Showing papers in "Geophysics in 1989"

Amplitude‐versus‐offset variations in gas sands

Abstract: Seismic reflections from gas sands exhibit a wide range of amplitude‐versus‐offset (AVO) characteristics. The two factors that most strongly determine the AVO behavior of a gas‐sand reflection are the normal incidence reflection coefficient R0 and the contrast in Poisson’s ratio at the reflector. Of these two factors, R0 is the least constrained. Based on their AVO characteristics, gas‐sand reflectors can be grouped into three classes defined in terms of R0 at the top of the gas sand. Class 1 gas sands have higher impedance than the encasing shale with relatively large positive values for R0. Class 2 gas sands have nearly the same impedance as the encasing shale and are characterized by values of R0 near zero. Class 3 sands have lower impedance than the encasing shale with negative, large magnitude values for R0. Each of these sand classes has a distinct AVO characteristic. An example of a gas sand from each of the three classes is presented in the paper. The Class 1 example involves a Hartshorn channel s...

Empirical relationships among seismic velocity, effective pressure, porosity, and clay content in sandstone

Abstract: We use a multivariate analysis to investigate the influence of effective pressure Pe, porosity ϕ, and clay content C on the compressional velocity Vp and shear velocity Vs of sandstones. Laboratory measurements on water‐saturated samples of 64 different sandstones provide a large data set that was analyzed statistically. For each sample, relationships between effective pressure and Vp and Vs have been determined. All samples were well fit by relationships that have an exponential increase in velocity at low Pe, tapering to a linear increase with Pe for Pe greater than 0.2 kbar. There are differences in the pressure dependences of velocity for different rocks, particularly at very low pressures; however, the differences cannot be attributed to ϕ or C. For the combined set of measurements from all samples, the best fitting formulations are Vp=5.77-6.94ϕ-1.73C+0.446(Pe-e-16.7Pe) and Vs=3.70-4.94ϕ-1.57C+0.361(Pe-e-16.7Pe). While this is admittedly a very simplified parameterization, it is remarkable how well ...

Depth of investigation in electromagnetic sounding methods

Abstract: The time or frequency at which the electromagnetic (EM) response of a buried inhomogeneity can first be measured is determined by its depth of burial and the average conductivity of the overlying section; it is relatively independent of the type of source or receiver and their separation. The ability to make measurements at this time or frequency, however, depends on the sensitivity and accuracy of the instrumentation, the signal strength, and the ambient noise level. These factors affect different EM sounding systems in surprisingly different ways.For the magnetotelluric (MT) method, it is possible to detect a buried half-space under about 1.5 skin depths of overburden. The maximum depth of investigation is virtually unbounded because of high signal strengths at low frequencies. Transient electromagnetic (TEM) soundings, on the other hand, have a limited depth of penetration, but are less affected by static shift errors. For TEM, a buried inhomogeneity can be detected under about one diffusion depth of overburden. For conventional near-zone sounding in which induced voltage is measured (impulse response), the depth of investigation is proportional to the 1/5 power of the source moment and ground resistivity. By contrast, if the receiver is a magnetometer (step response system), the depth of investigation is proportional to the 1/3 power of source moment and is no longer a function of resistivity. Magnetic-field measurements may, therefore, be superior for exploration in conductive areas such as sedimentary basins. Far-zone, or long-offset, TEM soundings are traditionally used for deep exploration. The depth of investigation for a voltage receiver is proportional to the 1/4 power of source moment and resistivity and is inversely proportional to the source-receiver separation. Magnetic-field measurements are difficult to make at long offsets because instrumental accuracy limits the measurement of the very slow decay of the magnetic field.Frequency-domain controlled-source systems are ideally suited for sounding at the very shallow depths needed for engineering, archaeological, and groundwater applications because of the relative ease of extending the measurements to arbitrarily high frequencies, and also because geometric soundings can be made at low induction numbers.

A new method for the automatic interpretation of Schlumberger and Wenner sounding curves

Abstract: A fast iterative method for the automatic interpretation of Schlumberger and Wenner sounding curves is based on obtaining interpreted depths and resistivities from shifted electrode spacings and adjusted apparent resistivities, respectively. The method is fully automatic. It does not require an initial guess of the number of layers, their thicknesses, or their resistivities; and it does not require extrapolation of incomplete sounding curves. The number of layers in the interpreted model equals the number of digitized points on the sounding curve. The resulting multilayer model is always well‐behaved with no thin layers of unusually high or unusually low resistivities. For noisy data, interpretation is done in two sets of iterations (two passes). Anomalous layers, created because of noise in the first pass, are eliminated in the second pass. Such layers are eliminated by considering the best‐fitting curve from the first pass to be a smoothed version of the observed curve and automatically reinterpreting i...

A calculus for finely layered anisotropic media

Abstract: Matrix algebra and group theory combine to offer a formalism for the simple calculation of the elastic, anisotropic, homogeneous medium which is equivalent, in the long-wavelength limit, to a heterogeneous distribution of fine layers, each layer itself an elastic anisotropic medium. The properties of each anisotropic constituent in a set of fine layers map to an element of a commutative group. A reverse mapping returns the material properties of the constituent. Adding group elements gives the group element for the homogeneous medium equivalent to a heterogeneous set of layers. Addition of an inverse element--that is subtraction--provides the means to remove a set of layers from an anisotropic medium; then, if the remaining layer is a stable anisotropic medium, a valid decomposition of the original medium into anisotropic constituents is obtained. Within the group structure, eight subgroups corresponding to eight types of elastic symmetry systems may be identified, immediately yielding the symmetry of the equivalent medium, given the symmetry of its constituents. Sets of parallel fractures or aligned microcracks are also represented as group elements, allowing fractures and anisotropic rocks to be manipulated in a consistent and uniform manner. These group elements depend on at most six fracture parameters and are independent of the properties of the material in which the fractures are embedded. Multiple sets of fractures are easily taken into account by rotating (back in model space) a rock to a coordinate system appropriate to each fracture system, and then adding the appropriate fracture system group element.

Velocity analysis by iterative profile migration

Abstract: In conventional seismic processing, velocity analysis is performed by using the normal moveout (NMO) equation which is based on the assumption of flat, horizontal reflectors. Imaging by migration (either before or after stack) is done normally in a subsequent step using these velocities. In this paper, velocity analysis and imaging are combined in one step, and migration itself is used as a velocity indicator. Because, unlike NMO, migration can be formulated for any velocity function, migration‐based velocity analysis methods are capable of handling arbitrary structures, i.e., those with lateral velocity variations. In the proposed scheme, each shot gather (profile) is migrated with an initial depth‐velocity model. Profile migration is implemented in the (x, ω) domain, but the actual implementation of profile migration is not critical, as long as it is not done in a spatial‐wavenumber domain, which would preclude handling of lateral velocity variations. After migration with an initial velocity model, the ...

Inversion = migration + tomography

Abstract: Seismic inversion, broadly enough defined, is equivalent to doing migration and reflection tomography simultaneously. Diffraction tomography and inversion work best when sources and receivers surround the region of interest, as in medical imaging applications. Theoretical studies have shown that high vertical wavenumber velocity perturbations are resolved by inverting surface seismic reflection data, but the low vertical wavenumbers must be obtained using a separate step, such as velocity analysis or reflection tomography. I propose that a nonlinear iterative inversion that updates a varying background velocity obtains all wavenumbers that are resolvable separately by migration and tomography. The background velocity must contain reflectors to provide data on both upward and downward transmission paths through the earth and hence the low wavenumbers. By considering the downward transmission paths to be between surface sources and buried image geophones and the upward transmission paths to be between surfa...

Two‐dimensional joint inversion of magnetotelluric and dipole‐dipole resistivity data

Abstract: This paper describes 2-D joint inversion of MT and dipole‐dipole resistivity data with the emphasis on the computer algorithm. The algorithm produces a 2-D model composed of a large number of recta...

Depth of investigation of collinear symmetrical four-electrode arrays

Abstract: A study of collinear symmetrical four-electrode arrays and their tripotential variations indicates the existence of an electrode array for which all the tripotential arrangements have the same depth of investigation. Examination of computer-generated sounding curves confirms this result only when depth of investigation is defined as the median of the depth of investigation characteristic curve. The results lend support to this being the most practically useful definition of depth of investigation.

Crosshole seismic tomography

Abstract: Many tomographic interpretations of crosshole seismic traveltimes have approximated the raypaths with straight lines connecting the source and receiver. This approximation is valid where the velocity does not vary greatly, but in many regions of interest velocity variations of 10–20 percent or more are observed, causing significant ray curvature. Other work has taken this nonlinear effect into account, but there do not appear to be many cases of demonstrated success in its application to the crosshole seismic problem. We present here an iterative inversion scheme based on two‐dimensional ray tracing and its successful application to field data. The interpretation method iteratively ray traces and then updates the velocity model. Within each iteration, the differences between the data and the current model traveltimes obtained by ray tracing are related to the unknown velocity perturbations through a system of linear equations. A damped least‐squares method solves for the velocity perturbations which updat...

Wavelengths of Earth structures that can be resolved from seismic reflection data

Abstract: The aim of inverting seismic waveforms is to obtain the “best” earth model. The best model is defined as the one producing seismograms that best match (usually under a least‐squares criterion) those recorded. Our approach is nonlinear in the sense that we synthesize seismograms without using any linearization of the elastic wave equation. Since we use rather complete data sets without any spatial aliasing, we do not have the problem of secondary minima (Tarantola, 1986). Nevertheless, our gradient methods fail to converge if the starting earth model is far from the true earth (Mora, 1987; Kolb et al., 1986; Pica et al., 1989).

Imaging of near‐borehole structure using full‐waveform sonic data

Abstract: The full waveforms recorded by an array of receivers in a modern borehole sonic tool contain secondary arrivals that are reflected from near‐borehole structural features. These arrivals are used to form an image of the near‐borehole structural features in a manner similar to seismic migration. Possible uses of this technique include horizontal well logging; structural dip and contour determination; fault, salt dome, pinnacle reef, and fracture zone imaging; and EOR steam‐flood monitoring. Since both the source and the receivers pass through structures that cross the borehole, the downdip structure and the updip structure can be imaged separately. The technique involves a backprojection of the recorded data into a matrix of accumulation bins representing distances radially out from the borehole and along the borehole axis. Separate matrices are formed for the updip and for the downdip raypaths. The basic technique is illustrated with synthetic data, generated to approximate the case of a sonic tool logging...

Permeability and borehole Stoneley waves: Comparison between experiment and theory

Abstract: We performed laboratory experiments to evaluate theoretical models of borehole Stoneley wave propagation in permeable materials. A Berea sandstone and synthetic samples made of cemented glass beads were saturated with silicone oils. We measured both velocity and attenuation over a frequency band from 10 kHz to 90 kHz. Our theoretical modeling incorporated Biot theory and Deresiewicz-Skalak boundary conditions into a cylindrical geometry and included frequency-dependent permeability. By varying the viscosity of the saturating pore fluid, we were able to study both low-frequency and high-frequency regions of Biot theory, as well as the intermediate transition zone. In both low-frequency and high-frequency regions of the theory, we obtained excellent agreement between experimental observations and theoretical predictions. Velocity and attenuation (1/Q) are frequency-dependent, especially at low frequencies. Also at low frequencies, velocity decreases and attenuation increases with increasing fluid mobility (permeability/viscosity). More complicated behavior is observed at high frequencies. These results support recent observations from the oil field suggesting that Stoneley wave velocity and attenuation may be indicative of formation permeability.

Fracture evaluation using reflected Stoneley-wave arrivals

Abstract: We use the low-frequency reflected Stoneley-wave mode to locate permeable fractures intersecting a borehole and to estimate their effective apertures. Assuming a model in which the average aperture of the fracture is roughly constant, theoretical work relates the magnitude of the Stoneley-wave reflectivity to an effective fracture width. We treat both the case of a horizontal fracture and the case of a fracture crossing the borehole at an angle.Laboratory experiments verify the analytic solution for the case of a horizontal fracture. Full-waveform array sonic data were also acquired in a wellbore with a long recording time (25.5 ms) in order to capture the late Stoneley-wave arrivals. The data processing involves computation of the Stoneley-wave reflectivity response using the measured direct and reflected Stoneley-wave arrivals. A least-squares fit to the arrival time of the reflected-wave arrivals is used to estimate the locations of permeable fractures, and the effective width of the fractures is estimated by comparing the computed Stoneley-wave reflectivity to the theoretical response from a parallel-plate model. Test-well results are consistent with a borehole televiewer analysis.

Chirp subbottom profiler for quantitative sediment analysis

Abstract: A wide‐band, frequency‐modulated, subbottom profiling system (the chirp sonar) can remotely determine the acoustic attenuation of ocean sediments and produce artifact‐free sediment profiles in real time. The chirp sonar is controlled by a minicomputer which performs analog‐to‐digital and digital‐to‐analog conversion, correlation processing, and attenuation estimation in real time. The minicomputer generates an FM pulse that is phase‐ and amplitude‐compensated to correct for the sonar system response. Such precise waveform control helps suppress correlation noise and source ringing. The chirp sonar, which has an effective bandwidth of 5 kHz, can generate chirp (Klauder) wavelets with a tuning thickness (Rayleigh’s criterion for resolution) of approximately 0.1 ms. After each return is correlated, a computationally fast algorithm estimates the attenuation of subbottom reflections by waveform matching with a theoretically attenuated waveform. This algorithm obtains an attenuation estimate by minimizing the m...

Singularity removal: A refinement of resistivity modeling techniques

Abstract: Modeling techniques commonly exhibit errors of 3 to 10 percent or more in the calculation of apparent resistivities over earth models for which analytic solutions are easily available. A singularity occurs in the solution of any elliptic partial differential equation for which the forcing function is not smooth. The inability to adequately represent in discrete space a discontinuous function (in this case, the delta function describing the introduction of current at a point) commonly results in numerical error near the source of a modeled singularity.Inspection of an integrated finite-difference method for modeling the dc resistivity geophysical technique indicates much of the error encountered is of singular origin. A procedure is herein detailed by which the singularity is mathematically removed from the modeling process and reintroduced as a last step, thus preventing it from contributing to the numerical error. Using this procedure, the average error in apparent resistivity values for a model of a polar-dipole traverse over a nonconducting sphere is reduced by 40 percent. For a dipole-dipole traverse of a two-layer model the error decreases by 75 percent, and in the case of a Wenner profile of a model of a vertically faulted earth, the average error is diminished by 90 percent.

A boundary integral equation‐discrete wavenumber representation method to study wave propagation in multilayered media having irregular interfaces

Abstract: We present a method which combines boundary‐integral equation techniques with the discrete wavenumber Green’s function representation to study wave propagation in multilayered media having irregular interfaces. The approach is based on the representation of the interfaces by distributions of body forces, the radiation from which is equivalent to the scattered wave field produced by the diffracting boundaries. The Green’s functions are evaluated by the discrete wavenumber method. Propagator matrices are introduced to relate force distributions on neighboring interfaces. The solution then requires the inversion of a matrix at each interface. The dimensions of the linear system are independent of the number of layers considered, and the computation time varies linearly with the number of interfaces. We apply the method to calculate surface and vertical seismic profiles in the presence of synclinal or anticlinal structures.

The Fresnel zone and its interpretive significance

Abstract: Resolution of the reflection seismic record in the temporal (vertical) dimension has historically dominated the attention of geophysicists The ability to “resolve” two reflectors with close vertical spacing is of particular importance in stratigraphic analysis However, resolution is most completely understood when it is considered in a three‐dimensional sense To what extent can the seismic method resolve a point? If we can understand the resolution limits of seismic sections for expressing a point acoustic discontinuity in subsurface space, then the question of resolution is answered for any reflector configuration (since, under Huygens’ principle, all reflection surfaces may be represented as sufficiently dense arrays of point diffractors)

Aquifer boundaries explored by geoelectrical measurements in the coastal plain of Yemen; a case of equivalence

Abstract: Until 216 vertical electrical Schlumberger soundings were made, little was known about the subsurface geology at the Wadi Surdud area in the coastal plain of the Yemen Arab Republic. These soundings provide information on the distribution of aquifers, aquitards, and impermeable rocks, and the salinity of the groundwater. A first, qualitative interpretation of the geoelectrical sounding curves gives a general impression of the hydrogeologic setting and the aquifer boundaries. However, a more detailed, quantitative interpretation shows that no single resistivity model completely satisfies all the hydrogeologic and geologic facts and theories. Basically, two different models are possible: in one the thickness of the aquifer is fixed; in the other, its resistivity is fixed. Several geologic and hydrogeologic hypotheses may be put forward to support each model, varying from relying upon tectonic events and sedimentary depositional environments to influences of agriculture and water abstraction. Both models are...

A new approach to modeling the electromagnetic response of conductive media

Abstract: We introduce a new and potentially useful method for computing electromagnetic (EM) responses of arbitrary conductivity distributions in the earth. The diffusive EM field is known to have a unique integral representation in terms of a fictitious wave field that satisfies a wave equation. We show that this integral transform can be extended to include vector fields. Our algorithm takes advantage of this relationship between the wave field and the actual EM field. Specifically, numerical computation is carried out for the wave field, and the result is transformed back to the EM field in the time domain. The proposed approach has been successfully demonstrated using two‐dimensional (2‐D) models. The appropriate TE‐mode diffusion equation in the time domain for the electric field is initially transformed into a scalar wave equation in an imaginary q domain, where q is a time‐like variable. The corresponding scalar wave field is computed numerically using an explicit q‐stepping technique. Standard finite‐diffe...

A terracing operator for physical property mapping with potential field data

Abstract: The terracing operator works iteratively on gravity or magnetic data, using the sense of the measured field's local curvature, to produce a field comprised of uniform domains separated by abrupt domain boundaries. The result is crudely proportional to a physical-property function defined in one (profile case) or two (map case) horizontal dimensions. This result can be extended to a physical-property model if its behavior in the third (vertical) dimension is defined, either arbitrarily or on the basis of the local geologic situation. The terracing algorithm is computationally fast and appropriate to use with very large digital data sets. Where gravity and magnetic data are both available, terracing provides an effective means by which the two data sets can be compared directly.Results of the terracing operation somewhat resemble those of conventional susceptibility (or density) mapping. In contrast with conventional susceptibility mapping, however, the terraced function is a true step function, which cannot be depicted by means of contour lines. Magnetic or gravity fields calculated from the physical-property model do not, in general, produce an exact fit to the observed data. By intent, the terraced map is more closely analogous to a geologic map in that domains are separated by hard-edged domain boundaries and minor within-domain variation is neglected.The terracing operator was applied separately to aeromagnetic and gravity data from a 136 km X 123 km area in eastern Kansas. Results provide a reasonably good physical representation of both the gravity and the aeromagnetic data. Superposition of the results from the two data sets shows many areas of agreement that can be referenced to geologic features within the buried Precambrian crystalline basement. The emerging picture of basement geology is much better resolved than that obtained either from the scanty available drill data or from interpretation of the geophysical data by inspection.

A hybrid fast Hankel transform algorithm for electromagnetic modeling

Abstract: A hybrid fast Hankel transform algorithm has been developed that uses several complementary features of two existing algorithms: Anderson’s digital filtering or fast Hankel transform (FHT) algorithm and Chave’s quadrature and continued fraction algorithm. A hybrid FHT subprogram (called HYBFHT) written in standard Fortran-77 provides a simple user interface to call either subalgorithm. The hybrid approach is an attempt to combine the best features of the two subalgorithms in order to minimize the user’s coding requirements and to provide fast execution and good accuracy for a large class of electromagnetic problems involving various related Hankel transform sets with multiple arguments. Special cases of Hankel transforms of double‐order and double‐argument are discussed, where use of HYBFHT is shown to be advantageous for oscillatory kernel functions.

Induced-polarization effects in time-domain electromagnetic measurements

Abstract: Sign reversals in the coincident-loop transient response can be produced by employing a Cole-Cole model in numerical TEM modeling of polarizable conductors. These reversals may be thought of in terms of a polarization current which changes sign during the transient, passing from a charging current at early times to a discharging current at late times. In a layered earth, the relative strength of this current compared to the normally induced vortex current dictates whether or not a reversal is seen. If the earth is conductive, the effects of the polarization current may never be seen. If, however, the earth is only moderately conductive, the polarization current may dominate.In the case of a 3-D polarizable conductor in a conductive host, the addition of a host response serves to delay the time of any sign reversal in the transient. Reducing the host rock response by increasing its resistivity enables the polarization current to dominate earlier. By bringing the conductor closer to the surface, the amplitude of the negative response can be made greater and hence the sign reversal brought earlier in time. In such cases, moderate polarization parameters may cause substantial negative responses.It is possible to interpret TEM anomalies exhibiting sign reversals. The location and geometry of a discrete polarizable conductor can be correctly assessed, and a valid but approximate TEM time constant can be measured, from the positive part of the transient before the sign reversal.

Reference velocity model estimation from prestack waveforms: Coherency optimization by simulated annealing

Abstract: Coherency inversion, which consists of maximizing a semblance function calculated from unstacked seismic waveforms, has the potential of estimating reliable velocity information without requiring traveltime picking on unstacked data. In this work, coherency inversion is based on the assumption that reflectors’ zero‐offset times are known and that the velocity in each layer may vary laterally. The method uses a type of Monte Carlo technique termed the generalized simulated annealing method for updating the velocity field. At each Monte Carlo step, time‐to‐depth conversion is performed. Although this procedure is slow at convergence to the global minimum, it is robust and does not depend on the initial model or topography of the objective function. Applications to both synthetic and field data demonstrate the efficiency of coherency inversion for estimating both lateral velocity variations and interface depth positions.

Mapping a bedrock surface under dry alluvium with shallow seismic reflections

Abstract: Shallow seismic-reflection techniques were used to image the bedrock-alluvial interface, near a chemical evaporation pond in the Texas Panhandle, allowing optimum placement of water-quality monitor wells. The seismic data showed bedrock valleys as shallow as 4 m and accurate to within 1 m horizontally and vertically. The normal-moveout velocity within the near-surface alluvium varies from 225 m/s to 400 m/s. All monitor-well borings near the evaporation pond penetrated unsaturated alluvial material. On most of the data, the wavelet reflected from the bedrock-alluvium interface has a dominant frequency of around 170 Hz. Low-cut filtering at 24 dB/octave below 220 Hz prior to analog-to-digital conversion enhanced the amplitude of the desired bedrock reflection relative to the amplitude of the unwanted ground roll. The final bedrock contour map derived from drilling and seismic-reflection data possesses improved resolution and shows a bedrock valley not interpretable from drill data alone.

Concept of effective density: Key to gravity depth determinations for sedimentary basins

Abstract: The gravity anomaly observed over a layered sedimentary basin is close to that calculated over a basin with the same configuration and depth but filled by homogeneous sediments with density equal to the effective (weighted average) density of the real layered basin. Therefore, it is possible to calculate the effective density contrast of a basin from the residual gravity anomaly over it, if the depth of the basin is known at least at one point. Assuming a mathematical function for density‐depth dependence, an expression for the gravity‐depth dependence using the infinite slab (Bouguer) formula can be developed. This expression makes it possible to calculate the depth of a basin or an isopach map from gravity data. An exponential function and a new hyperbolic density‐depth function and their gravity‐depth functions are analyzed and implemented for determining the depths of basins from gravity data for the San Jacinto graben, California, and the Tucson basin, southern Arizona. The hyperbolic functions are m...

Stoneley-wave attenuation and dispersion in permeable formations

Abstract: The tube wave, or low‐frequency manifestation of the Stoneley wave, has been modeled previously using the quasi‐static approximation; I extend this method to include the effect of the formation matrix compressibility, which tends to marginally increase the tube‐wave attenuation. Using the Biot theory of poroelasticity, I develop a fully dynamic description of the Stoneley wave. The dispersion relation derived from Biot’s equations reduces in the low‐frequency limit to the quasi‐static dispersion relation. Comparisons of the quasi‐static and dynamic theories for typical sandstones indicate the former to be a good approximation to at least 1 kHz for oil and water infiltration. At higher frequencies, usually between 5 and 20 kHz for the formations considered, a maximum in the Stoneley Q is predicted by the dynamic theory. This phenomenon cannot be explained by the quasi‐static approximation, which predicts a constantly increasing Q with frequency. Instead, the peak in Q may be understood as a transition from...

Reduction to the pole at low latitudes by Wiener filtering

Abstract: Using simple estimates of the signal and noise power from gridded magnetic data, we design regulated frequency-domain operators for reduction to the pole at low magnetic latitudes. These operators suppress the artifacts along the direction of the magnetic declination associated with the conventional reduction-to-the-pole procedure, with negligible increase in computational load. The new procedure is applied to produce high-quality reductions to the pole for noisy low-latitude synthetic data and for magnetic data from the Dixon Seamount.

Geometrical ray theory; rays and traveltimes in seismic systems (second-order approximations of the traveltimes)

Abstract: In this paper, I establish general laws that represent global properties for the reflected and transmitted rays in seismic systems. The method used for establishing these laws is based on Hamilton's original ideas; the laws do not depend on the specific seismic system being used.The reflected and transmitted rays, as well as their traveltimes, are described by the same four 2 X 2 matrices. These matrices are interrelated by three equations that provide the basis for the general laws and also provide relations between reflected and transmitted rays that eventually will be exploitable for downward continuation and migration.The traveltimes of reflected and transmitted rays are summed up in single functions, giving the traveltime for any pair of source and receiver positions in a seismic system and corresponding to Hamilton's point characteristics in optics. The traveltimes of the gathers commonly used in seismic exploration are special cases of the general point characteristics.The consequences from the derived laws are directly available as a priori knowledge before any actual ray tracing and computation of traveltimes are done in any particular seismic system. The laws are also available as a general foundation for future treatments of (1) downward continuation, (2) amplitudes of reflected and transmitted events, (3) true amplitude migration, (4) examples of nonuniqueness of seismic interpretations, (5) focus phenomena, and (6) higher order approximations.

Gravity interpretation using correlation factors between successive least‐squares residual anomalies

Abstract: The correlation factors between successive least‐squares residual (or regional) gravity anomalies from a buried sphere, a two‐dimensional (2‐D) horizontal cylinder, and a vertical cylinder and the first horizontal derivative of the gravity from a 2‐D thin faulted layer are computed. Correlation values are used to determine the depth to the center of the buried structure, and the radius of the sphere or the cylinder and the thickness of the fault are estimated. The method can be applied not only to residuals but also to the Bouguer‐anomaly profile consisting of the combined effect of a residual component due to a purely local structure and a regional component represented by a polynomial of any order. The method is easy to apply and may be automated if desired. It can also be applied to the derivative anomalies of the gravity field. The validity of the method is tested on two field examples from the United States and Denmark.