Bailey+Groom_1987_SEGAbstract

Magnetotellurics Wednesday Morning Decomposition of the Magnetotelluric ImpedanceTensorWhich is Useful in the Presenceof Channeling M T 1 .1 R. C. Bailey and R. W Groom, University of Toronto, Canada Summary It is believed that there are many occasions when the magnetotellu ric impedance tensor is the result of local galvani c distortion of electric currents which arise from induction in a structure which is approximately two-dimensional on a regional scale. Procedures for ro- tating the impedance tensor, such as minimizing the mean square modulus of the diagonal elements, cannot in general recover the principal axes of induction nor do they re cover the principal impedances but rather linear combinations of them. This paper presents a decomposition of the impedance tensor which separates the effects of cha nnelling from those of induction in these cases. That such a decomposition is possible in principle is implicit in the work of Bahr (1985). When the impedance tensor is actuall y the result of regional one or two-dimensional induction combined with local, frequenc y- independent telluric distortion this method correctl y recovers t,he principal axes of induction and except for a constant, in dependent of frequency, the principal impedances. Also obtained are two parameters (twist and shear) which partiall y describe the current distortion. The method is applied to field data from a site in northern Canada. A normalized residual error is used as an estimator of fit to the model. The ability to invert the data for channel ling parameters which are independent of frequency is a test of the hypothesis of cur rent distortion coupled with two-dimensional induction. The cantly less than that by the standard method. Introduction Z, = R C Zz RT If the Earth has a two-dimens ional conductive structure on a regional scale, then magnet otellurics assumes a source f ield such that e, = e, and h, represent the regional electric and mag netic fields parallel to the Earth’ s surface with respect to the principal axes of the two-dimensional structure. Local inhomogeneities will distort regional telluric currents and prohibit meaningfu l one or two-dimensi onal interpretations even if the large-scal e (regional) geologi- cal structure is not strongly three-dimensional. If these inhomogeneities are s mall compared to the electromag- netic skin-depth in either the host or the heterogeneity, dent distortion of the lar ge scale currents. In this case, the magnetotellur ic impedance tensor can be written, in the principal co-ordinate system, as z= czz where C is a real tensor (matrix) operator of rank two acting on the two-dimensional regional (large-scale) impedance tensor ( Z,). Zhang et a2(1986) have consid- ered the special case where C is due to “ two-dimensi onal channelling” and thus is symmetric. The channelling tensor, C, will be frequency independent over some range. It is worth noting that it is only when C is diagonal that the channelling produces what is normally described as a “static shift” of the apparent resistivities. .4 mathematic al decomposition of the channellin g tensor is developed which separates the effects of channelling into three p hysical ly meaningful parameters (shear, twist and channel ling anisotropy). A non-linear system of equations governin g the relation between the measured impedance tensor and physical parameters (induction and channelling) is derived. A discussion of the method for so lving the non-li near system and the prac- tical appli cation of the method follows, illustrated by actual field data. As well, a comparison of results with the more standard decomposition method is given. Factorization of the Channelling Tensor: The model of current channelling coupled with at most two-dimensi onal induction decomposes the measured impedance tensor, Z, , as where R represents the rotati on from the measu red co-ordinat e system to the principal co-ordinate system. This decomposition is clearly not unique as there are nine real par ameters but only eight real equations. A more useful dec omposition is outlined below. As well, a means of testing the hypothesis of model is given and the det ermined parameters are at least approximat ely interpretable. It can be shown (Bailey and Groom, 1986), for all channelli ng tensors, there is a factorizati on C=gTSA where T, S and A are tensors and g is a scaling factor. The effects of these tensors are illustrated in Figure 1. 154 Magnetotellu rics Wednesday Morning Decomposition of the Magnetotelluric Impedance Tensor Which is Useful in the Presence of Channeling MT R. C. Bailey and R. W Groom, University o f Toronto, Canada Summary It is believed that there are many occasions when the magnetotelluric impedance tensor is the result of local galvanic distortion of electric currents which arise from induction in a structure which is approximately two-dimensional on a regional scale. Procedu res for ro tating the impedance tensor, such as minimizing the mean square modulus of the diagonal elements, cannot in general recov er the principa l axes of induction nor do they recover the principal impedances bu t rather line ar combinations of them. This paper presents a decomposition of the imped ance tensor which separates the effects of channelling from those of induction in these cases. That such a decomposition is possible in principle is implicit in the work of Bahr (1985). When the impedance tensor is actually th e result of regional one or two-dimensional induction combined with local, frequency- independent telluric distortion this method correctly recovers the principal axes of induction an d except for a con s tant , in dependent of frequ ency, the principal impedances. Al so obtained are two parameters (twist an d shear) which partially describe the current distortion. The method is applied to field data from a site in northern Canada. A normalized residu al error i s used as an estima tor of fit to th e model. Th e ability to invert the data for channelling pa.rameters which are indepen dent of frequency is a test of the hypothesis of current distortion coupled with two-dimensional induction . Th e error of f it, by thi s technique, was found to be signifi cantly less than that by the standard method. Introduction I f the Earth has a two-dimensional conductive struc ture on a regional scale, then magnetotellurics assumes a source field such that e r and hr represent the regional electric and mag netic fields parallel to the Earth's surface with respect to the principal axes of the two-dimensional structure. Local inhomogeneities will distort regional telluric cur rents an d prohibit meaningful one or two-dimensional inter preta tions even if the large-scale (regional) geologi cal structure is not strongly three-dimensional. I f these inhomogeneities are small compared to the electromag netic skin-depth in either the host or the heterogeneity, 154 the effect will mostly be a galvanic, frequency indepen dent dis tortion of the large sc ale currents. In this cas e, the magnetotelluric impedance tensor can be written, in the principal co-ordinate system, as where C is a real tensor (matrix) operator of rank tw o act ing on the two-dimensional re gional (large-scale) impedance tensor ( Z2). Zhang et al (1986) have consid ered the special cas e where C is due to "two-dimensional channelling" and thus is symmetric. The channelling tensor, C, will be frequency independent over some range. It is worth noting that it is only when C is di agonal that the channelling produces what is normally described as a "static shift" of the apparent resistivities. A mathematical decomposition of the channelling tensor is developed which sep arate s th e effect s of chan nelling into three physically meaningful parameter s (shear, twist and channelling anisotropy). A non-linear system of equations governing the relation between the mea sured impedance tensor an d physical parameters (in duction an d channelling) is derived. A discussion of th e method for solving th e non-linear system and the prac tical application of the method follows, illustrated by actual field data. As well, a comparison of results with the more standard decomposition method is given. Factorization of the Channelling Tensor: The model of current channelling coupled with at most two-dimensional induction decomposes the mea sured impedance tensor, Zm, as where R represents the rotati on from the measured c o ~ o r d i n a t e system to the principal co-ordinate system. This decomposition is clearly not unique as there are nine real parameters bu t only eight real equations. A more useful decomposition is outlined below. As well, a means of testing th e hypothesis of model is given and the determined parameters are at least approximately interpretable. It can be shown (Bailey an d Groom, 1986), for all C=gTSA where T, S an d A are tensors and 9 is a scaling factor. The effects of these tensors are illustrated in Figure 1.

Bailey+Groom_1987_SEGAbstract

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