A multiscale method for subsurface inverse modeling: Single-phase transient flow

High-resolution geologic models that incorporate observed state data are expected to effectively enhance the reliability of reservoir performance prediction. One of the major challenges faced is how to solve the large-scale inverse modeling problem, i.e., to infer high-resolution models from the given observations of state variables that are related to the model parameters according to some known physical rules, e.g., the flow and transport partial differential equations. There are typically two difficulties, one is the high-dimensional problem and the other is the inverse problem. A multiscale inverse method is presented in this work to attack these problems with the aid of a gradient-based optimization algorithm. In this method, the model responses (i.e., the simulated state data) can be efficiently computed from the high-resolution model using the multiscale finite-volume method. The mismatch between the observations and the multiscale solutions is then used to define a proper objective function, and the fine-scale sensitivity coefficients (i.e., the derivatives of the objective function with respect to each node’s attribute) are computed by a multiscale adjoint method for subsequent optimization. The difficult high-dimensional optimization problem is reduced to a one-dimensional one using the gradient-based gradual deformation method. A synthetic single-phase transient flow example problem is employed to illustrate the proposed method. Results demonstrate that the multiscale framework presented is not only computationally efficient but also can generate geologically consistent models. By preserving spatial structure for inverse modeling, the method presented overcomes the artifacts introduced by the multiscale simulation and may enhance the prediction ability of the inverse-conditional realizations generated.     

An explicit four-dimensional variational data assimilation method based on the proper orthogonal decomposition: Theoretics and evaluation

The proper orthogonal decomposition (POD) method is used to construct a set of basis functions for spanning the ensemble of data in a certain least squares optimal sense. Compared with the singular value decomposition (SVD), the POD basis functions can capture more energy in the forecast ensemble space and can represent its spatial structure and temporal evolution more effectively. After the analysis variables are expressed by a truncated expansion of the POD basis vectors in the ensemble space, the control variables appear explicitly in the cost function, so that the adjoint model, which is used to derive the gradient of the cost function with respect to the control variables, is no longer needed. The application of this new technique significantly simplifies the data assimilation process. Several assimilation experiments show that this POD-based explicit four-dimensional variational data assimilation method performs much better than the usual ensemble Kalman filter method on both enhancing the assimilation precision and reducing the computation cost. It is also better than the SVD-based explicit four-dimensional assimilation method, especially when the forecast model is not perfect and the forecast error comes from both the noise of the initial filed and the uncertainty of the forecast model. Supported by the National Natural Science Foundation of China (Grant No. 40705035), National High Technology Research and Development Program of China (Grant No. 2007AA12Z144), Knowledge Innovation Project of Chinese Academy of Sciences (Grant Nos. KZCX2-YW-217 and KZCX2-YW-126-2), and National Basic Research Program of China (Grant No. 2005CB321704)     

A multiscale adjoint method to compute sensitivity coefficients for flow in heterogeneous porous media

A multiscale adjoint (MSADJ) method is developed to compute high-resolution sensitivity coefficients for subsurface flow in large-scale heterogeneous geologic formations. In this method, the original fine-scale problem is partitioned into a set of coupled subgrid problems, such that the global adjoint problem can be efficiently solved on a coarse grid. Then, the coarse-scale sensitivities are interpolated to the local fine grid by reconstructing the local variability of the model parameters with the aid of solving embedded adjoint subproblems. The approach employs the multiscale finite-volume (MSFV) formulation to accurately and efficiently solve the highly detailed flow problem. The MSFV method couples a global coarse-scale solution with local fine-scale reconstruction operators, hence yielding model responses that are quite accurate at both scales. The MSADJ method is equally efficient in computing the gradient of the objective function with respect to model parameters. Several examples demonstrate that the approach is accurate and computationally efficient. The accuracy of our multiscale method for inverse problems is twofold: the sensitivity coefficients computed by this approach are more accurate than the traditional finite-difference-based numerical method for computing derivatives, and the calibrated models after history matching honor the available dynamic data on the fine scale. In other words, the multiscale based adjoint scheme can be used to history match fine-scale models quite effectively.     

An explicit four-dimensional variational data assimilation method

A new data assimilation method called the explicit four-dimensional variational (4DVAR) method is proposed. In this method, the singular value decomposition (SVD) is used to construct the orthogonal basis vectors from a forecast ensemble in a 4D space. The basis vectors represent not only the spatial structure of the analysis variables but also the temporal evolution. After the analysis variables are ex-pressed by a truncated expansion of the basis vectors in the 4D space, the control variables in the cost function appear explicitly, so that the adjoint model, which is used to derive the gradient of cost func-tion with respect to the control variables, is no longer needed. The new technique significantly simpli-fies the data assimilation process. The advantage of the proposed method is demonstrated by several experiments using a shallow water numerical model and the results are compared with those of the conventional 4DVAR. It is shown that when the observation points are very dense, the conventional 4DVAR is better than the proposed method. However, when the observation points are sparse, the proposed method performs better. The sensitivity of the proposed method with respect to errors in the observations and the numerical model is lower than that of the conventional method.     

Parameter identification in particle models

For the simulation of the transport of dissolved matter particle models can be used. In this paper a technique is developed for the identification of uncertain parameters in these models. This model calibration is formulated as an optimization problem and is solved with a gradient based algorithm. Here adjoint particle tracks are used for the calculation of the gradient of the cost function. The performance of the calibration method is illustrated by simulations and an application to a river Rhine water quality calamity in November 1986.     

Parameter identification in particle models

For the simulation of the transport of dissolved matter particle models can be used. In this paper a technique is developed for the identification of uncertain parameters in these models. This model calibration is formulated as an optimization problem and is solved with a gradient based algorithm. Here adjoint particle tracks are used for the calculation of the gradient of the cost function. The performance of the calibration method is illustrated by simulations and an application to a river Rhine water quality calamity in November 1986.     

时间域航空电磁2.5维非线性共轭梯度反演

对于时间域航空电磁法二维和三维反演来说,最大的困难在于有效的算法和大的计算量需求.本文利用非线性共轭梯度法实现了时间域航空电磁法2.5维反演方法,着重解决了迭代反演过程中灵敏度矩阵计算、最佳迭代步长计算、初始模型选取等问题.在正演计算中,我们采用有限元法求解拉式傅氏域中的电磁场偏微分方程,再通过逆拉氏和逆傅氏变换高精度数值算法得到时间域电磁响应.在灵敏度矩阵计算中,采用了基于拉式傅氏双变换的伴随方程法,时间消耗只需计算两次正演,从而节约了大量计算时间.对于最佳步长计算,二次插值向后追踪法能够保证反演迭代的稳定性.设计两个理论模型,检验反演算法的有效性,并讨论了选择不同初始模型对反演结果的影响.模型算例表明:非线性共轭梯度方法应用于时间域航空电磁2.5维反演中稳定可靠,反演结果能够有效地反映地下真实电性结构.当选择的初始模型电阻率值与真实背景电阻率值接近时,能得到较好的反演结果,当初始模型电阻率远大于或远小于真实背景电阻率值时反演效果就会变差.     

Data assimilation (4D-VAR) to forecast flood in shallow-waters with sediment erosion

In this paper, the four-dimensional variational data assimilation technique (4D-VAR) is presented as a tool to forecast floods. Our study is limited to purely hydrological flows and supposes that the weather, here a big rain, has been already forecasted by meteorological services. The technique consists in minimizing, in the sense of Lagrange, the cost function: a measure of the difference between calculated data and available observations, here the water level. This is done under constraints that are the equations of the physical model. In our case, we modified the shallow-water equations to include a simplified sediment transport model. The steepest descent algorithm is then used to find the minimum. This is made possible because we can compute analytically the gradient of the cost function by using the adjoint equations of the model. As an application of the 4D-VAR technique, the overflowing of the Chicoutimi River at the Chute-Garneau dam, during the 1996 flood, is investigated. It is found that the 4D-VAR method reduces the error in the water height forecast even when the erosion model is not activated. In terms of Lyapunov exponents, we estimate the predictability horizon of such an event to be about half-an-hour after a big rain. However, this limit of predictability can be increased by using more observations or by using a finer computational grid.     

Estimating open boundary conditions from coastal tidal observations by adjoint approach

A numerical study for estimating the tidal open boundary conditions of a shelf current modrl from tb coastal tidal observations is presented. The method is based on the optimal control/adjoint method. A lrast square fitting of the model state to simulated data is used. Two ideal domains and coastlines are considered. Using the IAP shallow. water model and its adjoint model, some identical twin experiments are carried out to test efficiency and lirnilsd of the method. The results show that the adjoint method can efficiently estimate the open boundary conditions well for gulf/bay like domains. The adjoint method seems to have great potential to improve the accuracy of tide and shelf current modeling in coastal regions. Project supported hy the National Natural Science Fuundation of China (Grant No. 49376256)     

改进的伴随状态法初至波走时层析成像方法

目前,有关伴随状态法初至波走时层析成像方法的文献,基本上都是基于面积分来定义目标函数,由此得到的伴随方程也都依赖于地表的法向量.这样,一方面会因为伴随变量计算的不准确而造成梯度的不合理,另一方面也无法合理地处理井中观测问题.本文从理论或数值试验角度指出了这些问题,并提出了不依赖地表法向量的改进的伴随状态法走时层析成像方...     

NUMERICAL MODELING OF SILT SEDIMENTATION AND RELEVANT ENGINEERING PROBLEMS

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Adjoint method for the optimum planning of industrial pollutant sources

The deterioration of air quality is threatening the life and health of people. Scientists in China and other countries have done a great deal of research work on the details of air pollution and the methods of preven-tion and control during the past decades. Up to now, most of the achievements are concentrated on the techniques of controlling pollutant sources and the programs of reduction, which focus on the improve-ment of air quality and the restoration of environment. The techniques of con…     

Variational Data Assimilation Technique in Mathematical Modeling of Ocean Dynamics

Problems of the variational data assimilation for the primitive equation ocean model constructed at the Institute of Numerical Mathematics, Russian Academy of Sciences are considered. The model has a flexible computational structure and consists of two parts: a forward prognostic model, and its adjoint analog. The numerical algorithm for the forward and adjoint models is constructed based on the method of multicomponent splitting. The method includes splitting with respect to physical processes and space coordinates. Numerical experiments are performed with the use of the Indian Ocean and the World Ocean as examples. These numerical examples support the theoretical conclusions and demonstrate the rationality of the approach using an ocean dynamics model with an observed data assimilation procedure.     

Analysis on observing optimization for the wind-driven circulation by an adjoint approach

The adjoint approach is a variational method which is often applied to data assimilation widely in meteorology and oceanography. It is used for analyses on observing optimization for the wind-driven Sverdrup circulation. The adjoint system developed by Thacker and Long (1992), which is based on the GFDL Byran-Cox model, includes three components, i. e. the forward model, the adjoint model and the optimal algorithm. The GFDL Byran-Cox model was integrated for a long time driven by a batch of ideal wind stresses whose meridional component is set to null and zonal component is a sine function of latitudes in a rectangle box with six vertical levels and 2 by 2 degree horizontal resolution. The results are regarded as a "real" representative of the wind-driven Sverdrup circulation, from which the four dimensional fields are allowed to be sampled in several ways, such as sampling at the different levels or along the different vertical sections. To set the different samples, the fields of temperature, salinity and velocities function as the observational limit in the adjoint system respectively where the same initial condition is chosen for 4D VAR data assimilation. By examining the distance functions which measure the misfit between the circulation field from the control experiment of the adjoint system with a complete observation and those from data assimilation of adjoint approach in these sensitivity experiments respectively, observing optimizations for the wind-driven Sverdrup circulation will be suggested under a fixed observational cost.     

Laboratory sandbox validation of pollutant source location methods

Inverse methods can be used to recover the pollutant source location from concentration data. In this paper, the relative effectiveness of two proposed methods, simultaneous release function and source location identification (SRSI) and backward probability model based on adjoint state method (BPM-ASM) are evaluated using real data collected by using experimental equipment. The device is a sandbox that reproduces an unconfined aquifer in which all the variables are controlled. A numerical model was calibrated using experimental observations. The SRSI is a stochastic procedure which finds the source location and the release history by means of a Bayesian geostatistical approach (GA). The BPM-ASM provides the backward probability location of the pollutant detected at a monitoring point by means of a reverse transport simulation. The results show that both methods perform well. While the simultaneous release function and SRSI method requires a preliminary delineation of a probable source area and some weak hypotheses about the statistical structure of the unknown release function, the backward probability model requires some hypothesis about the contaminant release time. A case study was performed using two observation points only, and despite the scarcity of data, both methodologies were able to accurately reconstruct the true source location. The GA has the advantage to recover the release history function too, whilst the backward probability model works well with fewer data. If there are many observations, both methodologies may be computationally heavy. A transfer function approach has been adopted for the numerical definition of the sensitivity matrix in the SRSI method. The reliability of the experimental equipment was tested in previous laboratory works, conducted under several different conditions.     

Direct and Inverse Problems in a Variational Concept of Environmental Modeling

A concept of environmental forecasting based on a variational approach is discussed. The basic idea is to augment the existing technology of modeling by a combination of direct and inverse methods. By this means, the scope of environmental studies can be substantially enlarged. In the concept, mathematical models of processes and observation data subject to some uncertainties are considered. The modeling system is derived from a specially formulated weak-constraint variational principle. A set of algorithms for implementing the concept is presented. These are: algorithms for the solution of direct, adjoint, and inverse problems; adjoint sensitivity algorithms; data assimilation procedures; etc. Methods of quantitative estimations of uncertainty are of particular interest since uncertainty functions play a fundamental role for data assimilation, assessment of model quality, and inverse problem solving. A scenario approach is an essential part of the concept. Some methods of orthogonal decomposition of multi-dimensional phase spaces are used to reconstruct the hydrodynamic background fields from available data and to include climatic data into long-term prognostic scenarios. Subspaces with informative bases are constructed to use in deterministic or stochastic-deterministic scenarios for forecasting air quality and risk assessment. The results of implementing example scenarios for the Siberian regions are presented.     

Geoid and Sea Surface Topography Derived from ERS-1 Altimeter Data by the Adjoint Method

A method for splitting sea surface height measurements from satellite altimetry into geoid undulations and sea surface topography is presented. The method is based on a combination of the information from altimeter data and a dynamic sea surface height model. The model consists of geoid undulations and a quasi-geostrophic model for expressing the sea surface topography. The goal is the estimation of those values of the parameters of the sea surface height model that provide a least-squares fit of the model to the data. The solution is accomplished by the adjoint method which makes use of the adjoint model for computing the gradient of the cost function of the least-squares adjustment and an optimization algorithm for obtaining improved parameters. The estimation is applied to the North Atlantic. ERS-1 altimeter data of the year 1993 are used. The resulting geoid agrees well with the geoid of the EGM96 gravity model.     

Image‐warping waveform tomography

Imaging the change in physical parameters in the subsurface requires an estimate of the long wavelength components of the same parameters in order to reconstruct the kinematics of the waves propagating in the subsurface. One can reconstruct the model by matching the recorded data with modeled waveforms extrapolated in a trial model of the medium. Alternatively, assuming a trial model, one can obtain a set of images of the reflectors from a number of seismic experiments and match the locations of the imaged interfaces. Apparent displacements between migrated images contain information about the velocity model and can be used for velocity analysis. A number of methods are available to characterize the displacement between images; in this paper, we compare shot‐domain differential semblance (image difference), penalized local correlations, and image‐warping. We show that the image‐warping vector field is a more reliable tool for estimating displacements between migrated images and leads to a more robust velocity analysis procedure. By using image‐warping, we can redefine the differential semblance optimization problem with an objective function that is more robust against cycle‐skipping than the direct image difference. We propose an approach that has straightforward implementation and reduced computational cost compared with the conventional adjoint‐state method calculations. We also discuss the weakness of migration velocity analysis in the migrated‐shot domain in the case of highly refractive media, when the Born modelling operator is far from being unitary and thus its adjoint (migration) operator poorly approximates the inverse.     

巴西构造应力场的遗传有限单元法反演

基于观测的主应力方位资料,应用遗传有限单元法对巴西及邻区边界位移和板内力源进行了反演,确认了前人的研究结果：板块边界作用是控制区域应力场的主要因素,山脉和海岸地形造成的扩展力是控制应力场的一个重要因素．同时显示了板块在垂直载荷下的弯曲对局部水平应力场造成相当的影响．反演结果也表明,除了过去公认的东西向挤压作用外,巴西还受到相当明显的南北向构造作用．它的存在与最近ＧＰＳ资料的分析是一致的．研究还表明遗传有限单元法是偏微分方程反演问题求解的一种有效方法．     

An automated algorithm for river ice monitoring over the Susquehanna River using the MODIS data

Reliable and prompt information on river ice condition and extent is needed to make accurate hydrological forecasts to predict ice jams breakups and issue timely flood warnings. This study presents a technique to detect and monitor river ice using observations from the MODIS instrument onboard the Terra satellite. The technique incorporates a threshold‐based decision tree image classification algorithm to process MODIS data and to determine the extent of ice. To differentiate between ice‐covered and ice‐free pixels within the riverbed, the algorithm combines observations in the visible and near‐infrared spectral bands. The developed technique presents the core of the MODIS‐based river ice mapping system, which has been developed to support National Oceanic and Atmospheric Administration NWS's operations. The system has been tested over the Susquehanna River in northeastern USA, where ice jam events leading to spring floods are a frequent occurrence. The automated algorithm generates three products: daily ice maps, weekly composite ice maps and running cloud‐free composite ice maps. The performance of the system was evaluated over nine winter seasons. The analysis of the derived products has revealed their good agreement with the aerial photography and with in situ observations‐based ice charts. The probability of ice detection determined from the comparison of the product with the high‐resolution Landsat imagery was equal to 91%. A consistent inverse relationship was found between the river discharge and the ice extent. The correlation between the discharge and the ice extent as determined from the weekly composite product reached 0.75. The developed CREST River Ice Observation System has been implemented at National Oceanic and Atmospheric Administration–Cooperative Remote Sensing Science and Technology Center as an operational Web tool allowing end users and forecasters to assess ice conditions on the river. Copyright © 2012 John Wiley & Sons, Ltd.