W.A. Mulder

A comparison between one-way and two-way wave-equation migration

Results for wave-equation migration in the frequency domain using the constant-density acoustic two-way wave equation have been compared to images obtained by its one-way approximation. The two-way approach produces more accurate reflector amplitudes and provides superior imaging of steep flanks. However, migration with the two-way wave equation is sensitive to diving waves, leading to low-frequency artifacts in the images. These can be removed by surgical muting of the input data or iterative migration or high-pass spatial filtering. The last is the most effective. Iterative migration based on a least-squares approximation of the seismic data can improve the amplitudes and resolution of the imaged reflectors. Two approaches are considered, one based on the linearized constantdensity acoustic wave equation and one on the full acoustic wave equation with variable density. The first converges quickly. However, with our choice of migration weights and high-pass spatial filtering for the linearized case, a real-data migration result shows little improvement after the first iteration. The second, nonlinear iterative migration method is considerably more difficult to apply. A real-data example shows only marginal improvement over the linearized case. In two dimensions, the computational cost of the twoway approach has the same order of magnitude as that for the one-way method. With our implementation, the two-way method requires about twice the computer time needed for one-way wave-equation migration.

Automatic velocity analysis by differential semblance optimization

We present a method for automatic velocity analysis of seismic data based on differential semblance optimization (DSO). The data are mapped for each offset from the time domain to the depth domain by a Born migration scheme using ray tracing with the efficient wavefront construction method. The DSO cost functional is evaluated by taking differences of the migration images for neighboring offsets. The gradient of this functional with respect to the underlying velocity model is obtained by a first‐order approximation of the adjoint‐state method, leading to an optimal complexity: the cost of evaluating the gradient is about the same as that of evaluating the functional.The method has been applied to a marine line. Multiples turned out to be a problem, but were handled effectively by incorporating a multiple filter inside the DSO cost functional.

Finite-difference Iterative Migration By Linearized Waveform Inversion In the Frequency Domain

We present an iterative migration technique for mapping seismic data to reflector amplitudes, obtained by formulating migration as an optimization problem. The method can be based on any kind of seismic modeling and provides true-amplitude images in a natural way. In this paper, we use a finite-difference solution of the linearized constant-density wave equation in the frequency domain. Because the constant-density acoustic equation cannot handle impedance contrasts, we use its linearized form, which is equivalent to the Born approximation. Examples are given for synthetic and real data and show that the iterative technique improves the amplitudes and resolution of reflectors.

On the role of reflections, refractions and diving waves in full-waveform inversion

Full-waveform inversion suffers from local minima, due to a lack of low frequencies in data. A reflector below the zone of interest may, however, help in recovering the long-wavelength components of a velocity perturbation, as demonstrated in a paper by Mora. With the Born approximation for a perturbation in a reference model consisting in two homogeneous isotropic acoustic half-spaces and the assumption of infinitely large apertures available in the data, analytic expressions can be found that describe the spatial spectrum of the recorded seismic signal as a function of the spatial spectrum of the inhomogeneity. Diving waves can be included if the deeper part of the homogeneous model is replaced by one that has a vertical velocity gradient. We study this spectrum in more detail by separately considering scattering of direct, reflected and head waves, as well as singly and multiply reflected diving waves for a gradient model. Taking the reflection coefficient of the deeper reflector into account, we obtain sensitivity estimates for each wavetype. Although the head waves have a relatively small contribution to the reconstruction of the velocity perturbation, compared to the other waves, they contain reliable long-wavelength information that can be beneficial for full-waveform inversion. If the deeper part has a constant positive velocity gradient with depth, all the energy eventually returns to the source-receiver line, given a sufficiently large acquisition aperture. This will improve the sensitivity of the scattered reflected and refracted wavefields to perturbations in the background model. The same happens for a zero velocity gradient but with a very high impedance contrast between the two half-spaces, which results in a large reflection coefficient.

Velocity analysis with multiples — NMO modeling for layered velocity structures

Several method exist to invert for a velocity model from seismic data. Most commonly used methods are defined in the image domain. However, some methods that are defined in the data domain also exist. In this paper we review and compare a few methods on synthetic data, using the convolutional model and NMO traveltimes to model the data. Special attention will be given to the behavior of the different methods in the presence of multiples. In traditional MVA methods these usually pose a problem since multiples are not flattened or focused for the correct velocity model. In the data domain multiples do focus for the correct velocity model if they are correctly modeled. This is illustrated with a simple example. Another simple example illustrates how ideas from waveform inversion and data-domain velocity analysis can be combined to obtain the correct velocity model and reflectivity from synthetic data with multiples.

Hysteresis in the nonmonotonic electric response of homogeneous and layered unconsolidated sands under continuous flow conditions with water of various salinities, 100 kHz to 2 MHz

We measured the electric parameters for four different configurations of unconsolidated homogeneous and layered sands as a function of frequency, water saturation, and salinity under fluid flow conditions. Our objective is to determine if the effect of heterogeneities at scales much smaller than the skin depth can be captured by introducing effective frequency?dependent electrical values whose behavior can be described by simple functions. We employed the parallel plate capacitor technique to measure the complex impedance over a broad frequency range, from 100 kHz up to 3 MHz. We conducted main drainage and secondary imbibition cycles at atmospheric pressure and temperatures between 21°C and 22°C. The hysteretic effect in the real part of the effective complex permittivity at higher concentrations of NaCl is more pronounced for the homogeneous configurations than for the heterogeneous samples. Effective medium theory works well for dry and saturated layered sand, when the NaCl solution concentration is 1 mmol/l. It fails for fully saturated layered sands at salinities of 10 mmol/l or more. It also does not work for partially saturated sands, independent of salinity. A description of the electric properties of a layered sand at all saturation levels by means of an effective homogeneous medium will therefore require a dependence on frequency, saturation level, and salinity of the pore fluid. An extended version of the Cole?Cole model fits the nonmonotonic behavior of the real part of permittivity versus saturation.

A Variable Projection Method for Waveform Inversion

Waveform inversion can be cast into an optimization problem in two parameters: the reflectivity and the background velocity model. The problem is linear with respect to the reflectivity and strongly nonlinear with respect to the background velocity. By re

Data-correlation For Velocity Inversion

We distinguish two main classes of methods to automatically obtain a velocity model from seismic data via optimization. Waveform inversion attempts to fit the measured data with synthetic data. The bestknown example of this approach is least-squares (LS) inversion. Because the LS functional exhibits local minima, this approach is not suitable for finding velocity macro models via gradient-based optimization. Migration based methods rely on measuring the amount of focusing of a migrated image or the flatness of image gathers. These methods perform well, to the extent that the underlying migration algorithm represents the physics reasonably well. For strong multiple reflections, these methods usually fail. In this paper we propose a method that uses the focusing principle of the migration based methods in the data-domain. A possible advantage is the straightforward inclusion of multiples in the data-domain. The discussion is limited to the convolutional model for 1D velocity models. We also present results on real data.

Inversion of 3D time‐domain electromagnetic data: The effect of time‐weighting

In order to mitigate the airwave problem, caused by the interaction between source-excited electromagnetic fields and the air, controlled-source electromagnetic surveys on land are almost exclusively implemented as a transient electromagnetics system (TEM), typically measuring step-off or step-on responses. In this way, the earth response can be well separated from the air response, as the latter primarily arrives at very early times and is then followed by the earth response. Because the air response carries no information about deeper targets, the early-time data are often considered useless and are removed in processing and inversion. In this paper, we show that simply muting the earlytime TEM data may lead to unsatisfactory inversion results. Without the early-time response, inversion cannot retrieve the resistivity of the near surface. Due to the diffusive nature of the electromagnetic fields, this also affects the recontruction of the resistivity at larger depths. We illustrate this with a synthetic example.

A Study of Land CSEM Reservoir Monitoring in a Complex 3D Model

We studied the effect on land CSEM measurements of resistivity changes after oil production by running numerical simulations for several configurations in a complex 3D model. We include estimates of noise levels that can be expected in a field experiment,