Next: Conclusion Up: Optimization of Non-Linear Gravity Previous: Gravity Modeling and Inversion
Next: Conclusion
Up: Optimization
of Non-Linear Gravity Previous: Gravity
Modeling and Inversion
Example 1. We consider at first a synthetic example with the following
features: the medium is divided into 10 blocks, of constant density 
in six blocks, and 
for the remainder four blocks. Each rectangular block is 2km wide
and its height is 1km. We have 10 data points at the surface, which
means that the problem is determined. The location of the blocks are known,
so that the only unknown is the density distribution. Note that this problem
is linear, and that we just want to make a didactical test, and
compare the result via the GSA with a standard method for linear inverse
problems. For the forward and inverse modeling we consider the following
relation
where 
is the data vector (vertical gravity anomaly), H is the kernel matrix
and 
is the model parameter vector (residual density). The vertical gravity
anomaly is given in page 525 by [25]:
where the elements (distances and angles) can be seen in [25],
is the gravitational constant, and x is the location of the observation
point. We have used singular value decomposition, SVD, with all ten singular
values, and the inversion gave exactly the true model: from block 1 (left)
to 10 (right) the density is
, less for the blocks 3, 4, 7 and 8 the density is
. The GSA led to the same values, although requiring a much higher computation
time. The parametrization of the medium in blocks is a common approach
although it has the disadvantage of having in general a large number of
blocks, and of course, there is always the problem of ambiguity. This approach
can easily be extended to 3-D density distributions, as done in [26]
where the Levenberg-Marquardt approach with SVD. As in this example, in
their work the only unknown is the density distribution, i.e., the problem
is linear because the coordinates of all blocks are assumed to be known.
Example 2. In order to show the influence of the statistics for the
visiting function (and of course of the inicial control parameter), used
to find the global minimum of a cost function, we choose a simple gravity
model. Consider a single gravity anomaly with spherical shape. The spherical
shape is particularly useful as a first approximation in the interpretation
of three-dimensional anomalies which are approximately symmetrical [20].
For this body the vertical gravity field 
is given by [25], (pp. 517):
where
is again the gravitational constant, R is the sphere radius, 
is the density contrast, (0,0,z) is the sphere centre coordinates,
and x is the location of the measure. Of course,
will be gravity anomaly and not the total gravity field. Note that the
mass M is equal to 
. For a negative density contrast equal to 
, the maximum of the corresponding gravity anomaly is -1,07mgal
, if the sphere has a radius of 610m, and with its center 1220m
below the surface. Given the mass and the coordinates, the gravity anomaly
is easily obtained by using the above equation (forward modeling). However,
the inverse problem is non-linear if we want to recover the mass and the
location of coordinates (z will be the only unknown coordinate)
from the gravity anomaly distribution. As shows the Figure
1, good solutions are found faster when 
(with 
), according to the proposed limits by Tsallis et al. [27],
and with privileged regions for the initial temperature. We observe that
for this particular case, the GSA algorithm points out new machines, like
the ( 
) one, faster than the usual ones used in the conventional SA (Cauchy and
Boltzmann machines). To obtain the Figure 1, we
limited the number of iterations to 100,000. We exactly recovered the true
model after 350,000 iterations.
Example 3. We took this example from [28]. Consider a set of seven 2-D blocks, where each block is 2km wide, but the heights are different as shown in Figure 2. The upper limit of all blocks are assumed to be known from other sources, like for instance information from a seismic reflection survey. The density contrast is also assumed to be known, and in this case was considered a positive contrast of -0.25g/cm3 . By establishing the bottom limits, as shown as the solid line in Figure 2, we can perform the forward modeling by slightly adapting the modeling equation of the first example, so that the gravity anomaly is shown in Figure 3. For the inverse procedure we adopted that the only unknown are the seven bottom limits, since we have fixed the upper limits, as well as the density contrast. Thus the inverse problem is again non-linear, and we have used the GSA, in order to recover the bottom limits from the gravity anomaly field collected in 14 stations, that is, the problem is overdetermined. Although the inversion via GSA required a large number of simulations, we obtained a result which is closer to true model, in fact closer than the result presented in [28].
For the solution of non-linear inverse problems there are basically two ways, or two family of methods. The methods of linearized inversion are very popular, but in general they require a good starting model (prior information), otherwise the algorithm may be trapped in local minima during the search process, and these minima have no physical sense. Global search models do not require good starting model but they are in general computationally very expensive. A third way that conciliates the two approaches is a combination of two, i.e., a hybrid method is computationally less expensive that the global search methods, since it utilizes for instance the prior information from some linearized method. Liu et al. [29] have combined simulated annealing with the downhill simplex method for the solution of acoustical inversion and residual statics.
