Bayesian Analysis of the Discovery Process Model Using Markov Chain Monte Carlo

Based upon the Bayesian framework for analyzing the discovery sequence in a play, a Markov chain Monte Carlo sampler—the Metropolis–Hastings algorithm, is employed to sample model parameters and pool sizes from their joint posterior distribution. The proposed sampling scheme ensures that the parameter space of changing dimension can be traversed in spite of the unknown number of pools. The equal sample weights make it easy to obtain the confidence intervals and assess the statistical error in the estimates, so that the statistical behaviors of the discovery process modeling can be well understood. Two application examples of the Halten play in Norwegian Sea and the Bashaw reef play in the Western Canada Basin show that, the computational advantage of this method to the simple Monte Carlo integration is considerable. In order to increase the convergence speed of the sample chains to the posterior distributions, several parallel simulations with different starting values are recommended.     

Bayesian Discovery Process Modeling of the Lower and Middle Jurassic Play of the Halten Terrace,Offshore Norway,as Compared with the Previous Modeling

A Bayesian version of the discovery process model was applied to the pre-rift Lower and Middle Jurassic play of the Halten Terrace, Mid-Norway. The Bayesian approach estimates the lognormal parameters, the discoverability parameter, and the distribution of sizes of the undiscovered fields as well as the play potential, conditioned on a discovery sequence averaging for all possible prior choices weighted by their likelihood. This approach avoids the problem of having to make arbitrary choices for the parameters. The estimates of parameters and play potential based upon the present methodology compares well with previous estimates, if the play is divided into two sub-plays representing the overpressured and normally pressured zones. These sub-plays have been estimated independently and aggregated in order to get the total undiscovered resource potential. This study estimates that the expected remaining play potential is 100 × 10⁶Sm³ o.e., about 9% of the total resources in the play. There is however a 90% chance that the remaining potential ranges from 13 to 282× 10⁶ Sm³ o.e. and a 5% possibility of exceeding this value.     

The Coefficient of Localization—An Addendum

Geographical data sets sometimes contain missing observations that need to be estimated. A statistical approach to the problem is discussed for multivariate normal spatial data sets satisfying the first-order spatial Markov property with constant mean, where the information at neighboring or contiguous observed sites is used to estimate the missing values. The completed data are used to estimate the parameters of the distribution. The procedure is iterative. The approach is a special case of the Orchard and Woodbury missing information principle. The paper concludes with an illustrative empirical example using rainfall data from an area of Kansas and Nebraska. The quality of the estimates for different sites are compared.     

A STATISTICAL APPROACH TO THE PROBLEM OF MISSING SPATIAL DATA USING A FIRST-ORDER MARKOV MODEL*

Geographical data sets sometimes contain missing observations that need to be estimated. A statistical approach to the problem is discussed for multivariate normal spatial data sets satisfying the first-order spatial Markov property with constant mean, where the information at neighboring or contiguous observed sites is used to estimate the missing values. The completed data are used to estimate the parameters of the distribution. The procedure is iterative. The approach is a special case of the Orchard and Woodbury missing information principle. The paper concludes with an illustrative empirical example using rainfall data from an area of Kansas and Nebraska. The quality of the estimates for different sites are compared.     

Estimating number and field size distribution in frontier sedimentary basins using a Pareto model

Estimating the number of deposits likely to be found and their size distribution is important in exploration planning. In a frontier basin the geologic information for conducting a resource assessment is mostly limited to the regional level. Some methods, such as the play analysis approach (Crovelli and Balay, 1986) and the conceptual play model (Lee and Wang, 1983), can be used in conceptual plays or frontier basins, but these methods require the knowledge of the number of prospects, pool data characterizing the conditional field size distribution, and information documenting the exploration risk. For unexplored sedimentary basins, such as those in southern Africa, there are neither sufficient data covering reservoir volumetric parameters nor a number of mapped prospects that can be used to conduct such an assessment. In such a case a volumetric method using geologic analogy is applicable, by which a point estimate of hydrocarbon potential can be estimated, but this estimate provides no information on the likely size distribution. As a complement to the volumetric method, we discuss how to use the empirical Pareto law for estimating the number of fields and their size distribution in such a frontier region.     

LUCC驱动力模型研究综述

驱动力研究是土地利用变化研究中的核心问题。土地利用变化驱动力模型是分析土地利用变化原因和结果的有力工具，模型通过情景分析可为土地利用规划与决策提供依据。基于不同理论的驱动力研究方法很多，论文选取了几种国内外应用较多的LUCC驱动力模型进行综述，分析了每个模型的优缺点及适用范围，最后得出结论：1) 基于过程的动态模型更适于研究复杂的土地利用系统。2) 基于经验的统计模型能弥补基于过程的动态模型的不足。3) 基于不同学科背景的模型进一步集成将是LUCC驱动力模型未来的发展趋势。     

A Bayesian approach to estimating tectonic stress from seismological data

Earthquakes are conspicuous manifestations of tectonic stress, but the non-linear relationships between the stresses acting on a fault plane, its frictional slip, and the ensuing seismic radiation are such that a single earthquake by itself provides little information about the ambient state of stress. Moreover, observational uncertainties and inherent ambiguities in the nodal planes of earthquake focal mechanisms preclude straightforward inferences about stress being drawn on the basis of individual focal mechanism observations. However, by assuming that each earthquake in a small volume of the crust represents a single, uniform state of stress, the combined constraints imposed on that stress by a suite of focal mechanism observations can be estimated. Here, we outline a probabilistic (Bayesian) technique for estimating tectonic stress directions from primary seismological observations. The Bayesian formulation combines a geologically motivated prior model of the state of stress with an observation model that implements the physical relationship between the stresses acting on a fault and the resultant seismological observation. We show our Bayesian formulation to be equivalent to a well-known analytical solution for a single, errorless focal mechanism observation. The new approach has the distinct advantage, however, of including (1) multiple earthquakes, (2) fault plane ambiguities, (3) observational errors and (4) any prior knowledge of the stress field. Our approach, while computationally demanding in some cases, is intended to yield reliable tectonic stress estimates that can be confidently compared with other tectonic parameters, such as seismic anisotropy and geodetic strain rate observations, and used to investigate spatial and temporal variations in stress associated with major faults and coseismic stress perturbations.     

Discovery process modeling—A sensitivity study

Lognormal discovery process modeling characterizes oil and gas discovery as sampling from a lognormal parent distribution with probability proportional to size and without replacement. In this article, we present a sensitivity study that is based on simulated discovery sequences with different assumptions regarding discovery efficiency, exploration status, and the shape of the parent field size distribution. The results indicate that lognormal discovery process modeling provides good overall estimates of the lognormal parameters if the parent field size distribution is lognormal. If the parent field size distribution is Pareto, an underestimation of the play potential may occur if a lognormal discovery process model is applied. Failure of the likelihood value converging to a maximum is more frequent when sample size is small and/or discovery efficiency is low.     

Uncovering Influences on the Form of Oil and Gas Field Size Distributions

A detailed understanding of the processes that led to empirical oil and gas field size distributions, especially the dynamic character of the discovery process, is needed to improve the quality of forecasts of oil and gas resources. An empirical distribution results from a complex interaction of economic, technical, and social factors with geology in the form of a distribution of deposits. These factors may cause an empirical distribution to mutate nonrandomly through time. Changes in the price of oil, the cost of exploration and development, technology, and access to prospects influence the discovery process. Failure to recognize and account for them in the modeling process can result in serious bias in estimates of the number and volume of future discoveries. In addition, the broad range of some forecasts for a given region may be explained by differences in perspective of those involved in the process. Geologists who understand the basic processes and collect the data may be scientific determinists. Statisticians who model and analyze the data are trained to think in terms of random variables and stochastic processes.     

Seismic scattering in the upper crystalline crust based on evidence from sonic logs

Telluric distortion occurs when electric charges accumulate along near-surface inhomogeneities. At low frequencies, the electric currents associated with these charges can be neglected compared to currents induced deeper in the Earth. At higher frequencies, the magnetic fields associated with these currents may be significant. Some parameters describing the distortion magnetic fields can be estimated from measured magneto-telluric impedance matrices. For regional magnetic fields aligned with regional strike directions, parameters associated with the distortion magnetic field component parallel to the regional magnetic field are undeterminable, whereas parameters associated with the distortion magnetic field component perpendicular to the regional magnetic field can be estimated. Optimal estimates are straightforward even for the realistic case of measurement errors that are correlated between elements of a measured impedance matrix. In a simple example of a 1-D anisotropic model with anisotropy direction varying with depth, the modelling of distortion magnetic fields results in regional impedance estimates corresponding more closely to the responses of uncoupled isotropic models, allowing sensible interpretation of an additional one and a half decades of data.     

A practical solution for linear inference computations

Summary. In this paper, we present a matrix form of Backus' theory of linear inference with multiple predictions. The Bayesian approach used by Backus allows the treatment of erroneous data and the imposition of the essential a priori bound on the model norm. We introduce a modification which involves translating the problem in accordance with an estimated model. Such a model may be known a priori or it may be constructed from the data. We are effectively able to bound the norm of all acceptable models from above and below and this results in more confining estimates of the predictions than provided by just an upper bound. In addition, the model construction approach allows us to make maximum use of the data in the inference computation. Our algorithm is robust and efficient, and estimates comparable to to those obtained from linear programming techniques have been achieved.     

Joint three-dimensional inversion of coupled groundwater flow and heat transfer based on automatic differentiation: sensitivity calculation, verification, and synthetic examples

Inverse methods are useful tools not only for deriving estimates of unknown parameters of the subsurface, but also for appraisal of the thus obtained models. While not being neither the most general nor the most efficient methods, Bayesian inversion based on the calculation of the Jacobian of a given forward model can be used to evaluate many quantities useful in this process. The calculation of the Jacobian, however, is computationally expensive and, if done by divided differences, prone to truncation error. Here, automatic differentiation can be used to produce derivative code by source transformation of an existing forward model. We describe this process for a coupled fluid flow and heat transport finite difference code, which is used in a Bayesian inverse scheme to estimate thermal and hydraulic properties and boundary conditions form measured hydraulic potentials and temperatures. The resulting derivative code was validated by comparison to simple analytical solutions and divided differences. Synthetic examples from different flow regimes demonstrate the use of the inverse scheme, and its behaviour in different configurations.     

Prediction of categorical spatial data via Bayesian updating

This study introduces a transition probability-based Bayesian updating (BU) approach for spatial classification through expert system. Transition probabilities are interpreted as expert opinions for updating the prior marginal probabilities of categorical response variables. The main objective of this paper is to provide a spatial categorical variable prediction method which has a solid theoretical foundation and yields relatively higher classification accuracy compared with conventional ones. The basic idea is to first build a linear Bayesian updating (LBU) model that corresponds to an application of Bayes’ theorem. Since the linear opinion pool is intrinsically suboptimal and underconfident, the beta-transformed Bayesian updating (BBU) model is proposed to overcome this limitation. Another type of BU approach, conditional independent Bayesian updating (CIBU), is derived based on conditional independent experts. It is shown that traditional Markovian-type categorical prediction (MCP) is equivalent to a particular CIBU model with specific parameters. As three variants of the BU method, these techniques are illustrated in synthetic and real-world case studies, comparison results with both the LBU and MCP favor the BBU model.     

Oil and gas resource assessment: The linkage between geological analysis and discovery process model results

Two widely-used techniques to estimate the volume of remaining oil and gas resources are discovery process modeling and geologic assessment. Both were used in a recent national assessment of oil and gas resources of the United States. Parallel estimates were obtained for 27 provinces. Geological-based estimates can typically see into areas not available to discovery process models (that is areas with little or no exploration history) and thus, on average, yield higher estimates. However, a linear relation does exist between the mean estimates obtained from these two methods. In addition, other variables were found in a multiple regression model that explained much of the difference. Thus, it is possible to perform discovery process modeling and adjust the estimates to yield results that might be expected from geological-based assessments.     

Low detection capability of global earthquakes after the occurrence of large earthquakes: investigation of the Harvard CMT catalogue

We investigated the detection capability of global earthquakes immediately after the occurrence of a large earthquake. We stacked global earthquake sequences after occurrences of large earthquakes obtained from the Harvard centroid-moment tensor catalogue, and applied a statistical model that represents an observed magnitude–frequency distribution of earthquakes to the stacked sequence. The temporal variation in model parameters, which corresponds to the detection capability of earthquakes, was estimated using a Bayesian approach. We found that the detection capability of global earthquakes is lower than normal for several hours after the occurrence of large earthquakes; for instance, the duration of lowered detection capability of global earthquakes after the occurrence of an earthquake with a magnitude of seven or larger is estimated to be approximately 12 hr.     

Application of discovery process models in estimating petroleum resources at the play level in China

Discovery process modeling has gained wide acceptance in the Chinese exploration community. In recent years, a variety of discovery process models have been applied to the prediction of undiscovered petroleum resources at the play level in sedimentary basins in China. However, challenging problems have been encountered, particularly when one method alone has been applied to small plays in nonmarine sedimentary basins or in plays with an unusual order of discovery wells. This paper presents results gotten by using the lognormal discovery process model of the Geological Survey of Canada and the geoanchored method for three petroleum plays in basins with different geologic settings. Although the predicted shapes of the parentsize distributions which use these two models, were not always similar, the expected values of the total resources and the number of fields (pools) to be discovered are comparable. The combined use of two discovery process models in the same play compensates for the weaknesses in one method compared with the other and vice versa. Thus, more reliable estimates are the result.     

Lithology-derived structure classification from the joint interpretation of magnetotelluric and seismic models

Magnetotelluric and seismic methods provide complementary information about the resistivity and velocity structure of the subsurface on similar scales and resolutions. No global relation, however, exists between these parameters, and correlations are often valid for only a limited target area. Independently derived inverse models from these methods can be combined using a classification approach to map geologic structure. The method employed is based solely on the statistical correlation of physical properties in a joint parameter space and is independent of theoretical or empirical relations linking electrical and seismic parameters. Regions of high correlation (classes) between resistivity and velocity can in turn be mapped back and re-examined in depth section. The spatial distribution of these classes, and the boundaries between them, provide structural information not evident in the individual models. This method is applied to a 10 km long profile crossing the Dead Sea Transform in Jordan. Several prominent classes are identified with specific lithologies in accordance with local geology. An abrupt change in lithology across the fault, together with vertical uplift of the basement suggest the fault is sub-vertical within the upper crust.     

Use of closure area and resources per unit area for assessing undiscovered petroleum resources in part of the Cooper Basin,South Australia

Undiscovered petroleum resources may be assessed using a computer program that simulates drilling each trap in a play and estimates the size of each potential discovery by multiplying together estimates of area of closure (A _cles) and resources per unit area (V _A ). There are several ways of determining values forA _cles andV _A so that the method can be widely applied. As an example, the sales gas potential of part of the Cooper Basin in South Australia is assessed using statistical projections of loglinear models to determine values forA _cles andV _A . A discontinuity in these particular data appears to result from an improvement in the quality and density of the seismic surveying, demonstrating that an assessment based on widely spaced or poor-quality seismic surveys can underestimate sales gas potential.