Earth Sciences Division
Fundamental and Exploratory Research Program
Annual Report 1998-1999
Stochastic GILD Coupled Modeling and Inversion Research
Objectives
the acoustic velocity, electric conductivity and flow conductivity, etc. The Ganquan Xie and Jianhua Li Seismic, electromagnetic and advantages of our SGILD method are hydrological modeling and inversion that: (1) we use the global integral Contact: Ganquan Xie are important to the field of oil exploequation and local differential equa(510) 486-7134, g_xie@lbl.gov ration, environmental sciences, CO2 tions to compensate each other to sequestration and atmospheric sciimprove the ill-posed property and ences. Many imaging approaches in geophysical research areas obtain high resolution; (2) replacing the artificial approximate are based on deterministic and traditional nonlinear inversion. boundary condition, we use an exact integral equation on the Because the data are often incomplete and contaminated by ranboundary to greatly reduce numerical noise; (3) we decompose dom noise, the deterministic approach does not provide accuthe full matrix into several small sparse matrices and one small rate or stable solutions. During the iteration of the traditional full matrix, we solve these small matrices, in parallel, and greatly nonlinear inversion, inexact reflections of the artificial boundary reduce computational time and storage; (4) we can use the condition enter the inversion domain; these degrade resolution SGILD method for large-scale oil exploration, marine exploand even cause divergence.The discretization of the volume interation, and atmosphere prediction.The basic frame of our SGILD gral equation is ill posed and its full matrix needs very large storcoupled inversion is the GILD modeling and inversion method. age and lengthy computational time. The conjugate gradient (CG) method requires a long computation time and often termiNew Magnetic integral equations in the time domain: nates in local minima, thus producing a wrong result. Our objecBecause most hydrological data is measured in the time domain, tive is to develop a new parallel stochastic Global Integral and we develop the joint inversion in the time domain.We derive a new integral equation Local Differential (SGILD) electromagnetic (EM), seismic and hydrological modeling and coupled inversion. This method can σb σ overcome the shortcomings of traditional deterministic nonlin1 −ε t 1 − εb t ∗t − e H(r, t ) = H b ( r, t ) − ℵ ∫ e (1) ear inversion. We have derived new integral equations for mag εb Vs ε netic, acoustic and flow field in the frequency domain (Xie and Li, 1995 and 1997) and time domain (Xie et al., 1999) We couple M ∇ × H ∗ t ∇ × G b dr ', the integral equation on the boundary and differential equation in the interior of the domain to develop SGILD modeling. We and differential equation couple the Jacobian volume integral equation on a small selected sub-domain and variation differential equation on the remaining σ (2) large sub-domain to develop SGILD inversion for updating the 1 −ε t ∂H ∗ t ∇×H +µ =- µMδ' (t )δ (r',r ), ∇× e electric conductivity/permittivity, seismic velocity and hydrologε ∂t ical conductivity. We use a new hybrid direct-iteration algorithm and a regularizing method associated with the constitutive law for the magnetic field in the time (Xie et al., 1999) to solve the equation. where ℵ is the unit that makes the integral quantity unit of the M Approach right-hand side of the equation a magnetic unit, G b (r ', r ) is the M magnetic Green tensor, E b (r',r ) is the electric tensor, and *t is Stochastic GILD modeling and inversion: The outline of a convolution. the SGILD method is as follows: (1) we decompose the domain Similarly, we derive the integral equation for seismic and into the two subdomains; (2) we use a new stochastic moment hydraulic inversion in the time domain. The new integral equaintegral equation on the boundary and moment differential equation (1) will be important progress in electromagnetic theory tion in the interior of the domain to calculate the moments of and application. the wave field in the modeling; (3) we use new Jacobian volume We use Bayes theory, the new integral equations in the moment integral equations in the small selected subdomain and selected subdomain and differential equations in the remainder variation differential equations in the remainder large subdosubdomain to develop the SGILD geophysical and hydrological main for the joint inversion. Supposing that the parameters and modeling and coupled inversion in the time domain. data are random variables, we derive the new integral and differential equation system for the statistical moments of the mean, GILD regularizing: We find that the ill-posed pro p e rty of covariance, and standard deviation. We use our parallel SGILD the inve rse pro blem is re l a t i ve to the constitutive law.We use a algorithm to solve the moment integral and differential equan ew constitutive term to construct the new GILD re g u l a rizing tions and obtain the high resolution imaging of the mean impedoperator. ance, covariance standard deviations and confidence interval for
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