Efficient spreading strategy for global solution search using a trust-region optimization method: An application to the common-reflection-surface stacking problem

In the last few years new imaging methods have been introduced. One of the most important and powerful procedure for seismic time imaging is Common Reflection Surface stack (CRS stack) method. In contrast to the conventional methods like NMO/DMO stack and pre-stack time migration which are based on velocity information, the CRS stack method is independent of a velocity model. In this study we perform a stacked section without any velocity analysis using CRS move out formula. In contrast to the conventional stacking process, signal-to-noise ratio and continuity of signal along the events are improved significantly using CRS stack method. In this study we use this method for 2D land data by inhomogeneous layers.

We present a method to obtain a true-amplitude<br>migration and amplitude-versus-angle (AVA) at selected<br>points using the attributes generated by the<br>Common Reflection Surface (CRS) Stack. Our approach<br>combines the CRS stack/inversion process<br>applied to multicoverage data, together with the<br>use of a kinematic Kirchhoff migration, to achieve<br>true-amplitudes (TA) at assigned depth points of<br>the migrated images. The proposed method consists<br>of the following steps: (i) apply the CRS process<br>to the given multicoverage data; the obtained<br>CRS attributes are next used to produce a simple<br>macro-velocity depth model; (ii) perform an unweighted<br>Kirchhoff migration for imaging purposes<br>only; for selected points on target reflectors in the<br>migrated image, we use the macro-velocity model<br>to determine, by ray tracing, common-reflectionpoint<br>(CRP) gathers that belong to the input data;<br>for these rays, we compute the incident angles and<br>the geometrical spreadings; (iii) go back to CRP<br>gathers and compensate the amplitudes for geometrical<br>spreading. The results permit to construct<br>AVA curves on the assigned CRPs. In summary,<br>our method is designed to aggregate amplitude information<br>on selected points of a reflector, after a<br>purely kinematic image (migration) has been obtained.<br>The method is tested on a synthetic inhomogeneous<br>layered model with good results.

In experimental separations, the acquired signal can have four common but unwanted elements: (i) high-frequency noise, (ii) drift, (iii) occasional periodic and pink noise, and (iv) a mixture of components traveling with close velocities. Additionally, separation scientists observe partially resolved peaks in chromatographic or capillary electrophoresis detector output. A protocol for digitally recovering the "true data" is provided with clear-cut details and open-access codes for formulating denoising, baseline correction, and peak resolution problems as linear or nonlinear optimizations. The smoother of Bohlmann-Whittaker denoises and preserves the total area of the signal. Fourier transform methods to detect and remove periodic noise are also proposed for erratic periodic noise problems. A peak width preserving filter, called the bilateral filter, is also given. This procedure is followed by two reliable baseline correction algorithms based on asymmetric reweighted least squares or curvature-based weights to deal with low-frequency signal variations. Once low- and high-frequency noise components are removed, a convenient peak mode location method is introduced, relying upon numerically stable second derivative calculation from Savitzky-Golay (SG) filter. Peak locations, area, height, and higher statistical moments can be derived from iterative curve fitting. The protocol shows two peak models that can model left or right skewed peaks. The new peak models are (i) a numerically stable version of the bi-directional exponentially modified Gaussian (EMG), suitable for a wide variety of real chromatograms, and (ii) the twice generalized normal model with third- and fourth-moment parameters built into it. The nonlinear least squares regression approach uses the trust-region reflective optimization method. The algorithm converges to a solution even with multiple peaks and approximate guesses of peak parameters. The protocol can be generalized to fit more than 200 peak functions in separation sciences available in literature by minor adaptations of the provided codes in the MATLAB environment.

This study introduces a Common Offset (CO) extension of the Common Reflection Surface (CRS) method to address seismic imaging challenges in complex geological settings and with noisy data. This CO-CRS approach aims to enhance the signal-to-noise ratio and overcome the limitations of conventional preconditioning techniques that rely on accurate parameterization. Building upon established work on zero-offset CRS (ZO-CRS), the CO method generates regularized prestack data suitable for both time- and depth-domain processing by interpolating missing offsets using a local hyperbolic approximation. Ultimately, this study utilizes CO-CRS for enhanced velocity analysis and data preconditioning prior to performing prestack time migration (PSTM). In this study, the CO-CRS is then used for velocity analysis and prestack time migration. The results show that prestack CO-CRS data yield improved time-migrated seismic images, and we suggest extending the application to the depth domain. To achieve a reliable velocity model for imaging, recursive seismic inversion (RSI) is applied to derive the velocity model using the PSTM stack and a velocity interval time, based on CRS semblance velocity analysis. Furthermore, the prestack depth migration (PSDM) is then tested. The depth-imaging results are reliable, and it can be concluded that combining the benefits of the CRS noise-reduction feature with more accurate velocity analysis and prestack migration can provide enhanced capabilities.

Response surface methodology (RSM) is a widely used method for simulation optimization. Its strategy is to explore small subregions of the decision space in succession instead of attempting to explore the entire decision space in a single attempt. This method is especially suitable for complex stochastic systems where little knowledge is available. Although RSM is popular in practice, its current applications in simulation optimization treat simulation experiments the same as real experiments. However, the unique properties of simulation experiments make traditional RSM inappropriate in two important aspects: (1) It is not automated; human involvement is required at each step of the search process; (2) RSM is a heuristic procedure without convergence guarantee; the quality of the final solution cannot be quantified. We propose the stochastic trust-region response-surface method (STRONG) for simulation optimization in attempts to solve these problems. STRONG combines RSM with the classic trust-region method developed for deterministic optimization to eliminate the need for human intervention and to achieve the desired convergence properties. The numerical study shows that STRONG can outperform the existing methodologies, especially for problems that have grossly noisy response surfaces, and its computational advantage becomes more obvious when the dimension of the problem increases.

The common-reflection-surface (CRS) stacking method produces stacked zero-offset (ZO) sections with a high signal-to-noise ratio, a good definition of the lateral continuity of seismic reflections as well as of reflections with steep dips. The stacking operator of the CRS method depends on three kinematic attributes that are determined through optimization algorithms from the prestack data. In spite of the good results obtained so far with this method, there are still problems to be bypassed, such as the correct simulation of events with conflicting dips and diffraction events. In this work we present one application of the CRS stacking operator in the prestack time migration, i.e., instead of simulating a ZO stacked section, the result is a time migrated section. To demonstrate the efficiency and robustness, we apply this new prestack time migration approach based on CRS method in a synthetic and real land dataset.

The main goal of 3D CRS (Common Reflection Surface) stack is the improvement of the image with respect to NMO/DMO processing route. However, also the set of parameters defining the CRS stacking surface are a precious source of information on subsurface properties. In 3D, the CRS second order traveltime trajectory is defined by eight parameters, that can be divided into three groups: KN, KNIP (each one is a 2x2 matrix with three independent parameters) and angles (azimuth and emergence). KN are mainly related to the subsurface reflectors geometry, KNIP to velocity; angles describe the normal ray emergence. The NIP-wave tomography method described in Duvenek, 2004 uses KNIP and angles to reconstruct a smooth velocity model in depth, suited for a PSDM and/or for petrophysical studies . In this paper, the first application to a real 3D data-set (from West Africa) of the method is presented. The comparison among common image gathers obtained from a reference velocity model and the ones coming out using NIP-wave tomography shows that a reliable velocity model can be reconstructed in a very cost-effective way directly from CRS processing.

The simulation of a zero-offset (ZO) seismic section from multi-coverage seismic data is a standard imaging method widely used in seismic processing to reduce the amount of data and increases the signal-to-noise ratio. In the last years, have appeared diverse stacking methods as an extension of the conventional imaging method, the Common-midpoint (CMP) stacking, to simulate ZO sections. Examples of such methods are the well-known Common-Reflection-Surface (CRS) stack, Multifocusing stack and Non-Hyperbolic CRS stack. Recently an implicit traveltime approximation for monotypic waves in anisotropic media was proposed to simulate ZO sections. This new approximation depends on the same three parameters as the conventional CRS method. In this paper, we tested the performance of this approximation to simulate ZO seismic sections by considering primary and symmetric multiple reflections.

The Common-Reflection-Surface (CRS) stack is a well-established time imaging method that provides high quality stacks of three or two-dimensional seismic data and also important kinematic wavefield attributes as a byproduct. In the present work, we build from the CRS a common diffraction surface (CDS) and migrate multicoverage seismic data to zero-offset. The migration to zero offset CRS-MZO operator is defined by a second order (hyperbolic) travel time approximation that depends on two wavefield attributes (or CRS parameters): the emergence angle of the normal ray and the radius of curvature of the normal incidence point (NIP) wave. This parameter estimation is carried out by means of optimization procedures using as objective function the coherence (semblance) of the seismic traces along the CRSMZO operator. The CRS-MZO method is tested using the 2D Marmousi synthetic data and finally applied to 2D land seismic data of the Tacutu Basin (Brazil).

Trust-region methods have yielded state-of-the-art results in policy search. A common approach is to use KL-divergence to bound the region of trust resulting in a natural gradient policy update. We show that the natural gradient and trust region optimization are equivalent if we use the natural parameterization of a standard exponential policy distribution in combination with compatible value function approximation. Moreover, we show that standard natural gradient updates may reduce the entropy of the policy according to a wrong schedule leading to premature convergence. To control entropy reduction we introduce a new policy search method called compatible policy search (COPOS) which bounds entropy loss. The experimental results show that COPOS yields state-of-the-art results in challenging continuous control tasks and in discrete partially observable tasks.

Shape distances are an important measure to guide the task of shape classification. In this chapter we show that the right choice of shape similarity is also important for the task of image segmentation, even at the absence of any shape prior. To this end, we will study three different shape distances and explore how well they can be used in a trust region framework. In particular, we explore which distance can be easily incorporated into trust region optimization and how well these distances work for theoretical and practical examples.

The trust region problem, minimization of a quadratic function subject to a spherical trust region constraint, occurs in many optimization algorithms. In a previous paper, the authors introduced an inexpensive approximate solution technique for this problem that involves the solution of a two-dimensional trust region problem. They showed that using this approximation in an unconstrained optimization algorithm leads to the same theoretical global and local convergence properties as are obtained using the exact solution to the trust region problem. This paper reports computational results showing that the two-dimensional minimization approach gives nearly optimal reductions in then-dimension quadratic model over a wide range of test cases. We also show that there is very little difference, in efficiency and reliability, between using the approximate or exact trust region step in solving standard test problems for unconstrained optimization. These results may encourage the application of similar approximate trust region techniques in other contexts.

The common-reflection surface (CRS) method is a sophisticated alternative to the traditional common-midpoint stacking because its traveltime approximation allows for the use of more traces than the normal moveout. This in turn requires more parameters for the moveout description, thus increasing the computational burden of the parameter estimation. In the literature, a suboptimal strategy is often used, which decreases the complexity but, as we found in this work, compromises the accuracy of the parameters in some cases. To cope with this problem, in this work, we have devised detailed information for efficient estimation of the CRS parameters using the differential evolution (DE) global optimization algorithm. Because we used data sets with low fold and low signal-to-noise ratio, from which no reliable velocity analysis could be easily performed, we applied this algorithm in a fully automatic global search, i.e., without any velocity guide. The results for a 2D real data set from Brazil indicated that the global strategy yielded good results, both in terms of image quality as in the quality of the parameter volumes, especially the stacking velocity estimates, while keeping the computational costs relatively low. We also developed a convergence and a sensitivity analysis of the DE that shows its computational efficiency and the robustness of the optimization method with respect to the choice of the control parameters of the algorithm.

In this paper, we have proposed a new method to construct an auxiliary function for the discriminative training of HMMs in speech recognition. The new auxiliary function serves as a first-order approximation of the original objective function but more importantly it remains as a lower bound of the original objective function as well. Furthermore, the trust region (TR) method in [1] is applied to find the globally optimal point of the new auxiliary function. Due to its lower-bound property, the found optimal point is theoretically guaranteed to increase the original discriminative objective function. The proposed bounded trust region method has been investigated on two LVCSR tasks, namely WSJ-5k and Switchboard 60-hour subset tasks. Experimental results show that the bounded TR method yields much better convergence behavior than both the conventional EBW method and the original TR method.

Intravoxel incoherent motion (IVIM) modeling offers the parameters f, D and D* as biomarkers for different lesion types and cancer stages from diffusion MR signals. Challenges with the available optimization algorithms in fitting the model to the signals motive new studies for improved parameter estimations. In this study, one thousand value sets of f, D, D* for human breast are assembled and used to generate five thousand diffusion MR signals considering noise-free and noisy situations exhibiting signal-to-noise ratios (SNR) of 20, 40, 60 and 80. The estimates of f, D, D* are obtained using Levenberg-Marquardt (LM), trust-region (TR) and particle swarm (PS) algorithms. On average, the algorithms provide the highest fitting performance for the noise-free signals (R2adj=1.000) and great fitting performances on the noisy signals with SNR>20 (R2adj>0.988). TR algorithm performs slightly better for SNR=20 (R2adj=0.947). TR and PS algorithms achieve the highest parameter estimation performance for all the parameters while LM algorithm reveals the highest performance for f and D only on the noise-free signals (r=1.00). For the noisy signals, performances increase while SNR increases. All algorithms accomplish poor performances for D* (r=0.01-0.20) while TR and PS algorithms perform same for f (r=0.48-0.97) and D (r=0.85-0.99) but remarkably better than LM algorithm for f (r=0.08-0.97) and D (r=0.53-0.99). Overall, TR and PS algorithms demonstrate better but indistinguishable performances. Without requiring any user-given initial value, PS algorithm may facilitate improved estimation of IVIM parameters of the human breast tissue. Further studies are needed to determine its benefit in clinical practice.