A Method of Estimating the Block-Like Body Under the Ground from the Polarization Ellipse of the Magnetic Field

The feasibility of using the broad-band Magnetotelluric (MT) me­ thod for locating a two-dimensional (2-D) block-like body is studied by means of numerical models. An example of its applicability for locating a block-like anomalous body indicates that the sign reversal frequency of the tilt angle, or the ellipticity responses of magnetic polarization relates to the depth of the center of the anomalous body, and also to the resistivity contrast of the anomalous body to the host rock, based on the relationship of the few nomograms are developed, that provide a new approach to the interpretation of the magnetic polarization data.


INTRODUCTION
The application of numerical techniques to magnetotelluric modelling has been developed in recent years.The majority of these techniques are based on the finite element method (Rijo, 1977;Kaikkonen, 1977) and the finite differ ence method (Jones and Pascoe, 1971;Brewitt-Taylor and Weaver, 1976).In this work the finite element technique is used to compute the magnetotelluric responses ov er a two-dimensional block-like body.The element matrix equa tions are derived from the minimization of the electromagnetic energy (Coggon,  1971 ).The boundary values used in the global matrix equation are the same as those used by Pascoe and Jones (1972).The global matrix equation shown in the algorithm is solved using Gaussian elimination.
Using this algorithm, a series of theoretical MT response distributions have been computed for a variety of resistivity contrasts and burial depths.These responses of apparent resistivity or phase versus frequency are presented for both to TE (Transverse Electric) mode and the TM (Transverse Magnetic) mode.Finally the tilt angle and ellipticity of the magnetic field polarization (Smith and Ward, 1974) versus the frequency are evaluated.
From these pseudosections, it appears that the tilt angle and ellipticity will give more information abol,lt buried block-like anomalous bodies than the con ventional apparent resistivity and phase pseudosections will starting from the pseudosections for the tilt angle and ellipticity versus frequency, we construct the nomograms for interpretation of the magnetic polarization data.The prin cipal steps in this procedure are: (1) Preparing different pseudosections for the tilt angle and ellipticity versus frequency by (a) Varying the depth of the center of the anomalous body and computing the MT response, and (b) Varying the resistivity contrast between the anomalous body and the host rock and comput ing the MT response; (2) Constructing nomograms based on the relationship between the sign reversal frequency versus resistivity contrast and the depth of the buried body.A quantitative interpretation of the magnetic polarization response for these anomalous bodies using the nomograms will also be given in this study.

MODEL RESULTS
In this section, the results of modelling for two-dimensional block-like anomalous bodies are presented.From the comparison of the results shown in Figs. 1 and 2, it indicates that there is no obvious characteristic to be used to identify these type of underground structure, especially for the conductive overburden.for the TE mode can be used to identify the existence of the "block-like" body.

a. Effects of the overburden
Fig. 4 has the same type of geoelectric models as Fig. 2. As pointed out by Rijo  1977)) the presence of flatly-layered overburden has little effect on the value of the sign reversal frequency, the frequency of the� nodal point (circle mark as shown in Fig. 4), the ellipticity or tilt angle anomalies.an anomalous body will result in the decrease of the sign reversal frequency of ellipticity (or tilt angle).The linear frequency (in log space) dependence of the depth of the anomalous body can be successfully used to locate the underlying body center provided that the resistivity contrast is known in advance for no overburden case.

c. Effect of the size
The results shown in Figs. 3 and 6 indicate that the sign reversal frequency of ellipticity (or tilt angle) depends only on the center of the anomalous body, and is independent of the size of the "block-like" body.

d. Effect of resistivity contrast
As shown in Figs. 3 and 7, when the resistivity contrast of the anomaly to the host rock is a cons tant, the frequency at which the ellipticity (or tilt angle) reverses its sign depends on the skin depth of the host rock.As noted therein, the response attributable to a block-like anomalous body buried in a half space could be discriminated when the tilt angle and ellipticity  An example of a step-by-step interpretation procedure will be given in this section for the MT data by means of the newly-developed nomograms.The resistivity of the host the rock does affect the sign reversal frequency.First, we estimate the resistivity of the host rock from the apparent resistivity profile.In our example, •the resistivity of host rock is 200 ohm-m.
If we have the sign reversal frequency from the ellipticity pseudosection only, then nomogram sheet 1 is available, and the following steps are suggested.
(1) In this example, the sign reversal frequency estimate for the ellipticity is 0.06 Hz.On the transparent bilogarithmic paper, draw a horizontal line through a point in the vertical coordinate of the sign reversal frequency (0.06 Hz), and set the origin to be at 0.01 Hz, as shown in If we could find the sign reversal frequencies for both tilt angle and ellipticity from the observed pseudosections, then nomogram sheet 2 is available, the following steps are a generalized approach for this study.

Fig. 1
Fig.1illustrates the pseudosections of the apparent resistivity and phase versus frequencies for a case with a two-dimensional block-like body of a 3 x 4 unit area (2 km per unit).In this model the depth of the buried body and the resistivity contrast between the buried body and the host rock, have been assigned the values of 2 units and 20 ohm-m/100 ohm-m respectively.Fig. la is the geoelectric model, Fig. 11:J is the apparent resistivity plots f or the TE and TM modes and Fig. le is the phase anomalies for the TE and TM modes.From these pseudosections, anomalies are presented as conductivity inhomogeneity exists.Fig; 2 shows the MT responses of a horizontal layer over the model shown in Fig. 1.The difference between the model shown in Figs.2a and 2b is that the former model has a resistive overburden while the latter has a conductive overburden.From the comparison of the results shown in Figs.1 and 2, it indicates that there is no obvious characteristic to be used to identify these type of underground structure, especially for the conductive overburden.Fig. 3 shows the same geoelectric model as shown in Fig. la, it is obvious

Fig. 3
Fig.1illustrates the pseudosections of the apparent resistivity and phase versus frequencies for a case with a two-dimensional block-like body of a 3 x 4 unit area (2 km per unit).In this model the depth of the buried body and the resistivity contrast between the buried body and the host rock, have been assigned the values of 2 units and 20 ohm-m/100 ohm-m respectively.Fig. la is the geoelectric model, Fig. 11:J is the apparent resistivity plots f or the TE and TM modes and Fig. le is the phase anomalies for the TE and TM modes.From these pseudosections, anomalies are presented as conductivity inhomogeneity exists.Fig; 2 shows the MT responses of a horizontal layer over the model shown in Fig. 1.The difference between the model shown in Figs.2a and 2b is that the former model has a resistive overburden while the latter has a conductive overburden.From the comparison of the results shown in Figs.1 and 2, it indicates that there is no obvious characteristic to be used to identify these type of underground structure, especially for the conductive overburden.Fig. 3 shows the same geoelectric model as shown in Fig. la, it is obvious

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that the characteristics of sign reversal of the tilt angle and ellipticit y anomaly

I
Fig. 2. (a ) Effect of a highly resistive flat layer over the model shown in Fi g. la.(b) Effect of a highly conductive flat layer over the model shown in Fi g. la.

Fig. 7 .
Fig. 7. (Continued) Fig. lOa.(2) Superimpose the transparent paper on Fig. 9a and move one sheet with respect to the other, keeping the vertical axes parallel with each other until the origin of the transparent paper is coincident with a point 200, the value of the resistivity of the host rock on the vertical coordinate of nomogram sheet 1, as shown in Fig. 1 la.The horizontal line will cross a set of possible resistivity contrast curves related to different depths.If the information of the resistivity contrast is available, then the depth of the anomalous buried body can be estimated.

( 1 )
Fig. 8.The modelling results.(a) The relationship betwee_n �h � de�th o� the center of the anomaly and the sign reversal frequency of the ellipticity with differ ent resistivity contrasts.(b) The relationship between the sign reversal frequency of the tilt angle and ellipticity for different resistivity contrasts and the depth of the center of the anomalous body.

Folliplicity:Fig. 10 .
Fig. 10.Example for illustrating the use of nomograms. (a) When sign reve � sal frequency from the ellipticity pseudosection is available only.(b) When both sign reversal frequencies for both tilt angle and ellipticity are available.
Fig. 11.(a) The interpretation procedure when the sign reversal frequency of the ellipticity is available only.(b) The interpretation procedure when the sign reversal frequencies of the tilt angle and ellipticity are known.