We introduce a new family of N log N best basis search algorithms for functions of more than one variable. These algorithms search the collection of anisotropic wavelet packet and cosine packet bases and output a minimum entropy basis for a given function. These algorithms are constructed after treating the model problem of computing best Walsh packet bases. Several intermediate algorithms for conducting mixed isotropic/anisotropic best basis searches in the function's various coordinate directions are also presented.

Abstract-In this paper we describe a new technique for detecting and characterizing ellipsoidal shapes automatically from any type of image. This technique is a single pass algorithm which can extract any group of ellipse parameters or characteristics which can be computed from those parameters without having to detect all five parameters for each ellipsoidal shape. Moreover, the method can explicitly incorporate any a priori knowledge the user may have concerning ellipse parameters. The method is based on techniques from Projective Geometry and on the Hough Transform. This technique can significantly reduce interpretation and computation time by automatically extracting only those features or geometric parameters of interest from images and making exact use of a priori information.

We describe here a test of a new technology for successful drilling of horizontal wells in thin oil columns. We constructed a three-dimensional (3D) earth model of the overburden and of the target reservoir layers on the basis of predrilling data and updated this model in real time on the basis of logging-while-drilling (LWD) measurements transferred to remote locations using the World Wide Web. This strategy allowed us to check and update the planned drilling trajectory continuously with all the information available at any given time. We used uncertainties in the depth of markers observed in a number of offset vertical wells to determine the uncertainty in the thickness of layers in the earth model. This 3D model comprised best estimates of the thicknesses and a covariance matrix that quantified their initial uncertainties. We then drilled a pilot well. Trajectory data, LWD logs, and resistivity images from downhole measurement-while-drilling (MWD) and LWD tools were transmitted in real time from the Simpson No. 22 drill site in Indiana to a prototype application running in Connecticut. As we acquired new measurements in the pilot well, we compared log curves predicted by the model to the measured logs. Our prototype allowed an interpreter to update the location of markers as the well was drilled; an update of the entire 3D earth model and its uncertainty was then automatically computed in near-real time. Quantified uncertainties are key in this stage to ensure that the model update is in agreement with all the data considered previously. This procedure was repeated while drilling the horizontal drain hole, which was successfully steered within a dipping 6-ft-thick layer for 808 ft. Our prototype also allowed for remote collaboration: 3D model updates, LWD data, and resistivity images were available to collaborators who were connected to the network and simultaneously ran copies of the prototype at additional locations. In particular, the remote availability of real-time resistivity images was key to the successful well placement, as these images show how the well trajectory follows the layering. Remote collaboration means that drilling decisions can be made collaboratively by a globally distributed team in a secure network environment. This can be a key capability for geosteering, especially in remote locations or when staffing is constrained. Introduction High-angle and horizontal wells offer the opportunity to tap previously unrecoverable hydrocarbons that occur in thin columns. To achieve this objective it is often necessary to precisely position the well with respect to fluid contacts and/or sedimentary layers. Frequently there is uncertainty in the disposition of these features, so that the target cannot be geometrically defined and real-time data on the position of the features are required to geosteer the well.1 We wished to test the application of novel completion technologies to drain oil from a very thin (originally 13-ft-thick) oil column in the Mount Vernon Unit of the Lamott Consolidated field, Posey County, Indiana (Fig. 1). To do this, we needed to drill an 808-ft-long horizontal well in oil-bearing sandstone.2 The East Mount Vernon Unit is operated by Team Energy and produces oil from the Tar Springs and Cypress sandstones. Most production is from the Mississippian Cypress sandstone reservoir. The previously existing vertical wells produce at a very high water cut (~ 95%) because of the thin nature of the Cypress reservoir oil-column. The Simpson No. 22 well was drilled first as a deviated pilot well, to penetrate the Cypress sandstone close to the planned heel of the horizontal section, and subsequently as a smooth build section (< 4°/100 ft) leading to a horizontal section in the Cypress reservoir (Fig. 2).

The data delivered by a new reservoir mapping while drilling (RMWD) tool provides more geological information than that from any other logging-while-drilling (LWD) technology previously available in the oil field. Its answer product images the surrounding formation structure, and the resulting maps can be used by the geoscientists to improve their understanding of the subsurface, the well placement and the reservoir.To take advantage of the richness of the measurements and deep depth of investigation across multiple formation boundaries, an automatic stochastic inversion has been developed that combines approximately a hundred phase and attenuation measurements at various frequencies and transmitter-to-receiver distances. This efficient Bayesian model-based stochastic inversion runs in parallel with multiple independent search instances that randomly sample hundreds of thousands of formation models using a Markov chain Monte Carlo method. All samples above a quality threshold over the solution space are used to generate the distribution of formation models that intrinsically contain the information for model uncertainties.RMWD is a highly nonlinear problem; inverting for a unique solution is analytically difficult due to the well-known local minima issue. The stochastic method addresses that by sampling thousands of possible formation models and outputting a distribution of layered models that are consistent with the measurements. Statistical distributions are displayed for formation resistivity, anisotropy and dip at each logging point. Additionally, the median formation models for resistivity are shown along the well trajectory as a curtain section plot. This provides an intuitive interpretation for the entire reservoir formation around the tool. The inversion curtain section plot can be overlaid with the seismic formation model for combined interpretation. Furthermore, the curtain plot provides graphical information for dip and distance to boundary, which are critical for field applications such as landing, geosteering, remote fluid contact identification, etc. The stochastic-sampling-based answer product has been intensively field tested and has proven to provide reliable estimation of the formation geometries and fluid distributions in many locations and geological environments worldwide.Field applications and simulated examples of the stochastic inversion include remote detection of the reservoir to enable accurate landing, navigating multilayered reservoirs, remote identification of fluid contacts and reservoir characterization in the presence of faults. The stochastic inversion samples the formation properties randomly and provides the distribution of formation properties based on a large number of samples, instead of providing only the most likely solution as is typical for deterministic inversions. A statistical method of presenting inversion results in formation space provides an instant and intuitive understanding of the formation surrounding the tool. Quantifying the non-uniqueness of the inverted fo...

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