A new global search method for inversion of non-linear methods, applied to geophysical inversion studies.

1. Global optimization algorithm:Search
for the Lost Valley (SLV)
Manuel Abarca
Urubamba, Perú
Email: manuel.z.abarca@outlook.es
A new global search algorithm is proposed here. This technique to find a
minimum (or maximum) of an objective function begins with a population of
models randomly generated . The global minimum of this function must be at
some point of the variables space. Search for that minimum uses of an arithmetic
mean (centroid) between neighbourhoods models. There is also a similar tactic to
bring outside of a local minimum. The new algorithm is tested with a seismological
inversion problem, modelling the Earth through receiver function. First with
synthetic data and finally with real data of receiver functions.
Keywords: computation of global search; optimization; geophysical inversion; seismology;
receiver function
Received ; revised
1. INTRODUCTION
The global search algorithms are computational tools
used in many fields of science and industry; the goal is
to solve a problem through the minimization (or maxi-
mization) of an objective function (or error function).
The proposal of an algorithm to solve an also known as
optimization problem means we know two things: one,
the way to sampling the variables space and second, an
search mechanism. The focus of our research is on the
development of a new search mechanism.
There are some kind of search mechanisms based in
the gradient analysis (steepest descent [1];conjugate
gradient [2]). Another search mechanisms are purely
random or controlled random (Monte Carlo [5] , CRS
[7]). The process by which molten metal crystallizes
and reaches equilibrium in the process of annealing
can, for instance, gives place to a method of optimiza-
tion called Simulated Annealing [4]. A new criteria
in optimization were introduced by evolutionary algo-
rithms (Genetic Algorithm [3] ) which uses operations
as cross-over and mutation to find the best fitted indi-
vidual.
Our algorithm implements a search strategy similar
to the Simplex [6] but combined with a tracking in the
neighbourhood. The main difference respect to Simplex
is that this uses the point of centroid over all population
in variables space. While our logical reasoning is to find
the centroid around a (transitory) minimum point plus
two points in their neighbourhood. The idea below this
search mechanism is that one point of minimum in a
surface representing an error function could mean one
of this things: it is a local minimum or it is a point
near the global minimum. In the first case we have
a tactic to bring outside the local minimum, will be
described in the next section. In the second case we
have a mechanism to track in the vicinity of that point
as if it were located in a valley, until to reach the bottom
of the valley (in the hope that is the global minimum).
The name of our algorithm explains this search for an
unknown valley and the tracking by their surface until
to find the global minimum, Search for the Lost Valley
(SLV).
2. METHOD
First than all, some definitions here. We call ”model”
to the group of variables able to describe a physical
situation. For example the variables x1, x2, ..., xp can
represent seismic velocities and thicknesses of layers
in a sedimentary basin. One model explaining the
geophysical relationship between strata in the basin
could be ⃗
m1(x1
1, x1
2, ..., x1
p); another model would be
⃗
m2(x2
1, x2
2, ..., x2
p); where p is the number of variables in
the model.
We call ”forward problem” to the physical-
mathematical equations which relates a ”model”
with an ”measurable” response in some point of the
space; in a general form is a f(⃗
m). This function of
the model has unique solution, so, for each model ⃗
mi
there is one and only one set of observed responses in
the space f( ⃗
mi).
The ”real” data or ”observed” data ⃗
do are the mea-
sures of some physical magnitude in points of the space
(generally on surface earth).
We call ”inverse problem” to the mathematical pro-
cedure to find the model which best fit the observed

2. 2 M. Abarca
data. This is made minimizing an objective (or error)
function,
τ(⃗
m) = f(⃗
m) − ⃗
do
2
2
(1)
for instance we are using here the least square norm.
The inverse procedure is generally dealing with ill
posed problems. This is because the noise (natural,
instrumental, anthropogenic) included in the real
data. Then to stabilize the solution and to get
uniqueness in the inverse problem we have to apply
some regularization criteria. We choose the Tijonov
criteria ([9]), which implies to add a-priori information
to the objective function,
τ(⃗
m) = f(⃗
m) − ⃗
do
2
2
+ λ2
∥W · ⃗
m∥
2
2 (2)
⃗
do : real data;
W : regularization function;
λ : equalization factor .
Another measure necessary to our search strategy is
the ”distance” between models,
s̄ =
q
(x1
1 − x1)2 + (x1
2 − x2)2 + ... + (x1
p − xp)2 (3)
Now, we want to minimize the objective function τ
applying the SLV algorithm:
1. Create a random population of models,
⃗
m1, ⃗
m2, ..., ⃗
mn, with n number of elements; insert
a threshold value e.
2. Evaluate (2) for each model.
3. Sort the population of models by the magnitude
of error function. The lowest error is the best model,
until this step.
4. Measure the distance between best model and all
elements of population, (3).
5. Sort models by distance to the best model.
6. Choose 3 models with shortest distance; this group
includes the best (distance zero).
7. Take the centroid from 3 elements of group.
8. Evaluate (2) for the centroid.
8.1 Compare objective function of centroid with thresh-
old value; if τ is less to e the process ends. Another
stop criteria could be the number of iterations, defined
by n.
9. If the objective function of centroid is better than
τ of best model, then centroid takes the place of best
model; go to step 4.
10. Else, if the error of centroid is not better than best
model, then takes the next 2 models by distance and
the best one.
11. Go to step 7.
The tactic to bring outside a local minimum is given
in step 10. This search mechanism is not totally ran-
dom because we are using the best model as pivot.
If there is a lost valley in any region of the variables
space it will be possible to find with this mechanism.
We present the same algorithm in a flux diagram in
the next figure,
FIGURE 1. SLV algorithm for global search in
optimization problems. Begin (inicio), end (fin).
2.1. An improvement to search mechanism
When we evaluate the centroid over 3 models, in
mathematical terms we are taking the arithmetic mean,
⃗
mc = ( ⃗
m1 + ⃗
m2 + ⃗
m3)/3 (4)
so, we can refine our tracking by the surface of the
valley putting different weights to each one of this 3
models.
⃗
mc = (k1 · ⃗
m1 + k2 · ⃗
m2 + k3 · ⃗
m3)/6 (5)

3. 3
where,
k1 = 1; k2 = 2; k3 = 3 , for the first weighted mean;
k1 = 3; k2 = 1; k3 = 2 , for the second weighted mean;
k1 = 2; k2 = 3; k3 = 1 , for the third weighted mean.
Then, inside the step 8 of SLV we choose the centroid
with the lowest error function.
3. INVERSE PROBLEM IN GEOPHYSICS
We can take a geophysical example to show how to
apply the SLV algorithm in solution of inverse problems.
In a sedimentary basin the layers would be characterized
by parameters as seismic velocity, density of rocks
and thicknesses of strata. Knowing some physical-
mathematical formulae we are able to calculate the
travel time of a seismic wave, from the bottom of
sediments to top (air-earth interface). This is the
forward problem in geophysics,
T(⃗
m(xj, xj+1, ..., xp−1, xp)) (6)
where,
xj : seismic velocity in the first layer ;
xj+1 : depth of first layer;
xp−1 : seismic velocity of semi-infinite medium;
xp : depth trending to infinite (90000 m in table 1
means semi-infinite);
T : travel time.
But, in a real seismological investigation we got time
series (seismograms) registered by instruments (called
seismometers) where we can read arrival times of seis-
mic waves. In Receiver Function method (RF) we have
not just the travel time of one kind of seismic wave, but
also the times of converted waves (P to S) and rever-
berated waves; further we have the complete waveform
of RF. In any case the relevant parameter is time,
t0 observed in a seismogram or in a pseudo-seismogram.
We don’t describe the RF method in their seis-
mological intricacies for one reason: this is not a
seismological study. This a research with focus in the
area of optimization, or could be also in geophysical
inversion techniques, so our proposal is to make known
a new method in global search (or maybe in non linear
optimization).
The final subject of an inversion is to get a
realistic model of sedimentary basin from RF pseudo-
seismogram. This implies to minimize an objective
function,
τ(⃗
m) = ∥T(⃗
m) − t0∥
2
2 + λ2
∥W · ⃗
m∥
2
2 (7)
4. TESTING SLV WITH A SYNTHETIC RF
It is a theoretical model, with variables xj represent-
ing seismic velocity of P-wave and depth of layers
into a sedimentary basin , Last depth has a very large
[hhh]
TABLE 1. Seismic model of a sedimentary basin.
Depht m Vp m/s
1500 1800
4000 2500
90000 5000
value because it is representing a semi-inifinite medium.
Theoretical response of a seismic P-wave crossing this
pack o layers is given by a RF (Fig. 2)
FIGURE 2. Synthetic Receiver Function (RF), obtained
from the model of table 1.
The test consists in applying our SLV algorithm to
inverse problem; in that way we have to recovery the
same model (or approximately the same) given in table
1, from inversion of ”observed” RF (Fig. 2). Our RF
gets a more realistic shape including 5% of noise.
Minimization of( 7) will produce a best fitted RF
associated to a seismic model which must be similar
to the proposed in the premises of our test,
1.0
2.0
3.0
4.0
5.0
1000
2000
3000
5000
FIGURE 3. ”Real” waveform of RF and best fitted RF
(at the left); seismic model of a sedimentary basin result of
inversion (right side of figure).

4. 4 M. Abarca
Considering the difficulties of RF inversion, results of
the test are satisfactory.
5. EVALUATION OF GOODNESS OF SLV
ALGORITHM
SLV algorithm begins generating a random population
of models; the ideal number of models in this population
is in the range 50-60. Test of sensitivity shows that
an initial population too great no correlates with an
increasing in fitness of final result.
FIGURE 4. Surface of objective function from 512 models
randomly created.
Creation of random models open the possibility to
put an interval for each variable. This is a kind of soft
regularization.
The first best model (from initial random population)
is signed with a green square. In figure (Fig. 5) the path
of search by new local minima is indicated with orange
straight lines. When SLV found a general minima marks
this with a clear blue square. Note that the area around
the blue square is not revealed as a valley in the initial
population (Fig. 4).
inicio
final
FIGURE 5. Best model from initial population in green
square (inicio); final solution for inversion in blue square;
path of search in orange lines; the nodes of path are points
of centroid, this explains why there are not nodes in the
borders of surface of objective function.
6. TESTING SLV WITH A REAL GEOPHYS-
ICAL PROBLEM
The waveform of a RF is the subject of present
evaluation of SLV in their capacity to solve inverse
problems. This RF (Fig.6)was obtained in a study
developed over sedimentary basin of Parana river ([11]).
0 1 2 3 4 5
Tiempo - seg.
-0,2
-0,1
0
0,1
0,2
0,3
0,4
FR
amplitud
FR ajustada
FR observada
FIGURE 6. Best fitted RF in black line; observed RF in
green dotted line.
After running our program which realize the SLV
algorithm we got inversion of RF, obtaining following

5. 5
2000 2500 3000 3500 4000 4500 5000
Velocidad de P - m/s
-9000
-8000
-7000
-6000
-5000
-4000
-3000
-2000
-1000
0
Profundidad
-
m
FIGURE 7. Seismic model of sedimentary basin in Parana
(Brazil), obtained from inversion of RF.
2,5 3 3,5 4 4,5 5 5,5 6
Vp (km/s)
-7
-6
-5
-4
-3
-2
-1
0
Profundidade
(km)
Modelo sismico
Dois RF inversion - capb stn.
FIGURE 8. Result of RF inversion with GA in Parana
basin ([10]), using same seismic station and the same
waveform of RF used in this section.
results (Fig. 7).
In the doubt smoothes, says some geophysicist who’s
name I don’t remember. So, we apply smoothness
in the regularization function of our τ. This type
of regularization try to oblige the model to minimize
differences between velocities of layers; but abrupt
changes in velocity can appear if the data indicates
necessity of that jumps in velocity. This is a good proof
for our algorithm and the model in following figure is
signalling that strong differences in velocities are real.
In this case we have not a previously well known model
to compare with the result. However we have made an
study some years ago with the same RF . In that old
study we used a Genetic Algorithm (GA) in inversion of
RF, obtaining the model exhibited in Fig.8. Comparing
the resulting seismic models for Parana basin, obtained
with two different algorithms, SLV Fig. 7 and GA
Fig. 8, we can see many similarities. One of the
main goals of that old study was to determine depth
of sedimentary basin (until basement). In both models
the basement depth is located near 3500 m. A second
feature distinctive of Parana basin is the existence of
a surficial volcanic rocks cover; this basalts layer has
higher seismic velocity than sediments below. Both
algorithms were capable to define the high velocity
upper layer in the range 4250-4500 m/s for P-wave.
GA uses more layers to fit the RF, this is the reason
by which we can see a thin low velocity layer at the
top of basin; but SLV (using fewer layers) can fit the
waveform with an average-velocity layer in surface.
7. CONCLUSIONS
SLV algorithm meets two important proposals of all
global optimization strategy,. First one. is capable of
an uniform and complete sampling of variables space;
this quality is given by a good random numbers gen-
erator. And the second one, has a search mechanism
able to track the variables space until to find any local
minimum, and probably the general minima. Looking
in the neighbourhood of a transitory best model with 4
centroids increase probability to get a new best model.
It is a notable feature of algorithm the pivoting on each
new best model, re-beginning the search for a new min-
ima, or jumping outside this local minimum but track-
ing again all the space.
The new global search algorithm SLV passes satisfacto-
rily two tests. The first test with synthetic RF, and the
second test inverting a real RF.
DATA AVAILABILITY
The data underlying this article will be shared on
reasonable request to the corresponding author.
ACKNOWLEDGEMENTS
I have to thank to mi sister Martha by her financial
support during research epoch.
REFERENCES
[1] R. Fletcher, M. J. D. Powell (1963) A Rapidly
Convergent Descent Method for Minimization. The
Computer Journal, 6-2, 163–168.
[2] R. Fletcher, C. M. Reeves (1964) Function minimiza-
tion by conjugate gradients. The Computer Journal,
7-2, 149–154.
[3] J.H. Holland (1975) Adaptation in Natural and
Artificial Systems. University of Michigan Press, Ann
Arbor, Michigan, re-issued by MIT Press.
[4] Scott Kirkpatrick, C. D. Gelatt, Jr., and M. P. Vecchi
(1983) Optimization by simulated annealing. Science,
220(4598), 671–680.
[5] Nicholas Metropolis, S. Ulam (1949) The Monte
Carlo Method. Journal of the American Statistical
Association, 44, No. 247, 335–341.
[6] Nelder, John A.; R. Mead (1965) A simplex method for
minimization. The ComputerJournal , 7, 308–313.
[7] W. L. PRICE (1983) Global Optimization by Con-
trolled Random Search. Journal of optimization theory
and applications, 40, 333–348.

6. 6 M. Abarca
[8] Tarantola, A. (2005) Inverse problem theory and
methods for model parameter estimation. Society for
Industrial and Applied Mathematics, Philadelphia.
[9] Tijonov,Andrei Nikolayevich (1963) Solution of in-
correctly formulated problems and the regularization
method. Soviet Mathematics , 4, 1035–1038.
[10] Zevallos Abarca, Ivan (2004) Modelamento da Bacia
do Parana - reservatorio Capivara - atraves da inversao
conjunta de Funcao do Receptor e de sondagem
Magnetotelurica. Universidade de Sao Paulo, Sao
Paulo.
[11] Iván Zevallos, Marcelo Assumpc̃ão, A. L. Padilha.
(2009) Inversion of teleseismic receiver function and
magnetotelluric sounding to determine basement depth
in the Paraná basin, SE Brazil. Journal of Applied
Geophysics , 68, 231–242.