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The problem of reconstructing a multi-dimensional field from noisy, limited projection measurements is approached using an object-based stochastic field model. Objects within a cross-section are characterized by a finite-dimensional set of parameters, which are estimated directly from limited, noisy projection measurements using maximum likelihood estimation. In Part I, the computational structure and performance of the ML estimation procedure are investigated for the problem of locating a single object in a deterministic background; simulations are also presented. In Part H, the issue of robustness to modeling errors is addressed. L INTRODUCTIONThe problem of reconstructing an n-dimensional function from its (n-1)-dimensional projections arises, typically in the context of cross-sectional imaging, in a diversity of disciplines. In the two-dimensional (2D) problem, let f(x) represent the value of the cross-sectional function (for example x-ray attenuation coefficient) at a point specified by the vector x -(xl x 2 )'. The projection of f(x) at any angle 9 is a ID function denoted as g(t,9) shown in Figure 1. For a given value of projection angle 9, the projection evaluated at the point t is the integral g(t,9) -f ff(x)8(t-x') dxIdx 2 -f(x)ds X [Rf](t,9) along the line This work was conducted at the M.I.T. Laboratory for Information and Decision Systems, with support from the National Science Foundation under Grant ECS-8012668. Figure 2. In (1), 8(t) is the Dirac delta function. Equation (1) corresponds to the Radon transformation, which maps the 2D function f: R 2 -R into the function on a half-cylinder g: Y-'R; g(t,O) is called the Radon transform of f(x) [1][2][3], and is also denoted by [Rf](t, ). The reconstruction problem, determining the function f from its projections g, is an inverse problem, since the measurements are specified by the integral equation in (1) which must be inverted to recover an estimate of the original function.Reconstruction from projections has been employed successfully in radio astronomy, electron microscopy, medical CAT scanning and other applications. Recently, it has been suggested that these techniques be similarly applied to a number of technologically demanding and novel applications, for example real-time monitoring of high production rate manufacturing processes, mesoscale oceanographic thermal mapping, quality control nondestructive evaluation, and 'stop action' imaging of very rapidly changing media [4,51. Virtually all of these applications, as well as many current applications, are characterized by a limited availability of measurement data, due to: * economic constraints that limit the total number of measurement transducers--e.g. oceanographic transducers are sophisticated, low power units that are costly to build, place and maintain [5].* time constraints that limit measurement quality --e.g. limited measurement time interval in "stop action' imaging of very rapid events, or in nondestructive testing of high production rate processes such as steel manufactu...

The problem is considered of determining the shape of an object embedded within a medium from noisy tomographic projection measurements. In particular, the issue is addressed of how accurately coarse features of object geometry --size, elongation and orientation --can be characterized from noisy projection data. A Maximum Likelihood parameter estimation formulation is used and estimation performance is analyzed by evaluation of the Cramer-Rao lower bound on the error variances of the estimates. It is demonstrated that for measurements available at all projection angles and at a given noise level (1) object size and orientation are more accurately determined than is the degree of object elongation, and (2) reliable orientation estimation requires a minimum degree of object elongation, and the required degree of elongation is inversely related to the measurement signal-to-noise ratio (SNR). ABSTRACTThe problem is considered of determining the shape of an object embedded within a medium from noisy tomographic projection measurements. In particular, the issue is addressed of how accurately coarse features of object geometry -size, elongation and orientation -can be characterized from noisy projection data. A Maximum Likelihood parameter estimation formulation is used and estimation performance is analyzed by evaluation of the CramerRao lower bound on the error variances of the estimates. It is demonstrated that for measurements available at all projection angles and at a given noise level (1) object size and orientation are more accurately determined than is the degree of object elongation, and (2) reliable orientation estimation requires a minimum degree of object elongation, and the required degree of elongation is inversely related to the measurement signal-to-noise ratio (SNR).

TX 75083-3836, U.S.A., fax 01-972-952-9435. AbstractOptimum management of oil and gas reservoirs is a continuous, iterative process which encompasses monitoring the reservoir, interpreting the monitoring data, and deciding from the results how best to continue reservoir development and executing those decisions. Monitoring data vary widely in time and space scales.Temporally, they range from continuous to infrequent, episodic measurements; spatially, they range from local well-centric to global reservoir measurements.The reservoir management workflow similarly operates at multiple, parallel time-space scales. A "fast" workflow loop handles continuous well and surface network data (e.g. pressure, temperature, and rate), using fast data handling and fast decision-making to optimize hydrocarbon delivery. A "slow" workflow loop assimilates episodic reservoir data (e.g. time-lapse seismic and borehole reservoir measurements) to optimize reservoir drainage.Reservoir monitoring data are assimilated at the most appropriate time into the reservoir shared earth model, which feeds both the "fast" and "slow" workflow loops. A continuing industry challenge is to determine the best way to do this, since the types of monitoring data are diverse and the volume of data to assimilate is often vast.This paper begins with a review of reservoir monitoring data that are available today, with a focus on the range of time-space scales. A reservoir management workflow is introduced which has multiple time scales appropriate for these data. The paper concludes with a review of key challenges: (1) to develop improved interpretation technologies to unify and integrate the fast well-network centric and slow reservoir-centric workflow loops for faster conversion of measurement signals into information, and (2) to provide fuller support for uncertainties, including determining how the level of uncertainty in the reservoir model changes when assimilating monitoring data.

ABSTRACTzero), and a single object (N=1) is situated at a known objThe problem of detecting, locating and charaprojectioerizing location; see [1][2][3] for a discussion of ML localization of objects in a 2D cross-section from noisy projection data an object from noisy projection measurements. has been considered recently [1][2][3], in which objects are characterized by a finite number of parameters, whichIn the present analysis, a specific parameterization of are estimated directly from noisy projection object size and shape is chosen, and the performance of measurements. In this paper, the problem of maximum ML estimation of the geometry parameters in y is likelihood (ML) estimation of those parameters evaluated. To begin, consider a circularly-symmetric characterizing the geometry of an object (e.g. size and normalized (i.e. unit-sized) object located at the origin, orientation) is considered, and estimation performance denoted fo(x) (f (r) as a function of the radial polar is investigated. coordinate r). Denote the object Radon transform as go (t), which is independent of the projection angle 0, and its Radon transform energy as