Regularized conjugate gradient method

parameter while keeping the others constant. From the forward modeling study, it is found that the imaginary resistivity has the biggest change when changing parameters.

That is why we can calculate the model parameter based on the imaginary resistivity data only.

2.2 Regularized conjugate gradient method

The purpose of inversion is to recover model parameters from measured data. An inverse problem can be formulated as the solution of the operator equation:

(2.1) where is some function (or vector) describing the model parameters, d is the data set and A(m) is the operator of forward modeling.

The problems in this research are ill-posed and unstable. To find the stable solution for the minimization problem, we have to consider the regularized minimization of the Tikhonov parametric functional:

(2.2) where is some stabilizing functional. The regularization ensures that a unique and stable solution is obtained from the measured data. The Regularized Conjugate-Gradient (RCG) method is utilized in this research period. Derivations and numerical schemes of both methods can be found in Zhdanov (2002).

The RCG method is an iterative solver, which updates the model parameters on each iteration using conjugate gradient directions ̃, according to the following formula:

̃ ̃ (2.3) First, the direction of regularized steepest ascent is used:

̃ (2.4) The next direction is a linear combination of the regularized steepest ascent in this step, and the conjugate gradient direction in the previous step:

̃ ̃ (2.5) On the (n+1) step

̃ ̃ (2.6) The steps ̃ are selected based on the minimization of the parametric functional:

̃ ̃ ̃ (2.7) minimization of this functional gives the following best estimation for the length of the step using a linear line search:

̃

̃ ̃

̃

‖ ̃ ‖ ‖ ̃ ‖ (2.8) The coefficients are determined by the following formula:

‖ ‖ ‖ ‖ (2.9) The final numerical scheme for the RCG method can be summarized as follows:

(2.10)

The initial regularization parameter α is calculated using the equation 2.11,

^{‖ ‖}

‖ ‖ (2.11) We also implement the adaptive regularization, therefore, becomes

(2.12) where:

(2.13) The iterative process is terminated when the misfit reaches the given level or the iterative number reaches the maximum.

‖ ‖ (2.14) In addition, I also used the extensive line search method in my study. The basic idea of this method has three steps. First, find all possible value ranges of the model parameters and determine and value; secondly, divide the possible parameter values and calculate misfit by using all possible parameters combinations; finally, find the minimum misfit value and determine the model parameters. Both numerical schemes have been implemented using a code written in MATLAB.

2.3 Inversion: synthetic data

The used synthetic data set is obtained considering one disseminated phase that occurs at one grain size. The spherical model has proven to offer good results in a previous study (Phillips, 2010). In this example, I focus on the randomly oriented ellipsoidal GEMTIP model, the grains of the inclusions are approximated in shape by a prolate spheroid. The value of ellipticity is 4. It comprises a rock matrix with the

resistivity, 200Ωm. The volume fraction of pyrite in the sample is 10%. The known values of the relaxation parameter and time constant are 0.8 and 0.1, respectively. The equal radius of the approximated prolate spheroid is 1mm. All the parameters used in the two-phase synthetic model are summarized in Table 2.6. With these variables defined, the complex effective resistivity can be determined by two-phase spherical GEMTIP model. Figure 2.6 shows both the real effective resistivity and the imaginary effective resistivity plotted as a function of frequency. The two data sets shown are observed and predicted data. The observed data came from using the true model parameters value, while the predicted data were determined by minimizing the parametric functional using the regularized conjugated gradient method. It is obvious that the original observed data fit the predicted data very well. The panel (c) shows the plot of misfit against iteration number. After 100 iterations, the correct model parameters were recovered, the final misfit is 0.01%.

Table 2.6 GEMTIP two-phases synthetic data inversion result Parameter True model Initial model Two-phases

200 200 -

e 4 4 -

10 50 10

C 0.8 0.1 0.8

0.1 0.5 0.1

Figure 2.6. Two-phases synthetic data inversion result. (a) Real effective resistivity. (b) Imaginary effective resistivity. (c) Misfit vs. iteration.

2.3.2 Model 2: three-phases ellipsoidal GEMTIP model

Figure 2.7 represents both the observed and predicted resistivity curves for three- phases ellipsoidal GEMTIP model. The synthetic data are obtained from the forward modeling, considering a model formed by a homogeneous host rock filled with two types of grains and two grain sizes. All the parameters are summarized in Table 2.7. It clearly shows that there are two maximum IP responses in the imaginary part for the three-phases model. The observed curve came from using the true model parameter value, while the predicted data were obtained by minimizing the misfit functional using the extensive search method. The predicted data fit the original observed data very well. The final misfit is 0.3%.

Figure 2.7. Three-phases synthetic data inversion result. (a) Real effective resistivity. (b) Imaginary effective resistivity.

Table 2.7 Three-phases synthetic data inversion result Parameter True

model Three-phases

result Three-phases with 5% noise result

200 200 202

1 1 1.9

4 3.9 3.7

15 15 13

10 10 11

0.9 0.88 0.88

0.9 0.88 0.88

0.01 0.009 0.009

0.9 0.88 0.89

2.3.3 Model 3: three-phases ellipsoidal GEMTIP model with 5% random noise

Now I have added 5% random noise to the data while all the parameters are kept the same as in the previous three-phases synthetic model. The observed and predicted resistivity variations with frequency are illustrated in Figure 2.8. The recovered parameters are shown in Table 2.7. The final misfit is 4.3%. From the figure, the predicted curve fit the observed data very well. The comparison results between three- phases synthetic data and three phases synthetic data with noise are shown in Figure 2.9.

From the synthetic data with noise study, I conclude that the inversion code is reliable for the work in the following chapter.

Figure 2.8. Three-phases synthetic sample with 5% noise inversion result. (a) Real effective resistivity. (b) Imaginary effective resistivity.

Figure 2.9. Comparison inversion result of three-phases synthetic data and three-phases synthetic data with 5% noise. (a) Real effective resistivity. (b) Imaginary effective resistivity.