Abstract: A method can include receiving locations of microseismic events associated with a fracturing operation performed in a geologic environment; determining an individual correlation exponent for one of the microseismic events based at least in part on distances where each of the distances is between the location of the one microseismic event and a location of another one of the microseismic events; and, based at least in part on the individual correlation exponent, associating the one of the microseismic events with a fracture generated or activated by the fracturing operation.

Abstract: The present invention relates to the processing of well data obtained from a completed wellbore. This processing comprises receiving initial well data comprising at least one lithologic segment, each segment representing a lithologic layer each of which comprises a line of inclination or a surface of inclination. This processing furthermore comprises converting the initial well data into restored well data; and delivering the restored well data. The conversion of the initial well data comprises, for each lithologic segment, projecting said lithologic segment onto a normal to the line of inclination or to the surface of inclination to form a projected lithologic segment, and the concatenation of at least one projected lithologic segment to form restored well data.

Abstract: Computing device, computer instructions and method for determining an image of a surveyed subsurface. The method includes receiving recorded wave fields D recorded with seismic sensors over the subsurface; generating a series of modified recorded wave fields Dn based on the recorded wave fields D; iteratively applying an objective function Fi to (1) one element Di of the series of modified recorded wave fields Dn and (2) predicted wave fields Pmi, where “i” is an index associated with a given iteration; calculating with a computing device an updated velocity model mi+1 based on a previous velocity model mi and a step length; and producing the image of the subsurface based on the recorded wave fields D and the updated velocity model mi+1. The predicted wave fields Pmi are predicted by the previous velocity model mi.

Abstract: The present invention relates to a method of assistance in geological modeling by grouping meshes together in a geological model comprising a plurality of meshes, each mesh of said model being associated with at least one digital parameter representing a geological property. For at least one column of at least n meshes of the model, a plurality of decompositions of said column into m subsets of adjacent meshes can be determined. Then, for each composition determined, one determines, for each one of the m subsets, a new digital parameter associated with said subset depending on the digital parameters associated with the meshes of said subset. Finally, one can determine a dispersion value of said decomposition depending, at least, on the new digital parameters associated with said subsets.

Abstract: Techniques are disclosed relating to reducing noise in geophysical marine survey data. According to some embodiments, a complex-valued, directional, multi-resolution (CDM) transform may be applied to marine survey input data and a noise template. Global adaptation constraints and, optionally, local adaptation constraints may be generated dependent on the transformed marine survey data and transformed noise template. The transformed noise template may be adapted dependent upon the global (and, optionally, local) constraints, and the adapted transformed noise template may be subtracted from the transformed marine survey data to remove noise. An inverse CDM transform may be performed on the resulting data to generate reduce-noise marine survey data in the input domain.

Abstract: A system for processing seismic data for a geologic formation generated by an array of acoustic transducers responsive to an acoustic source may include a seismic data storage device and a processor. The processor may cooperate with the seismic data storage device to use a volatility measurement model to generate current correlations of data from the array of acoustic transducers based upon a current estimate for at least one of density and velocity of the geologic formation, and compare the current correlations to a threshold. When the current correlations are below the threshold, the processor may update the current estimate for at least one of density and velocity of the geologic formation, and repeat use of the volatility measurement model to generate updated correlations of data from the array of acoustic transducers based upon the updated estimate for at least one of density and velocity of the geologic formation.

Abstract: Disclosed is a method of modelling a subsurface volume, and a corresponding computer program and apparatus. The method comprises defining a regular planar grid of regularly arranged locations in the two dimensions substantially parallel to the surface, and calculating a plurality of one dimensional arrays, each one dimensional array extending from a different one of the regularly arranged locations, in the direction from surface to subsurface. Each of the one dimensional arrays comprises a plurality of one dimensional cells, wherein the cells are delineated according to actual geological features of the subsurface volume.

Abstract: A method is described that includes receiving one or more seismic measurements corresponding to a plurality of source and receiver locations and providing an earth model for a geologic medium having a heterogeneous tilted symmetry axis. The earth model includes a nonzero shear velocity in the direction of the symmetry axis. The one or more seismic measurements are propagated over a plurality of time-steps in accordance with the earth model and a set of energy-conservative self-adjoint pseudoacoustic equations. The set of energy-conservative self-adjoint pseudoacoustic equations describes one or more seismic wavefields.

Abstract: A system and method for managing location-based information associated with the oil and gas industry. The system, in one embodiment, comprises a computing device connected to a location-based service, wherein the location-based information comprises energy operations data which is associated with a specific geographic location, and wherein the computing device is associated with a graphical map interface. The graphical map interface is configured to display location based information. The graphical map interface can also be configured to display an indication of the frequency of an event. The location based information can be associated with a geofence.

Abstract: Visualization of three-dimensional data without loss of desired characteristics is enabled by receiving mapping data pertaining to a physical object in the world, the physical object represented by a series of segments, the three-dimensional data including a predetermined number of measurements of the series of segments, storing the mapping data in data storage and generating an average measurement value for each of the series of segments. The process further involves generating a first generalization of the mapping data by selecting, for each of the series of segments, a desired number of measurements, fewer than the predetermined number of measurements, that represent the largest deviations from the respective average measurement, storing the selected measurements as the first generalization of the mapping data in the data storage, and upon request, providing access to the first generalization of the mapping data for use in rendering the mapping data as three-dimensional data.

Abstract: Systems and methods are provided for detecting subterranean properties associated with a geological domain. One example method may comprise obtaining waveform information corresponding to a geological domain. The method may further comprise performing a global inversion on the waveform information. Furthermore, the method may comprise performing a deterministic inversion on the waveform information. The method may also comprise determining one or more subterranean properties associated with the geological domain based at least in part on the global inversion and the deterministic inversion.

Abstract: Microseismic mapping using buried arrays with the integration of passive and active seismic surveys provides enhanced microseismic mapping results. The system is initially set up by recording seismic data with the buried array installation while shooting a significant portion of the 3D surface seismic survey. The 3D surface seismic survey provides the following data: shallow 3D VSP data from the buried arrays; P-wave and converted wave data for the area covered by the buried array that benefits from the planned data integration processing effort; and microseismic data and associated analysis.

Abstract: The subject matter of this specification can be embodied in, among other things, a system that includes a three-dimensional fabricator. An image processing module acquires an image of a rock sample having a network of pores, a transformation module transforms the image into a binary matrix and determine a set of statistical moments of the binary matrix, a layer generation module generates a first representation of a first stochastic layer based on the set and emulative of the rock sample and generates a second representation of a second stochastic layer based on the set and emulative of the rock sample. An arrangement module arranges the first representation and the second representation as adjacent layers of a three-dimensional model emulative of the rock sample, and provides the first representation and the second representation to the 3D fabricator for fabrication as a physical three-dimensional fluid micromodel emulative of the rock sample.

Abstract: A method for generating an improved stack from wide azimuth data. This is accomplished by displaying gather traces in a geometrically sensible way that depends on gather and processing type. Once a gather is available in such a manner, a geoscientist can use various geometric or freehand shapes to indicate which traces from the gather to stack and which to ignore. Given sufficient computing power, this can be done interactively, allowing a geoscientist to determine a set of traces that create an optimal stack at a single location in the output cube. Multiple locations can be improved in this manner and a final stack can be generated that contains a blended agglomeration of all of the improvements.

Abstract: Method for simultaneously inverting full-wavefield seismic data (51) for multiple classes of physical property parameters (e.g., velocity and anisotropy) by computing the gradient (53), i.e. search direction (54), of an objective function for each class of parameters, then applying (preferably exhaustive) first-pass independent line searches to each parameter class to obtain the corresponding step size (55) along the search direction for each parameter class; then without yet updating the model, using the step sizes to define a relative scaling between gradients of all parameter classes. Next, each scaled search direction is recombined to form a new search direction (56), and a new second-pass line search is performed along the new search direction (57), and all parameters are simultaneously updated with the resulting step size (58). Alternatively to the preceding alternating two-pass embodiment, the model may be updated after each first-pass line search, and no second-pass line search is performed.

Abstract: Described herein is an apparatus and method for allocating hardware-based processing units for a wave modeling computation. The method generates a computational domain representing a physical region in which wave amplitudes for physical waves are determined. The domain is divided into a plurality of sub-domains, whereafter the method determines which sub-domains of the plurality of sub-domains are active. Further, the method, for each computational step of a plurality of computational steps and in each active sub-domain, executes a first portion of the wave modeling computation corresponding to the active sub-domain by at least one available hardware-based processing unit and determines whether the active sub-domain will remain active in a next computational step of the plurality of computational steps. When the active sub-domain is determined to become inactive in the next computational step, the method allocates the at least one available hardware-based processing unit to other computational tasks.

Abstract: Methods, systems, and computer program products for designing a horizontal well in a hydrocarbon field having naturally fractured reservoir integrates workflows from multiple oilfield related disciplines, including a geophysics workflow, geomechanics workflow, and completion and production workflow to achieve an a dynamic and integrated solution.

Abstract: An example method for tomographic migration velocity analysis may include collecting seismographic traces from a subterranean formation and using an initial velocity model to generate common image gathers and a depth image volume based, at least in part, on the seismographic traces. A structure tensor may be computed with the depth image volume for automated structural dip and azimuth estimation. A semblance may be generated using said plurality of common image gathers and said structure tensor. Image depth residuals may be automatically picked from said semblance. A ray tracing computation may be performed on said initial velocity models using said structure tensor. An updated velocity model may be generated with a tomographic inversion computation, wherein said tomographic inversion computation uses said plurality of image depth residuals and said ray tracing computation.

Abstract: A method for migrating three dimensional seismic data in tilted transversely isotropic media (“TTI”) is based on generating numerical solutions to an exact relationship for a three dimensional dispersion relationship of seismic energy traveling through TTI media. The numerical solutions use selected input values of polar angle and azimuth angle of a transverse isotropic axis of symmetry, and selected values of Thomsen anisotropic parameters to generate tables of numerical values. Based on the tables of numerical values, optimized coefficients for a finite-difference relationship are estimated along multiple splitting directions. The seismic data are then migrated using a three dimensional Fourier finite difference extrapolation algorithm.

Abstract: A full wave inversion (FWI) may utilize an Amplitude-Frequency-Differentiation (AFD) or a Phase-Frequency-Differentiation (PFD) operation to form a velocity model of a subterranean formation utilizing recovered low wavenumber data. Received seismic data is processes to isolate two data signals at different frequencies. In an AFD operation, the two data signals are summed and the data of the envelope of the summed function is used for the FWI. In a PFD operation, the phase data of the quotient of the two data signals is used for the FWI. The FWI proceeds iteratively utilizing either the AFD or PFD data or with single frequency data until the cost function of the AFD or PFD is satisfied.

Abstract: A fully automated method for correcting errors in one interpretation (13) of seismic data based on comparison to at least one other interpretation (14) of the same subsurface region. The errors may occur in any feature of the seismic data volume, for example a horizon, surface, fault, polyline, fault stick, or geo-body. In some embodiments of the invention, an error may be a hole in a horizon (53), and the whole is patched by a piece of a horizon in another interpretation (55). In an alternative embodiment of the invention, a single interpretation may be used to repair itself, for example by identifying similarly shaped, adjacent horizons (67), and merging them (68).

Abstract: Apparatus and methods for controlling equipment to recover hydrocarbons from a reservoir including constructing a collection of reservoir models wherein each model represents a realization of the reservoir and comprises a subterranean formation measurement, estimating the measurement for the model collection, and controlling a device wherein the controlling comprises the measurement estimate wherein the constructing, estimating, and/or controlling includes a rolling flexible approach and/or a nearest neighbor approach.

Abstract: Method for multi-parameter inversion using elastic inversion. This method decomposes data into offset/angle groups and performs inversion on them in sequential order. This method can significantly speed up convergence of the iterative inversion process, and is therefore most advantageous when used for full waveform inversion (FWI). The present inventive approach draws upon relationships between reflection energy and reflection angle, or equivalently, offset dependence in elastic FWI. The invention uses recognition that the amplitudes of small angle (near offset) reflections are largely determined by acoustic impedance alone (1), independent for the most part of Vp/Vs. Large angle (middle and far offset) reflections are affected by Ip, Vp/Vs (2) and other earth parameters such as density (3) and anisotropy.

Abstract: A roadmap for a field development strategy for optimal recovery is provided in a high quality 3D geological model. This geological model combines geological attributes, pore and rock properties for an optimum 3D representation of the reservoir thousands of feet beneath the surface. The model is based on the pertinent geological facies, derived from well core description and detailed studies of rock, as well as fluid and pore properties (Full Pore System) obtained from laboratory analyses of core material and well log data. These data differentiate various important pore throat and pore body regions and relationships, i.e., macroporosity and microporosity. Understanding hydrocarbon volumes in the various pore type groups and then establishing proper recovery techniques through focused laboratory studies yields a field development strategy that can significantly increase hydrocarbon recovery from a reservoir.

Abstract: A technique includes receiving data indicative of a wavefield in a processor-based system and processing the data on the processor-based system to generate a modeled wavefield. The processing includes varying boundary conditions of the modeled wavefield with respect to time to regulate coherent boundary reflections in the modeled wavefield.

Abstract: Example embodiments involve a system, computer-readable storage medium storing at least one program, and computer-implemented method for graphical modeling and editing of database query statements. The method may include accessing a database query statement, and generating an abstract syntax tree (AST) representing a syntactic structure of the database query statement. The method may further include generating a graphical model of the database query statement based on the AST, and rendering the graphical model.

Abstract: A corridor modelling system models a corridor to be constructed along an existing terrain. The system obtains an electronic representation of the existing terrain, an electronic representation of a typical cross-section of the corridor, and an electronic representation of a 3D baseline of the corridor that aligns the typical cross-section relative to the existing terrain. The system also generates a digital surface model of the corridor based on the electronic representations. The system does so by extruding the typical cross-section along the 3D baseline and fundamentally modeling continuous surfaces of the corridor resulting from the extrusion as a collection of non-planar parametrically defined surfaces, rather than as a group of discrete facets formed from a triangulated irregular network.

Abstract: The present disclosure provides a system and method for automatically identifying and classifying seismic terminations within a seismic data volume. A set of surfaces is obtained (step 303) describing the seismic data volume. A plurality of seismic terminations is identified within the set of surfaces (step 307). Based upon seismic attributes or geometric criterion, a termination direction can be determined (step 309) for at least one termination.

Abstract: In one embodiment, a method includes facilitating a real-time display of drilling-performance data for a current well. The method further includes receiving new channel data for the current well from a wellsite computer system. The method also retrieving input data including historical drilling-performance data for an offset well relative to the current well. In addition, the method includes computing calculated data for the current well based on the channel data and the input data. Moreover, the method includes updating the real-time display with the calculated data.

Abstract: Method for adaptive weighting of geophysical data types in iterative joint inversion to speed convergence and aid escape from local minima of the penalty (objective) function. Two or more geophysical data sets (11) representing a region of interest are obtained, and are jointly inverted to infer models of the physical properties that affect the particular types of data used. The misfit for each data type is a weighted tem in the penalty function (13). The invention involves changing the weights (51) as the iteration cycles progress when the iteration convergence criteria are satisfied (15), to see if they remain satisfied (52) with the modified penalty function.

Abstract: Hydrocarbon recovery strategy development. At least some of the illustrative embodiments are methods including: selecting a set of attributes regarding a first undeveloped hydrocarbon reservoir; identifying a first group of developed hydrocarbon reservoirs, the identifying based on the set of attributes of the first undeveloped hydrocarbon reservoir, and the identifying by a computer system; calculating a first set of statistical data based on the first group of developed hydrocarbon reservoirs, the calculating by the computer system; scaling the first set of statistical data in reference to the set of attributes of the first undeveloped hydrocarbon reservoir to create a second set of statistical data, the scaling by the computer system; and implementing at least one hydrocarbon recovery strategy for the first undeveloped hydrocarbon.

Abstract: Methods and systems for generating a stable reverse time demigration (RTDM) equation for predicting wave phenomena such as reflections, refractions and multiples are described. A coupling term is added to an RTDM equation and reflectivity associated with the coupling term is replaced with a pseudo-density function derived from a nonlinear equation. The resultant coupled and stabilized RTDM equation is then used to predict the desired wave phenomena based on the applied seismic image.

Abstract: Spatial data sequences are extracted in a frequency-space domain at selected frequencies in selected data windows in seismic data. A signal model and residuals are iteratively constructed for each extracted spatial data sequence. Each data sample in each extracted spatial data sequence is assessed to determine if the data sample is noisy. Each noisy data sample in each data sequence is replaced by a corresponding signal model value. The earth's subsurface is imaged using the noise-attenuated seismic data.

Abstract: Embodiments of a method for correcting velocity models for complex topographies are disclosed herein. In general, embodiments of the method utilize velocity corrections based on geomechanical effects to correct a velocity model to take into account complex surface topographies. In particular, embodiments of the method use a nucleus strain theory to determine the velocity corrections. Further details and advantages of various embodiments of the method are described in more detail herein.

Abstract: A method for quantitative analysis of time-lapse seismic data of a reservoir, including: obtaining a plurality of compressional and shear velocities from a seismic inversion analysis; selecting a rock physics model based on a property of the reservoir; calculating a transform function using the rock physics model, where the transform function transforms variations in the plurality of compressional and shear velocities into variations in saturation and pore pressure; calculating a transform grid performing a domain transformation of the transform function; obtaining a plurality of cloud points from the seismic inversion analysis and the transform grid; and overlaying the plurality of cloud points onto the transform grid to estimate a plurality of reservoir parameters of the reservoir.

Abstract: Systems, methods, and computer-readable media for modeling a fracture network are provided. The method includes mapping a sub-seismic data set to a portion of a seismic-resolution data set, and defining a fractal region of the seismic-resolution data set containing the portion thereof to which the sub-seismic data is mapped. The method also includes generating a training image for sub-seismic scale characteristics of the one or more fracture networks of the seismic-resolution data set using the portion of the seismic-resolution data, and modeling the sub-seismic scale characteristics of the one or more fracture networks of the fractal region of the seismic-resolution data set outside of the portion, using the training image.

Abstract: A method for modeling a subsurface anomalous density zone including the steps of forming a density model, computing a response to the density model, inverting the response to arrive at a geometric model of the anomalous density zone, and applying a sharpening function to boundary regions of the geometric model to distinguish between the anomalous density zone and a surrounding region.

Abstract: A method of visually enhancing at least one geologic feature in 3D seismic survey data, comprising the steps of: (a) generating at least one attribute volume definable in Cartesian space and comprising at least one attribute derivable from said 3D seismic survey data; (b) generating a first Radon data volume from data resulting from a transaxial Radon Transform of said at least one attribute volume with respect to a first Cartesian axis; (c) generating a second Radon data volume from data resulting from a transaxial Radon Transform of said first Radon data volume with respect to a second Cartesian axis; (d) generating a third Radon data volume from data resulting from exponentiating a characteristic parameter of each one of a plurality of voxels forming said second Radon data volume to a predetermined first power value, and (e) applying a first Inverse Radon Transform to said third Radon data volume with respect to said second Cartesian axis, and a subsequent second Inverse Radon Transform to the resulting da

Abstract: Integrated formation modeling systems and methods are described. An example method of performing seismic analysis of a subterranean formation includes obtaining seismic data of the formation, obtaining fluid from the formation and analyzing at least some of the fluid to determine a fluid parameter. The example method additionally includes generating a model of the formation based at least on the seismic data and modifying the model based on the fluid parameter.

Abstract: A method for generating synthetic seismic traces. The method includes receiving a hybrid model that includes geological and seismic data, where the geological data corresponds to a zone of interest and the seismic data corresponds to the zone of interest and adjacent areas of the zone of interest. The method may determine a plurality of reflection properties along a path for a trace location using data from the hybrid model. The method may then generate a synthetic seismic trace for the trace location using the plurality of reflection properties.

Abstract: Automatic detection and accurate time picking of weak events embedded in strong noise such as microseismicity induced by hydraulic fracturing is accomplished by: a noise reduction step to separate out the noise and estimate its spectrum; an events detection and confidence indicator step, in which a new statistical test is applied to detect which time windows contain coherent arrivals across components and sensors in the multicomponent array and to indicate the confidence in this detection; and a time-picking step to accurately estimate the time of onset of the arrivals detected above and measure the time delay across the array using a hybrid beamforming method incorporating the use of higher order statistics. In the context of hydraulic fracturing, this could enhance the coverage and mapping of the fractures while also enabling monitoring from the treatment well itself where there is usually much higher and spatially correlated noise.

Abstract: Seismic visibility analysis of selected subsurface structures is employed to determine surface locations offering high visibility of target events. These locations can then be used as a basis for acquiring additional seismic survey data and/or selecting existing traces for re-migration with more sophisticated migration methods. With either usage, the newly migrated data is expected to offer enhanced images of the target event. In some embodiments, the visibility determination includes using a wave equation based propagator to find, for each of multiple simulated shots, a reflection wavefield from the target event in a seismic model; and to calculate, for each of multiple receiver positions, a contribution signal from each reflection wavefield. The visibility determination further includes converting each contribution signal into a source-receiver visibility value. Because data acquisition and/or re-migration is limited to the selected region, the imaging effort for the target event is significantly reduced.

Abstract: A method and system includes determining a subsurface fluid seismic attribute comprising recording a signal at a wellhead related to pumping fracture stimulation fluid to obtain a pressure pulse pump signal. A deconvolution operator is determined from the obtained pressure pulse pump signal. Seismic data are acquired from a plurality of sensors. Travel time differences are computed for the seismic data between the plurality of sensor locations and the subsurface position. Seismic data are deconvolved with the deconvolution operator to obtain a plurality of deconvolution coefficients associated with the subsurface position. The computed travel time differences are used to sum the plurality of deconvolution coefficients associated with the subsurface position to obtain a subsurface fluid seismic attribute at the subsurface position.

Abstract: A system for and computer implemented method for analysis of data representative of subsurface properties of a subsurface region. The method includes transforming the data representative of subsurface properties of the subsurface region into transformed data in accordance with a selected criterion. A three dimensional window geometry to be applied to the transformed data is selected, based, at least in part, on expected feature sizes present, data sampling density and a size of the subsurface region. A plurality of values for a three dimensional lacunarity statistic are calculated by applying the selected three dimensional window geometry to randomly selected regions of the subsurface region, and correlating the calculated values to the subsurface properties of the subsurface region.

Abstract: A computer-implemented method for processing a seismic trace of a subterranean region comprising: (a) deriving a seismic wavelet for a time-window of seismic trace; (b) generating a model for the subterranean region corresponding to that time-window comprising a plurality of seismic reflectors by comparing the seismic wavelet with the time-window of the seismic trace to identify locations for the seismic reflectors, wherein the locations of the seismic reflectors are subject to a separation constraint requiring neighboring seismic reflectors to be separated by at least a minimum separation threshold which is greater than a separation corresponding to a sampling interval for the seismic trace; (c) defining a model seismic wavelet having a model seismic wavelet amplitude spectrum with a frequency bandwidth which is greater than that of the seismic wavelet amplitude spectrum determined for the time-window of the seismic trace; and (d) generating a model of the time-window of the seismic trace for the subterranea

Abstract: Embodiments described herein use stochastic inversion (460) in lower dimensions to form an initial model (458) that is to be used in higher-dimensional gradient-based inversion (466). For example, an initial model may be formed from 1.5-D stochastic inversions, which is then processed (464) to form a 3-D model. Stochastic inversions reduce or avoid local minima and may provide an initial result that is near the global minimum.

Abstract: A system and a computer implemented method for determining a best well location from a plurality of possible well locations are described herein. The method includes drawing a plurality of earth models from a posterior distribution, wherein the posterior distribution is generated by stochastic inversion of existing data; calculating a well production at a plurality of proposed well locations within an earth model in the plurality of earth models using a relationship between the well production and earth parameters; calculating from the plurality of earth models, cost distributions using the relationship between well cost and the earth parameters; and calculating probability weighted values for the proposed well locations using probabilities from location dependent stochastic inversions as weights.

Abstract: There is provided a seismic imaging technology for imaging a subsurface structure by processing measured data reflected from the subsurface structure after a wave from a source wave has been propagated to the subsurface structure. According to an aspect, there is provided a seismic imaging method for obtaining imaging data of a subsurface structure through waveform inversion using a macro-velocity model as an initial value, wherein the macro-velocity model which is used as the initial value for the waveform inversion is calculated by: calculating a velocity difference function which is a difference between a real velocity and an initial velocity model, wherein the velocity difference function is a ratio of measured seismic scattered energy and modeled scattered energy based on the initial velocity model; and calculating the macro-velocity model by updating the initial velocity model with the velocity difference function.

Abstract: Method for stabilizing the updated model (13) in iterative seismic data inversion so that the model-simulated data for the next iteration does not “blow up.” A Projection Onto Convex Sets (“POCS”) operator is defined that converts the matrix corresponding to the model to a positive semi-definite matrix. The stability projection operator may be looped with physical constraint projection operators until the model converges (15). The resulting stable and constrained model is then used to simulate seismic data in the next cycle of the outer iteration loop (16).

Abstract: Apparatus, computer instructions and method for processing seismic data related to a subsurface of a body of water. The method includes receiving input data for a vertical direction and radial direction and/or from a hydrophone, applying a radon transform to the input data, separating primary signals from ghosts signals based on the vertical and radial components, applying an inverse radon transform to determine up-going and down-going wave fields in a time-distance domain, and separating interfering up-going and down-going wave fields that are recorded by the same receivers.