able to register
reflections only near the source and consequently travel
times have been discarded along the edges of the domain
(Fig. 7).
Conclusions
Usually seismic models are described by two parameters
of diferent nature: wave velocities on rocks and interfaces
between layers. Such situation is a source of ambiguity
in travel time seismic inversion procedures. A strategy to
mitigate such problem is to consider such model as an
unique object: a velocity field parameterized by a sigmoidal
series that is able to simulate geometrical discontinuities at
interfaces by a continuous and fast variation. Sigmoidal
function representation make stronger the coupling or
relationship between lenghts and velocities, reducing, then,
model parameters indeterminancy. Such advantages are
explored and illustrated in this proposed work in order
to improve results as shown in the numerical inversion
performed experiments.
(a) MH, a = 10
(b) MH, a = 100
(c) VFSA, a = 100
Figure 3: Velocity fields and ray paths obtained by inverting
the travel times of the rays shown in Fig. 4: (a) MH with a
= 10; (b) MH with a = 100; (c) VFSA with a = 100.

AZEVEDO ET AL. 5
(a) MH, a = 10
(b) MH, a = 100
(c) VFSA, a = 100
Figure 4: Comparison between observed (ref) and
calculated (sig) traveltimes in the horizontally layered
model (M1) considering slope parameters a = 10 and a
= 100: global methods MH (a)-(b) and VFSA (c). The
sigmoidal parameterization with a = 100 had a good
approximation with the MH while the VFSA performed
better, however with a very high CPU time.
Figure 5: Reference solutions for the high-velocity intrusive
model (M2): paths of 40 rays with take-off angles uniformly
distributed in [−45◦,45◦].
(a) a = 10
(b) a = 100
Figure 6: Ray paths from sigmoidal representations of the
inverted intrusive model (M2) with a = 10 (a) and a = 100
(b).

(a) a = 10
(b) a = 100
Figure 7: Comparison between observed (ref) and
calculated (sig) traveltimes of 200 rays in the high-velocity
intrusive model (M2) considering slope parameters a = 10
(a) and a = 100 (b). The calculated traveltimes have been
found by the MH method.
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