Douglas E. Miller

A new slant on seismic imaging: Migration and integral geometry

A new approach to seismic migration formalizes the classical diffraction (or common-tangent) stack by relating it to linearized seismic inversion and the generalized Radon transform. This approach recasts migration as the problem of reconstructing the earth’s acoustic scattering potential from its integrals over isochron surfaces. The theory rests on a solution of the wave equation with the geometrical-optics Green function and an approximate inversion formula for the generalized Radon transform. The method can handle both complex velocity models and (nearly) arbitrary configurations of sources and receivers. In this general case, the method can be implemented as a weighted diffraction stack, with the weights determined by tracing rays from image points to the experiment’s sources and receivers. When tested on a finite-difference simulation of a deviated-well vertical seismic profile (a hybrid experiment which is difficult to treat with conventional wave-equation methods), the algorithm accurately reconstructed faulted-earth models. Analytical reconstruction formulas are derived from the general formula for zero-offset and fixed-offset surface experiments in which the background velocity is constant. The zero-offset inversion formula resembles standard Kirchhoff migration. Our analysis provides a direct connection between the experimental setup (source and receiver positions, source wavelet, background velocity) and the spatial resolution of the reconstruction. Synthetic examples illustrate that the lateral resolution in seismic images is described well by the theory and is improved greatly by combining surface data and borehole data. The best resolution is obtained from a zero-offset experiment that surrounds the region to be imaged.

An exact inversion for anisotropic moduli from phase slowness data

The problem of recovering density-normalized elastic moduli of a transversely isotropic anisotropic medium from data consisting of qP or qSV phase velocities measured in multiple directions is addressed. Previous studies have used linear fitting methods with approximate forms of the dispersion relation. Here, it is shown that with algebraic manipulation, and a prior estimate of the squared shear velocity along the symmetry axis (A55), it is possible to use simple linear methods with an exact alternative form of the anisotropic dispersion relation. The method is demonstrated with an application to data from a walkaway vertical seismic profile (VSP) experiment and then used as a tool to address several questions raised by that experiment. It is shown that given data with realistically achievable accuracy, the prior estimate of A55 cannot be improved by optimizing the fit to qP data. It is shown that a near perfect fit by a transversely isotropic medium with a vertical symmetry axis (TIV) model to qP data collected in a single vertical plane does not rule out azimuthal anisotropy. Finally, it is shown that a variation of the method, combined with an algorithm suggested by Hood and Schoenberg, suggests a practical way to determine from walkaway VSP data, all the parameters of an orthorhombic medium formed by adding vertical fractures to a transversely isotropic medium.

Spatial Resolution of Migration Algorithms

This paper presents a systematic approach to the description of spatial resolution of seismic experiments and migration (or inversion) algorithms.

Walk‐away VSP using drill noise as a source

We describe a method for extracting and deconvolving a signal generated by a drill bit and collected by an array of surface geophones. The drill‐noise signature is reduced to an effective impulse by means of a multichannel Wiener deconvolution technique, producing a walk‐away reverse vertical seismic profile (VSP) sampled almost continuously in depth. We show how the multichannel technique accounts for noise and for internal drill‐string reflections, automatically limiting the deconvolved data to frequencies containing significant energy. We have acquired and processed a data set from a well in Germany while drilling at a depth of almost 4000 m. The subsurface image derived from these data compares well with corresponding images from a 3-D surface seismic survey, a zero‐offset VSP survey, and a walk‐away VSP survey acquired using conventional wireline techniques. The effective bandwidth of the deconvolved drill‐noise data is comparable to the bandwidth of surface seismic data but significantly smaller tha...

Multichannel Wiener deconvolution of vertical seismic profiles

We describe a technique for performing optimal, least‐squares deconvolution of vertical seismic profile (VSP) data. The method is a two‐step process that involves (1) estimating the source signature and (2) applying a least‐squares optimum deconvolution operator that minimizes the noise not coherent with the source signature estimate. The optimum inverse problem, formulated in the frequency domain, gives as a solution an operator that can be interpreted as a simple inverse to the estimated aligned signature multiplied by semblance across the array. An application to a zero‐offset VSP acquired with a dynamite source shows the effectiveness of the operator in attaining the two conflicting goals of adaptively spiking the effective source signature and minimizing the noise. Signature design for seismic surveys could benefit from observing that the optimum deconvolution operator gives a flat signal spectrum if and only if the seismic source has the same amplitude spectrum as the noise.

On the measurability of orbits in Borel actions

We replace measure with category in an argument of G. W. Mackey to characterize closed subgroups H of a totally nonmeager, 2nd countable topological group G in terms of the quotient Borel structure G/H. As a corollary, we obtain an improved version of a theorem of C. Ryll- Nardzewski on the Borel measurability of orbits in continuous actions by Polish groups. In (9), G. W. Mackey gave an argument using Haar measures to prove the following: Assume G is locally compact topological group. If H is a subgroup such that the space G/H, formed by giving the set of (left) H-cosets the quotient Borel structure, is countably separated, then H is closed in G. We will show that this result can be obtained by an analogous argument using the theory of Baire category. The category version has a wider application and shows that the above statement remains true under the weaker assumption that G is totally nonmeager. In particular, it holds whenever G is topologically complete (in the sense of Cech). As a corollary, we will prove that a well-known theorem of C. Ryll-Nardzewski on the Borel measurability of orbits in continuous actions by Polish (separable, completely metrizable) groups holds for Borel actions as well. We will also show that a recent result of R. Vaught on decompositions

Field testing of modular borehole monitoring with simultaneous distributed acoustic sensing and geophone vertical seismic profiles at Citronelle, Alabama

A modular borehole monitoring concept has been implemented to provide a suite of well-based monitoring tools that can be deployed cost effectively in a flexible and robust package. The initial modular borehole monitoring system was deployed as part of a CO2 injection test operated by the Southeast Regional Carbon Sequestration Partnership near Citronelle, Alabama. The Citronelle modular monitoring system transmits electrical power and signals, fibre-optic light pulses, and fluids between the surface and a reservoir. Additionally, a separate multi-conductor tubing-encapsulated line was used for borehole geophones, including a specialized clamp for casing clamping with tubing deployment. The deployment of geophones and fibre-optic cables allowed comparison testing of distributed acoustic sensing. We designed a large source effort (>64 sweeps per source point) to test fibre-optic vertical seismic profile and acquired data in 2013. The native measurement in the specific distributed acoustic sensing unit used (an iDAS from Silixa Ltd) is described as a localized strain rate. Following a processing flow of adaptive noise reduction and rebalancing the signal to dimensionless strain, improvement from repeated stacking of the source was observed. Conversion of the rebalanced strain signal to equivalent velocity units, via a scaling by local apparent velocity, allows quantitative comparison of distributed acoustic sensing and geophone data in units of velocity. We see a very good match of uncorrelated time series in both amplitude and phase, demonstrating that velocityconverted distributed acoustic sensing data can be analyzed equivalent to vertical geophones. We show that distributed acoustic sensing data, when averaged over an interval comparable to typical geophone spacing, can obtain signal-to-noise ratios of 18 dB to 24 dB below clamped geophones, a result that is variable with noise spectral amplitude because the noise characteristics are not identical. With vertical seismic profile processing, we demonstrate the effectiveness of downgoing deconvolution from the large spatial sampling of distributed acoustic sensing data, along with improved upgoing reflection quality. We conclude that the extra source effort currently needed for tubing-deployed distributed acoustic sensing vertical seismic profile, as part of a modular monitoring system, is well compensated by the extra spatial sampling and lower deployment cost as compared with conventional borehole geophones.

Analysis of multiazimuthal VSP data for anisotropy and AVO

Multioffset vertical seismic profile (VSP) experiments, commonly referred to as walkaways, enable anisotropy to be measured reliably in the field. The results can be fed into modeling programs to study the impact of anisotropy on velocity analysis, migration, and amplitude versus offset (AVO). Properly designed multioffset VSPs can also provide the target AVO response measured under optimum conditions, since the wavelet is recorded just above the reflectors of interest with minimal reflection point dispersal. In this paper, the multioffset VSP technique is extended to include multioffset azimuths, and a multiazimuthal multiple VSP data set acquired over a carbonate reservoir is analyzed for P-wave anisotropy and AVO. Direct arrival times down to the overlying shale and reflection times and amplitudes from the carbonate are analyzed. Data analysis involves a three‐term fit to account for nonhyperbolic moveout, dip, and azimuthal anisotropy. Results indicate that the overlying shale is transversely isotropi...

Imaging radar maps underground objects in 3-D

City streets cover a complex array of underground electric, gas, and communication lines. Effective maintenance, expansion, and new installation of these networks require accurate information regarding the location of the conduits, cables, and other structures that lie beneath the surface. Underground maps, if they exist, are often inaccurate, incomplete, or out of date, and attempts to find underground lines or obstacles using metal locators often prove disappointing. To help companies create accurate maps of subsurface networks, researchers have developed a new ground-penetrating imaging radar (GPIR) system that creates sharp, three-dimensional (3-D) images of underground lines and objects. Schlumberger Corporation, in conjunction with the Electric Power Research Institute (EPRI) and the Gas Research Institute, has developed a GPIR system that detects, locates, and produces 3D maps of underground features. The new underground imaging system holds the potential to reduce utility operating and maintenance costs by avoiding unneeded excavation and by reducing incidences of costly damage such as ruptured gas lines. Field demonstrations in New York City, San Diego, and other utility locations have proven the ability of the new mapping system to create accurate images of objects in crowded urban areas at depths as great as 10 ft (3 m).

Ultrasonic-to-seismic Measurements of Shale Anisotropy In a North Sea Well

An extensive data set was collected in a North sea well for the purposes of characterizing the anisotropic elastic properties of the shales. The data set included extensive VSP (Verticle Seismic Profile) surveys, a full logging suite including DSI (Dipole Shear Sonic Imager), and whole core analysis. A walkway VSP survey collected using Schlumberger`s ASI (Array Seismic Imager) tool, was used to estimate the anisotropic elastic properties of the North Sea shale. DSI waveforms were analyzed for depth continuous compressional and shear wave velocities. The core was analyzed for anisotropic elastic properties at in situ overburden and pore pressures. The measurements were taken at ultrasonic frequencies using transducers with a bandwidth approximately from 100 to 900 kHz. Comparisons were made of the P and S wave velocities determined at ultrasonic frequencies in the lab with the sonic frequency DSI measurements and seismic frequency P and S wave velocities estimated from the rig source VSP. Finally, traveltimes acquired with the walkway VSP survey were inverted for the TI elastic parameters of the North Sea shale and compared with the laboratory estimations.