Comparative Research On The 1-D MT Inversion Algorithms

We inverse the electronic parameters underground from the apparent resistivity on the ground in 1-D MT. Inversion is to construct an appropriate object function of errors and to search the minimum of the function, which is a mathematic optimization in essence. The 2 main kinds of the inverse algorithms of 1-D MT are linear and non-linear. The geophysics inversion problems are always non-linear, namely the object functions are non-linear functions.The linear inversion algorithm makes a certain spot as an initial solution and expands the object function to the low order Taylor polynomials near the spot. We can solve the next better solution through the Taylor polynomials, which makes an iterative spot sequence. The limit spot of the convergence spot sequence is the optimization solution. The characters, principles, calculating results and calculating speed of some linear inversion algorithms such as gradient algorithm, Gauss-Newton algorithm, Marquardt algorithm, generalized inverse matrix algorithm, Newton-Raphson algorithms, are compared mutually. The final solution depends on how much the initial solution is, because the next new solution is derived from the last one, which results in that the final solution is the local extreme spot, instead of the global optimization solution in maximal probability. The accuracy and reliability of the inversion results decreases.Though the non-linear algorithm also makes an initial spot, the next new solution is searched randomly and directly in the solution space, instead of searching through the Taylor polynomials of the object function, thus the next new solution does not depend on the last old one. Whether the new solution is accepted is determined by an appropriate condition, after that we can get the iterative spot sequence. The limit spot of the convergence spot sequence is the optimization solution. The final solution does not depend on the start solution or the probability of dependence is low, which makes the probability of that the solution of the non-linear algorithms is the global optimization is more than the one of the linear algorithms is. The characters, principles, calculating results and calculating speed of some linear inversion algorithms such as Monte Carlo, simulated annealing and genetic algorithms are compared mutually and with the linear algorithms'. A new algorithm combining the non-linear ones to the linear ones and the specific improvements on the efficiency of the reproduction, crossover, mutation operators are presented in this paper.Besides the pure mathematics views on searching the minimum of the object function of errors, there are other algorithms such as OCCAM algorithm which fits to the real distribution of the electronic parameters underground satisfied with a certain constraint condition. OCCAM algorithms construct some physical quantities describing the distribution of the electronic parameters and search its minimum among the models which fit to an acceptant precision of the errors object function value, not just searching the minimum of the errors object function. This inversion algorithm is more fit to the reality of the layers than the ones which just search the minimum of the object function. It is from the reality, instead of pure mathematics theory and abandoning the reality. It gets the faster calculating speed and better solution in combining the mathematics theory to the reality to inverse. The OCCAM algorithm use the fact that the real electronic parameters of the layers change continuously instead of abrupt change to construct the roughness of the distribution of the parameters and search the minimum of the roughness function among the models whose object function fit to a certain accuracy. Along the views of OCCAM, we can present other physical value to describe the real layers. The theory and calculating efficiency of OCCAM and the algorithm which just search the minimum of the object function are compared mutually.