Magnetic field transforms with low sensitivity to the direction of source magnetization and high centricity

Magnetic data interpretation faces difficulties due to the various shapes of magnetic anomalies and the positions of their extrema with respect to the causative bodies for different directions of the source magnetization. The well‐known transforms — reduction to the pole, pseudogravity field, and analytic signal (total gradient) — help in reducing the problem. Another way to achieve the required effect is the transformation of magnetic data, ΔT or Z, into values of the anomalous magnetic intensity T. In this respect, we have found some transforms based on differential operators such as the gradient of T and its modulus R = |?T|, the Laplacian L = ?²T, the product T ?²T and its square root Q, and the Laplacian ?²(T²) and its square root E, to be useful. They are slightly sensitive to the magnetization orientation and their extrema occur above the sources. For a 2D anomaly of a homogeneous causative body, the proposed transforms do not depend on the inclination of magnetization. In the 3D case, such independence does not exist even for the elementary field of a point dipole. The influence of the magnetization direction is estimated by an integral coefficient of sensitivity. This coefficient takes values of up to 2.0 for ΔT or Z anomalies, while their transforms T, R, E, Q and L have values of less than 0.28, 0.29, 0.24, 0.16 and 0.07, respectively, i.e. on average, 10 times less. The estimation of the centricity is carried out using the relative deviation of the principal extremum of the anomaly or its transforms from the epicentre of the model body at a depth equal to 100 units. For a ΔT anomaly this deviation is up to 67%; for the L transform it is less than 8%; for Q, E, R and T it is less than 10%, 15%, 20% and 25%, respectively. The proposed transforms take only non‐negative values. With respect to their shape, the peripheral magnetic extrema are removed, the anomalous configuration is simplified and the resolution of complicated interference patterns is improved. Their calculation does not require additional data for the direction of magnetization, which is an essential advantage over the reduction‐to‐the‐pole and pseudogravity‐field transforms. A joint analysis of the measured field and its transforms T, E and L offers possibilities for more confident separation of the anomalous effects and direct correlation to their sources. The model tests performed and the 3D field applications to real magnetic data confirm the useful properties of the transforms suggested here.     

Review Paper: Historical development of the total normalized gradient method in profile gravity field interpretation

In the Russian school, the total normalized gradient method belongs to the most wide-spread of direct interpretation methods for potential field data. This method was also used and partly developed by many experts from abroad. The main advantage of the total normalized gradient method is its relative independence of parameters such as the expected differential density of interpreted structures. The method is built from a construction of a specially transformed field (total normalized gradient) on a section crossing the potential field sources. The special properties of this transformed field allow it to be used to detect the source positions. From the 1960s, the mathematical basis of the method underwent enormous development and several modifications of the method have been elaborated. The total normalized gradient operator itself represents a relatively complicated, non-linear band-pass filter in the spectral domain. The properties of this operator can be handled by means of several parameters that act to separate the information about field sources at different depth levels. In this contribution, we describe the development of the method from its very beginning (based mostly on qualitative interpretation of simple total normalized gradient sections) through to more recent numerical improvements to the method. These improvements include the quasi-singular points method, which refines the filter properties of the total normalized gradient operator and defines an objective criteria (so called criterion ‘α’ and ‘Г') for the definition of source depths in the section. We end by describing possibilities for further development of the method in the future.     

Short note: A simple nomogram for interpretation due to magnetic horizontal cylinders

The magnetic anomaly ΔF due to a long horizontal cylinder yields three points related to its extrema. A nomogram is presented for the determination of some source parameters of the causative body.     

Depth from extreme points method for gravity gradient tensor data

The ‘depth from extreme points’ method is an important tool to estimate the depth of sources of gravity and magnetic data. In order to interpret gravity gradient tensor data conveniently, formulas for the tensor data form regarding depth from the extreme points method were calculated in this paper. Then, all of the gradient tensor components were directly used to interpret the causative source. Beyond the g_zz component, also the g_xx and g_yy components can be used to obtain depth information. In addition, the total horizontal derivative of the depth from extreme points of the gradient tensor can be used to describe the edge information of geologic sources. In this paper, we investigated the consistency of the homogeneity degree calculated by using the different components, which leads to the calculated depth being confirmed. Therefore, a more integrated interpretation can be obtained by using the gradient tensor components. Different synthetic models were used with and without noise to test the new approach, showing stability, accuracy and speed. The proposed method proved to be a useful tool for gradient tensor data interpretation. Finally, the proposed method was applied to full tensor gradient data acquired over the Vinton Salt Dome, Louisiana, USA, and the results are in agreement with those obtained in previous research studies.     

Estimating magnetic dike parameters using a non‐linear constrained inversion technique: an example from the Särna area,west central Sweden

In this paper, we describe a non‐linear constrained inversion technique for 2D interpretation of high resolution magnetic field data along flight lines using a simple dike model. We first estimate the strike direction of a quasi 2D structure based on the eigenvector corresponding to the minimum eigenvalue of the pseudogravity gradient tensor derived from gridded, low‐pass filtered magnetic field anomalies, assuming that the magnetization direction is known. Then the measured magnetic field can be transformed into the strike coordinate system and all magnetic dike parameters – horizontal position, depth to the top, dip angle, width and susceptibility contrast – can be estimated by non‐linear least squares inversion of the high resolution magnetic field data along the flight lines. We use the Levenberg‐Marquardt algorithm together with the trust‐region‐reflective method enabling users to define inequality constraints on model parameters such that the estimated parameters are always in a trust region. Assuming that the maximum of the calculated g_zz (vertical gradient of the pseudogravity field) is approximately located above the causative body, data points enclosed by a window, along the profile, centred at the maximum of g_zz are used in the inversion scheme for estimating the dike parameters. The size of the window is increased until it exceeds a predefined limit. Then the solution corresponding to the minimum data fit error is chosen as the most reliable one. Using synthetic data we study the effect of random noise and interfering sources on the estimated models and we apply our method to a new aeromagnetic data set from the Särna area, west central Sweden including constraints from laboratory measurements on rock samples from the area.     

Age of the source of the Jarrafa gravity and magnetic anomalies offshore Libya and its geodynamic implications

The interpretation of the Jarrafa magnetic and gravity highs, NW Libyan offshore, suggests that it may be caused by a body of high-density and high magnetization. Analysis of their power spectra indicates two groups of sources at: (1) 2.7 km depth, probably related to the igneous rocks, some of which were penetrated in the JA-1 borehole, (2) 5 km depth, corresponding to the top of the causative body and (3) 10 km depth, probably referring to the local basement depth. The boundary analysis derived from applied horizontal gradient to both gravity and magnetic data reveals lineaments many of which can be related to geological structures (grabens, horsts and faults).The poor correlation between pseudogravity fields for induced magnetization and observed gravity fields strongly suggests that the causative structure has a remanent magnetization (D = −16°, I = 23°) of Early Cretaceous age, fitting with the opening of the Neo Tethys 3 Ocean.Three-dimensional interpretation techniques indicate that the magnetic source of the Jarrafa magnetic anomaly has a magnetization intensity of 0.46 A/m, which is required to simulate the amplitude of the observed magnetic anomaly. The magnetic model shows that it has a base level at 15 km.The history of the area combined with the analysis and interpretation of the gravity and magnetic data suggests that: (1) the source of the Jarrafa anomaly is a mafic igneous rock and it may have formed during an Early Cretaceous extensional phase and (2) the Jarrafa basin was left-laterally sheared along the WNW Hercynian North Graben Fault Zone, during its reactivation in the Early Cretaceous.     

Inversion of potential field data using the structural index as weighting function rate decay

Nonparametric inverse methods provide a general framework for solving potential‐field problems. The use of weighted norms leads to a general regularization problem of Tikhonov form. We present an alternative procedure to estimate the source susceptibility distribution from potential field measurements exploiting inversion methods by means of a flexible depth‐weighting function in the Tikhonov formulation. Our approach improves the formulation proposed by Li and Oldenburg (1996, 1998) , differing significantly in the definition of the depth‐weighting function. In our formalism the depth weighting function is associated not to the field decay of a single block (which can be representative of just a part of the source) but to the field decay of the whole source, thus implying that the data inversion is independent on the cell shape. So, in our procedure, the depth‐weighting function is not given with a fixed exponent but with the structural index N of the source as the exponent. Differently than previous methods, our choice gives a substantial objectivity to the form of the depth‐weighting function and to the consequent solutions. The allowed values for the exponent of the depth‐weighting function depend on the range of N for sources: 0 ≤N≤ 3 (magnetic case). The analysis regarding the cases of simple sources such as dipoles, dipole lines, dykes or contacts, validate our hypothesis. The study of a complex synthetic case also proves that the depth‐weighting decay cannot be necessarily assumed as equal to 3. Moreover it should not be kept constant for multi‐source models but should instead depend on the structural indices of the different sources. In this way we are able to successfully invert the magnetic data of the Vulture area, Southern Italy. An original aspect of the proposed inversion scheme is that it brings an explicit link between two widely used types of interpretation methods, namely those assuming homogeneous fields, such as Euler deconvolution or depth from extreme points transformation and the inversion under the Tikhonov‐form including a depth‐weighting function. The availability of further constraints, from drillings or known geology, will definitely improve the quality of the solution.     

Interpretation of magnetic and gravity gradient tensor data using normalized source strength – A case study from McFaulds Lake,Northern Ontario,Canada

In this paper, we present a case study on the use of the normalized source strength (NSS) for interpretation of magnetic and gravity gradient tensors data. This application arises in exploration of nickel, copper and platinum group element (Ni‐Cu‐PGE) deposits in the McFaulds Lake area, Northern Ontario, Canada. In this study, we have used the normalized source strength function derived from recent high resolution aeromagnetic and gravity gradiometry data for locating geological bodies. In our algorithm, we use maxima of the normalized source strength for estimating the horizontal location of the causative body. Then we estimate depth to the source and structural index at that point using the ratio between the normalized source strength and its vertical derivative calculated at two levels; the measurement level and a height h above the measurement level. To discriminate more reliable solutions from spurious ones, we reject solutions with unreasonable estimated structural indices. This method uses an upward continuation filter which reduces the effect of high frequency noise. In the magnetic case, the advantage is that, in general, the normalized magnetic source strength is relatively insensitive to magnetization direction, thus it provides more reliable information than standard techniques when geologic bodies carry remanent magnetization. For dipping gravity sources, the calculated normalized source strength yields a reliable estimate of the source location by peaking right above the top surface. Application of the method on aeromagnetic and gravity gradient tensor data sets from McFaulds Lake area indicates that most of the gravity and magnetic sources are located just beneath a 20 m thick (on average) overburden and delineated magnetic and gravity sources which can be probably approximated by geological contacts and thin dikes, come up to the overburden.     

Three‐dimensional potential field data inversion with L0 quasinorm sparse constraints

The quantitative explanation of the potential field data of three‐dimensional geological structures remains one of the most challenging issues in modern geophysical inversion. Obtaining a stable solution that can simultaneously resolve complicated geological structures is a critical inverse problem in the geophysics field. I have developed a new method for determining a three‐dimensional petrophysical property distribution, which produces a corresponding potential field anomaly. In contrast with the tradition inverse algorithm, my inversion method proposes a new model norm, which incorporates two important weighting functions. One is the L0 quasi norm (enforcing sparse constraints), and the other is depth‐weighting that counteracts the influence of source depth on the resulting potential field data of the solution. Sparseness constraints are imposed by using the L0 quasinorm on model parameters. To solve the representation problem, an L0 quasinorm minimisation model with different smooth approximations is proposed. Hence, the data space (N) method, which is much smaller than model space (M), combined with the gradient‐projected method, and the model space, combined with the modified Newton method for L0 quasinorm sparse constraints, leads to a computationally efficient method by using an N × N system versus an M × M one because N ? M. Tests on synthetic data and real datasets demonstrate the stability and validity of the L0 quasinorm spare norms inversion method. With the aim of obtaining the blocky results, the inversion method with the L0 quasinorm sparse constraints method performs better than the traditional L2 norm (standard Tikhonov regularisation). It can obtain the focus and sparse results easily. Then, the Bouguer anomaly survey data of the salt dome, offshore Louisiana, is considered as a real case study. The real inversion result shows that the inclusion the L0 quasinorm sparse constraints leads to a simpler and better resolved solution, and the density distribution is obtained in this area to reveal its geological structure. These results confirm the validity of the L0 quasinorm sparse constraints method and indicate its application for other potential field data inversions and the exploration of geological structures.     

Interpretation of regional aeromagnetic data by the scaling function method: the case of Southern Apennines (Italy)

A complex aeromagnetic anomaly in Southern Apennines (Italy) is analysed and interpreted by a multiscale method based on the scaling function. We use multiscale methods allowing analysis of a potential field along ridges, which are lines defined by the position of the extrema of the field at the considered scales. The method developed and applied in this paper is based on the study of the scaling function of the total magnetic field. It allows recovering of source parameters such as depth and structural index. The studied area includes a Pleistocene volcanic structure (Mt. Vulture) whose intense dipolar anomaly is superimposed on a longer wavelength regional anomaly. The interpretation of ridges of the modulus of the analytic signal at different altitude ranges allows recognition of at least three distinct sources between about 5 km and 20 km depth. Their interpretation is discussed in light of borehole data and other geophysical constraints. A reasonable geological model for these sources indicates the presence of intrusions, probably linked to the past activity of Mt. Vulture.     

Gravity interpretation of nonoutcropping sedimentary basins in which the density contrast decreases parabolically with depth

The decrease in density contrast of sedimentary rocks with depth in many sedimentary basins can be approximated by a parabolic density function. Analytical gravity expression of an outcropping two-dimensional vertical step along which the density contrast decreases parabolically with depth is derived in the space domain. A modification ofBott's (1960) method of gravity interpretation is proposed by considering two outcropping vertical steps on either side of the first and last observation points in addition toN outcropping vertical prisms in order to interpret the gravity anomalies of nonoutcropping basins. The thicknesses of the two outcropping vertical steps are made equal to the thicknesses of the two outcropping vertical prisms placed below the first and last observation points. The initial depth estimates of the sedimentary basin are calculated by the infinite slab formula ofVisweswara Rao et al. (1993). The gravity effects of theN outcropping prisms and the two outcropping vertical steps are calculated at each anomaly point and the depth to the floor of the basin are adjusted based on the differences between the observed and calculated anomalies. A gravity anomaly profile of Los Angeles basin, California is interpreted.     

Analysing two‐dimensional effects in central loop transient electromagnetic sounding data using a semi‐synthetic tipper approach

We present a simple and feasible approach to analyse and identify two‐dimensional effects in central loop transient electromagnetic sounding data and the correspondingly derived quasi two‐dimensional conductivity models. The proposed strategy is particularly useful in minimising interpretation errors. It is based on the calculation of a semi‐synthetic transient electromagnetic tipper at each sounding and for each observational transient time point. The semi‐synthetic transient electromagnetic tipper is derived from the measured vertical component of the induced voltage and the synthetically calculated horizontal component. The approach is computationally inexpensive and involves one two‐dimensional forward calculation of an obtained quasi two‐dimensional conductivity section. Based on a synthetic example, we demonstrate that the transient electromagnetic tipper approach is applicable in identifying which transient data points and which corresponding zones in a derived quasi two‐dimensional subsurface model are affected by two‐dimensional inhomogeneities. The one‐dimensional inversion of such data leads to false models. An application of the semi‐synthetic transient electromagnetic tipper to field data from the Azraq basin in Jordan reveals that, in total, eight of 80 investigated soundings are affected by two‐dimensional structures although the field data can be fitted optimally using one‐dimensional inversion techniques. The largest semi‐synthetic tipper response occurs in a 300 m‐wide region around a strong lateral resistivity contrast. The approach is useful for analysing structural features in derived quasi two‐dimensional sections and for qualitatively investigating how these features affect the transient response. To avoid misinterpretation, these identified zones corresponding to large tipper values are excluded from the interpretation of a quasi two‐dimensional conductivity model. Based on the semi‐synthetic study, we also demonstrate that a quantitative interpretation of the horizontal voltage response (e.g. by inversion) is usually not feasible as it requires the exact sensor position to be known. Although a tipper derived purely from field data is useful as a qualitative tool for identifying two‐dimensional distortion effects, it is only feasible if the sensor setup is sufficiently accurate. Our proposed semi‐synthetic transient electromagnetic tipper approach is particularly feasible as an a posteriori approach if no horizontal components are recorded or if the sensor setup in the field is not sufficiently accurate.     

TSVD analysis of Euler deconvolution to improve estimating magnetic source parameters: An example from the Åsele area,Sweden

In this paper, I introduce a new approach based on truncated singular value decomposition (TSVD) analysis for improving implementation of grid-based Euler deconvolution with constraints of quasi 2D magnetic sources. I will show that by using TSVD analysis of the gradient matrix of magnetic field anomaly (reduced to pole) for data points located within a square window centered at the maximum of the analytic signal amplitude, we are able to estimate the strike direction and dip angle of 2D structures from the acquired eigenvectors. It is also shown that implementation of the standard grid-based Euler deconvolution can be considerably improved by solving the Euler's homogeneity equation for source location and structural index, simultaneously, using the TSVD method. The dimensionality of the magnetic anomalies can be indicated from the ratio between the smallest and intermediate eigenvalues acquired from the TSVD analysis of the gradient matrix. For 2D magnetic sources, the uncertainty of the estimated source location and structural index is significantly reduced by truncating the smallest eigenvalue.Application of the method is demonstrated on an aeromagnetic data set from the Åsele area in Sweden. The geology of this area is dominated by several dike swarms. For these dolerite dikes, the introduced method has provided useful information of strike directions and dip angles in addition to the estimated source location and structural index.     

Response to three‐component seismic motion of arbitrary direction

This paper presents a response spectrum analysis procedure for the calculation of the maximum structural response to three translational seismic components that may act at any inclination relative to the reference axes of the structure. The formula GCQC3, a generalization of the known CQC3‐rule, incorporates the correlation between the seismic components along the axes of the structure and the intensity disparities between them. Contrary to the CQC3‐rule where a principal seismic component must be vertical, in the GCQC3‐rule all components can have any direction. Besides, the GCQC3‐rule is applicable if we impose restrictions to the maximum inclination and/or intensity of a principal seismic component; in this case two components may be quasi‐horizontal and the third may be quasi‐vertical. This paper demonstrates that the critical responses of the structure, defined as the maximum and minimum responses considering all possible directions of incidence of one seismic component, are given by the square root of the maximum and minimum eigenvalues of the response matrix R , of order 3×3, defined in this paper; the elements of R are established on the basis of the modal responses used in the well‐known CQC‐rule. The critical responses to the three principal seismic components with arbitrary directions in space are easily calculated by combining the eigenvalues of R and the intensities of those components. The ratio r_max/r_SRSS between the maximum response and the SRSS response, the latter being the most unfavourable response to the principal seismic components acting along the axes of the structure, is bounded between 1 and √(3γ_a²/(γ_a² + γ_b² + γ_c²)), where γ_a?γ_b?γ_c are the relative intensities of the three seismic components with identical spectral shape. Copyright © 2001 John Wiley & Sons, Ltd.     

SOME IMPROVEMENTS IN THE INTERPRETATION OF VERTICAL ELECTRIC SOUNDING CURVES*

The process of VES interpretation is discussed, including the following points. (a) Preliminary interpretation by means of master curves. It is shown that the positions of the auxiliary points K and Q depend on the resistivity of the substratum. The interpretation is improved if the auxiliary points are determined separately for each master curve. (b) The individual parts of the measured curves are shifted in overlapping MN electrode positions so that the total sum of squares of the shifts is minimal. (c) The ambiguity may be reduced by means of supplementary information or assumptions on the resistivities. Fixing the resistivities is not always possible because discrepancies may arise between the ground measurement and the well-logging data. The simultaneous interpretation of several VES curves is recommended assuming constant resistivities. This assumption may be subsequently verified by means of the F-test. (d) A nonlinear algorithm is proposed for the determination of confidence intervals. As the multi-dimensional confidence intervals are often very complicated, it is recommended to construct only one-dimensional confidence intervals for the estimable parametric functions. (e) A ‘double-least-squares’ optimization technique is presented. The optimization is performed on the estimable parametric functions, and the individual parameters are determined so that the solution remains near the initial guess. This technique is faster than the singular value decomposition.     

Testing the Universal Soil Loss Equation‐MB equation in plots in Central and South Italy

Planning soil conservation strategies requires predictive techniques at event scale because a large percentage of soil loss over a long‐time period is due to relatively few large storms. Considering runoff is expected to improve soil loss predictions and allows relation of the process‐oriented approach with the empirical one, furthermore, the effects of detachment and transport on soil erosion processes can be distinguished by a runoff component. In this paper, the empirical model USLE‐MB (USLE‐M based), including a rainfall‐runoff erosivity factor in which the event rainfall erosivity index EI₃₀ of the Universal Soil Loss Equation (USLE) multiplies the runoff coefficient Q_R raised to an exponent b₁ > 1 is tested by the measurements carried out for the Masse (10 plots) and Sparacia (22 plots) experimental stations in Italy. For the Masse experimental station, an exponent b₁ > 1 was also estimated by tests carried out by a nozzle‐type rainfall simulator. For each experimental site in fallow conditions, the effect of the sample size of the plot soil loss measurements on the estimate of the b₁ coefficient was also studied by the extraction of a fixed number N of randomly obtained pairs of the normalized soil loss and runoff coefficient. The analysis showed that the variability of b₁ with N is low and that 350 pairs are sufficient to obtain a stable estimate of b₁. A total of 1,262 soil loss data were used to parameterize the model both locally and considering the two sites simultaneously. The b₁ exponent varied between the two sites (1.298–1.520), but using a common exponent (1.386) was possible. Using a common b₁ exponent for the two experimental areas increases the practical interest for the model and allows the estimation of a baseline component of the soil erodibility factor, which is representative of the at‐site soil intrinsic and quasi‐static properties. Development of a single USLE‐MB model appears possible, and sampling other sites is advisable to develop a single USLE‐MB model for general use.     

A NEW APPROACH TO MAGNETIC PROFILE INTERPRETATION*

By using two components of anomalous magnetic fields and a formulation including complex numbers it is possible to calculate the position parameters of thick plates and both magnetization and position of thin plates directly from any two or three points of anomaly profiles. The formulae (interpretation operators) allow automatic topographic corrections to be made. The new two-component operators give more reliable results than the conventional methods of interpretation. The variance of the parameter values obtained with subsequent points of an anomaly measures directly, the total error of interpretation. The application of infinite thin plate operators to a long profile results in characteristic patterns, from which the estimation of the number of plates and their approximate position is possible.     

Daily variability of river concentrations and fluxes: indicators based on the segmentation of the rating curve

The variability of water chemistry on a daily scale is rarely addressed due to the lack of records. Appropriate tools, such as typologies and dimensionless indicators, which permit comparisons between stations and between river materials, are missing. Such tools are developed here for daily concentrations (C), specific fluxes or yields (Y) and specific river flow (q). The data set includes 128 long‐term daily records, for suspended particulate matter (SPM), total dissolved solids (TDS), dissolved and total nutrients, totalling 1236 years of records. These 86 river basins (10³–10⁶ km²) cover a wide range of environmental conditions in semi‐arid and temperate regions. The segmentation—truncation of C–q rating curves into two parts at median flows (q₅₀) generates two exponents (b_50inf and b_50sup) that are different for 66% of the analysed rating curves. After segmentation, the analysis of records results in the definition of nine major C–q types combining concentrating, diluting or stable patterns, showing inflexions, chevron and U shapes. SPM and TDS are preferentially distributed among a few types, while dissolved and total nutrients are more widely distributed. Four dimensionless indicators of daily variability combine median (C₅₀, Y₅₀), extreme (C₉₉, Y₉₉) and flow‐weighted (C*, Y*) concentrations and yields (e.g. C₉₉/C₅₀, Y*/Y₅₀). They vary over two to four orders of magnitude in the analysed records, discriminating stations and river material. A second set of four indicators of relative variability [e.g. (Y*/Y₅₀)/(q*/q₅₀)], takes into account the daily flow variability, as expressed by q*/q₅₀ and q₉₉/q₅₀, which also vary over multiple orders of magnitude. The truncated exponent b_50sup is used to describe fluxes at higher flows accounting for 75% (TDS) to 97% (SPM) of interannual fluxes. It ranges from ? 0·61 to + 1·86 in the database. It can be regarded as the key amplificator (positive b_50sup) or reductor (negative b_50sup) of concentrations or yields variability. C₅₀, Y₅₀, b_50sup can also be estimated in discrete surveys, which provides a new perspective for quantifying and mapping water quality variability at daily scale. Copyright © 2011 John Wiley & Sons, Ltd.     

Thermal nature of the source of the Indian gravity anomaly

The non-hydrostatic part of the second zonal harmonic term has been included in the Indian gravity anomaly. The interpretation of the potential, gravity and deflection of the vertical has been produced by a new method. The depth of the source is 580 km and the solution satisfies the temperature gradient of 1°K/km. The ratio Δ V_p/Δp > 0 shows the thermal nature of the density distribution near 670 km.     

A THEOREM FOR DIRECT CURRENT REGIMES AND SOME OF ITS CONSEQUENCES *

For any direct current regime, the theorem holds, where φ_p is the total measured or calculated potential at any point P, φ is the potential distribution known a priori, r is the distance between P and any volume element dV, the gradients are evaluated at the element, and the current sources and sinks have finite dimensions. Thus, each space element behaves as a dipole of moment (1/4π) ?φdV and contributes its share of signal or potential accordingly. By suitable summation or integration, the contribution from any assigned portion of space to the total measured signal can be determined. Except for the chargeability factor m, the formula also establishes Seigel's initial postulate for the time domain induced polarisation theory. The contribution depends on the potential gradient, not the current density, and the integration extends over the entire space. Although an insulating target carries no current, it contributes a signal that is in general larger than normal by virtue of its higher potential gradient, and thus helps in creating an overall positive anomaly or resistivity high. On the other hand, an infinitely conducting target—even though it supports a larger amount of current than normal—contributes nothing to the measured signal as the potential gradient is zero everywhere inside. Thus, by contributing less than normal, a conducting target promotes the creation of what is usually a resistivity low. In all cases, the contributions from the space elements add up exactly to the measured or total calculated value. Some other consequences of the theorem are also discussed, especially in relation to a simple two-layer earth. For instance, the contribution from the upper half-space (air) turns out to be equal to that from the lower (real ground), for all observation points on the ground surface and for any ground configuration.