Fair Function Minimization for Direct Interpretation of Residual Gravity Anomaly Profiles Due to Spheres and Cylinders

Abstract

A new interpretative approach is proposed to interpret residual gravity anomaly profiles in order to determine the depth, the amplitude coefficient and the geometric shape factor of simple spherical and cylindrical buried structures. This new approach is based on both Fair function minimization and on stochastic optimization modeling. The validity of this interpretative approach is demonstrated through studying and analyzing two synthetic gravity anomalies, using simulated data generated from a known model with different random noises components and a known statistical distribution. Being theoretically proven, this new approach has been applied on three real field gravity anomalies from Sweden, Senegal and the United States. The agreement between the results obtained by the proposed method and those obtained by other interpretation methods is good and comparable.

Appendix

1.1 The Adaptive Simulated Annealing Algorithm

Simulated annealing techniques are implemented for finding global minimum (or maximum) of a target function in parameter space. The techniques are adopted from the physical annealing procedure where a liquid is cooled down in order to obtain a minimum energy formation. These techniques where developed due to the fact that stochastic and non linear systems are extremely difficult to be minimized. Stochastic methods such as adaptive simulated annealing (very fast simulated re-annealing) have been found to be extremely useful tools for a wide variety of minimization problems of large non linear systems.

Adaptive simulated annealing is a powerful stochastic optimisation method applicable to a wide range of problems, especially for multi-modal, discrete, non linear and non differentiable target functions. The major advantage of adaptive simulated annealing over other methods is its ability to avoid becoming trapped at local minima. The algorithm employs a random search, which dos not only accept changes that decrease the objective function, but also accept some changes that increase it, at least temporarily.

We will now illustrate the adaptive simulated annealing random search algorithm for solving the following multi-variables unconstrained problem:

$$ \begin{gathered} Minimize\,\phi (v) \hfill \\ Subject\,to\,v \in {\mathbb R}^{n} \hfill \\ \end{gathered} $$

where the numerical function Φ(v) is called the objective (target) function of the problem and $v = (v_{1} , \ldots ,v_{n} ) \in {\mathbb R}^{n}$ is the vector of model parameters (decision variables).

Using function minimization for illustrative purposes, the algorithm proceeds as follows:

The algorithm

Initialization:/*Definition of initial temperatures, radius of sphere, initial solution, iteration control parameter*/. Let

User-defined control parameters: $\alpha_{0} > 0,r > 0,t^{0} \in {\mathbb R}_{ + }^{n}$

A user-defined initial solution: $v^{0} \in {\mathbb R}^{n}$

A user-defined small positive real number close to zero: $\varepsilon$

An initial number of iterations: i = 0

Main procedure: Repeat until (α _i  < ε)

Step 1:/*Applied a random perturbation to each decision parameter*/

for j = 1 to n do

{

Generate a random number u between 0 and 1: u = random[0, 1]/*by the continuous uniform distribution*/

Set $\theta = \text{sgn} \left( {u - 0.5} \right)\,t_{j}^{i} \,\left[ {\left( {1 + {\frac{1}{{t_{j}^{i} }}}} \right)^{{\left| {2\,u - 1} \right|}} - 1} \right]$/*new random generator*/

Set $\hat{v}_{j} = v_{j}^{i} + \theta \,{\frac{r}{\sqrt n }}$/* r is the radius of sphere centred at the point v ⁱ*/

}

Step 2:/*Acceptance or rejection of the changes made on the decision parameters*/

If $\phi (\hat{v}) < \phi (v^{i} )$ then set $v^{i + 1} = \hat{v}$ and go to step 3

else calculate the probability $p = {\frac{1}{{1 + e^{{{\frac{{\phi (\hat{v}) - \phi (v^{i} )}}{{\alpha_{i} }}}}} }}}$/*Boltzmann distribution*/

if p > γ = random[0, 1] then set $v^{i + 1} = \hat{v}$

else set v ⁱ⁺¹ = v ⁱ

Step 3:/*Modification of temperatures and iteration control parameter*/

for j = 1 to n do

{

Set $t_{j}^{i + 1} = {\frac{{t_{j}^{0} }}{i + 1}}$

}

Set $\alpha_{i + 1} = {\frac{{\alpha_{0} }}{i + 1}},\,i = i + 1$ and go to step 1.

This algorithm has a good robustness and is easy to put into a code. It does not require differentiability of the objective function with respect to the decision variables. Ingber and Rosen (1992), Ingber (1989, 1996), and Sen and Stoffa (1996) provide more details about this algorithm.