P K Gupta
Articles written in Journal of Earth System Science
Volume 89 Issue 1 March 1980 pp 43-49
The paper presents a computational algorithm designed for efficient modelling of apparent resistivity over complex geological structures, using finite element method. The algorithm can be used to study variations of apparent resistivities using any electrode configuration at any point on the earth’s surface, not necessarily regular. A Schlumberger apparent resistivity sounding curve over a buried anticline, is presented here as an example and compared with the corresponding analytical curve, to demonstrate the correctness of the FEM algorithm.
The various potential derivatives required for the computation of apparent resistivities evaluated through different electrode configurations have been obtained by calculating the ‘influence coefficients’ using reciprocal theorems, an approach successfully applied in structural engineering. In essence, a set of self balancing nodal currents, obtained from the appropriate derivative(s) of the shape functions of the elements contributing to the point of observation, is applied as the load vector.
The resulting quantities corresponding to the potential distribution in traditional finite element method, then, turn out to be the potential derivatives at the point of observation for different positions of the current electrodes. These are known as influence coefficients.
The continuum nature of the domain beyond the region of interest has been modelled by using ‘infinite elements’ across which the potential is assumed to decay exponentially.
Volume 103 Issue 3 September 1994 pp 401-411
We estimate the distribution of slip in the dip section of the causative fault for the 1905 Kangra earthquake by applying the minimum norm inversion technique to differences in pre- and post-earthquake levelling data collected along the Saharanpur-Dehradun-Mussoorie highway. For this purpose it is assumed that the causative fault of the 1905 Kangra earthquake was planar with a dip of 5° in the northeast direction and that it had a depth of 6 km at the southern limit of the Outer Himalaya in Dehradun region. The reliably estimated maximum slip on the fault is 7.5 m under the local northern limit of the Outer Himalaya. Using the inverted slip distribution we estimate that the maximum permanent horizontal and vertical displacements at the surface due to the Kangra earthquake were about 4 m and 1.5m respectively. The maximum transient displacements at the surface should have exceeded these permanent displacements. These estimates of maximum slip on the causative fault and the resultant maximum permanent and transient displacements at the surface during the Kangra earthquake may be taken tentatively as being representative of the great Himalayan earthquakes.
Volume 105 Issue 4 December 1996 pp 413-429
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 of
Volume 107 Issue 2 June 1998 pp 149-154
This paper reviews the validity of earlier models obtained after quantitative interpretation of GDS data and presents a fresh model using the inversion scheme EM2INV. The 2-D inversion of data is more objective than the earlier interpretation performed by using trial and error method. The inversion results indicate that the present model differs from the earlier ones. The reason could be that available GDS data are sufficient only for deriving the horizontal variation of subsurface resistivity. In order to study the vertical resistivity variation additional MT sounding data would be required. It would therefore be desirable to carry out MT survey in the specified area. A more comprehensive/appropriate model could be derived from joint inversion of GDS and MT data.
Volume 108 Issue 4 December 1999 pp 233-253
The 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.
Volume 114 Issue 5 October 2005 pp 523-531
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.
Volume 117 Issue 3 June 2008 pp 189-200
Magnetotelluric investigations have been carried out in the Garhwal Himalayan corridor to delineate the electrical structure of the crust along a proﬁle extending from Indo-Gangetic Plain to Higher Himalayan region in Uttarakhand,India.The proﬁle passing through major Himalayan thrusts:Himalayan Frontal Thrust (HFF),Main Boundary Thrust (MBT)and Main Central Thrust (MCT),is nearly perpendicular to the regional geological strike.Data processing and impedance analysis indicate that out of 44 stations MT data recorded,only 27 stations data show in general,the validity of 2D assumption.The average geoelectric strike, N70°W, was estimated for the proﬁle using tensor decomposition.2D smooth geoelectrical model has been presented,which provides the electrical image of the shallow and deeper crustal structure.The major features of the model are (i)a low resistivity (> 50𝛺 m),shallow feature interpreted as sediments of Siwalik and Indo-Gangetic Plain,(ii)highly resistive (< 1000 𝛺m)zone below the sediments at a depth of 6 km,interpreted as the top surface of the Indian plate,(iii)a low resistivity (> 10𝛺 m) below the depth of 6 km near MCT zone coincides with the intense micro-seismic activity in the region. The zone is interpreted as the partial melting or ﬂuid phase at mid crustal depth.Sensitivity test indicates that the major features of the geoelectrical model are relevant and desired by the MT data.