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Опубликовано в Binary options in germany | Октябрь 2, 2012

“Hay fondos de inversión general que nunca han invertido en energía En lo que va de año, 40 firmas de petróleo y gas obtuvieron Se cuenta periódicamente con estadísticas de comercio exterior de crudo y derivados, reintegros para gastos en moneda local, y registros anuales de inversión. Effect of the HLD parameter on the oil/ water emulsion inversion point. Scholarships in Brazil Scientific Initiation. Rodrigo Berini Tamaki.
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The same holds when a line is inverted in a circle, it is inverted to either a circle or a line. It is convenient not to distinguish between a circle and a line, and instead use a so-called generalized circle , i. An angle at a point on a circle is measured from the tangent through that point. There are always two supplementary angles between two generalized lines. When you invert three points in a circle, their orientation is reversed, just as when reflecting in a line.

We will not prove the theorem for all cases but sketch out the proof for the case where the angle is formed by two circles. This is shown by using the same steps as in Exercise 4. Other cases of angles between circles or lines can be proved in a similar way. Note that when two circles are reflected in a circle, the angles are preserved but since the orientation of points is reversed the orientation of angles is also reversed.

When forming the angles between two circles you can always do it so that exactly one of the supplementary angles is inside both circles. After inversion the two angles are reversed with regards to which angle that is inside both circles. As a consequence of angle preservation, objects that are tangent to each other, are inverted to objects that are also tangent to each other.

A geodesic is the hyperbolic equivalence of a Euclidean line. The reason why perpendicular generalized arcs are used as hyperbolic lines to model hyperbolic geometry, is because of the properties of perpendicular circles when doing circle inversion.

For this part of the proof we will use that a tangent to a circle through a point is perpendicular to the radius through the same point. We have that. When inverting in a circle, angles are preserved but distances are not. There is, however, a relation between distances that is preserved even after a circle inversion. In order for the hyperbolic distance to have the same functionality as Euclidean distance, it must have a number of properties.

Thales theorem states that if two vertices of a triangle are the endpoints of a diameter of a circle, and if the third vertex also lies on the circle, then the angle at the third vertex is a right angle. You don't have to do this. Anton Petrunin - Euclidean and Hyperbolic Planes, A minimalistic introduction with metric approach pdf. Inversion in Circle. The model generated is of higher quality, and does not suffer from tuning and interference caused by the wavelet. CSSI transforms seismic data to a pseudo-acoustic impedance log at every trace.

Acoustic impedance is used to produce more accurate and detailed structural and stratigraphic interpretations than can be obtained from seismic or seismic attribute interpretation. In many geological environments acoustic impedance has a strong relationship to petrophysical properties such as porosity, lithology , and fluid saturation. A good CSSI algorithm will produce four high-quality acoustic impedance volumes from full or post-stack seismic data: full-bandwidth impedance, bandlimited Impedance, reflectivity model, and low-frequency component.

Each of these components can be inspected for its contribution to the solution and to check the results for quality. To further adapt the algorithm mathematics to the behavior of real rocks in the subsurface, some CSSI algorithms use a mixed-norm approach and allow a weighting factor between minimizing the sparsity of the solution and minimizing the misfit of the residual traces. Pre-stack inversion is often used when post-stack inversion fails to sufficiently differentiate geologic features with similar P-impedance signatures.

While many geologic features can express similar P-impedance characteristics, few will share combined P-impedance and S-impedance traits allowing improved separation and clarity. Often a feasibility study using the wells logs will indicate whether separation of the desired lithotype can be achieved with P-impedance alone or whether S-impedance is also required.

This will dictate whether a pre- or post-stack inversion is needed. Simultaneous Inversion SI is a pre-stack method that uses multiple offset or angle seismic sub-stacks and their associated wavelets as input; it generates P-impedance, S-impedance and density as outputs although the density output resolution is rarely as high as the impedances. This helps improve discrimination between lithology, porosity and fluid effects. For each input partial stack, a unique wavelet is estimated.

All models, partial stacks and wavelets are input to a single inversion algorithm — enabling inversion to effectively compensate for offset-dependent phase, bandwidth, tuning and NMO stretch effects. The inversion algorithm works by first estimating angle-dependent P-wave reflectivities for the input-partial stacks. Next, these are used with the full Zoeppritz equations or approximations, such as Aki—Richards, for some algorithms to find band-limited elastic reflectivities.

These are in turn merged with their low-frequency counterparts from the model and integrated to elastic properties. This approximate result is then improved in a final inversion for P-impedance, S-impedance and density, subject to various hard and soft constraints. One constraint can control the relation between density and compressional velocity; this is necessary when the range of angles is not great enough to be diagnostic of density.

An important part in the inversion procedure is the estimation of the seismic wavelets. This is accomplished by computing a filter that best shapes the angle-dependent well log reflection coefficients in the region of interest to the corresponding offset stack at the well locations. Reflection coefficients are calculated from P-sonic, S-sonic and density logs using the Zoeppritz equations.

The wavelets, with amplitudes representative of each offset stack, are input directly into the inversion algorithm. Since a different wavelet is computed for each offset volume, compensation is automatically done for offset-dependent bandwidth, scaling and tuning effects. A near-stack wavelet can be used as the starting point for estimating the far-angle or offset wavelet. No prior knowledge of the elastic parameters and density beyond the solution space defined by any hard constraints is provided at the well locations.

This makes comparison of the filtered well logs and the inversion outputs at these locations a natural quality control. The lowest frequencies from the inversion are replaced with information from the geologic model since they are poorly constrained by the seismic data. When applied in global mode a spatial control term is added to the objective function and large subsets of traces are inverted simultaneously. The simultaneous inversion algorithm takes multiple angle-stacked seismic data sets and generates three elastic parameter volumes as output.

The resulting elastic parameters are real-rock properties that can be directly related to reservoir properties. The more advanced algorithms use the full Knott—Zoeppritz equations and there is full allowance for amplitude and phase variations with offset. This is done by deriving unique wavelets for each input-partial stack. The elastic parameters themselves can be directly constrained during the seismic inversion and rock-physics relationships can be applied, constraining pairs of elastic parameters to each other.

Final elastic-parameter models optimally reproduce the input seismic, as this is part of the seismic inversion optimization. Geostatistical inversion integrates high resolution well data with low resolution 3-D seismic, and provides a model with high vertical detail near and away from well control. This generates reservoir models with geologically-plausible shapes, and provides a clear quantification of uncertainty to assess risk.

Highly detailed petrophysical models are generated, ready for input to reservoir-flow simulation. Geostatistics differs from statistics in that it recognizes that only certain outcomes are geologically plausible. Geostatistical inversion integrates data from many sources and creates models that have greater resolution than the original seismic, match known geological patterns, and can be used for risk assessment and reduction.

Seismic, well logs and other input data are each represented as a probability density function PDF , which provides a geostatistical description based on histograms and variograms. Together these define the chances of a particular value at a particular location, and the expected geological scale and composition throughout the modeled area. Unlike conventional inversion and geomodeling algorithms, geostatistical inversion takes a one-step approach, solving for impedance and discrete property types or lithofacies at the same time.

Taking this approach speeds the process and improves accuracy. Individual PDFs are merged using bayesian inference techniques, resulting in a posterior PDF conditioned to the whole data set. The algorithm determines the weighting of each data source, eliminating potential bias. The posterior PDF is then input to a Markov chain Monte Carlo algorithm to generate realistic models of impedance and lithofacies, which are then used to co-simulate rock properties such as porosity.

These processes are typically iterated until a model emerges that matches all information. Even with the best model, some uncertainty remains. Uncertainty can be estimated using random seeds to generate a range of realizations. This is especially useful when dealing with parameters that are sensitive to change; an analysis of this sort enables greater understanding of development risk. The output model realizations are consistent with well log information, AVO seismic data, and honor rock property relationships found in the wells.

The algorithm also simultaneously produces elastic properties P-impedance, S-impedance and density and lithology volumes, instead of sequentially solving for lithology first and then populating the cell with impedance and density values. Because all output models match all input data, uncertainty can be quantitatively assessed to determine the range of reservoir possibilities within the constraining data.

It is thus possible to exploit "informational synergies" to retrieve details that deterministic inversion techniques blur out or omit. As a result, geoscientists are more successful in reconstructing both the overall structure and the fine details of the reservoir.

The use of multiple-angle-stack seismic volumes in AVA geostatistical inversion enables further evaluation of elastic rock properties and probable lithology or seismic facies and fluid distributions with greater accuracy. The process begins with a detailed petrophysical analysis and well log calibration. The calibration process replaces unreliable and missing sonic and density measurements with synthesized values from calibrated petrophysical and rock-physics models.

Well log information is used in the inversion process to derive wavelets, supply the low frequency component not present in the seismic data, and to verify and analyze the final results. Next, horizon and log data are used to construct the stratigraphic framework for the statistical information to build the models. In this way, the log data is only used for generating statistics within similar rock types within the stratigraphic layers of the earth.

Hyperbolic distance is defined in such a way that it is preserved when reflecting in a geodesic. The hyperbolic distance between two points is not the same as the Euclidean distance. By using inversion in circle as a hyperbolic version of the Euclidean reflection in a line, we will be able to construct hyperbolic tools such as midpoint, perpendicular line and perpendicular bisector.

We will start by defining circle inversion and then show all the properties of circle inversion that are needed to construct hyperbolic tools. Inversion in a circle is a transformation that flips the circle inside out. It is possible to construct the inverted point using a ruler and compass. In order to show the properties of circle inversion that are needed, we need to find relations between points and inverted points.

When a circle is inverted in a circle it is inverted to either a circle or a line. The same holds when a line is inverted in a circle, it is inverted to either a circle or a line. It is convenient not to distinguish between a circle and a line, and instead use a so-called generalized circle , i. An angle at a point on a circle is measured from the tangent through that point.

There are always two supplementary angles between two generalized lines. When you invert three points in a circle, their orientation is reversed, just as when reflecting in a line. We will not prove the theorem for all cases but sketch out the proof for the case where the angle is formed by two circles. This is shown by using the same steps as in Exercise 4.

Other cases of angles between circles or lines can be proved in a similar way. Note that when two circles are reflected in a circle, the angles are preserved but since the orientation of points is reversed the orientation of angles is also reversed. When forming the angles between two circles you can always do it so that exactly one of the supplementary angles is inside both circles. After inversion the two angles are reversed with regards to which angle that is inside both circles.

As a consequence of angle preservation, objects that are tangent to each other, are inverted to objects that are also tangent to each other. A geodesic is the hyperbolic equivalence of a Euclidean line. The reason why perpendicular generalized arcs are used as hyperbolic lines to model hyperbolic geometry, is because of the properties of perpendicular circles when doing circle inversion. For this part of the proof we will use that a tangent to a circle through a point is perpendicular to the radius through the same point.

An example of a post-stack seismic resolution inversion technique is the constrained sparse-spike inversion CSSI. This assumes a limited number of reflection coefficients, with larger amplitude. The inversion results in acoustic impedance AI , which is the product of rock density and p-wave velocity.

Unlike seismic reflection data which is an interface property AI is a rock property. The model generated is of higher quality, and does not suffer from tuning and interference caused by the wavelet. CSSI transforms seismic data to a pseudo-acoustic impedance log at every trace. Acoustic impedance is used to produce more accurate and detailed structural and stratigraphic interpretations than can be obtained from seismic or seismic attribute interpretation.

In many geological environments acoustic impedance has a strong relationship to petrophysical properties such as porosity, lithology , and fluid saturation. A good CSSI algorithm will produce four high-quality acoustic impedance volumes from full or post-stack seismic data: full-bandwidth impedance, bandlimited Impedance, reflectivity model, and low-frequency component.

Each of these components can be inspected for its contribution to the solution and to check the results for quality. To further adapt the algorithm mathematics to the behavior of real rocks in the subsurface, some CSSI algorithms use a mixed-norm approach and allow a weighting factor between minimizing the sparsity of the solution and minimizing the misfit of the residual traces. Pre-stack inversion is often used when post-stack inversion fails to sufficiently differentiate geologic features with similar P-impedance signatures.

While many geologic features can express similar P-impedance characteristics, few will share combined P-impedance and S-impedance traits allowing improved separation and clarity. Often a feasibility study using the wells logs will indicate whether separation of the desired lithotype can be achieved with P-impedance alone or whether S-impedance is also required.

This will dictate whether a pre- or post-stack inversion is needed. Simultaneous Inversion SI is a pre-stack method that uses multiple offset or angle seismic sub-stacks and their associated wavelets as input; it generates P-impedance, S-impedance and density as outputs although the density output resolution is rarely as high as the impedances.

This helps improve discrimination between lithology, porosity and fluid effects. For each input partial stack, a unique wavelet is estimated. All models, partial stacks and wavelets are input to a single inversion algorithm — enabling inversion to effectively compensate for offset-dependent phase, bandwidth, tuning and NMO stretch effects.

The inversion algorithm works by first estimating angle-dependent P-wave reflectivities for the input-partial stacks. Next, these are used with the full Zoeppritz equations or approximations, such as Aki—Richards, for some algorithms to find band-limited elastic reflectivities. These are in turn merged with their low-frequency counterparts from the model and integrated to elastic properties.

This approximate result is then improved in a final inversion for P-impedance, S-impedance and density, subject to various hard and soft constraints. One constraint can control the relation between density and compressional velocity; this is necessary when the range of angles is not great enough to be diagnostic of density.

An important part in the inversion procedure is the estimation of the seismic wavelets. This is accomplished by computing a filter that best shapes the angle-dependent well log reflection coefficients in the region of interest to the corresponding offset stack at the well locations. Reflection coefficients are calculated from P-sonic, S-sonic and density logs using the Zoeppritz equations.

The wavelets, with amplitudes representative of each offset stack, are input directly into the inversion algorithm. Since a different wavelet is computed for each offset volume, compensation is automatically done for offset-dependent bandwidth, scaling and tuning effects. A near-stack wavelet can be used as the starting point for estimating the far-angle or offset wavelet.

No prior knowledge of the elastic parameters and density beyond the solution space defined by any hard constraints is provided at the well locations. This makes comparison of the filtered well logs and the inversion outputs at these locations a natural quality control.

The lowest frequencies from the inversion are replaced with information from the geologic model since they are poorly constrained by the seismic data. When applied in global mode a spatial control term is added to the objective function and large subsets of traces are inverted simultaneously. The simultaneous inversion algorithm takes multiple angle-stacked seismic data sets and generates three elastic parameter volumes as output. The resulting elastic parameters are real-rock properties that can be directly related to reservoir properties.

The more advanced algorithms use the full Knott—Zoeppritz equations and there is full allowance for amplitude and phase variations with offset. This is done by deriving unique wavelets for each input-partial stack. The elastic parameters themselves can be directly constrained during the seismic inversion and rock-physics relationships can be applied, constraining pairs of elastic parameters to each other.

Final elastic-parameter models optimally reproduce the input seismic, as this is part of the seismic inversion optimization. Geostatistical inversion integrates high resolution well data with low resolution 3-D seismic, and provides a model with high vertical detail near and away from well control.

This generates reservoir models with geologically-plausible shapes, and provides a clear quantification of uncertainty to assess risk. Highly detailed petrophysical models are generated, ready for input to reservoir-flow simulation. Geostatistics differs from statistics in that it recognizes that only certain outcomes are geologically plausible.

Geostatistical inversion integrates data from many sources and creates models that have greater resolution than the original seismic, match known geological patterns, and can be used for risk assessment and reduction. Seismic, well logs and other input data are each represented as a probability density function PDF , which provides a geostatistical description based on histograms and variograms.

Together these define the chances of a particular value at a particular location, and the expected geological scale and composition throughout the modeled area. Unlike conventional inversion and geomodeling algorithms, geostatistical inversion takes a one-step approach, solving for impedance and discrete property types or lithofacies at the same time.

Taking this approach speeds the process and improves accuracy. Individual PDFs are merged using bayesian inference techniques, resulting in a posterior PDF conditioned to the whole data set. The algorithm determines the weighting of each data source, eliminating potential bias. The posterior PDF is then input to a Markov chain Monte Carlo algorithm to generate realistic models of impedance and lithofacies, which are then used to co-simulate rock properties such as porosity.

These processes are typically iterated until a model emerges that matches all information. Even with the best model, some uncertainty remains. Uncertainty can be estimated using random seeds to generate a range of realizations.

This is especially useful when dealing with parameters that are sensitive to change; an analysis of this sort enables greater understanding of development risk. The output model realizations are consistent with well log information, AVO seismic data, and honor rock property relationships found in the wells. The algorithm also simultaneously produces elastic properties P-impedance, S-impedance and density and lithology volumes, instead of sequentially solving for lithology first and then populating the cell with impedance and density values.

Because all output models match all input data, uncertainty can be quantitatively assessed to determine the range of reservoir possibilities within the constraining data. It is thus possible to exploit "informational synergies" to retrieve details that deterministic inversion techniques blur out or omit. As a result, geoscientists are more successful in reconstructing both the overall structure and the fine details of the reservoir.

The use of multiple-angle-stack seismic volumes in AVA geostatistical inversion enables further evaluation of elastic rock properties and probable lithology or seismic facies and fluid distributions with greater accuracy. The process begins with a detailed petrophysical analysis and well log calibration.