DERIVATION OF LAYER PARAMETERS OF AN ELASTIC MEDIUM FROM REFLECTION COEFFICIENT MATRICES*

In this paper we develop a recursive algorithm to obtain the layer parameters of an elastic medium (density, P-wave velocity, S-wave velocity) from reflection coefficient matrices in terms of energy flux ratios for a non-normal incidence case. We define a layer impedance matrix, analogous to the impedance of an acoustic medium. Next we derive a matrix relationship between the layer impedance matrix of the n+ 1st layer and the reflection coefficient and parameter matrices of the nth layer. This relationship leads to recursively computing the parameters of the subsurface. We show that the elastic case—unlike the acoustic case—allows one to recover the layer parameters from the impedance matrix for non-normal incidence. The results of this work play a key role in the solution of the inverse problem with non-normal-incidence plane-wave seismic data when using a downward continuation technique.     

Data acquisition and pre-processing required for simultaneous P-SV inversion

The goal of seismic reflection surveys is the derivation of petrophysical subsurface parameters from surface measurements. Today's well established technique in data acquisition, as well as processing terms, is based on the acoustic approximation to the real world's wave propagation. In recent years a lot of work has been done to extend the technique to the elastic approximation. There was especially an important trend towards elastic inversion techniques operating on plane-wave seismograms, called simultaneous P-SV inversion (or short P-SV inversion) within this paper. Being still under investigation, some important aspects of P-SV inversion concerning data acquisition as well as pre-processing, should be pointed out. To fit the assumptions of P-SV inversion schemes, at least a two-dimensional picture of the reflected wavefield with vertical and in-line horizontal receivers has to be recorded. Moreover, the theoretical work done suggests that in addition to a survey with a compressional wave source, a second survey should be done using sources radiating vertically polarized shear waves, is needed. Finally, proper slant stacking must be performed to get plane-wave seismograms. The P/S separated plane-wave seismograms are then well prepared for feeding into the inversion algorithms. P/S separated planewave seismograms are then well prepared for feeding into the inversion algorithm.s In this paper, a tutorial overview of the data acquisition and pre-processing in accordance with the P-SV inversion philosophy is given and illustrated using synthetic seismograms. A judgement on the feasibility of the P-SV inversion philosophy must be left to ongoing research.     

TIME-DOMAIN COMPUTATION OF NON-NORMAL INCIDENCE WAVEFIELDS IN PLANE LAYERED MEDIA1

A new time-domain method is introduced for the calculation of theoretical seismograms which include frequency dependent effects like absorption. To incorporate these effects the reflection and transmission coefficients become convolutionary operators. The method is based on the communication theory approach and is applicable to non-normal incidence plane waves in flat layered elastic media. Wave propagation is simulated by tracking the wave amplitudes through a storage vector inside the computer memory representing a Goupillaud earth model discretized by equal vertical transit times. Arbitrary numbers of sources and receivers can be placed at arbitrary depth positions, while the computational effort is independent of that number. Therefore, the computation of a whole plane-wave vertical seismic profile is possible with no extra effort compared to the computation of the surface seismogram. The new method can be used as an aid to the interpretation of plane-wave decomposed reflection data where the whole synthetic vertical seismic profile readily gives the interpreter the correct depth position of reflection events.     

APPLICATION OF ELASTIC MODELLING IN PROCESSING AND INTERPRETATION OF VSP DATA: A CASE HISTORY1

Data from offshore Norway is used to study applications of elastic VSP modelling in detecting shear waves and observing the effects of successive mode conversion in field-recorded VSP data. The shear-wave velocities and densities from log data are used in conjunction with compressional wave velocities determined from surface seismic and log data in the VSP modelling. The time domain non-normal incidence elastic VSP modelling technique of Aminzadeh and Mendel is used as the modelling algorithm. Two surface seismograms are computed first. One is the vertical component and the other is the horizontal component for plane waves that have specified incident angles. A downward continuation method is then applied to generate seismograms at different depth points. The collection of these seismograms constitutes non-normal incidence VSPs. Both vertical and horizontal components of VSP data can be obtained by this procedure. In this paper non-normal incidence VSPs are generated for a 12.5° incident plane wave. The modelling results of layered earth systems of thin layers and thick layers are both compared with field data, and the effect of mode conversions in thin layers is observed. Several events in the field data can be explained by this elastic VSP modelling. Comparison of the model data and field data enabled a probable tube wave or out-of-plane event to be identified, the removal of which significantly improved the final VSP section. This study also shows how the VSP data helped the interpretation of the surface 3D data.     

SYNTHETIC SEISMOGRAMS AT NON-VERTICAL INCIDENCE*

Synthetic seismograms are usually computed for reflections from vertical incidence of P waves for a horizontally layered medium. In actual practice the angle of incidence departs from the vertical, as receivers are usually located at some distance from the source. At angles other than the vertical, the conversion of P- to S-wave energy and changes in the reflection coefficient affect the shape of the synthetic seismograms. The effect of non-vertical incidence on synthetic seismograms is examined in this paper. Seismograms at non-vertical incidence have been computed using the plane-wave approach of Haskell (1953) for a layered medium. The use of plane waves is an approximation to the actual case of spherical wavefronts from a surface source. Using plane-wave theory, the expected wave forms as a function of angle of incidence were computed numerically for several simple models. The results indicate that the synthetic seismograms do not change significantly for angles of incidence between o and 25 degrees. For larger angles the changes in the wave forms may be severe. The effect is more pronounced for high-velocity layers than for low-velocity layers.     

NUMERICAL RAY GENERATION IN THE COMPUTATION OF SYNTHETIC VERTICAL SEISMIC PROFILES1

Ray theories are a class of methods often chosen to compute synthetic seismograms due to their efficiency and ability to deal with complex, three-dimensional inhomogeneous media. To deal with the large number of rays needed to compute synthetic seismograms, a ray generation algorithm is given which is capable of generating a numerical code describing each ray. The code describes a subset of all possible rays by considering only pre-critical reflections. In a horizontally plane-layered medium the generation of rays and computation of amplitudes and traveltimes can be efficiently accomplished by grouping the rays into reflection order and dynamic analogue groups. Expressions summing all unconverted rays and rays with a single mode conversion are given for source and receiver located at arbitrary positions within the medium. Examples of zero-offset synthetic VSPs obtained by this method are given.     

Generalized migration in frequency-wavenumber domain (MGF-K) in anisotropic media

In this paper, the background of MGF-K migration in dual domain (wavenumber-frequency K-F and space-time) in anisotropic media is presented. Algorithms for poststack (zero-offset) and prestack migration are based on downward extrapolation of acoustic wavefield by shift-phase with correction filter for lateral variability of medium’s parameters. In anisotropic media, the vertical wavenumber was determined from full elastic wavefield equations for two dimensional (2D) tilted transverse isotropy (TTI) model. The method was tested on a synthetic wavefield for TTI anticlinal model (zero-offset section) and on strongly inhomogeneous vertical transverse isotropy (VTI) Marmousi model. In both cases, the proper imaging of assumed media was obtained.     

COMPLETE INVERSION OF ZERO-OFFSET SEISMIC DATA*

The one-dimensional seismic inverse problem consists of recovering the acoustic impedance (or reflectivity function) as a function of traveltime from the reflection response of a horizontally layered medium excited by a plane-wave impulsive source. Most seismic sources behave like point sources, and the data must be corrected for geometrical spreading before the inversion procedure is applied. This correction is usually not exact because the geometrical spreading is different for primary and multiple reflections. An improved algorithm is proposed which takes the geometrical spreading from a point source into account. The zero-offset reflection response from a stack of homogeneous layers of variable thickness is used to compute the thickness, velocity and density of each layer. This is possible because the geometrical spreading contains additional information about the velocities.     

FILTER DESIGN FOR SUPPRESSION OF SURFACE MULTIPLES IN A NON-NORMAL INCIDENCE SEISMOGRAM*

In this paper we design a non-linear filter to suppress surface multiples in a non-normal incidence plane wave seismogram of a horizontally-layered elastic (or acoustic) earth model. Our filter is optimal in a least-squares sense and is very efficient in suppressing surface multiples, especially for small incidence angles. The design is based on an extension of earlier work by Mendel on normal incidence Bremmer series decomposition to the case of non-normal incidence, and relies heavily on work by Aminzadeh on a non-normal incidence state space model.     

Exact elastic impedance tensor for isotropic media

The existing expressions of elastic impedance,as the generalized form of acoustic impedance,represent the resistance of subsurface media to seismic waves of non-normal incidence,and thus include information on the shear-wave velocity.In this sense,conventional elastic impedance is an attribute of the seismic reflection and not an intrinsic physical property of the subsurface media.The derivation of these expressions shares the approximations made for reflectivity,such as weak impedance contrast andisotropic or weakly anisotropic media,which limits the accuracy of reflectivity reconstruction and seismic inversion.In this paper,we derive exact elastic impedance tensors of seismic P-and S-waves for isotropic media based on the stress-velocity law.Each componentof the impedance tensor represents a unique mechanical property of the medium.Approximations of P-wave elastic impedance tensor components are discussed for seismic inversion and interpretation.Application to synthetic data and real data shows the accuracy and robust interpretation capability of the derived elastic impedance in lithology characterizations.     

利用伪谱法模拟横向各向同性介质中的波

介质的弹性常数为三维四阶张量的分量,共有８１个,由于应力张量和应变张量的对称性及能量密度是应变的二次函数,一般各向异常性介质的独立弹性常数可减为２１个,如果介质具有较高的对称性,独立弹性常数的数目会更少。 对于地壳和上地幔,具有５个独立弹性常数的横向各向同性介质是一个非常好的近似,本研究中横向各向同性介质的对称轴方向可以是任意的（即对称轴可以不平等于铅直方向）,在此情况下,需要进行坐标变换,如果已知介质在某一坐标系（其坐标轴平行或垂直于介质的对称轴）中的弹性常数,我们能够容易地利用变换公式得到变换后新坐标系中的弹性常数。 本文提出了一种方案,利用伪谱法既能模拟横向各向同性介质中的平面波,也能模拟点源激发的波场。在勘探地球物理和地震学中,模拟横向各向同性介拮中传播的平面波及区域源产生的波是最重要的研究课题之一。然而在一般各向异性介质中,很难或不可能确定弹性波的相速度和偏振方向,但在横向各向同性介质中,则可以通过坐标变换来实现,这里我们所提出的方法可以用于横向各向同性介质中弹性波的模拟。     

WAVEFRONT DIVERGENCE,MULTIPLES, AND CONVERTED WAVES IN SYNTHETIC SEISMOGRAMS*

Synthetic seismograms can be very useful in aiding understanding of wave propagation through models of real media, verification of geologic models derived from interpretation of field seismic data, and understanding the nature and complexity of wave phenomena. If meaningful results are to be obtained from synthetic seismograms, the method of their computation must, in general, include three-dimensional geometrical spreading of wavefronts associated with highly concentrated (i.e., point) sources. The method should also adequately represent the seismic response of solid-layered media by including enough primaries, multiples, and converted phases to accurately approximate the total wavefield. In addition to these features, it is also very helpful, although not always essential, if the method of seismogram computation provides for explicit identification of wave type and ray path for each arrival. Various seismograms, computed via asymptotic ray theory and an automatic ray generation scheme, are presented for a highly simplified North Sea velocity structure. This is done to illustrate the importance of the above features and to demonstrate the inadequacy of the plane-wave synthesis method of seismogram computation for point sources and the limitations of acoustic models of solid-layered media.     

A NONITERATIVE PROCEDURE FOR INVERTING PLANE-WAVE REFLECTION DATA AT SEVERAL ANGLES OF INCIDENCE USING THE RICCATI EQUATION*

Various exact methods of inverting the complete waveform of vertical seismic reflection data to produce acoustic impedance profiles have been suggested. These inverse methods generally remain valid for nonvertical, plane-wave data, provided total reflection does not occur. Thus, in principle, the “seismogram” at each ray parameter in a slant stack can be interpreted separately. Rather than invert each plane-wave seismogram separately, they can all be interpreted simultaneously and an “average” model thus obtained. Inversion for both the velocity and the density also becomes possible when two or more plane-wave seismograms are simultaneously inverted. The theory for a noniterative inversion method, based on the time-domain Riccati equation, is discussed. Numerical examples of inversions using this technique on synthetic data demonstrate its numerical stability and the advantage of simultaneous inversion of several seismograms to reduce the effect of noise in the data and increase the stability of the inversion process.     

DOWNHOLE SYNTHETIC SEISMIC PROFILES IN ELASTIC MEDIA1

A multichannel lattice filter structure is utilized to represent seismic waves propagating in adjacent layers in an elastic medium. Using this model, an explicit time-domain solution for arbitrary source and receiver locations is obtained as an ARMA (AutoRegressive and Moving-Average) process. The lattice and ARMA structures have given rise to an effective algorithm for the calculation of offset/downhole synthetic seismograms. A large range of recently developed offset/downhole seismic survey geometries, such as the ‘Yo-Yo’ arrangement, can thus be simulated. In addition, the explicit solutions for upgoing and downgoing waves provide new insight into the properties of general downhole seismic signals, including wave-mode conversion effects and multiple reflections. Furthermore, offset/downhole seismograms generated by a line source (i.e. 2D point source) can also be constructed by superposition of plane waves with different incidence angles. Synthetic seismograms generated using a different source-receiver arrangement indicate that the properties especially associated with offset/downhole seismic signals can be predicted by this modelling method. These properties include arrival times, amplitude attenuation and wave-mode conversion effects. Finally, utilizing this numerical modelling method to a real downhole survey with Yo-Yo geometry may lead to a proper data acquisition and processing procedure, and improves the interpretation confidence of the field section.     

Seismic resonances of spherical acoustic cavities

We study the interaction of a seismic wavefield with a spherical acoustic gas‐ or fluid‐filled cavity. The intention of this study is to clarify whether seismic resonances can be expected, a characteristic feature that may help in detecting cavities in the subsurface. This is important for many applications, in particular the detection of underground nuclear explosions, which are to be prohibited by the Comprehensive Test Ban Treaty. To calculate the full seismic wavefield from an incident plane wave that interacts with the cavity, we considered an analytic formulation of the problem. The wavefield interaction consists of elastic scattering and the wavefield interaction between the acoustic and elastic media. Acoustic resonant modes caused by internal reflections in the acoustic cavity show up as spectral peaks in the frequency domain. The resonant peaks coincide with the eigenfrequencies of the un‐damped system described by the particular acoustic medium bounded in a sphere with stiff walls. The filling of the cavity could thus be determined by the observation of spectral peaks from acoustic resonances. By energy transmission from the internal oscillations back into the elastic domain, the oscillations experience damping, resulting in a frequency shift and a limitation of the resonance amplitudes. In case of a gas‐filled cavity, the impedance contrast is still high, which means low damping of the internal oscillations resulting in very narrow resonances of high amplitude. In synthetic seismograms calculated in the surrounding elastic domain, the acoustic resonances of gas‐filled cavities show up as persisting oscillations. However, due to the weak acoustic–elastic coupling in this case, the amplitudes of the oscillations are very low. Due to a lower impedance contrast, a fluid‐filled cavity has a stronger acoustic–elastic coupling, which results in wide spectral peaks of lower amplitudes. In the synthetic seismograms derived in the surrounding medium of fluid‐filled cavities, acoustic resonances show up as strong but fast decaying reverberations.     

LEVINSON INVERSION OF EARTHQUAKE GEOMETRY SH-TRANSMISSION SEISMOGRAMS IN THE PRESENCE OF NOISE*

The Kunetz-Claerbout equation for the acoustic transmission problem in a layered medium in its original form establishes the relation between the transmission and the reflec tion response for P-waves in an horizontally layered medium and with vertical incidence. It states that the reflection seismogram due to an impulsive source at the surface, is one side of the autocorrelation of the seismogram due to an impulsive source at depth and a surface receiver. By adapting Claerbout's formulation to the transmission of SH-waves, the Kunetz-Claerbout equation also holds for reflection and transmission coefficients dependent on the incident angle. Thus, earthquake geometry SH-transmission seismograms can be used to caculate corresponding pseudoreflection seismograms which, in turn, can be inverted for the impedance structure using the Levinson algorithm. If the average incidence angle is known, a geometrical correction on the resulting impedance model can improve the resolution of layer thicknesses. In contrast to the inversion of reflection seismograms, the Levinson algorithm is shown to yield stable results for the inversion of transmission seismograms even in the presence of additive noise. This noise stabilization is inherent to the Kunetz-Claerbout equation. Results of inverted SH-wave microearthquake seismograms from the Swabian Jura, SW Germany, seismic zone obtained at recording site Hausen im Tal have been compared with sonic-log data from nearby exploration drilling at Trochtelfingen. The agreement of the main structural elements is fair to a depth of several hundred metres.     

HYBRID SEISMIC MODELLING: A TECHNIQUE TO COMBINE PHYSICAL AND COMPUTER METHODS FOR VERTICAL WAVE INCIDENCE*

A hybrid seismic modelling technique has been developed to investigate complex geological phenomena. Those parts of a geological structure which are too complicated to be treated theoretically are studied by two-dimensional physical models; other sections of the structure which can be treated theoretically, i.e., inhomogeneities in the vertical direction, are modelled by computer methods. A feedback process is used to combine the results of both physical and computer modelling. Horizontally layered coal-seam models are presented to test the hybrid modelling technique for normal incidence. A comparison of the hybrid seismograms with pure synthetic seismograms shows an acceptable conformity for normal incidence. A hybrid zero-offset section is shown to investigate a complex geological structure in the Ruhr coalfield in Germany.     

FULL WAVE EQUATION DOWNWARD CONTINUATION OF SEISMIC REFLECTION DATA*

A new method for suppressing multiple reflections in seismograms is developed. It is based on a downward continuation procedure which uses the full acoustic wave equation (hyperbolic form) as a downward continuation operator. We demonstrate that the downward continuation of the recorded wave field maps a reflectivity function without multiply reflected events. The method is applied successfully to individual traces of plane-wave decomposed (slant-stacked) synthetic and field data.     

Seismogram Synthesis in Multi-layered Half-space Part Ⅰ. Theoretical Formulations

In the past two decades numerous studies were made to develop and improve the theory and practical computation techniques of synthesizing theoretical seismograms for the model of multi-layered half-space. Today, synthesizing theoretical seismograms in multi-layered half-space is an important tool for understanding the structure of the Earth's interior as well as the seismic source process from well-recorded seismograms data. As part of a review of the state-of-the-art, in this article I shall present a systematic and self-contained theory of elastic waves in multi-layered half-space media based on the developments in the past two decades.     

THEORETICAL REFLECTION SEISMOGRAMS FOR ELASTIC MEDIA*

Theoretical seismograms for an explosive source in a multilayered elastic medium are constructed by Fourier synthesis and plane wave superposition. The calculation scheme which builds up a reflection matrix layer by layer in the frequency and wave number domain allows the inclusion of attenuation and a choice of the level of internal multiples in each layer. Comparative calculations of theoretical seismograms for an elastic model and in the acoustic approximation, neglecting shear, show that the main differences arise at large offsets. The inclusion of shear waves leads to lower reflected P wave amplitudes at the end of the spread but only small amounts of converted phases.