Analysis of gravity source moment inversion. Part I: Using noa priori information about the source

A generic gravity source moment is an integral, over the source volume, of the product of the density distribution by a polynomia in the Cartesian coordinates of a point belonging to this volume. We obtained a formal expression for a generic moment in terms of integrals involving the gravity anomaly and the gravity potential. By analyzing the conditions under which this expression is valid, we conclude that, without usinga priori information regarding the sources, it is possible to determine, from the gravity anomaly, any moment or linear combination of moments whose associated polynomial has null Laplacian and depends only on the coordinates defining the measurement plane. Additionally, no moment whose associated polynomial has a nonnull laplacian can be determined without usinga priori information of the source.     

Inversion of controlled‐source electromagnetic data using a model‐based approach

A two‐and‐half dimensional model‐based inversion algorithm for the reconstruction of geometry and conductivity of unknown regions using marine controlled‐source electromagnetic (CSEM) data is presented. In the model‐based inversion, the inversion domain is described by the so‐called regional conductivity model and both geometry and material parameters associated with this model are reconstructed in the inversion process. This method has the advantage of using a priori information such as the background conductivity distribution, structural information extracted from seismic and/or gravity measurements, and/or inversion results a priori derived from a pixel‐based inversion method. By incorporating this a priori information, the number of unknown parameters to be retrieved becomes significantly reduced. The inversion method is the regularized Gauss‐Newton minimization scheme. The robustness of the inversion is enhanced by adopting nonlinear constraints and applying a quadratic line search algorithm to the optimization process. We also introduce the adjoint formulation to calculate the Jacobian matrix with respect to the geometrical parameters. The model‐based inversion method is validated by using several numerical examples including the inversion of the Troll field data. These results show that the model‐based inversion method can quantitatively reconstruct the shapes and conductivities of reservoirs.     

Inversion of waveform from the data of six moderate strong earthquakes in Chinese mainland to seismic moment tensors and source mechanism

Based on Generalized Seismic Ray Theory (Helmberger, 1968), a new quickly linear inversion method from the data of seismic waveform to seismic moment tensor and source mechanism for domestic earthquake is studied in this paper. Six moderately strong earthquakes which occurred in Chinese mainland in the past few years are studied. The seismic source parameters of these earthquakes, seismic moment tensors, scalar seismic moments, fault plane solutions and source time functionsetc, are obtained. The Chinese version of this paper appeared in the Chinese edition ofActa Seismologica Sinica,15, 261–268, 1993.     

Joint inversion of Rayleigh‐wave dispersion data and vertical electric sounding data: synthetic tests on characteristic sub‐surface models

In the traditional inversion of the Rayleigh dispersion curve, layer thickness, which is the second most sensitive parameter of modelling the Rayleigh dispersion curve, is usually assumed as correct and is used as fixed a priori information. Because the knowledge of the layer thickness is typically not precise, the use of such a priori information may result in the traditional Rayleigh dispersion curve inversions getting trapped in some local minima and may show results that are far from the real solution. In this study, we try to avoid this issue by using a joint inversion of the Rayleigh dispersion curve data with vertical electric sounding data, where we use the common‐layer thickness to couple the two methods. The key idea of the proposed joint inversion scheme is to combine methods in one joint Jacobian matrix and to invert for layer S‐wave velocity, resistivity, and layer thickness as an additional parameter, in contrast with a traditional Rayleigh dispersion curve inversion. The proposed joint inversion approach is tested with noise‐free and Gaussian noise data on six characteristic, synthetic sub‐surface models: a model with a typical dispersion; a low‐velocity, half‐space model; a model with particularly stiff and soft layers, respectively; and a model reproduced from the stiff and soft layers for different layer‐resistivity propagation. In the joint inversion process, the non‐linear damped least squares method is used together with the singular value decomposition approach to find a proper damping value for each iteration. The proposed joint inversion scheme tests many damping values, and it chooses the one that best approximates the observed data in the current iteration. The quality of the joint inversion is checked with the relative distance measure. In addition, a sensitivity analysis is performed for the typical dispersive sub‐surface model to illustrate the benefits of the proposed joint scheme. The results of synthetic models revealed that the combination of the Rayleigh dispersion curve and vertical electric sounding methods in a joint scheme allows to provide reliable sub‐surface models even in complex and challenging situations and without using any a priori information.     

On dynamo action in the high-conductivity limit

Abstract

This paper builds on a speculation by Moffatt (1979) on an apparent conflict between two results of dynamo theory in the high conductivity limit. Firstly, the finding by Bondi and Gold (1950) on the boundedness of the magnetic dipole moment of a perfectly conducting fluid body is, for a sphere, extended to all magnetic multipole moments. Secondly, a refined version is considered of the simple spherical mean-field dynamo model proposed by Krause and Steenbeck (1967). Some constraints on the mean electromotive force near the boundary of the conducting body are taken into account, which have not been recognized up to now. In the framework of the second order correlation approximation it is shown that it is just these constraints that ensure the boundedness of the magnetic multipole moments in the high conductivity limit. Thus the apparent conflict is resolved. In this context another possible source of error in mean-field dynamo models is pointed out. The present theory also adds insight into dynamo process in cosmical objects, in a way that is briefly discussed.     

Seismic spectra on soil with uncertain properties

Expected spectral ordinates and their standard deviations are computed for ground motions produced by SH waves at the surface of a homogeneous soil formation resting on a half-space of rock. Wave transmission is idealized as linear and one-dimensional. Uncertainties about soil properties are taken into account using various approximations: a Taylor series expansion of the Fourier spectrum magnification factors with different numbers of terms and different intervals for numerical differentiation; discretization of the joint probability density function of the soil properties so as to match its moments up to those of third order including the crossed moments; a discretization that matches up to some fifth-order moments; and a fine numerical integration for the marginal density function of the modulus of rigidity, discretizing the rest of the density function. The latter approach gives sufficiently accurate results for practical purposes. Effects of even moderate uncertainty about soil properties on the statistics of spectral ordinates are found to be decisive.     

Forward computation of the magnetic field of a 3D body with arbitrary boundary and continually varying magnetization

The forward computation of the gravitational and magnetic fields due to a 3D body with an arbitrary boundary and continually varying density or magnetization is an important problem in gravitational and magnetic prospecting. In order to solve the inverse problem for the arbitrary components of the gravitational and magnetic anomalies due to an arbitrary 3D body under complex conditions, including an uneven observation surface, the existence of background anomalies and very little or no a priori information, we used a spherical coordinate system to systematically investigate forward methods for such anomalies and developed a series of universal spherical harmonic expansions of gravitational and magnetic fields. For the case of a 3D body with an arbitrary boundary and continually varying magnetization, we have also given the surface integral expressions for the common spherical harmonic coefficients in the expansion of the magnetic field due to the body, and a very precise numerical integral algorithm to calculate them. Thus a simple and effective method of solving the forward problem for magnetic fields due to 3D bodies of this kind has been found, and in this way a foundation is laid for solving the inverse problem of these magnetic fields. In addition, by replacing the parameters and unit vectors in the spherical harmonic expansion of a magnetic field by gravitational parameters and a downward unit vector, we have also derived a forward method for the gravitational field (similar to that for the magnetic case) of a 3D body with an arbitrary boundary and continually varying density.     

Piecewise‐continuous velocity inversion

We suggest a new method to determine the piecewise‐continuous vertical distribution of instantaneous velocities within sediment layers, using different order time‐domain effective velocities on their top and bottom points. We demonstrate our method using a synthetic model that consists of different compacted sediment layers characterized by monotonously increasing velocity, combined with hard rock layers, such as salt or basalt, characterized by constant fast velocities, and low velocity layers, such as gas pockets. We first show that, by using only the root‐mean‐square velocities and the corresponding vertical travel times (computed from the original instantaneous velocity in depth) as input for a Dix‐type inversion, many different vertical distributions of the instantaneous velocities can be obtained (inverted). Some geological constraints, such as limiting the values of the inverted vertical velocity gradients, should be applied in order to obtain more geologically plausible velocity profiles. In order to limit the non‐uniqueness of the inverted velocities, additional information should be added. We have derived three different inversion solutions that yield the correct instantaneous velocity, avoiding any a priori geological constraints. The additional data at the interface points contain either the average velocities (or depths) or the fourth‐order average velocities, or both. Practically, average velocities can be obtained from nearby wells, whereas the fourth‐order average velocity can be estimated from the quartic moveout term during velocity analysis. Along with the three different types of input, we consider two types of vertical velocity models within each interval: distribution with a constant velocity gradient and an exponential asymptotically bounded velocity model, which is in particular important for modelling thick layers. It has been shown that, in the case of thin intervals, both models lead to similar results. The method allows us to establish the instantaneous velocities at the top and bottom interfaces, where the velocity profile inside the intervals is given by either the linear or the exponential asymptotically bounded velocity models. Since the velocity parameters of each interval are independently inverted, discontinuities of the instantaneous velocity at the interfaces occur naturally. The improved accuracy of the inverted instantaneous velocities is particularly important for accurate time‐to‐depth conversion.     

Anisotropic azimuthal surface-wave inversion for a vertically fractured and transverse isotropy medium: Theory and examples from a controlled field experiment

Fractures in elastic media add compliance to a rock in the direction normal to the fracture strike. Therefore, elastic wave velocities in a fractured rock will vary as a function of the energy propagation direction relative to the orientation of the aligned fracture set. Anisotropic Thomson–Haskell matrix Rayleigh-wave equations for a vertically transverse isotropic media can be used to model surface-wave dispersion along the principal axes of a vertically fractured and transversely isotropic medium. Furthermore, a workflow combining first-break analysis and azimuthal anisotropic Rayleigh-wave inversion can be used to estimate P-wave and S-wave velocities, Thomsen's ε, and Thomsen's δ along the principal axes of the orthorhombic symmetry. In this work, linear slip theory is used to map our inversion results to the equivalent vertically fractured and transversely isotropic medium coefficients. We carried out this inversion on a synthetic example and a field example. The synthetic data example results show that joint estimation of S-wave velocities with Thomsen's parameters ε and δ along normal and parallel to the vertical fracture set is reliable and, when mapped to the corresponding vertically fractured and transversely isotropic medium, provides insight into the fracture compliances. When the inversion was carried out on the field data, results indicated that the fractured rock is more compliant in the azimuth normal to the visible fracture set orientation and that the in situ normal fracture compliance to tangential fracture compliance ratio is less than half, which implies some cementation may have occurred along the fractures. Such an observation has significant implications when modelling the transport properties of the rock and its strength. Both synthetic and field examples show the potential of azimuthal anisotropic Rayleigh-wave inversion as the method can be further expanded to a more general case where the vertical fracture set orientation is not known a priori.     

Joint Electrical and Time Domain Electromagnetism (TDEM) Data Inversion Applied to the Super Sauze Earthflow (France)

In order to evaluate the risk of an earthflow to evolve abruptly into torrential surge, knowledge of its internal structure is necessary. This study deals with the internal structure of the Super Sauze earthflow developed in black marls in the southern French Alps. Difficulties in this study area are a rough topography, surface heterogeneities and a large thickness variability of the earthflow mass. These conditions hamper the application of geotechnical methods as a preferred investigation mean. Moreover, they pose problems to geophysical investigations and their interpretation.This paper shows the advantage offered by the joint inversion of Time Domain ElectroMagne-tism data (TDEM) and data obtained from Direct Current soundings (DC). The results of the joint inversions are checked using geotechnical data. The internal structure of the earthflow interpreted on the basis of joint inversion data is comparable to that obtained from geotechnical results. Moreover, contrary to separate electrical and TDEM inversions, a satisfactory joint inversion model can be derived without supplying additional a priori information.     

Response analysis of hysteretic multi-storey frames under earthquake excitation

Local incremental stiffness relations are formulated for a class of elasto-plastic beam elements. The earthquake acceleration is modelled as a filtered white noise process. The Itǒ differential equations of the integrated system made up of the structural system and the excitation process are then formulated. Instead of the original system an equivalent nonlinear system is considered, in which the drift vector is given by a series expansion of order n ≧ 1, where n = 1 represents the well-known case of equivalent linearization. Only components of the drift vector representing the non-analytieal constitutive equations are replaced by a polynomial expansion. The coefficients of this expansion are determined from a least mean square criterion, and are sequentially updated. Especially an equivalent system with a cubic expansion to the drift vector is investigated. The hierarchy of statistical moment equations is closed by a cumulant neglect closure scheme. The method has been applied to a two-storey frame. The results are compared to those of numerical simulation, and provide substantial improvements compared to equivalent linearization.     

On utilization ofa priori knowledge in inversion of remote sensing models

Satellite remote sensing deals with a complex system coupling atmosphere and surface. Any physical model with reasonable precision needs several to tens of parameters. Without a priori knowledge of these parameters, Proposition 3 of Verstraete et al. requires the number of independent observations to be greater than the number of unknown parameters. This requirement can hardly be satisfied even in the coming EOS era. As Tarantola pointed out, the inversion problems in geoscience are always underdetermined in some sense. In order to make good use of every kind of a priori knowledge for effectively extracting information from remote sensing observations, the right question to set is as follows:Given an imperfect model and a certain amount ofa priori information on model parameters, in which sense should one modify thea priori information, given the actual observation with noise?A priori knowledge of physical parameters can be presented in different ways such as physical limits, global statistical means and variance fora certain landcover type, or previous statistics and temporal variation of a specific target. When sucha priori knowledge can be expressed as joint probability density. Bayessian theorem can be used in the inversion to obtain posterior probability densities of parameters using newly acquired observations. There is no prerequirement on how many independent observations must be made, and the knowledge gained merely depends on the information content of the new observations. Some specific problems about knowledge accumulation and renewal are also discussed.     

Use of the Fourier-Laplace transform and of diagrammatical methods to interpret pumping tests in heterogeneous reservoirs

Advances in computer power and in reservoir characterization allow simulation of pressure transients in complex reservoirs generated stochastically. Generally, interpretation of these transient gives useful information about the reservoir hydraulic properties: a major goal is to interpret these transients in the stochastic context. First we ensemble average the pressure over all the random permeability field realizations to derive an equation which drives the ensemble averaged pressure. We use the Fourier transform in space and the Laplace transform in time, in conjuction with a perturbation series expansion in successive powers of the permeability fluctuations to obtain an explicit solution. The Nth order term of this series involves the hydrodynamic interaction between N permeability heterogeneities and after averaging we obtain an expansion containing correlation functions of permeability fluctuations of increasing order.Next, Feynman graphs are introduced allowing a more attractive graphical interpretation of the perturbation series. Then series summation techniques are employed to reduce the graph number to be summed at each order of the fluctuation expansion. This in turn gives useful physical insights on the homogenization processes involved. In particular, it is shown that the sum of the so-called ‘one-particle irreducible graphs’ gives the kernel of a linear integro-differential equation obeyed by the ensemble average pressure. All the information about the heterogeneity structure is contained in this renormalized kernel, which is a limited range function.This equation on its own is the starting point of useful asymptotic results and approximations. In particular it is shown that interpretation of pumping tests yields the steady-state equivalent permeability after a sufficiently long time for an infinite reservoir, as expected.     

An enhanced method for source parameter imaging of magnetic data collected for mineral exploration

We have developed a method for imaging magnetic data collected for mineral exploration to yield the following structural information: depth, model type (structural index) and susceptibility. The active nature of mineral exploration data requires we derive the structural information from a robust quantity: we propose that the first‐ or second‐order analytic‐signal amplitude is suitably stable. The procedure is to normalize the analytic‐signal amplitude by the peak value and then use non‐linear inversion to estimate the depth and the structural index for each anomaly. In our field example, different results are obtained depending on whether we inverted for the first‐ or second‐order analytic‐signal amplitude. This is probably because the two‐dimensional contact, thin sheet or horizontal cylinder models we have assumed are not appropriate. In cases such as these, when our model assumptions are not correct, the results should not be interpreted quantitatively, but they might be useful for giving a qualitative indication of how the structure might vary. With a priori information, it is possible to assume a model type (i.e. set the structural index) and generate estimates of the depth and susceptibility. These data can then be gridded and imaged. If a contact is assumed, the susceptibility contrast is estimated; for the dike model, the susceptibility‐thickness is estimated; for the horizontal cylinder, the susceptibility‐area is estimated. To emphasize that the results are dependent on our assumed model, we advocate prefixing any derived quantity by the term ‘apparent’.     

BAYESIAN ESTIMATION IN SEISMIC INVERSION. PART II: UNCERTAINTY ANALYSIS1

A parameter estimation or inversion procedure is incomplete without an analysis of uncertainties in the results. In the fundamental approach of Bayesian parameter estimation, discussed in Part I of this paper, the a posteriori probability density function (pdf) is the solution to the inverse problem. It is the product of the a priori pdf, containing a priori information on the parameters, and the likelihood function, which represents the information from the data. The maximum of the a posteriori pdf is usually taken as a point estimate of the parameters. The shape of this pdf, however, gives the full picture of uncertainty in the parameters. Uncertainty analysis is strictly a problem of information reduction. This can be achieved in several stages. Standard deviations can be computed as overall uncertainty measures of the parameters, when the shape of the a posteriori pdf is not too far from Gaussian. Covariance and related matrices give more detailed information. An eigenvalue or principle component analysis allows the inspection of essential linear combinations of the parameters. The relative contributions of a priori information and data to the solution can be elegantly studied. Results in this paper are especially worked out for the non-linear Gaussian case. Comparisons with other approaches are given. The procedures are illustrated with a simple two-parameter inverse problem.     

Adjustment of regularization in ill-posed linear inverse problems by the empirical Bayes approach

Regularization is the most popular technique to overcome the null space of model parameters in geophysical inverse problems, and is implemented by including a constraint term as well as the data‐misfit term in the objective function being minimized. The weighting of the constraint term relative to the data‐fitting term is controlled by a regularization parameter, and its adjustment to obtain the best model has received much attention. The empirical Bayes approach discussed in this paper determines the optimum value of the regularization parameter from a given data set. The regularization term can be regarded as representing a priori information about the model parameters. The empirical Bayes approach and its more practical variant, Akaike's Bayesian Information Criterion, adjust the regularization parameter automatically in response to the level of data noise and to the suitability of the assumed a priori model information for the given data. When the noise level is high, the regularization parameter is made large, which means that the a priori information is emphasized. If the assumed a priori information is not suitable for the given data, the regularization parameter is made small. Both these behaviours are desirable characteristics for the regularized solutions of practical inverse problems. Four simple examples are presented to illustrate these characteristics for an underdetermined problem, a problem adopting an improper prior constraint and a problem having an unknown data variance, all frequently encountered geophysical inverse problems. Numerical experiments using Akaike's Bayesian Information Criterion for synthetic data provide results consistent with these characteristics. In addition, concerning the selection of an appropriate type of a priori model information, a comparison between four types of difference‐operator model – the zeroth‐, first‐, second‐ and third‐order difference‐operator models – suggests that the automatic determination of the optimum regularization parameter becomes more difficult with increasing order of the difference operators. Accordingly, taking the effect of data noise into account, it is better to employ the lower‐order difference‐operator models for inversions of noisy data.     

Attenuation and source parameters at Deception Island (South Shetland Islands,Antarctica)

Deception Island is the most active volcano of the South Shetland Islands-Antarctic Peninsula group, experiencing eruptions in 1967, 1969 and 1970. Local attenuation derived from coda analysis and source parameters derived from Brune's model, for well located seismic events, have been studied in order to complement the available geophysical information. Results show abnormally lowQ ₀ values and an abnormally high frequency dependence, as well as large dispersion. These factors are strongly dependent on the path travelled by the seismic wave. Retrieved values of the source parameters (stress drop, seismic moment and source radius), are again abnormally low compared to world-wide average values, for example, those obtained for the Oroville, California aftershock series between June and September, 1976. These results are consistent with some aspects of the geology of Deception Island, such as the very high degree of fracturing and faulting, and the existence of a strong hydrothermal alteration affecting most of the subaerial volcanic rocks. Moreover, the pattern defined for the lateral variations ofQ ₀ shows minimum values in the inner bay of the island, close to the most recent eruption vents. A large reduction in spectral amplitudes over a particular frequency range occurs in several observations, corresponding to the path crossing the zone of highest attenuation. This observation suggests the existence of a hot magmatic intrusion produced during the most recent eruption, and coincides with the superficial low density mass distribution obtained from the gravimetric model and the long wave magnetic field component obtained from magnetic surveys. The width of this intrusion is estimated to be about 200 m, in agreement with the previous results obtained analyzing residuals of the location of seismic events.     

Predicting return periods of hydrological droughts using the Pearson 3 distribution: a case from rivers in the Canadian prairies

Abstract

The standardized series of monthly and weekly flow sequences, referred to as standardized hydrological index (SHI) series, from five rivers in the Canadian prairies were subjected to return period (T_r) analysis of drought length (L). The SHI series were truncated at drought probability levels q ranging from 0.5 to 0.05 with the intention of deducing drought events and corresponding drought lengths. The values of L were fitted to the Pearson 3, the gamma (2-parameter), the exponential (1-parameter), the Weibull 3 and the Weibull (2-parameter) probability density functions (pdfs). A priori assignment of one week or one month for the location parameter in the Pearson 3 pdf proved logical and also facilitated the rapid estimation of other parameters using either the method of moments or the method of maximum likelihood. The Pearson 3 turns out to be the most suitable pdf to describe and to estimate return periods of drought lengths. At the monthly and weekly time scales, it was inferred that the sample size (T, months or weeks) of SHI series could be treated equivalent to the return period of the largest recorded drought length. At the annual time scale, however, the sample size (T, years) should be modified using either the Hazen or the Gringorten plotting position formula to reflect the actual return period of the largest recorded drought length in years.

Editor D. Koutsoyiannis; Associate editor E. Gargouri     

Interpretation of magnetic data using analytic signal derivatives

This paper develops an automatic method for interpretation of magnetic data using derivatives of the analytic signal. A linear equation is derived to provide source location parameters of a 2D magnetic body without a priori information about the nature of the source. Then using the source location parameters, the nature of the source can be ascertained. The method has been tested using theoretical simulations with random noise for two 2D magnetic models placed at different depths with respect to the observation height. In both cases, the method gave a good estimate for the location and shape of the sources. Good results were obtained on two field data sets.