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DATA ACQUISITION

Electrode Arrays and Geometric factors

Field DataBasic Principles The relationship between the electrical resistivity, current and the electric potential is governed by Ohm's law

To calculate the potential in a continuous medium, the form of Ohms law combined with conservation of current as given by Poisson's equation is normally used

= Resistivity = Potential jc = Current DensityThe current and voltage measurements are then converted into an apparent resistivity

K= Geometric Factor5What is an inverse problem?

Model mData dForward Problem Inverse ProblemGm = dm= G-1 d

Arrangement of model blocks

Least Squares

Weighted least-squares

Singular value decomposition (SVD)

Damped least-squares (Marquadt-Levenberg) method

Smoothness constrained (Occam) inversion

Inversion with a priori information

The simulated annealing technique

Gauss-Newton & Quasi- Newton methodsAlgorithms used for inversion of 2D dataLeast Squares / Gauss-Newton method It was quite sensitive into the choice of the initial parameter vector

If initial property choice is too far its leading to divergence

Sometimes assumption is linear

Weighted least-squares

Does not improve the inverse problem considerably

Take the observation errors and constraints

Non-linear least-squares methods have been reported to be quite sensitive into the choice of the initial parameter vector x0, and as suggested by Box and Kanemasu (1972), if the initial parameter choice is poor it has sometimes been found that severe oscillations of the solution vector occur through the iterations.

The initial property choice is too far away from the true solution, the correction vector dx may become too large and hence the assumption of linearity becomes invalid, leading to divergence

Sometimes the linearity assumption causes the procedure to choose the local minimum nearest to the initial model, hence rendering the results dependent on the initial parameter choice

Weighted least-squares does not improve the conditioning of the inverse problem considerably, but can take into account the observation errors as well as other physical constraints (i.e. positive properties).

9Singular Value Decomposition (SVD)Ill-condition of the problem due to linear combinations of unknown parameters

Presence of data noise contaminate the entire solution

Identify these combinations and ignore them

SVD regulates the inverse problem into decompose The ill-conditioning is due to linear combinations of the unknown parameters which are weakly represented in the measurement system. Under the presence of data noise, these weakly represented linear combinations of parameters not only fail to convey any useful information, but also "contaminate" the entire solution

One way to avoid this problem is to identify these combinations (by getting a numerical estimate of the system's eigenvalues) and subsequently ignore them. This can be achieved by the use of the SVD technique

10Damped least-squares (Marquadt-Levenberg) method Introduce constraints into the optimization instead rejecting small values

It is more used due to its stability and fast convergence

The main disadvantages are

Final results strongly dependent on initial model choice

Produce unnecessarily complex solutions One way for treating ill-conditioning is to introduce constraints into the optimization procedure instead of directly rejecting the "small" eigenvalues

Marquadt's algorithm has been used extensively for the solution of the 2-D inverse resistivity problem mainly due to its stability and its fast convergence (quadratic)

Two main disadvantages of the algorithm have been reported:

a) The final result is strongly dependant on the initial model choice xo (Smith and Vozoff, 1984) and

b) In some cases the technique produces unnecessarily complex solutions unnecessary in the sense that although they are mathematically correct, they are not physically reasonable

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Inversion results using Marquadt algorithm

Smoothness constrained (Occam) inversion Introduced smoothness constraint

It stabilize poorly conditioned problems

Produce a simplified model with exact representation of subsurface

Its guarantees stability

Produce the model based on user but not by initial guess

Another way to tackle the instability of the resistivity inverse problem is to impose a smoothness constraint

The use of such a constraint to stabilize poorly conditioned problems belongs to a general class of methods known as regularization techniques (Tikhonov, 1963)

The technique has been proposed for geophysical purposes by Constable et al. (1987) who called it Occam's inversion

It is hoped that it will produce a simplified model which is a reasonable representation of the subsurface

The smoothness constraint guarantees stability and, most importantly, produces a model which is based on a characteristic that the user has chosen (the pattern or nature of the smoothness) and not on some arbitrary initial guess

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Inversion Results using Occam scheme

The Simulated Annealing technique SA is a stochastic non-linear optimization technique

Also known as Monte-Carlo methods

Based on ides of statistical mechanics, thermodynamics and probability

Its progress is not determined by conditioning of the controlling matrices

It is slower than the techniques involving matrix inversionSimulated annealing (SA) is a stochastic non-linear optimization technique which belongs to the class of methods known as Monte-Carlo methods

It is based on ideas from statistical mechanics, thermodynamics and multivariable probability theory

An advantages of the technique is that its progress is not determined by the conditioning of the controlling matrices, and that it enables the search of a greater proportion of the possible model parameter configurations (Dittmer and Szymanski, 1995)

On the other hand, it has to be noted that it is significantly slower (almost by a factor of 30 ) than the techniques involving matrix inversion

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Inversion Result using SA

For small resistivity contrasts there is no significant difference between two

For large resistivity contrast Gauss-newton gives accurate results

Combined inversion is faster and gives satisfactory resultsComparison of Gauss-newton and quasi-newton methodsFor models with small resistivity contrasts, thereare no significant differences in the results obtained with the GaussNewton and quasi-Newton methods

For large resistivity contrasts, the GaussNewtonmethod gives significantly more accurate results than the quasi-Newton method

The combined inversion method is faster than the GaussNewton method, it represents a satisfactory compromise between speed and accuracy for many data sets

17It is believed that currently the smoothness constrained inversion is the most preferable technique for practical data interpretation. It comprises numerous advantages:

a) it is physically reasonableb) it is robustc) it can cope well with noise and does not produce artifacts d) it produces solutions which have properties that the interpreter has chosen and are not a product of an arbitrary initial choice

For these reasons the smoothness constrained inversion will choose as the platform for developing a 2-D inversion algorithmConclusion