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Results
Nonlinear Shaping Regularization In Geophysical Inverse Problems
Fomel, Sergey (The University of Texas)
SUMMARY Shaping regularization is a general method for imposing constraints on the estimated model in the process of solving an inverse problem. In this paper, I extend the concept of shaping regularization to the case of nonlinear operators and show its connection to the nonlinear Landweber iteration and related iterative inversion methods. An example application is 1-D seismic inversion that extracts an interval velocity model from plane-wave (tau-p) moveout analysis. I develop a nonlinear inversion scheme that utilizes local seismic event slopes and apply it to a synthetic data example to demonstrate an application of nonlinear shaping regularization. Different regularization strategies produce smooth or blocky (layered) models. INTRODUCTION Regularization is an essential part of inversion methods that operate with incomplete or insufficient data. Regularization makes estimation problems well-posed by adding indirect constraints on the estimated model (Engl et al., 1996; Zhdanov, 2002). Developed originally by Tikhonov (1963) and others, the method of regularization has become an indispensable part of the inverse problem theory and has found many applications in geophysical problems: traveltime tomography (Osypov and Scales, 1996; Bube and Langan, 1999), migration velocity analysis (Woodward et al., 1998; Zhou et al., 2003), high-resolution Radon transform (Trad et al., 2003), spectral decomposition (Portniaguine and Castagna, 2004), etc. In an earlier work (Fomel, 2007c), I introduced a method of shaping regularization in the context of linear least-squares estimation. The meaning of a shaping operator is an explicit mapping of the estimated model to the space of acceptable models. Shaping gets embedded in each step of an iterative estimation algorithm and thus provides required regularization of the solution. Shaping regularization has a number of advantages compared with the traditional Tikhonov''s regularization, including an easier control on properties of the estimated model and, in some cases, significantly faster iterative convergence. It has been applied to the definition of local seismic attributes (Fomel, 2007b; Fomel and Jin, 2007; Fomel et al., 2007) and to nonstationary filtering (Fomel, 2007a). In this paper, I extend the idea of shaping regularization to nonlinear inversion and show its connection to nonlinear Landweber iteration and its variants, in particular the R algorithm described by Goldin (1986) and the sparseness-constrained inversion scheme of Daubechies et al. (2004). As an example application, I consider a 1-D prestack seismic inverse problem for interval velocity estimation using local seismic event slopes. I use experiments with synthetic data to demonstrate the effectiveness of nonlinear shaping regularization. SHAPING REGULARIZATION A sufficient condition for the convergence is that the operator on the right side of equation (2) is compressive-its spectral radius being less than one (Collatz, 1966). When B is taken as the adjoint of F (in the linear case) or the adjoint of the Fréchet derivative of F (in the nonlinear case), iteration (2) is known as the Landweber iteration (Landweber, 1951) and has been studied extensively in the inverse problems literature (Hanke, 1991; Hanke et al., 1995; Engl et al., 1996; Bertero and Boccacci, 1998). The Landweber iteration solves the system of normal equations and converges to the least-squares estimate of m.
- Geophysics > Seismic Surveying > Seismic Processing (1.00)
- Geophysics > Seismic Surveying > Seismic Modeling > Velocity Modeling > Seismic Inversion (0.90)
One-dimensional Prestack Seismic Waveform Inversion Using Ensemble Kalman Filter
Jin, Long (The University of Texas) | Sen, Mrinal. K. (The University of Texas) | Stoffa, Paul L. (The University of Texas)
Summary We propose an Ensemble Kalman Filter based onedimensional prestack seismic waveform inversion method for estimating elastic parameters. The basic idea is that the offset or incident angle dependent data are inverted sequentially, which is similar to the process of time dependent data being used sequentially in petroleum engineering or groundwater hydrology. The proposed method is tested with a synthetic data using both flat and good initial models. Introduction The estimation of petrophysical properties by prestack seismic waveform inversion is an active area of research. Prestack seismic waveform inversion is a challenging task because of its nonlinear and non-unique nature. Both local (Mora, 1987; Wood, 1993; Sen and Roy 2003) and global optimization (Sen and Stoffa, 1991, Stoffa and Sen, 1991) methods have been reported to solve this problem. The advantage of local optimization over global optimization is its efficiency. The limitation of this method lies in two parts. One is that local optimization method needs gradient information, which is difficult to be computed for nonlinear forward modeling operator. The gradient can also be computed by numerical method, such as finite differences. However, this kind of method is not efficient for problems with a large number of parameters which are common for prestack seismic waveform inversion. The second limitation of local optimization is that a good initial model is needed. The global optimization is designed to overcome the limitations of local optimization. No gradient and good initial model are needed for this kind of method, such as simulated annealing (SA) and genetic algorithms (GA). The limitation for global optimization is the requirement of a large number of forward model evaluations. In the recent years, the ensemble Kalman filter based sequential inversion method has become a promising tool to address the limitation of gradient based linear inversion and global inversion methods. The ensemble Kalman filter (EnKF) is a recursive filter suitable for problems with a large number of variables. It has been successfully applied for assimilating data in weather forecasting (Evensen, 1994; Evensen and van Leeuwen, 1996; Evensen, 1997, 2003), the groundwater hydrology (Reichle et al., 2002; Chen and Zhang, 2006) and petroleum engineering (Naeval et al, 2002, 2005; Gu and Oliver ,2004; Skjervheim , 2005, 2006). The EnKF is similar to global optimization method in the sense that no gradient information is needed. At the same time, it is a sequential inversion, which assimilates data sequentially. In this paper, we propose a EnKF based prestack waveform inversion. The main idea is that the offset or incident angle dependent information is inverted sequentially, similar to the time dependent data used sequentially in petroleum engineering or groundwater hydrology. Wavelet transform parameterization is also used to reduce the number of model parameters. We use a synthetic data which uses a well log data from Gulf of Mexico to demonstrate the feasibility of our proposed method. The results show the EnKF based inversion method works well even when the initial model is only a constant mean. The byproduct of this EnKF method is that multiple results are obtained simultaneously.
- Geophysics > Seismic Surveying > Seismic Processing > Seismic Migration (1.00)
- Geophysics > Seismic Surveying > Seismic Modeling > Velocity Modeling > Seismic Inversion (1.00)
SUMMARY Structural information is the most important content of seismic images. I introduce a numerical algorithm for spreading information in 3-D volumes according to the local structure of seismic events. The algorithm consists of two steps. First, local, spatially-variable slopes of seismic events are estimated (inline and crossline in 3-D) by the plane-wave-destruction method. Next, a seed trace is inserted in the volume, and the information contained in that trace is spread inside the volume, thus automatically "painting" the data space. Immediate applications this technique include automatic horizon picking and flattening in applications to both prestack and post-stack seismic data. Synthetic and field data tests demonstrate the effectiveness of predictive painting. INTRODUCTION Structural information is the most important content of seismic images. One way to characterize structure is to assign a dominant local slope attribute to all elements in a volume. Claerbout (1992) proposed the method of plane-wave destruction for detecting local slopes of seismic events. Closely related ideas were developed in the differential semblance optimization framework (Symes, 1994; Kim and Symes, 1998). Planewave destruction finds many important applications in seismic data analysis, including data regularization, noise attenuation, etc. (Fomel, 2002, 2007; Fomel et al., 2007). The main principle of plane-wave destruction is prediction: each seismic trace gets predicted from its neighbors that are shifted along the event slopes, and the prediction error gets minimized to estimate optimal slopes. In this paper, I propose to extract the prediction operator from the plane-wave destruction process and to use it for recursive spreading of information inside the volume. I call this spreading predictive painting. One particular kind of information that becomes meaningful when spread in a volume is relative geologic age, in the terminology of Stark (2004): seismic layers arranged according to the relative age of sedimentation. Once relative geological age is established everywhere in the volume, it is possible to flatten seismic images by extracting stratal slices (Zeng et al., 1998a) without manual picking of horizons. Even though flattened seismic horizons do not necessarily correspond to equivalent true geologic ages, flattening improves the interpreter’s ability to understand and quantify the structural architecture of sedimentary layers (Zeng et al., 1998b). The idea of using local shifts for automatic picking was introduced by Bienati and Spagnolini (2001) and Lomask et al. (2006). Stark (2003) presented an alternative approach involving instantaneous phase unwrapping. The predictive painting method, introduced in this paper, provides yet another alternative, with superior computational performance. DESTRUCTION AND PREDICTION OF PLANEWAVES Plane-wave destruction (Fomel, 2002) originates from a local plane-wave model for characterizing seismic data. The mathematical basis is the local plane differential equation. Prediction of a trace from a distant neighbor can be accomplished by simple recursion. .I call this recursive operator predictive painting. Once the elementary prediction operators in equation (4) are determined by plane-wave destruction, predictive painting can spread information from a given trace to its neighbors recursively by following seismic structure. The next section illustrates the painting concept using 2-D examples.
Summary It has been shown that boosting the more greatly attenuated higher frequencies (blue part) within the seismic band in order to match well-log-derived reflectivity can improve the resolution of seismic data. This method, known as spectral blueing includes designing and applying one or several operators to post-stack seismic data in order to enhance attenuated high frequencies within the frequency band. In this study we show that applying the blueing operator to prestack data also improves seismic resolution and may produce more consistent results and cause fewer ringing artifacts. Prestack blueing, poststack blueing and no blueing results are compared. 3D seismic data from the Ketzin CO2 sequestration site are used to illustrate this method. Introduction Because conventional seismic data are band limited, they provide limited subsurface geological information. Moreover, higher frequencies within the band are more attenuated. Spectral blueing and colored inversion methods boost the seismic spectra to a level controlled by the well-derived reflectivity spectrum or acoustic impedance spectrum, respectively. Although these methods enhance spectra only within the seismic band and do not go beyond the band limits, it has been shown that they can improve seismic resolution by recovery of attenuated frequencies within the band (Blache-Fraser and Neep, 2004). An assumption involved in spectral blueing is that the seismic data spectrum remains unchanged in the analysis time window. However attenuation results in continuous change of the spectrum and for larger windows this assumption may be far from reality. To overcome this problem, Neep (2007), designed a series of time-variant spectral blueing operators which are basedon log properties at the corresponding time. Another consideration in spectral blueing is that the higher frequencies added to the spectrum may result in a ringing appearance in poststack application. In this paper we examine results of spectral blueing in aprestack flow rather than in a poststack flow as presented by Blache-Fraser and Neep (2004) and Neep (2007). The spectral blueing operator is incorporated in the processing flow in a way similar to that of zero-phasing operators and the result is compared with the poststack method. We use 3D seismic data from the Ketzin CO2 injection site in Germany to illustrate the application of this strategy to real data. Data The data were acquired using overlapping templates with 5 receiver lines containing 48 active channels in each template. Nominally, 200 source points were activated in each template using an accelerated-weight-drop source giving a nominal fold of 25. Data quality is generally good with the uppermost 1000 m being well imaged (Juhlin et al., 2007). The area geologically contains several successions of shale and sandstones layers, in which some layers are not thick enough to be resolved by conventional seismic data. The processing flow originally used was relatively simple (Table 1). We used the same processing flow and compared results with the spectrally blueing operator included in the flow. We show the ability of the prestack spectral blueing to improve seismic resolution. The well used in this study is the Ktzi 200/2007 well, drilled as the CO2 injection well.
- North America > United States > Mississippi > Marion County (0.66)
- North America > United States > Texas (0.16)
Rock-physics Joint Inversion of Resistivity-log And Seismic Velocity For Hydrate Characterization
DeAngelo, Mike (The University of Texas) | Hardage, Bob (The University of Texas) | Murray, Paul (The University of Texas) | Sava, Diana (The University of Texas)
Summary We present a method for joint inversion of electrical resistivity measurements and velocity data for estimating gas-hydrate concentration in deep-water environments. Our technique is based on a Bayesian approach and combines rock-physics elastic theories and empirical relations for electrical resistivity with stochastic simulations to account for the natural variability of the petrophysical parameters involved in the inversion. Most gas-hydrate systems found in deep-water, nearseafloor strata in the Gulf of Mexico have to be described with limited data because the intervals over which companies acquire logs and cores involve only reservoirs below the gas hydrate stability zone (GHSZ). The usual well-log information acquired over the GHSZ is restricted to gamma-ray and electrical resistivity logs. Also, only sparse geotechnical data are available from which porosity and lithology information can be obtained for near-seafloor strata. When we estimate gas-hydrate concentration in deep-water environments, we must take into account the inherent uncertainty associated with our predictions because of these data limitations. Our method allows us to estimate not only the hydrate concentration from simultaneous inversion of electrical resistivity log and seismic velocity, but also provides a measure of the uncertainty associated with our predictions. By combining electrical resistivity and seismic velocity we can better constrain hydrate concentration and distribution within sediments, and we can reduce the inherent uncertainty associated with our predictions. We illustrate the methodology using examples from Green Canyon, GOM. Introduction Gas hydrates increase both the elastic moduli and the electrical resistivity of the sediments in which they occur (Collett, 2001). However, the relation between hydrate concentration, resistivity and velocity of strata containing hydrates is non-unique and uncertain. Some of these sources of uncertainty are related to data-measurement errors, limited availability of data (such as no density or neutron-porosity) , poor understanding of how hydrate is distributed among sediment grains, unexpected spatial variability of rock properties, and inadequate understanding of numerous other physical conditions and processes associated with hydrate systems. Therefore, by combining quantitatively the various types of hydrate-sensitive information we can better constrain our predictions about gas hydrate distribution. Our methodology for joint inversion uses a Bayesian (Bayes, 1783) approach and combines rock-physics theories and empirical relations with stochastic simulations. We show examples of estimating gas-hydrate concentration and the uncertainty associated with the estimates using electrical resistivity logs and 4C OBC seismic data at calibration wells. Forward modeling of C GH -R -V P joint relation In this section we discuss the forward modeling problem, which has as an outcome the joint theoretical relation between hydrate concentration, electrical resistivity, and velocity of sediments. Based on this joint theoretical relation, calibrated to our study area, we can then estimate hydrate concentrations using actual electrical resistivity and seismic velocity data at well locations. Both electrical resistivity and elastic properties of hydratebearing sediments depend on sediment porosity (?) and on hydrate concentration (CGH) in pores. Therefore, we can model the joint relation between hydrate concentration, resistivity and velocities using Archie Equation and the rock physics elastic model for unconsolidated sediments with load-bearing hydrates (Helgerud et al., 1999; Sava and Hardage, 2006).
- Geophysics > Borehole Geophysics (1.00)
- Geophysics > Seismic Surveying > Seismic Modeling > Velocity Modeling > Seismic Inversion (0.35)
SUMMARY There exists a clear distinction between seismic diffractions and reflections in the post-migration dip-angle gather domain. We analyze this distinction and show the possibility of using it for separating and imaging seismic diffractions with only single-offset data as input. When observed in dip-angle gathers, diffraction events are also significantly more sensitive to velocity errors, which opens up the possibility of using them for velocity analysis. We demonstrate the proposed technique on synthetic and real-data examples. INTRODUCTION Diffracted waves contain valuable information about small objects such as faults, pinchouts, fractures, etc. (Landa et al., 1987; Kanasewich and Phadke, 1988; Liu et al., 1997; Landa and Keydar, 1998; Bansal and Imhof, 2005). Diffraction analysis is a challenging problem because the energy retained by these events is typically one or two orders of magnitude weaker than the energy retained by the reflections. Several authors have suggested that diffractions should be separated from reflectionsbefore analysis (Harlan et al., 1984; Khaidukov et al., 2004). Correct identification and use of diffraction events are important also for velocity estimation, which can be carried out in the prestack as well as in the poststack domain (Sava et al., 2005; Taner et al., 2006; Fomel et al., 2007). The post-migration dip-angle domain has gained some attention lately and was shown to be of great importance to the quality of depth imaging in complex geological areas (Audebert et al., 2002; Reshef and Ruger, 2005). In this paper, we propose to use the dip-angle domain for development of methods to extract and analyze diffraction data. In particular we suggest using diffraction data in this domain for velocity analysis, in both time and depth migration. We describe first dip-angle-domain data decomposition after migration and show how diffraction and reflection events behave in this domain. The ability to use single offset to generate dip-angle common image gathers (CIGs) is also demonstrated. We then present, using synthetic and real-data examples, the separation and imaging of seismic diffractions and the influence of velocity errors on the appearance of migrated diffractions in the dip-angle domain. SIMPLE THEORY OF REFLECTIONS AND DIFFRACTIONS IN MIGRATED DIP-ANGLE GATHERS To explain the difference between reflections and diffractions in dip angle gathers, we consider, for simplicity, the case of post-stack migration in a constant velocity medium. A similar analysis can be extended to prestack migration. SEPARATION AND IMAGING OF SEISMIC DIFFRACTIONS IN MIGRATED DIP-ANGLE GATHERS Figure 6a shows a dip-angle gather created by prestack depth migration with a correct velocity of the Sigsbee synthetic dataset. This gather is situated at one of the artificial diffraction points inserted in the model. The flatness of the diffraction events is clearly visible. Taking advantage of the local dip discrepancy between reflection and diffraction events, we separated them using plane-wave destruction (Fomel, 2002). The separated sections are displayed in Figures 6b and 6c. When the velocity used for the migration is the correct one, the migrated diffractor will be horizontal only at the gather located right above it.
- Geophysics > Seismic Surveying > Seismic Processing > Seismic Migration (1.00)
- Geophysics > Seismic Surveying > Seismic Modeling > Velocity Modeling (1.00)
Adaptive Multiple Subtraction Using Regularized Nonstationary Regression
Fomel, Sergey (The University of Texas)
ABSTRACT Stationary regression is the backbone of different seismic data processing algorithms including match filtering, which is commonly applied for adaptive multiple subtraction. However, the assumption of stationarity is not always adequate for describing seismic signals. I present a general method of nonstationary regression and show its application to nonstationary filtering. The key idea is the use of shaping regularization for constraining the variability of nonstationary regression coefficients. Using benchmark synthetic data examples, I demonstrate successful applications of this method to the problem of adaptive multiple subtraction.
Seismic Data Analysis With One-dimensional Seislet Frame
Fomel, Sergey (The University of Texas) | Liu, Yang (The University of Texas)
SUMMARY We introduce 1-D seislet frame - a novel domain for analyzing data that are composed primarily of sinusoidal signals (in 1-D) or plane waves (in 2-D). In the case of a single component, the 1-D seislet transform is a wavelet-like transform that breaks the input data into multiple scales. What makes the digital wavelet transform (DWT) different is that elementary prediction and update operations follow a sinusoid with a particular frequency. In these terms, the classic DWT is simply a 1-D seislet transform for a zero frequency. In the case of multiple components, the transform turns into a frame - an invertible overcomplete representation - and becomes suitable for analyzing data with multiple frequencies or multiple plane-wave slopes. Using simple synthetic and field data examples, we show that 1-D seislet frame can provide better compression quality for sinusoidal signals than either DWT or the digital Fourier transform. This superior compression property indicates the potential of the new domain for applications such as seismic data regularization or noise attenuation. INTRODUCTION In seismic data analysis, it is common to represent signals as sums of sinusoids (in 1-D) or plane waves (in 2-D) with the help of the digital Fourier transform (DFT). Some methods of data regularization, such as the anti-leakage Fourier transform of Xu et al. (2005) or the Fourier reconstruction method of Zwartjes and Gisolf (2006, 2007) and Zwartjes and Sacchi (2007), rely on the ability to represent signals sparsely in the transform domain. The digital wavelet transform (DWT) is often preferred to the Fourier transform for characterizing digital images because of its ability to localize events in both time and frequency domains (Mallat, 1999). However, DWT may not be optimal for describing data that consist of individual sinusoids or plane waves. In this paper, we attempt to bridge the gap between DFT and DWT by introducing a new transform domain. Our construction follows the recipe for the seislet transform (Fomel, 2006; Liu and Fomel, 2008) but adapts it to the 1-D case. We employ the lifting scheme (Sweldens, 1995), a general method for constructing wavelet transforms, but replace elementary prediction and update operations with analogous operations tuned to particular frequencies. The 1-D seislet transform, defined in this way, is a generalization of DWT in the sense that DWT is simply a 1-D seislet transform tuned to the zero frequency. When more than one frequency (in 1-D) or more than one plane-wave slope (in 2-D) is used for analysis, the transform turns into an overcomplete representation, or a frame. If the analysis frequencies are chosen appropriately by autoregressive spectral analysis (Burg, 1975; Marple, 1987), the 1-D seislet frame can provide a optimally sparse representation of sinusoidal or plane-wave signals. ALGORITHM The lifting scheme A convenient general recipe for digital wavelet transforms is the lifting scheme of Sweldens (1995). A digital wavelet transform consists of data approximation at the coarsest level and residuals from all levels. Constructing such a transform is an efficient operation.
Antialias Spatial Filtering: A Slowness-frequency Approach
Dev, Ashwani (The University of Texas) | McMechan, George A. (The University of Texas)
SUMMARY A rigorous, explicit antialias spatial filter is designed and applied to remove energy above the first Nyquist wavenumber in the horizontal slowness-frequency domain. The antialias filter removes the spatially aliased frequencies selectively at each slowness; conventional antialias lowpass frequency filtering under- or over-corrects for spatial aliasing at all slownesses. A seismic gather can be spatially dealiased only at the expense of wavelet spectral changes; it does not preserve amplitude variations with offset. INTRODUCTION Spatial aliasing is a consequence of undersampling seismic data in space during acquisition. Aliasing can be associated with steep structural dips, low interval velocities or low surface wave velocities (Claerbout, 1985; Yilmaz, 2001; Yu et al., 2007). Spatial aliasing is a problem for prestack processes like dip moveout and migration (Peacock, 1982; Bardan, 2004; Yu et al., 2007). Thus, there exists a need for a method that can separate aliased and unaliased energy. Spitz''s (1991), Claerbout''s (1992) and Gulunay''s (2003) Fourier transform-based dealiasing interpolation methods are robust, but require equally spaced input traces, whereas local slantstack methods (Turner, 1990; Marfurt et al., 1996; Abma and Kabir, 2005) have no restriction for input trace spacing but are sensitive to the interpolation operator. Yu et al. (2007) dealiased and interpolated seismic data in the wavelet-Radon domain but require unaliased slownesses to be present in the data, and the signal must be consistent across wavelet scales. All the interpolation-based dealiasing methods provide qualitatively satisfactory, but indirect, treatments of aliasing. In the present work, a direct dealias spatial filter is proposed in the horizontal wave-slownesss (px) and frequency (f ) domain. SYNTHETIC EXAMPLE A synthetic aliased acoustic (x-t) common shot gather with 37 traces (Figure 1a) is generated for a model with two flat reflectors. The geophone spacing is 112 m (so the first Nyquist wavenumber = 4.46 s/km) and the time sampling interval is 1 ms (so the Nyquist frequency = 500 Hz). In Figure 1a, the first (shallowest) reflection has more aliased energy as it is steeper (i.e. has higher p values) than the second reflection. The input (Figure 1a), the px-tau (Figure 2a), and the px- f (Figure 2d) gathers show that aliased energy is present at both small and large p values. In the plane wave decomposition (Figure 2a), a curved reflection event is formed by constructive interference of a large number of plane waves, so the dominant energy of each reflection has a range of p>x values. The curved reflection trajectories need to be divided into different offset windows because not all px values are aliased at all offsets and there is an overlap between the unaliased energy at some p values and the aliased energy at the same p values generated at the other offsets. The procedure to spatially dealias the synthetic aliased input gather is the following; first, the input data are divided into non-overlapping offset windows based on the change in the px values across the gather; the optimal window width is a Fresnel zone.
Reservoir Modeling Accounting For the Scale And Precision of Seismic Data - Application to a Carbonate Reservoir
Sen, Mrinal K. (The University of Texas) | Srinivasan, Sanjay (The University of Texas)
Reservoir modeling accounting for the scale and precision of seismic data - application to a carbonate reservoir Mrinal K. Sen and Sanjay Srinivasan, The University of Texas at Austin, USA Integration of seismic data accounting for the scale and precision of the available information is crucial for synthesizing reliable reservoir models. Seismic attributes are related to rock attributes such as lithofacies distributions and porosity through a complex transfer function. The parameters of the transfer function model cannot be uniquely determined leading to a probabilistic relationship between the spatial variability of reservoir attributes and seismic data. Reservoir modeling approaches have to take into account this complex relationship within a probabilistic framework such that the spatial distribution of modeled attributes reflect the relationship with seismic correctly. We present a probabilistic neural network based approach for calibrating the relationship between multi‐attribute seismic and the variation in facies distribution observed along well logs.