Pravin K. Gupta

Straightforward inversion of vertical electrical sounding data

A straightforward inversion scheme (SIS) has been developed to interpret vertical electrical sounding data. This scheme does not require quasi-linearization of the inverse resistivity problem and thereby dispenses with the iterative process and the necessity of guessing the number of layers and their resistivities and thicknesses. The entire solution domain is divided into uniform thickness layers, whose scale must be judiciously selected for the desired resolution. The apparent resistivity formula can now be posed as an underdetermined matrix equation whose minimum norm solution is downward continued to obtain the reflection coefficients which, in turn, yield the vertical resistivity distribution. A recurrence relation has been developed especially for this purpose. In general, when data are expected to be noisy, a regressed minimum norm solution is used. Exhaustive tests of the algorithm have established its numerical efficiency. Results of six typical synthetic models, representing diverse geological conditions, as well as results of two field examples are included to demonstrate this claim.

Fast computation of Hankel Transform using orthonormal exponential approximation of complex kernel function

The computation of electromagnetic (EM) fields, for 1-D layered earth model, requires evaluation of Hankel Transform (HT) of the EM kernel function. The digital filtering is the most widely used technique to evaluate HT integrals. However, it has some obvious shortcomings. We present an alternative scheme, based on an orthonormal exponential approximation of the kernel function, for evaluating HT integrals. This approximation of the kernel function was chosen because the analytical solution of HT of an exponential function is readily available in literature. This expansion reduces the integral to a simple algebraic sum. The implementation of such a scheme requires that the weights and the exponents of the exponential function be estimated. The exponents were estimated through a guided search algorithm while the weights were obtained using Marquardt matrix inversion method. The algorithm was tested on analytical HT pairs available in literature. The results are compared with those obtained using the digital filtering technique with Anderson filters. The field curves for four types (A-, K-, H-and Q-type) of 3-layer earth models are generated using the present scheme and compared with the corresponding curves obtained using the Anderson sc heme. It is concluded that the present scheme is more accurate than the Anderson scheme

Straightforward inversion of MT data using a normalized impedance function

We present a new algorithm for 1D magnetotelluric (MT) data inversion. It inverts a normalized impedance response function derived from the classical Cagniard impedance function. The scheme transforms the nonlinear problem of estimating layer resistivities and thicknesses into a linear problem of estimating the coefficients of power series of the new response function. This is achieved by working with a model where each layer has a thickness of constant penetration. The first coefficient of the series provides top-layer resistivity, which, in conjunction with the constant penetration parameter, then provides the layer thickness. The scheme employs a recurrence relation developed between the coefficients of the power series of two successive layers. This relation is used to continue downward and estimate the remaining layer resistivities and thicknesses. The scheme has been tested on a synthetic model and on three well-studied data sets relating to deep, intermediate, and shallow exploration.

Normalized impedance function and the straightforward inversion scheme for magnetotelluric data

This paper investigates the performance of normalized response function obtained by normalizing the Cagniard impedance function by a suitable factor and then rotating the phase by 45‡ to make it purely real for homogeneous half-space and equal to the square root of the half-space resistivity. Two apparent resistivity functions based on respectively the real and imaginary parts of this response function are proposed. The apparent resistivity function using the real part contains almost the same information as that yielded by the Cagniard expression while the one using the imaginary part qualitatively works as an indicator of the number of interfaces in the earth model. The linear straightforward inversion scheme (SIS), developed by the authors employing the concept of equal penetration layers, has been used to validate the proposed apparent resistivity functions. For this purpose, several synthetic and field models have been examined. Five synthetic models are studied to establish the veracity of the new functions and two well-studied published field data sets are inverted through SIS for comparison. We noticed that the new function and SIS compliment each other and lead to better understanding of the data information and model resolution.

Straightforward inversion scheme (SIS) for one-dimensional magnetotelluric data

This paper presents a Straightforward Inversion Scheme (SIS) for interpreting one-dimensional magnetotelluric sounding data. The basic steps of SIS are (i) parameterization of the layered model such that the layer thickness, expressed in units of its skin depth, is a constant (α); (ii) expansion of the reflection function at each interface as a power series in parameter u = exp(-2(1 +j)α√f);(iii) development of a recurrence relation between the coefficients of the same powers ofu in the power series of reflection functions of any two successive layers; (iv) estimation of the impedance power series coefficients using regressed minimum norm estimator; and (v) evaluation of layer resistivities and thicknesses using the inverse recurrence relation. The power of SIS is established by inverting four synthetic data sets and two field data sets. The effect of noise is extensively studied on a synthetic data set, deliberately corrupted with increasing levels of Gaussian random noise up to 25%. It is found that the scheme can retrieve broad features of the true model even with noise levels as high as 25%. On the basis of findings of different experiments conducted on SIS, it is concluded that SIS is an efficient, robust algorithm with high resolving power. Further, being linear, it is non-iterative and it dispenses with the requirement of having to choose an initial guess model.

Geoelectric structure estimated from magnetotelluric data from the Uttarakhand Himalaya, India

Geoelectric strike and resistivity structure of the crust have been estimated from 37 magnetotelluric (MT) data sites along a profile from Roorkee to Gangotri in Uttarakhand Himalaya. Impedance decomposition schemes based on Bahr’s, Groom Bailey and Phase tensor were implemented in a MATLAB code for the average strike estimation. Geoelectric strike direction varies with period as well as in different litho-tectonic units along the profile. In the period band from 1 to 100 s average geoelectric strike in the southern end of the profile (Indo-Gangetic Plains) is N79°W, which is slightly rotated to the north in the Lesser Himalayan region and becomes N68°W whereas it is N81°W in the Higher Himalayan region. However, average strike is stabilized to N77°W for the entire profile in the long period band (100–1000 s). Geoelectrical structure of the crust has been obtained along the profile by 2D inversion of MT data. Major features of 2D resistivity model are: (i) southern part of the model is a low resistivity (<50 Om) zone at shallow depth (5–7 km) representing the loose sediments of the Indo-Gangetic Plains (IGP), whose thickness increases in the south; (ii) highly resistive (>1000 Om) layer below the IGP sediments is the basement rock, representing the resistivity of the top of the subducting Indian Plate; (iii) the Main Boundary Thrust (MBT) and the Main Central Thrust (MCT) zones can be seen in the electrical image. However, the Himalayan Frontal Thrust (HFT) could not be resolved and (iv) a low resistivity (<10 Om) feature in the MCT zone extending to the depth of 30 km is delineated. This low resistivity could be due to fluid-filled fractured rock matrix or partial melt zone. Hypocenters of many earthquakes are concentrated along the boundary of this low resistivity zone and relatively high resistivity blocks around it. The resulted model supports flat-ramp-flat geometry of the Main Himalayan Thrust along which the Indian Plate is subducting.

EM2INV — A finite difference based algorithm for two-dimensional inversion of geoelectromagnetic data

AbstractThe paper presents an efficient finite difference based 2D-inversion algorithm, EM2INV, for geoelectromagnetic data. The special features of the algorithm are• optimal grid generation based on grid design thumb rules,• finite domain boundary conditions,• interpolation matrix that permits generation of response at observation points different from grid points,• Gaussian elimination forward matrix solver, that enables reuse of already decomposed coefficient matrix,• super-block notion that reduces the number of blocks with unknown resistivities and, in turn, the size of Jacobian matrix and• bi-conjugate gradient matrix solver for inverse problem which circumvents the need of explicit Jacobian matrix computation. The algorithm is tested rigorously by setting up exercises of diverse nature and of practical significance. The stability of the algorithm is established by inverting the synthetic response corrupted with Gaussian noise. The inversion experiments are aimed at studying• relative performance of response functions,• inversion quality of E- and B-polarization data,• efficacy of single and multi-frequency data inversion,• minimum number of frequencies and observation points needed for successful data inversion. It has been observed that the Magneto-telluric data deciphers better the vertical position of the target and Geomagnetic Depth Sounding data deciphers the horizontal variations in a better way. The conductive and resistive bodies are better resolved by inversion of E- and B-polarization data respectively. The results of multi-frequency inversion imply that the increase in the number of frequencies does not necessarily enhance the inversion quality especially when the spread of observation points is sufficiently large to sense the target. The study of a minimum number of observation points highlights the importance of single point inversion that furnishes useful information about the inhomogeneity.

A MATLAB based 3D modeling and inversion code for MT data

Abstract The development of a MATLAB based computer code, AP3DMT , for modeling and inversion of 3D Magnetotelluric (MT) data is presented. The code comprises two independent components: grid generator code and modeling/inversion code. The grid generator code performs model discretization and acts as an interface by generating various I/O files. The inversion code performs core computations in modular form – forward modeling, data functionals, sensitivity computations and regularization. These modules can be readily extended to other similar inverse problems like Controlled-Source EM (CSEM). The modular structure of the code provides a framework useful for implementation of new applications and inversion algorithms. The use of MATLAB and its libraries makes it more compact and user friendly. The code has been validated on several published models. To demonstrate its versatility and capabilities the results of inversion for two complex models are presented.

Crustal electrical conductivity of the Indian continental subduction zone: New data from the profile in the Garhwal Himalaya

We present the results of studying the geoelectrical structure of the zone of continental subduction of the Indian lithospheric plate within the Gahrwal Himalaya. In the framework of the Russian–Indian project, the data of the broadband magnetotelluric soundings conducted by the Indian Institute of Technology Roorkee on the regional profile across the structures of the orogen were expanded, processed, and interpreted by the new program tools adapted for the measurements in the mountain conditions and for the presence of industrial noise. The constructed model of the deep electrical conductivity cross section for Garhwal revealed its two-dimensional (2D) features and more accurately delineated the location of the midcrustal conductor associated with the ramp structure of the detachment plane. The correlations with the regional distribution of the earthquake hypocenters and the seismotomographic images suggest a common, fluid-related nature of the seismic and geoelectrical anomalies in the crust of the Garhwal Tectonic Corridor and enabled the identification of the seismogenerating zones. Among the data of the expanded profile set of magnetotelluric and magnetovariational transfer functions, the response of a poorly explored deep conductive body is revealed. This object is located east of the profile and is probably associated with the activation of the ancient trans-Himalayan cratonic structures which prepares the segmentation of the Himalayan arc.

Reply by the authors to the discussion by Gildas Omnes

We are intrigued by Dr. Gildas Omnes’ reference to “misleading statements” in our above mentioned paper. It is a fact universally acknowledged that most inverse problems in geophysics are underdetermined and ill posed, possessing no unique solution. One is therefore left with the only option of seeking the “best” solution by heuristically minimizing certain norms or their weighted combinations. The “goodness” of the solution is qualified by the “closeness” of the inverse solution to the actual model parameters which can be tested using “difficult to invert” standard models. This is the basic premise on which the paper is based with claim to “goodness of the solution” and not uniqueness.