Two-term AVO inversion: Equivalences and new methods

Geophysics ◽  
2008 ◽  
Vol 73 (6) ◽  
pp. C31-C38 ◽  
Author(s):  
Charles P. Ursenbach ◽  
Robert R. Stewart
Keyword(s):  
Parameter Method ◽  
Angle Of Incidence ◽  
Amplitude Variation With Offset ◽  
New Methods ◽  
Avo Inversion ◽  
Parameter Inversion ◽  
Maximum Angle ◽  
Resource Exploration ◽  
New Formulation ◽  
Two Parameter

Most amplitude-variation-with-offset (AVO) studies use two-parameter inversion methods that are approximations of a more general three-parameter method based on the Aki-Richards approximation. Two-parameter methods are popular because the three-parameter inversion is often plagued by numerical instability. Reducing the dimensionality of the parameter space stabilizes the inversion. A variety of constraints can accomplish this, and these lead to the multiplicity of current two-parameter methods. It would be useful to understand relationships between various two-parameter methods. To this end, we derive formal expressions for inversion errors of each method. Using these expressions, conversion formulas are obtained that allow the flexibility to convert results of any two-parameter method to those of any other two-parameter method. The only requirement for the equivalence of methods is that the maximum angle of incidence be at least a few degrees less than the critical angle. In addition, error expressions result in a new formulation for a two-parameter AVO tool that combines strengths of two commonly used methods. The expressions also suggest a simple way to incorporate information from well-log calibration into legacy AVO inversions. These results should be helpful in resource exploration.

Three-term amplitude-variation-with-offset projections

Geophysics ◽  
2018 ◽  
Vol 83 (5) ◽  
pp. N51-N65 ◽  
Author(s):  
Vaughn Ball ◽  
Luis Tenorio ◽  
Christian Schiøtt ◽  
Michelle Thomas ◽  
J. P. Blangy
Keyword(s):  
Rock Physics ◽  
Projection Methods ◽  
Well Logs ◽  
Rock Properties ◽  
Inversion Method ◽  
Practical Implementation ◽  
Angle Of Incidence ◽  
Amplitude Variation ◽  
Amplitude Variation With Offset ◽  
Maximum Angle

A three-term (3T) amplitude-variation-with-offset projection is a weighted sum of three elastic reflectivities. Parameterization of the weighting coefficients requires two angle parameters, which we denote by the pair [Formula: see text]. Visualization of this pair is accomplished using a globe-like cartographic representation, in which longitude is [Formula: see text], and latitude is [Formula: see text]. Although the formal extension of existing two-term (2T) projection methods to 3T methods is trivial, practical implementation requires a more comprehensive inversion framework than is required in 2T projections. We distinguish between projections of true elastic reflectivities computed from well logs and reflectivities estimated from seismic data. When elastic reflectivities are computed from well logs, their projection relationships are straightforward, and they are given in a form that depends only on elastic properties. In contrast, projection relationships between reflectivities estimated from seismic may also depend on the maximum angle of incidence and the specific reflectivity inversion method used. Such complications related to projections of seismic-estimated elastic reflectivities are systematized in a 3T projection framework by choosing an unbiased reflectivity triplet as the projection basis. Other biased inversion estimates are then given exactly as 3T projections of the unbiased basis. The 3T projections of elastic reflectivities are connected to Bayesian inversion of other subsurface properties through the statistical notion of Bayesian sufficiency. The triplet of basis reflectivities is computed so that it is Bayes sufficient for all rock properties in the hierarchical seismic rock-physics model; that is, the projection basis contains all information about rock properties that is contained in the original seismic.

Analysis of the spectral hole in velocity versus depth resolution for reflection traveltimes with limited aperture

Geophysics ◽  
2005 ◽  
Vol 70 (3) ◽  
pp. U37-U45 ◽  
Author(s):  
Kenneth P. Bube ◽  
Robert T. Langan ◽  
Tamas Nemeth
Keyword(s):  
Taylor Expansion ◽  
Depth Resolution ◽  
Theoretical Explanation ◽  
Velocity Versus ◽  
Angle Of Incidence ◽  
Vertical Variation ◽  
Spectral Hole ◽  
Maximum Angle ◽  
Zero Offset ◽  
Horizontal Wavelength

It is difficult to resolve the ambiguity between velocity and reflector depth using reflection traveltimes when the aperture is small, as is common for deep reflectors. For velocity perturbations that are independent of the vertical variable, there is an even stronger velocity-versus-depth ambiguity at a horizontal wavelength of 2.5 times the reflector depth. We give a theoretical explanation of why this spectral hole occurs. When the maximum offset is small, there are velocity and reflector depth perturbations that cause almost cancelling traveltime perturbations; the net traveltime perturbations are second order in offset, making resolution between velocity and depth difficult at all wavelengths. But for the particular wavelength [Formula: see text] ≈ 2.565 times the reflector depth, an extra term in the Taylor expansion for traveltime near zero offset vanishes; the net traveltime perturbations are fourth order in offset. Thus velocity-versus-depth resolution degrades much sooner at this wavelength as the maximum offset gets small. We show in addition that this behavior extends to velocity perturbations that can depend on the vertical variable, and spectral holes in velocity-versus-depth resolution can appear at any horizontal wavelength. Velocity perturbations with very simple vertical variation are sufficient to create these spectral holes. This behavior is not limited to extremely small apertures; the effect of this spectral hole can be felt when the maximum angle of incidence is as large as 25°.

Anisotropic correction of sonic logs in wells with large relative dip

Geophysics ◽  
2008 ◽  
Vol 73 (1) ◽  
pp. E1-E5 ◽  
Author(s):  
Lev Vernik
Keyword(s):  
Reservoir Characterization ◽  
Low Frequency ◽  
P Wave ◽  
Amplitude Variation With Offset ◽  
Seismic Amplitude ◽  
Avo Inversion ◽  
Pore Pressure Prediction ◽  
Siliciclastic Rocks ◽  
Dip Angle ◽  
Sonic Logs

Seismic reservoir characterization and pore-pressure prediction projects rely heavily on the accuracy and consistency of sonic logs. Sonic data acquisition in wells with large relative dip is known to suffer from anisotropic effects related to microanisotropy of shales and thin-bed laminations of sand, silt, and shale. Nonetheless, if anisotropy parameters can be related to shale content [Formula: see text] in siliciclastic rocks, then I show that it is straightforward to compute the anisotropy correction to both compressional and shear logs using [Formula: see text] and the formation relative dip angle. The resulting rotated P-wave sonic logs can be used to enhance time-depth ties, velocity to effective stress transforms, and low-frequency models necessary for prestack seismic amplitude variation with offset (AVO) inversion.

Analytic Approximations to Nonlinear Boundary Value Problems Modeling Beam-Type Nano-Electromechanical Systems

2017 ◽  
Vol 72 (3) ◽  
pp. 201-206
Author(s):  
Li Zou ◽  
Songxin Liang ◽  
Yawei Li ◽  
David J. Jeffrey
Keyword(s):  
Boundary Value Problems ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Parameter Method ◽  
Nonlinear Boundary Value Problems ◽  
Electromechanical Systems ◽  
Adomian Decomposition ◽  
Nonlinear Boundary Value ◽  
Beam Type ◽  
Two Parameter

AbstractNonlinear boundary value problems arise frequently in physical and mechanical sciences. An effective analytic approach with two parameters is first proposed for solving nonlinear boundary value problems. It is demonstrated that solutions given by the two-parameter method are more accurate than solutions given by the Adomian decomposition method (ADM). It is further demonstrated that solutions given by the ADM can also be recovered from the solutions given by the two-parameter method. The effectiveness of this method is demonstrated by solving some nonlinear boundary value problems modeling beam-type nano-electromechanical systems.

Linearized amplitude variation with offset (AVO) inversion with supercritical angles

Geophysics ◽  
2006 ◽  
Vol 71 (5) ◽  
pp. E49-E55 ◽  
Author(s):  
Jonathan E. Downton ◽  
Charles Ursenbach
Keyword(s):  
Reflection Coefficient ◽  
Critical Angle ◽  
Waveform Inversion ◽  
Phase Variation ◽  
Reflection Coefficients ◽  
Amplitude Variation ◽  
Amplitude Variation With Offset ◽  
Zoeppritz Equations ◽  
Avo Inversion ◽  
Angle Data

Contrary to popular belief, a linearized approximation of the Zoeppritz equations may be used to estimate the reflection coefficient for angles of incidence up to and beyond the critical angle. These supercritical reflection coefficients are complex, implying a phase variation with offset in addition to amplitude variation with offset (AVO). This linearized approximation is then used as the basis for an AVO waveform inversion. By incorporating this new approximation, wider offset and angle data may be incorporated in the AVO inversion, helping to stabilize the problem and leading to more accurate estimates of reflectivity, including density reflectivity.

Phytotoxic Interaction Studies – Techniques for Evaluation and Presentation of Results

Weed Science ◽  
1981 ◽  
Vol 29 (2) ◽  
pp. 147-155 ◽  
Author(s):  
R. G. Nash
Keyword(s):  
Growth Response ◽  
Statistical Significance ◽  
Multiple Regression Equation ◽  
Parameter Method ◽  
Regression Estimate ◽  
Pesticide Concentration ◽  
Response Data ◽  
The Individual ◽  
Multiplicative Models ◽  
Two Parameter

Two or more pesticides together may produce a growth response in plants that is not predictable by their individual or independent toxicities. This unpredicted (dependent) response results from an interaction, a concept that usually is not easily interpreted. Dependent responses are further complicated by the fact that they can be either synergistic or antagonistic. Several methods exist for identifying and measuring phytotoxic interactions. Nearly all methods have certain shortcomings, however. Additive and multiplicative models (mathematical expressions) are the two basic approaches to determining pesticide interactions. The two-parameter, isobole, and calculus methods axe additive; whereas, Colby and regression estimate are multiplicative models. Regression estimate analysis considers deviations due to experimental errors, and a statistical significance can be attached to the interaction magnitude, thereby overcoming the deficiencies of the Colby method, but both methods seem to be limited to response data in which the combined pesticide concentration is the sum of the individual pesticide concentrations. The two-parameter method seems to be limited to response data in which the combined concentration is equal to the individual pesticide concentration and to response data in which a pesticide concentration necessary to produce a 50% of control value is interpolated rather than extrapolated. The calculus method is a mathematical expression of the growth response, and interaction is measured by derivation of the equation obtained. The calculus method is difficult to interpret and has a major weakness because it depends upon the multiple regression equation of the observed data. The regression estimate method is recommended as a reasonable approach to interpretation of interaction type data, with a SAS language computer program available from the author.

Introduction to this special section: AVO inversion

2019 ◽  
Vol 38 (10) ◽  
pp. 752-753
Author(s):  
Edward Townend ◽  
Michael Kemper
Keyword(s):  
Case Studies ◽  
Best Practice ◽  
Special Section ◽  
Leading Edge ◽  
Seismic Inversion ◽  
The Body ◽  
Amplitude Variation With Offset ◽  
Field Development ◽  
Avo Inversion ◽  
The Subject

It has been more than three years since The Leading Edge last published a special section on amplitude variation with offset (AVO) inversion, and interest in the subject remains strong. This past spring, SEG hosted a joint symposium in Houston, Texas, on the “Resurgence of seismic inversion,” and the body of talks and case studies demonstrated the method's continued relevance to making impactful drilling decisions. Despite this, and despite AVO inversion's position as a mature and well-established technique, there are an abundance of examples in which inaccurate AVO predictions have led to drastic failures at the drill bit. This highlights the challenges that still exist in the successful execution of such investigations and makes the subject occasionally controversial and certainly fraught with data-quality and best-practice considerations. In this vein, the special section presented here offers examples of the broad sweep of considerations and methods relevant to enabling successful AVO inversion and the interpretation of its products, as well as case studies that demonstrate how application of the technique can be impactful all the way through to appraisal and field development programs.

A Two-Parameter Method to Monitor and Characterize Tool Wear in End Milling Inconel 718

2012 ◽  
Author(s):  
W. Li ◽  
Y. B. Guo
Keyword(s):  
Tool Wear ◽  
Inconel 718 ◽  
Elevated Temperatures ◽  
Flank Wear ◽  
End Milling ◽  
Parameter Method ◽  
Manufacturing Cost ◽  
Cnc Machine ◽  
Coated Carbide ◽  
Two Parameter

Inconel 718 is among the most widely used superalloys in many industries. It is often used in very harsh conditions such as jet engines, combustors and nuclear reactors due to its high strength at elevated temperatures, high oxidation and corrosion resistance. Machining superalloy Inconel 718 has always been a challenging task due to its poor machinability including rapid work hardening, low thermal conductivity, and relatively short cutting tool life. The fast tool wear during cutting Inconel 718 results in longer production time, deteriorated surface integrity, and higher manufacturing cost. In this paper, an on-line optical tool monitoring system integrated with a CNC machine tool has been developed to examine tool wear evolutions in end milling Inconel 718 with PVD (Ti, Al) N/TiN-coated carbide insert. Three basic types of tool wear: flank wear, nose wear, and crater wear were examined and analyzed. A two-parameter method has been proposed to evaluate both flank wear and nose wear vs. cutting time.

Two-step joint PP- and PS-wave three-term amplitude-variation with offset inversion

2016 ◽  
Vol 4 (4) ◽  
pp. T613-T625 ◽  
Author(s):  
Qizhen Du ◽  
Bo Zhang ◽  
Xianjun Meng ◽  
Chengfeng Guo ◽  
Gang Chen ◽  
...  
Keyword(s):  
Reflection Coefficient ◽  
Wave Reflection ◽  
Coefficient Matrix ◽  
P Wave ◽  
Inversion Method ◽  
S Wave ◽  
Amplitude Variation ◽  
Amplitude Variation With Offset ◽  
Avo Inversion ◽  
Pp Wave

Three-term amplitude-variation with offset (AVO) inversion generally suffers from instability when there is limited prior geologic or petrophysical constraints. Two-term AVO inversion shows higher instability compared with three-term AVO inversion. However, density, which is important in the fluid-type estimation, cannot be recovered from two-term AVO inversion. To reliably predict the P- and S-waves and density, we have developed a robust two-step joint PP- and PS-wave three-term AVO-inversion method. Our inversion workflow consists of two steps. The first step is to estimate the P- and S-wave reflectivities using Stewart’s joint two-term PP- and PS-AVO inversion. The second step is to treat the P-wave reflectivity obtained from the first step as the prior constraint to remove the P-wave velocity related-term from the three-term Aki-Richards PP-wave approximated reflection coefficient equation, and then the reduced PP-wave reflection coefficient equation is combined with the PS-wave reflection coefficient equation to estimate the S-wave and density reflectivities. We determined the effectiveness of our method by first applying it to synthetic models and then to field data. We also analyzed the condition number of the coefficient matrix to illustrate the stability of the proposed method. The estimated results using proposed method are superior to those obtained from three-term AVO inversion.

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