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 Volume 44, Issue 2, 1996
Geophysical Prospecting  Volume 44, Issue 2, 1996
Volume 44, Issue 2, 1996


Estimation of effective source signatures from marine VSP data^{1}
Authors K. Hokstad, M. Landrø and R. MittetAbstractWe present a simple method for estimating an effective source wavelet from the first arrival in marine vertical seismic profiling (VSP) data. The method, which utilizes the free‐space Green's function of the Helmholz equation, is simple and very computer efficient. We show examples from synthetic and real offset and walkaway VSP data.
In the synthetic examples, we show that data modelled with the estimated wavelet give small residuals when subtracted from the reference data. In the real data examples, we show that when modelling with the wavelet estimated from the real data, in a smooth macromodel, we obtain a good fit between the first arrivals in the real and modelled data.



Near‐surface layer traveltime inversion: a synthetic example^{1}
By Robert BloorAbstractThe near‐surface layer is modelled as a constant‐velocity layer with varying thickness. The base of the layer is described by a B‐spline curve. The optimum model is calculated by minimizing, with respect to the model parameters, the difference between traveltimes predicted by the model and those observed in the data. Once a model has been produced, corrections that are dependent on the raypath geometry through the near‐surface layer can be calculated.
The effect of the near‐surface layer is normally considered to be consistent at each shot or geophone station for all traveltimes arriving at that location (the surface‐consistent approximation). This assumption linearizes the problem, allowing timeshifts to be calculated and the traveltimes corrected to a chosen datum, representing static corrections. The single correction at each point is an averaged correction, based on an assumption that is particularly inaccurate in the presence of lateral variations of velocity or thickness of the surface layer, in the presence of large surface layer velocities or in the presence of a thick surface layer. The method presented considers the non‐linear relationship between data and model explicitly, hence the correction that is dependent on the raypath. Linearization removes this dependence and reduces the problem to a surface‐consistent approximation.
The method is applied to synthetic data calculated from a model with surface layer variations. Comparisons are made between the corrected data resulting from the method described here and the conventional surface‐consistent approach. From these results it becomes apparent that the near‐surface layer inversion method presented here can reproduce accurate models and correct for near‐surface layer effects in cases where conventional methods encounter difficulties. Additionally the method can be readily extended to 3D.



Computation of apparent resistivity profiles from VLF‐EM data using linear filtering^{1}
Authors Michel Chouteau, Ping Zhang and Dominique ChapellierAbstractA simple filter is developed which transforms VLF‐EM real magnetic field transfer functions into apparent resistivities. It is based on the relationship between the horizontal derivative of the surface electric field and the vertical magnetic field at the surface of a two‐dimensional earth model. The performance of this simple autoregressive filter is tested for modelled and real survey data. The technique yields profiles of apparent resistivity very similar, both in magnitude and in wavelength, to those which would have been obtained using VLF‐EM resistivity measurements or d.c. resistivity profiling. This low‐pass filter has the advantage of reducing high‐wavenumber noise in the data; therefore only the major features of the VLF‐EM profile are displayed.



Interpretation of slingram conductivity mapping in near‐surface geophysics: using a single parameter fitting with 1D model^{1}
Authors R. Guérin, Y. Méhéni, G. Rakotondrasoa and A. TabbaghAbstractElectrical conductivity mapping is a prerequisite tool for hydrogeological or environmental studies. Its interpretation still remains qualitative but advantages can be expected from a quantitative approach. However a full 3D interpretation is too laborious a task in comparison with the limited cost and time which are involved in the majority of such field studies. It is then of value to define the situations where lateral variations are sufficiently smooth for a 1D model to describe correctly the underlying features.
For slingram conductivity measurements, criteria allowing an approximate 1D inversion are defined: these mainly consist of a limited rate of variation over three times the intercoil spacing.
In geological contexts where the weathering has generated a conductive intermediate layer between the underlying sound rock and the soil, this processing can be applied to determine the thickness of the conductive layer from the apparent resistivity map when the other geoelectrical parameters are known. The examples presented illustrate this application.



Two field applications of the rotating current EM method^{1}
By S.H. HallAbstractThe rotating current EM method has been applied to the delineation of two conductive orebodies, Elura near Cobar, NSW, and Thalanga near Charter's Towers, Queensland. The field data were collected in the form of observations of the vertical magnetic field strength ratio and phase difference using a Turam‐style receiver with twin vertical coils. By reconstituting this data back to the ring source field and phase, i.e. the observed Hz, phasor, it is possible to present contoured maps of the EM field. Anomaly phasors are obtained by subtracting theoretical phasors from the observed phasors in the complex plane of the Hz phasor. The theoretical phasors for the finite source are based on horizontally layered, half‐space earth models, computed at each point of the survey grids, then normalized to selected points of the observed fields. Use is made of the intrinsic circular symmetry of the method in X–Y plots of field versus source‐receiver distance to ascertain geoelectric parameters for the earth models. A steel picket fence at Thalanga is modelled by a line source grounded at each end and its Hz, phasor is removed by the same process.
A considerable improvement in anomaly delineation is gained over previous Turam‐style anomalies and the two survey examples illustrate the limitations of the method in the presence of a conductive overburden (Elura) and its abilities in the absence of a conductive overburden (Thalanga).



Optimum expression for computation of the magnetic field of a homogeneous polyhedral body^{1}, ^{2}
By M. IvanAbstractAn expression which is optimum with respect to the simplicity of the numerical computations is obtained for the magnetic field of a polyhedron with constant magnetization. The high accuracy of the results is illustrated using a realistic numerical model.
The existence of the magnetic field at points inside the source and on its boundary is discussed and related to real magnetic data modelling.



An iterative method for finding small magnetic objects in the subsurface by linear and non‐linear inversion^{1}
Authors M. Mirzaei and J.W. BredewoutAbstractA method to determine the position and magnetization vector of buried objects producing a magnetic anomaly is described. The data used were collected in boreholes. Since the anomaly is due to a number of objects, a ‘stripping’ procedure is employed for finding them, and therefore the process of inversion for finding all objects causing the anomaly consists of a few inversion steps.
In each inversion step, two dipoles are considered as a model which approximates an object. The position and magnetic moments of the dipoles are the unknown parameters. The initial parameters are optimized by minimization of an objective function. The optimization procedure consists of a combination of linear and non‐linear inversion. The solution of the linear inversion is obtained by singular value decomposition and that of the non‐linear inversion by a six‐dimensional simplex method (polytope algorithm). After finding one object, its effect is subtracted (‘stripped’) from the data and a new inversion step is started with new initial models and with a reduced data set. The inversion steps for finding different objects are continued until the absolute norm of the data becomes less than some adjustable value.
The data will also be inverted assuming a three‐dipole model in order to find the effect of using a more complex model in the inversion.
The efficiency of the method is demonstrated using synthetic and real borehole data.



Bayesian inference, Gibbs' sampler and uncertainty estimation in geophysical inversion^{1}
Authors Mrinal K. Sen and Paul L. StoffaAbstractThe posterior probability density function (PPD), σ(mdobs), of earth model m, where dobs are the measured data, describes the solution of a geophysical inverse problem, when a Bayesian inference model is used to describe the problem. In many applications, the PPD is neither analytically tractable nor easily approximated and simple analytic expressions for the mean and variance of the PPD are not available. Since the complete description of the PPD is impossible in the highly multi‐dimensional model space of many geophysical applications, several measures such as the highest posterior density regions, marginal PPD and several orders of moments are often used to describe the solutions. Calculation of such quantities requires evaluation of multidimensional integrals. A faster alternative to enumeration and blind Monte‐Carlo integration is importance sampling which may be useful in several applications. Thus how to draw samples of m from the PPD becomes an important aspect of geophysical inversion such that importance sampling can be used in the evaluation of these multi‐dimensional integrals. Importance sampling can be carried out most efficiently by a Gibbs' sampler (GS). We also introduce a method which we called parallel Gibbs' sampler (PGS) based on genetic algorithms (GA) and show numerically that the results from the two samplers are nearly identical.
We first investigate the performance of enumeration and several sampling based techniques such as a GS, PGS and several multiple maximum a posteriori (MAP) algorithms for a simple geophysical problem of inversion of resistivity sounding data. Several non‐linear optimization methods based on simulated annealing (SA), GA and some of their variants can be devised which can be made to reach very close to the maximum of the PPD. Such MAP estimation algorithms also sample different points in the model space. By repeating these MAP inversions several times, it is possible to sample adequately the most significant portion(s) of the PPD and all these models can be used to construct the marginal PPD, mean) covariance, etc. We observe that the GS and PGS results are identical and indistinguishable from the enumeration scheme. Multiple MAP algorithms slightly underestimate the posterior variances although the correlation values obtained by all the methods agree very well. Multiple MAP estimation required 0.3% of the computational effort of enumeration and 40% of the effort of a GS or PGS for this problem. Next, we apply GS to the inversion of a marine seismic data set to quantify uncertainties in the derived model, given the prior distribution determined from several common midpoint gathers.

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