Direct Sequential Co-simulation with Joint Probability Distributions

The practice of stochastic simulation for different environmental and earth sciences applications creates new theoretical problems that motivate the improvement of existing algorithms. In this context, we present the implementation of a new version of the direct sequential co-simulation (Co-DSS) algorithm. This new approach, titled Co-DSS with joint probability distributions, intends to solve the problem of mismatch between co-simulation results and experimental data, i.e. when the final biplot of simulated values does not respect the experimental relation known for the original data values. This situation occurs mostly in the beginning of the simulation process. To solve this issue, the new co-simulation algorithm, applied to a pair of covariates Z ₁(x) and Z ₂(x), proposes to resample Z ₂(x) from the joint distribution F(z ₁,z ₂) or, more precisely, from the conditional distribution of Z ₂(x ₀), at a location x ₀, given the previously simulated value z₁^(l)(x₀)z_{1}^{(l)}(x_{0}) (F(Z₂|Z₁=z₁^(l)(x₀)F(Z_{2}|Z_{1}=z_{1}^{(l)}(x_{0}) ). The work developed demonstrates that Co-DSS with joint probability distributions reproduces the experimental bivariate cdf and, consequently, the conditional distributions, even when the correlation coefficient between the covariates is low.     

Geostatistical Seismic Inversion with Self-Updating of Local Probability Distributions

Mathematical Geosciences - Three-dimensional subsurface elastic models inverted from seismic reflection data are the basis of the geo-modeling workflow. These models are often used to predict the...     

Updating Local Anisotropies with Template Matching During Geostatistical Seismic Inversion

Mathematical Geosciences - Iterative geostatistical seismic inversion methods are widely used to predict petro-elastic rock properties from seismic reflection data. When the model perturbation...     

Direct Pattern-Based Simulation of Non-stationary Geostatistical Models

Non-stationary models often capture better spatial variation of real world spatial phenomena than stationary ones. However, the construction of such models can be tedious as it requires modeling both statistical trend and stationary stochastic component. Non-stationary models are an important issue in the recent development of multiple-point geostatistical models. This new modeling paradigm, with its reliance on the training image as the source for spatial statistics or patterns, has had considerable practical appeal. However, the role and construction of the training image in the non-stationary case remains a problematic issue from both a modeling and practical point of view. In this paper, we provide an easy to use, computationally efficient methodology for creating non-stationary multiple-point geostatistical models, for both discrete and continuous variables, based on a distance-based modeling and simulation of patterns. In that regard, the paper builds on pattern-based modeling previously published by the authors, whereby a geostatistical realization is created by laying down patterns as puzzle pieces on the simulation grid, such that the simulated patterns are consistent (in terms of a similarity definition) with any previously simulated ones. In this paper we add the spatial coordinate to the pattern similarity calculation, thereby only borrowing patterns locally from the training image instead of globally. The latter would entail a stationary assumption. Two ways of adding the geographical coordinate are presented, (1) based on a functional that decreases gradually away from the location where the pattern is simulated and (2) based on an automatic segmentation of the training image into stationary regions. Using ample two-dimensional and three-dimensional case studies we study the behavior in terms of spatial and ensemble uncertainty of the generated realizations.     

Direct Sequential Simulation and Cosimulation

Sequential simulation of a continuous variable usually requires its transformation into a binary or a Gaussian variable, giving rise to the classical algorithms of sequential indicator simulation or sequential Gaussian simulation. Journel (1994) showed that the sequential simulation of a continuous variable, without any prior transformation, succeeded in reproducing the covariance model, provided that the simulated values are drawn from local distributions centered at the simple kriging estimates with a variance corresponding to the simple kriging estimation variance. Unfortunately, it does not reproduce the histogram of the original variable, which is one of the basic requirements of any simulation method. This has been the most serious limitation to the practical application of the direct simulation approach. In this paper, a new approach for the direct sequential simulation is proposed. The idea is to use the local sk estimates of the mean and variance, not to define the local cdf but to sample from the global cdf. Simulated values of original variable are drawn from intervals of the global cdf, which are calculated with the local estimates of the mean and variance. One of the main advantages of the direct sequential simulation method is that it allows joint simulation of N _v variables without any transformation. A set of examples of direct simulation and cosimulation are presented.     

Adding Local Accuracy to Direct Sequential Simulation

Geostatistical simulations are globally accurate in the sense that they reproduce global statistics such as variograms and histograms. Kriging is locally accurate in the minimum local error variance sense. Building on the concept of direct sequential simulation, we propose a fast simulation method that can share these opposing objectives. It is shown that the multiple-point entropy of the resulting simulation is related to the univariate entropy of the local conditional distributions used to draw simulated values. Adding local accuracy to conditional simulations does not detract much from variogram reproduction and can be used to increase multiple-point entropy. The methods developed are illustrated using a case study.     

反转构造模型的地震波模拟及成像

为了检验由地震剖面解释得出的反转构造的可靠性,探讨反转构造地区在地震数据采集,处理中应采用的有效方法,设计了一个典型的反转构造模型－突起构造。以该模型为基础,针对复杂构造地震波模拟中常见起源频散严重、人为边界反射吸收不理想两个问题,应用高阶差分法以及一种新的边界吸收条件,对反转模型进行了地震波场数值模拟和深度偏移成像处理。采用中心放炮两边接收的观测方式,共模拟了185炮地震记录。研究表明,波动方程高阶差分法是模拟复杂构造模型中地震波传播的有效方法,除了近于直立的断层因难以接收到其反射波而较难成像外,其余的反转断层及其组合在理论上是可以通过合适方法（如Fourier有限差分深度偏移）较好地成像的。     

Geostatistical Rock Physics Inversion for Predicting the Spatial Distribution of Porosity and Saturation in the Critical Zone

Mathematical Geosciences - Understanding the subsurface structure and function in the near-surface groundwater system, including fluid flow, geomechanical, and weathering processes, requires...     

Direct Multiple-Point Geostatistical Simulation of Edge Properties for Modeling Thin Irregularly Shaped Surfaces

Thin, irregularly shaped surfaces such as clay drapes often have a major control on flow and transport in heterogeneous porous media. Clay drapes are often complex, curvilinear three-dimensional surfaces and display a very complex spatial distribution. Variogram-based stochastic approaches are also often not able to describe the spatial distribution of clay drapes since complex, curvilinear, continuous, and interconnected structures cannot be characterized using only two-point statistics. Multiple-point geostatistics aims to overcome the limitations of the variogram. The premise of multiple-point geostatistics is to move beyond two-point correlations between variables and to obtain (cross) correlation moments at three or more locations at a time using training images to characterize the patterns of geological heterogeneity. Multiple-point geostatistics can reproduce thin irregularly shaped surfaces such as clay drapes, but this is often computationally very intensive. This paper describes and applies a methodology to simulate thin, irregularly shaped surfaces with a smaller CPU and RAM demand than the conventional multiple-point statistical methods. The proposed method uses edge properties for indicating the presence of thin irregularly shaped surfaces. Instead of pixel values, edge properties indicating the presence of irregularly shaped surfaces are simulated using snesim. This method allows direct simulation of edge properties instead of pixel properties to make it possible to perform multiple-point geostatistical simulations with a larger cell size and thus a smaller computation time and memory demand. This method is particularly valuable for three-dimensional applications of multiple-point geostatistics.     

Correction: Geostatistical Rock Physics Inversion for Predicting the Spatial Distribution of Porosity and Saturation in the Critical Zone

Mathematical Geosciences -     

测井约束地震反演分辨率及反演策略探讨

从地震反演的本质出发,分析了测井约束地震反演的分辨率与地震资料本身分辨率之间的关系,探讨了测井约束地震反演的基本策略,指出将地震属性技术、地震地层学研究方法应用于测井约束地震反演可以提高反演效果及可靠性。     

Layer-Stripping Full Waveform Inversion with Damped Seismic Reflection Data

Full waveform inversion(FWI) directly minimizes errors between synthetic and observed data.For the surface acquisition geometry,reflections generated from deep reflectors are sensitive to overburden structure,so it is reasonable to update the macro velocity model in a top-to-bottom manner.For models dominated by horizontally layered structures,combination of offset/time weighting and constant update depth control(CUDC) is sufficient for layer-stripping FWI.CUDC requires ray tracing to determine reflection t...     

Scales of Reservoir Heterogeneities and Impact of Seismic Resolution on Geostatistical Integration

Seismic measurements may be used in geostatistical techniques for estimation and simulation of petrophysical properties such as porosity. The good correlation between seismic and rock properties provides a basis for these techniques. Seismic data have a wide spatial coverage not available in log or core data. However, each seismic measurement has a characteristic response function determined by the source-receiver geometry and signal bandwidth. The image response of the seismic measurement gives a filtered version of the true velocity image. Therefore the seismic image cannot reflect exactly the true seismic velocity at all scales of spatial heterogeneities present in the Earth. The seismic response function can be approximated conveniently in the spatial spectral domain using the Born approximation. How the seismic image response affects the estimation of variogram. and spatial scales and its impact on geostatistical results is the focus of this paper. Limitations of view angles and signal bandwidth not only smooth the seismic image, increasing the variogram range, but also can introduce anisotropic spatial structures into the image. The seismic data are enhanced by better characterizing and quantifying these attributes. As an exercise, examples of seismically assisted cokriging and cosimulation of porosity between wells are presented.     

Integrating Seismic Data for Lithofacies Modeling: A Comparison of Sequential Indicator Simulation Algorithms

Clastic reservoir characterization starts typically with modeling lithofacies distribution and geometry. The architecture of the reservoir, governed by the lithofacies geometry, is a major source of heterogeneity in such clastic systems. Seismic data provide potentially valuable information about the areal distribution of different lithofacies, such as the averaged prior proportion of each lithofacies. However, seismic data are available only at coarse vertical resolution rather than the fine lithofacies sampling along wells, hence seismic is considered equivalent to 2D data while building 3D geological models. This scale difference between the seismic data and the lithofacies data available along the wells makes direct integration difficult. Different algorithms have been proposed to integrate the seismic data: (1) duplicate seismic data along the vertical line and use the prior proportions provided by the seismic data as prior local means; (2) integrate the 2D seismic data as collocated block averages; and (3) duplicate seismic data along the vertical line and integrate them using a Markov-Bayes algorithm. These three algorithms are applied on a data set originating from a real clastic reservoir. The results are compared with regard to how much kriging weight is applied to the seismic data and how well the information from seismic data is honored.     

Means and Covariance Functions for Geostatistical Compositional Data: an Axiomatic Approach

This work focuses on the characterization of the central tendency of a sample of compositional data. It provides new results about theoretical properties of means and covariance functions for compositional data, with an axiomatic perspective. Original results that shed new light on geostatistical modeling of compositional data are presented. As a first result, it is shown that the weighted arithmetic mean is the only central tendency characteristic satisfying a small set of axioms, namely continuity, reflexivity, and marginal stability. Moreover, this set of axioms also implies that the weights must be identical for all parts of the composition. This result has deep consequences for spatial multivariate covariance modeling of compositional data. In a geostatistical setting, it is shown as a second result that the proportional model of covariance functions (i.e., the product of a covariance matrix and a single correlation function) is the only model that provides identical kriging weights for all components of the compositional data. As a consequence of these two results, the proportional model of covariance function is the only covariance model compatible with reflexivity and marginal stability.     

大地电磁测深与地震初至波走时交叉梯度反演

在地球物理勘探过程中,观测数据有限且存在误差,导致反演结果存在非唯一解。为了解决非唯一性问题,笔者研究了交叉梯度数学理论,实现了用交叉梯度约束2D 大地电磁和地震初至波走时的联合反演,并建立了2个模型。通过各自模型实验和不同模型间的对比,证明了交叉梯度联合反演比单独反演修复几何形态和物性参数的效果好,且交叉梯度与岩石物性无关;利用得到的交叉梯度图进一步证明了联合反演模型相似度高,也从交叉绘图中得到联合反演的物性关系更好的结论,反向证明了联合反演的正确性。     

Solving the Seismic Inverse Problem Using an Ensemble of Models: A Case Study of Inversion of Receiver Functions

Doklady Earth Sciences - This paper proposes a new method for determining the seismic parameters of the subsurface below a seismic station using the waveforms of converted waves. The subsurface...     

可控震源地震勘探的数值模拟

用可控震源激发的线性扫频（Chirp）信号模拟了二维地震勘探,采用相关处理技术把反射Chirp地震信号转化为脉冲地震信号形成了相关地震剖面,把相关地震剖面与脉冲震源产生的合成地震记录剖面进行了对比。相关地震剖面同相轴波形的最大绝对值出现的位置反映了界面的真实位置信息。相关地震剖面比脉冲震源产生的合成地震记录剖面具有更高的分辨率。对震源扫频信号进行适当地锥化处理可以部分地压制相关地震剖面中反射地震信号的旁瓣。