A. J. W. Duijndam

Reconstruction of band-limited signals, irregularly sampled along one spatial direction

Seismic signals are often irregularly sampled along spatial coordinates, leading to suboptimal processing and imaging results. Least squares estimation of Fourier components is used for the reconstruction of band-limited seismic signals that are irregularly sampled along one spatial coordinate. A simple and efficient diagonal weighting scheme, based on the distance between the samples, takes the properties of the noise (signal outside the bandwidth) into account in an approximate sense. Diagonal stabilization based on the energies of the signal and the noise ensures robust estimation. Reconstruction for each temporal frequency component allows the specification of a varying spatial bandwidth dependent on the minimum apparent velocity. This parameterization improves the reconstruction capability for the lower temporal frequencies. In practical circumstances, the maximum size of the gaps in which the signal can be reconstructed is three times the (temporal frequency dependent) Nyquist interval. Reconstruction in the wavenumber domain allows a very efficient implementation of the algorithm, and takes a total number of operations a few times that of a 2-D fast Fourier transform corresponding to the size of the output data set. Quality control indicators of the reconstruction procedure can be computed which may also serve as decision criteria on in-fill shooting during acquisition. The method can be applied to any subset of seismic data with one varying spatial coordinate. Applied along the cross-line direction, it can be used to compute a 3-D stack with improved anti-alias protection and less distortion of the signal within the bandwidth.

Reconstruction of 3-D seismic signals irregularly sampled along two spatial coordinates

Seismic signals are often irregularly sampled along spatial coordinates, leading to suboptimal processing and imaging results. Least‐squares estimation of Fourier components is used for the reconstruction of band‐limited seismic signals that are irregularly sampled along two spatial coordinates. A simple and efficient diagonal weighting scheme, based on the areas surrounding the spatial samples, takes the properties of the noise (signal outside the bandwidth) into account in an approximate sense. Diagonal stabilization based on the energies of the signal and the noise ensures robust estimation. Reconstruction by temporal frequency component allows the specification of varying bandwidth in two dimensions, depending on the minimum apparent velocity. This parameterization improves the reconstruction capability for lower temporal frequencies. The shape of the spatial aperture affects the method of sampling the Fourier domain. Taking into account this property, a larger bandwidth can be recovered. The properti...

Parabolic Radon transform, sampling and efficiency

A good choice of the sampling in the transform domain is essential for a successful application of the parabolic Radon transform. The parabolic Radon transform is computed for each temporal frequency and is essentially equivalent to the nonuniform Fourier transform. This leads to new and useful insights in the parabolic Radon transform. Using nonuniform Fourier theory, we derive a minimum sampling interval for the curvature parameter and a maximum curvature range for which stability is guaranteed for general (irregular) sampling. A significantly smaller sampling interval requires stabilization. If diagonal stabilization is used, no gain in resolution is obtained. In contrast to conventional implementations, the curvature sampling interval is proposed to be inversely proportional to the temporal frequency. This results in improved quality of the transform and yields significant savings in computation time.

Ultrasonic velocity and shear‐wave splitting behavior of a Colton sandstone under a changing triaxial stress

Ultrasonic experiments on a dry Colton sandstone placed in a triaxial pressure machine show that effective stress changes lead to distinct anisotropic velocity changes in compressional waves and shear waves. The stress imprint can be recognized from the associated velocity pattern by relating the velocities to the three normal stress directions. The ultrasonic velocities indicate that the sensitivity of the different waves to stress predominantly depends on stresses applied in the polarization and propagation directions of the particular wave mode. Also, stress‐induced changes in shear‐wave splitting are observed.

Handling near surface effects in imaging by using common focal point technology.

Summary With the common focal point (CFP) technology the ef- fects of weathering layers can be integrated in prestack migration. CFP operators can be determined directly from the data through an updating procedure, independent from a velocity model. The updating of operators, which is not trivial in the presence of weathering layers, can be carried out by a semi-automatic procedure. With the aid of the CFP operators a clearly defined migrated section can be constructed. In this paper we consider the near surface as part of the imaging problem. The CFP imaging technology (Berkhout 1997) gives us a tool to describe the subsurface, including the weathered layer, in terms of a lateral distribution of operators. These op- erators can in principle be derived without knowing the correct macro model. This is an advantage over conventional methods since models of the near surface are often very complex. The synthetic results show that clearly defined time migrated images can be derived. Near surface imaging Common focal point migration Through reciprocity theorems a forward model can be derived which describes the measured seismic reflection data, backscat- tered from a depth level . Neglecting angle dependent reflectivity, the oneway integral representation for the primary upgoing scattered data at a receiver position ,a s result of a contrasting interface , and a point source at with source characteristic is given by,

Using focusing operators to estimate velocity models

Focusing operators are derived in the common focal point (CFP) formulation of prestack migration. Each operator corresponds essentially to the time-reversed Greens function in the actual medium, when the source is placed at the location to be imaged (the focal point). By applying these operators twice to the data, the downand upward propagation effects are removed and thus a migrated image is obtained. However by using the traveltime information in these operators we can also estimate a layered macro velocity model. The estimation is not limited to homogeneous layers and all layers can be estimated simultaneously. Several aspects, including uncertainty, with regard to this process are considered and illustrated with a number of examples.

Experimental verification of stress-induced anisotropy

RP1.7 Summary During experiments with a triaxial pressure machine in the laboratory a stress history in rock is induced. While loading and unloading the changes in elastodynamic wave behaviour are measured with ultrasonic transducers. These experiments give insight in the influence of a changing stress state on the wave propagation; the rock becomes anisotropic under differential stress, which expresses itself in an angular dependant change in velocities. A related effect of anisotropy is shear-wave splitting, which is made visible by using single pairs of broad-band shear wave transducers. (5)

Hydraulic fracture characterization with dispersion measurements of seismic waves

During hydraulic fracturing experiments in our laboratory the opening of hydraulic fractures is monitored with ultrasonic transducers. Transmitted and reflected signals show a distinct and strong dispersive behavior. An experiment with a controlled fracture width shows that although the width is very small, the fracture itself can explain the measured dispersion of compressional waves. The strong impedance contrast magnifies the one-way time delay. The ratio of dimensionless fracture width and impedance ratio determines the responses and jump in particle displacement. This is in agreement with the linear slip approximation when the fracture compliance is taken proportional to the fracture width. The product of dimensionless fracture width and impedance ratio acts as the dimensionless fracture density and is very small for hydraulic fractures. The jump in de tractions at the fracture interface can thus be neglected. The dispersion measurement approach can be used to monitor the width of hydraulic fractures and could therefore lead to an increased phenomenological understanding of hydraulic fracture growth.

Analysis and optimization of 3-D seismic acquisition geometries by focal beams

Summary The acquisition geometry of a 3-D seismic survey should be designed in such a way that it fulfils the imaging requirements while satisfying the economical constraints. Focal source and focal detector beams are defined which describe the focusing properties of the source and detector geometry. These beams can be computed and evaluated separately. The computation of these beams can be done very efficiently using a Fourier transform of the source and receiver sampling function. In the case of a stationary spread the potential resolution and accuracy of AVP information can be obtained from these beams. Using the combination rules (per template) for the focal beams a relation between the basic template layout and the roll along distance can be derived in order to optimize the quality of the focal functions. This property can be used to design cost effective acquisition geometries. Conventional marine acquisition geometries yield strong side lobes of the resolution function in the cross-line direction due to the large distance between the sail-lines. By changing the source configuration the side lobe level can be reduced by at least 10 dB.

Radon domain reconstruction of 3D irregularly sampled VSP data

Bandlimited wavefields, irregularly sampled along two spatial coordinates, can be reconstructed by estimating the Fourier components with a least squares procedure. A very efficient algorithm for this procedure has ben developed. Since many seismic processing schemes are carried out in the wave-number domain, the approach of the algorithm can be advantageously incorporated within several procedures. One of these procedures is the Radon transformation, which is commonly used in the further processing of VSP data. A good way of computing the Radon domain, taking into account both computational effort and accuracy, is applying a frequency dependent linear interpolation within the wave-number domain and transforming the result back to the time-domain. As the proposed reconstruction algorithm takes only a number of operations comparable to a modest amount of FFTs it is very worthwhile to implement the algorithm within the Radon transform, instead of collecting the traces within bins and stack them to a regular output grid to make it suitable for applying Fast Fourier Transforms. The method shows significant improvement over the binning and stacking method when applied to both synthetic as well as real data.