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 Volume 41, Issue 5, 1993
Geophysical Prospecting  Volume 41, Issue 5, 1993
Volume 41, Issue 5, 1993


MULTI‐OFFSET ACOUSTIC INVERSION OF A LATERALLY INVARIANT MEDIUM: APPLICATION TO REAL DATA^{1}
Authors JAN HELGESEN and PIERRE KOLBAbstractThe aim of seismic inversion methods is to obtain quantitative information on the subsurface properties from seismic measurements. However, the potential accuracy of such methods depends strongly on the physical correctness of the mathematical equations used to model the propagation of the seismic waves. In general, the most accurate models involve the full non‐linear acoustic or elastic wave equations. Inversion algorithms based on these equations are very CPU intensive. The application of such an algorithm on a real marine CMP gather is demonstrated. The earth model is assumed to be laterally invariant and only acoustic wave phenomena are modelled. A complete acoustic earth model (P‐wave velocity and reflectivity as functions of vertical traveltime) is estimated. The inversion algorithm assumes that the seismic waves propagate in 2D. Therefore, an exact method for transforming the real data from 3D to 2D is derived and applied to the data. The time function of the source is estimated from a vertical far‐field signature and its applicability is demonstrated by comparing synthetic and real water‐bottom reflections. The source scaling factor is chosen such that the false reflection coefficient due to the first water‐bottom multiple disappears from the inversion result. In order to speed up the convergence of the algorithm, the following inversion strategy is adopted: an initial smooth velocity model (macromodel) is obtained by applying Dix's equation to the result of a classical velocity analysis, followed by a smoothing operation. The initial reflectivity model is then computed using Gardner's empirical relationship between densities and velocities. In a first inversion step, reflectivity is estimated from small‐offset data, keeping the velocity model fixed. In a second step, the initial smooth velocity model, and possibly the reflectivity model, is refined by using larger‐offset data. This strategy is very efficient. In the first step, only ten iterations with a quasi‐Newton algorithm are necessary in order to obtain an excellent convergence. The data window was 0–2.8 s, the maximum offset was 250 m, and the residual energy after the first inversion step was only 5% of the energy of the observed data. When the earth model estimated in the first inversion step is used to model data at moderate offsets (900 m, time window 0.0–1.1 s), the data fit is very good. In the second step, only a small improvement in the data fit could be obtained, and the convergence was slow. This is probably due to the strong non‐linearity of the inversion problem with respect to the velocity model. Nevertheless, the final residual energy for the moderate offsets was only 11%.
The estimated model was compared to sonic and density logs obtained from a nearby well. The comparison indicated that the present algorithm can be used to estimate normal incidence reflectivity from real data with good accuracy, provided that absorption phenomena play a minor role in the depth interval considered. If details in the velocity model are required, large offsets and an elastic inversion algorithm should be used.



3D KINEMATIC INVERSION FROM A SET OF LINE PROFILES^{1}
Authors VALERY SORIN, SHEMER KEYDAR and EVGENY LANDAAbstractIn the case of 3D multilayered structures the 2D interval velocity analysis may be inaccurate. This fact is illustrated by synthetic examples.
The method proposed solves the 3D inverse problem within the scope of the ray approach. The solution, i.e. the interval velocities and the reflection interface position, is obtained using data from conventional 2D line profiles arbitrarily located and from normal incidence time maps. Although the input information is essentially limited, the method presented reveals only minor biased velocity estimates.
In order to implement the proposed 3D inversion method, we developed a processing procedure. The procedure performs the evaluation of reflection time and ray parameters along line profiles, 3D interval velocity estimation, and time‐to‐depth map migration. Tools to stabilize the 3D inversion are investigated.
The application of the 3D inversion technique to synthetic and real data is compared with results of the 2D inversion.



RESIDUAL STATICS ESTIMATION BY STACK‐POWER MAXIMIZATION IN THE FREQUENCY DOMAIN^{1}
By E. NØRMARKAbstractTraditionally, residual static corrections are based on timeshifts estimated for individual CMP sorted traces, which are later resolved into surface‐consistent statics. This is a stable and attractive procedure because the data flow is simple and the memory storage required is limited. An alternative station‐oriented method maximizing the stack‐power estimates surface‐consistent static corrections directly. The statics evaluation in this method involves several CMP gathers, which should improve the prediction of statics on noise‐contaminated data. In this paper the performance of the above methods will be compared using synthetic as well as real seismic data. Neither method is capable of estimating large statics compared to the dominating period, because local optimization might fail. Global Monte Carlo search by, for instance, simulated annealing has been used to overcome the cycle‐skipping problems when proper field statics are missing. Although this procedure is computationally very heavy, it may be the only way to deal with large residual statics. In order to enlarge the operational field for local optimization, it is suggested that the stack‐power in the frequency domain is maximized. This makes it easy to change the frequency band during the optimization. Making use of the frequency domain will also normally be faster than the traditional time‐domain optimization even for a limited number of iterations. Moreover, the main memory storage required can be significantly reduced, since it is only necessary to keep the frequency band in the memory, where the signal‐to‐noise ratio is good.



RESIDUAL STATICS ESTIMATION: SCALING TEMPERATURE SCHEDULES USING SIMULATED ANNEALING^{1}
Authors E. NØRMARK and K. MOSEGAARDAbstractLinearized residual statics estimation will often fail when large static corrections are needed. Cycle skipping may easily occur and the consequence may be that the solution is trapped in a local maximum of the stack‐power function. In order to find the global solution, Monte Carlo optimization in terms of simulated annealing has been applied in the stack‐power maximization technique. However, a major problem when using simulated annealing is to determine a critical parameter known as the temperature.
An efficient solution to this difficulty was provided by Nulton and Salamon (1988) and Andresen et al. (1988), who used statistical information about the problem, acquired during the optimization itself, to compute near optimal annealing schedules.
Although theoretically solved, the problem of finding the Nulton–Salamon temperature schedule often referred to as the schedule at constant thermodynamic speed, may itself be computationally heavy. Many extra iterations are needed to establish the schedule.
For an important geophysical inverse problem, the residual statics problem of reflection seismology, we suggest a strategy to avoid the many extra iterations. Based on an analysis of a few residual statics problems we compute approximations to Nulton–Salamon schedules for almost arbitrary residual statics problems. The performance of the approximated schedules is evaluated on synthetic and real data.



DETERMINATION OF ELECTRICAL PROPERTIES OF THE GROUND AT SHALLOW DEPTH WITH AN ELECTROSTATIC QUADRUPOLE: FIELD TRIALS ON ARCHAEOLOGICAL SITES^{1}
Authors ALAIN TABBAGH, ALBERT HESSE and RÉJEAN GRARDAbstractKnowledge of both the electrical resistivity ρ and the dielectric permittivity ɛ of the ground is important for the determination of characteristics such as granularity, porosity, moisture and salt content. Whereas the measurement of ρ is very common and can be achieved using either the d.c. resistivity method or a wide variety of EM devices, ɛ remains practically unknown in the low‐frequency domain between the IP domain and the high‐frequency domain. Following the principle of the quadrupole technique used in d.c. prospecting and of the quadrupolar probe for measuring the complex permittivity of the ionosphere, we propose a new approach which does not require any galvanic contact between the poles and the ground. The transfer impedance can be evaluated using the quasi‐static approximation for low frequency or the full EM theory for higher frequencies. The conditions under which both calculations apply are discussed. At low frequencies and for low resistivity ground, the electrostatic quadrupole measures ρ exactly as with the d.c. technique; for higher resistivities or frequencies the simultaneous measurement of both properties becomes possible. Examples in archaeological prospecting are presented and checked against independent d.c. resistivity measurements or excavation analyses.



STUDY OF TRAVELTIME AND AMPLITUDE TIME‐LAPSE TOMOGRAPHY USING PHYSICAL MODEL DATA^{1}
Authors M. LEGGETT, N. R. GOULTY and J. E. KRAGHAbstractIn seismic tomography the observed traveltimes or amplitudes of direct waves are inverted to obtain an estimate of seismic velocity or absorption of the section surveyed. There has been much recent interest in using cross‐well traveltime tomography to observe the progress of fluids injected into the reservoir rocks during enhanced oil recovery (EOR) processes. If repeated surveys are carried out, then EOR processes may be monitored over a period of time.
This paper describes the results of a simulated time‐lapse tomography experiment to image the flood zone in an EOR process. Two physical models were made out of epoxy resins to simulate an essentially plane‐layered sedimentary sequence containing a reservoir layer and simple geological structure. The models differed only in the reservoir layer, which was uniform in the ‘pre‐flood’ model and contained a flood zone of known geometry in the ‘post‐flood’ model. Data sets were acquired from each model using a cross‐well survey geometry. Traveltime and amplitude tomographic imaging techniques have been applied to these data in an attempt to locate the extent of the flood zone.
Traveltime tomography locates the flood zone quite accurately. Amplitude tomography shows the flood zone as a region of higher absorption, but does not image its boundaries as precisely. This is primarily because of multipathing and diffraction effects, which are not accounted for by the ray‐based techniques for inverting seismic amplitudes. Nevertheless, absorption tomograms could complement velocity tomograms in real, heterogeneous reservoirs because absorption and velocity respond differently to changes in liquid/gas saturations for reservoir rocks.



MIGRATION TO ZERO OFFSET (DMO) FOR A CONSTANT VELOCITY GRADIENT: AN ANALYTICAL FORMULATION^{1}
Authors MICHEL DIETRICH and JACK K. COHENAbstractMigration to zero offset (MZO) is a prestack partial migration process that transforms finite‐offset seismic data into a close approximation to zero‐offset data, regardless of the reflector dips that are present in the data. MZO is an important step in the standard processing sequence of seismic data, but is usually restricted to constant velocity media. Thus, most MZO algorithms are unable to correct for the reflection point dispersal caused by ray bending in inhomogeneous media.
We present an analytical formulation of the MZO operator for the simple possible variation of velocity within the earth, i.e. a constant gradient in the vertical direction. The derivation of the MZO operator is carried out in two steps. We first derive the equation of the constant traveltime surface for linear V(z) velocity functions and show that the isochron can be represented by a fourth‐degree polynomial in x, y and z. This surface reduces to the well‐known ellipsoid in the constant‐velocity case, and to the spherical wavefront obtained by Slotnick in the coincident source‐receiver case.
We then derive the kinematic and dynamic zero‐offset corrections in parametric form by using the equation of the isochron. The weighting factors are obtained in the high‐frequency limit by means of a simple geometric spreading correction. Our analytical results show that the MZO operator is a multivalued, saddle‐shaped operator with marked dip moveout effects in the cross‐line direction. However, the amplitude analysis and the distribution of dips along the MZO impulse response show that the most important contributions of the MZO operator are concentrated in a narrow zone along the in‐line direction. In practice, MZO processing requires approximately the same trace spacing in the in‐line and cross‐line directions to avoid spatial aliasing effects.



ON THE CONSTRUCTION OF THE 3D BAND‐LIMITED EXTRAPOLATION OPERATOR IN THE SPACE‐FREQUENCY DOMAIN^{1}
More LessAbstractBased on an expansion of the band‐limited 3D extrapolation operator in terms of orthogonal Chebyshev polynomials, a closed form expression of the space‐frequency response is presented. A key step is an evaluation of the (inverse) 2D Fourier transform of circularly symmetric functions, which is related to the (zero‐order) Hankel transform. Hankel transforms of individual members of the orthogonal set of polynomials are available from tables and summation of series; hence, the real and the imaginary parts of the space‐frequency response can be found in terms of cylindrical and spherical Bessel functions, respectively. The procedure permits an efficient and accurate evaluation of the space‐frequency response.

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