Integral Transforms in Geophysics

I Cauchy-Type Integrals in the Theory of a Plane Geopotential Field.- 1 Cauchy-Type Integral.- 1.1 Definition.- 1.1.1 Cauchy Integral Formula.- 1.1.2 Concept of the Cauchy-Type Integral.- 1.1.3 Piecewise Analytical Functions.- 1.2 Main Properties.- 1.2.1 Hölder Condition.- 1.2.2 Calculation of Singular Integrals in Terms of the Cauchy Principal Value.- 1.2.3 Sokhotsky-Plemelj Formulas.- 1.2.4 Generalization of the Sokhotsky-Plemelj Formulas for Piecewise Smooth Curves.- 1.2.5 Cauchy-Tpye Integrals Along the Real Axis.- 1.3 Cauchy and Hubert Integral Transforms.- 1.3.1 Integral Boundary Conditions for Analytical Functions.- 1.3.2 Determination of a Piecewise Analytical Function from a Specified Discontinuity.- 1.3.3 Inversion Formulas for the Cauchy-Type Integral (Cauchy Integral Transforms).- 1.3.4 Hilbert Transforms.- 2 Representation of Plane Geopotential Fields in the Form of the Cauchy-Type Integral.- 2.1 Plane Potential Fields and Their Governing Equations.- 2.1.1 Vector Field Equations.- 2.1.2 Concept of a Plane Field.- 2.1.3 Plane Field Equations.- 2.1.4 Plane Field Flow Function.- 2.2 Logarithmic Potentials and the Cauchy-Type Integral.- 2.2.1 Logarithmic Potentials.- 2.2.2 Logarithmic Potentials in Complex Coordinates.- 2.2.3 Cauchy-Type Integral as a Sum of the Logarithmic Potentials of Simple and Double Layers.- 2.3 Complex Intensity and Potential of a Plane Field.- 2.3.1 Concept of Complex Intensity of a Plane Field.- 2.3.2 Complex Intensity Equations.- 2.3.3 Representation of Complex Intensity in Terms of Field Source Density.- 2.3.4 Complex Potential.- 2.4 Direct Solution of the Equation for Complex Field Intensity.- 2.4.1 Two-Dimensional Ostrogradsky-Gauss Formula in Complex Notation.- 2.4.2 Pompei Formulas.- 2.4.3 Solution to the Equation for Complex Intensity.- 2.5 Representation of the Gravitational Field in Terms of the Cauchy-Type Integral.- 2.5.1 Complex Intensity of the Gravitational Field.- 2.5.2 Representation of the Gravitational Field of a Homogeneous Domain in Terms of the Cauchy-Type Integral.- 2.5.3 Representation of the Gravitational Field of a Domain with an Analytical Mass Distribution in Terms of the Cauchy-Type Integral.- 2.5.4 Case of Vertical or Horizontal Variations in the Density.- 2.5.5 Case of Linear Density Variations Along the Coordinate Axis.- 2.5.6 General Case of Continuous Density Distribution.- 2.5.7 Calculation of the Gravitational Field of an Infinitely Extended Domain.- 2.6 Representation of a Fixed Magnetic Field in Terms of the Cauchy-Type Integral.- 2.6.1 Complex Potential of a Polarized Source.- 2.6.2 Complex Intensities and Potential of a Magnetic Field.- 2.6.3 Representation of the Magnetic Potential of a Homogeneous Domain in Terms of the Cauchy-Type Integrals.- 2.6.4 General Case of Magnetization Distribution.- 2.6.5 Analytical Distribution of Magnetization.- 3 Techniques for Separation of Plane Fields.- 3.1 Separation of Geopotential Fields into External and Internal Parts Using Spectral Decomposition.- 3.1.1 Statement of the Problem of Plane Field Separation.- 3.1.2 Spectral Representations of Plane Fields.- 3.1.3 Determination of the External and Internal Parts of the Scalar Potential and Field (Gauss-Schmiedt Formulas).- 3.2 Kertz-Siebert Technique.- 3.2.1 Problem of Separation of Field Complex Intensity.- 3.2.2 Field Separation at Ordinary Points of the Line L.- 3.2.3 Field Separation at Corners of the Line L.- 3.2.4 Kertz-Siebert Formulas.- 3.2.5 Equivalence Between the Kertz-Siebert and the Gauss-Schmiedt Formulas.- 4 Analytical Continuation of a Plane Field.- 4.1 Fundamentals of Analytical Continuation.- 4.1.1 Taylor Theorem.- 4.1.2 Uniqueness of an Analytical Function.- 4.1.3 Concept of Analytical Continuation.- 4.1.4 Concept of the Riemann Surface.- 4.1.5 Weierstrass Continuation of an Analytical Function.- 4.1.6 Singular Points of an Analytical Function.- 4.1.7 Penleve Continuation of an Analytical Function (Principle of Continuity).- 4.1.8 Conformai Mapping.- 4.2 Analytical Continuation of the Cauchy-Type Integral Through a Path of Integration.- 4.2.1 Analytical Continuation of a Real Analytical Function of a Real Variable.- 4.2.2 Concept of an Analytical Arc; the Herglotz-Tsirulsky Equation for the Arc.- 4.2.3 Analytical Continuation of a Function Specified Along an Analytical Curve.- 4.2.4 Continuation of the Cauchy-Type Integral Through a Path of Integration; Singular Points of the Continued Field.- 4.3 Analytical Continuation of a Plane Magnetic Field into a Domain Occupied by Magnetized Masses.- 4.3.1 Analytical Continuation of a Magnetic Potential into a Domain of Analytically Distributed Magnetization.- 4.3.2 Continuation Through a One-Side Herglotz-Tsirulsky Analytical Arc.- 4.3.3 Analyticity Condition for the Boundary of a Domain Occupied by Magnetized Masses.- 4.3.4 Singular Points of Analytical Continuation of the Magnetic Potential.- 4.3.5 Determination of Complex Magnetization of a Body from its Magnetic Potential.- 4.4 Analytical Continuation of a Plane Gravitational Field into a Domain Occupied by Attracting Masses.- 4.4.1 Characteristics of the Gravitational Field of a Homogeneous Domain Bounded by an Analytical Curve.- 4.4.2 Continuation of the Gravitational Field into a Domain with an Analytical Density Distribution.- 4.4.3 Case of a Homogeneous Domain Bounded by a Piecewise Analytical Curve.- 4.4.4 Singular Points of the Continued Field, Lying on the Boundary of a Material Body.- 4.5 Integral Techniques for Analytical Continuation of Plane Fields.- 4.5.1 Forms of Analytical Continuation of Plane Fields in Geophysics.- 4.5.2 Reconstruction of a Function Analytical in the Upper Half-Plane from Its Real or Imaginary Part.- 4.5.3 Analytical Continuation of Plane Fields into a Horizontal Layer Using Spectral Decomposition of the Cauchy Kernel.- 4.5.4 Case of Field Specification on the Real Axis. The Zamorev Formulas.- 4.5.5 Downward Analytical Continuation of Functions Having Singularities Both in the Lower and in the Upper Half-Planes.- 4.5.6 Analytical Continuation into Domains with Curvilinear Boundaries.- 4.5.7 Bateman Formula; Continuation of Complex Intensity of the Field into the Lower Half-Plane Using Its Real Part.- II Cauchy-Type Integral Analogs in the Theory of a Three-Dimensional Geopotential Field.- 5 Three-Dimensional Cauchy-Type Integral Analogs.- 5.1 Three-Dimensional Analog of the Cauchy Integral Formula.- 5.1.1 Vector Statements of the Ostrogradsky-Gauss Theorem.- 5.1.2 Vector Statements of the Stokes Theorem.- 5.1.3 Analog of the Cauchy-Type Integral.- 5.1.4 Relationship Between the Three-Dimensional Analog and the Classical Cauchy Integral Formula.- 5.1.5 Gauss Harmonic Function Theorem.- 5.1.6 Cauchy Formula Analog for an Infinite Domain.- 5.1.7 Three-Dimensional Analog of the Pompei Formulas.- 5.2 Definition and Properties of the Three-Dimensional Cauchy Integral Analog.- 5.2.1 Concept of a Three-Dimensional Cauchy Integral Analog.- 5.2.2 Evaluation of Singular Integrals in Terms of the Cauchy Principal Value.- 5.2.3 Three-Dimensional Analogs of the Sokhotsky-Plemelj Formulas.- 5.3 Integral Transforms of the Laplace Vector Fields.- 5.3.1 Integral Boundary Conditions for the Laplace Field.- 5.3.2 Piecewise Laplace Vector Fields. Determination of a Piecewise Laplace Field from a Specified Discontinuity.- 5.3.3 Inversion Formulas for the Three-Dimensional Cauchy Integral Analog.- 5.3.4 Three-Dimensional Hilbert Transforms.- 5.4 Cauchy Integral Analogs in Matrix Notation.- 5.4.1 Matrix Representation of the Differentiation Operators of Scalar and Vector Fields.- 5.4.2 Matrix Representations of Three-Dimensional Cauchy Integral Analogs.- 6 Application of Cauchy Integral Analogs to the Theory of a Three-Dimensional Geopotential Field.- 6.1 Newton Potential and the Three-Dimensional Cauchy Integral Analog.- 6.1.1 Newton Potential.- 6.1.2 Newton Potential of Simple Field Sources.- 6.1.3 Newton Potential of Polarized Field Sources.- 6.1.4 Three-Dimensional Cauchy-Type Integral as a Sum of a Simple and a Double Layer Field.- 6.2 Representation of the Gravitational Field in Terms of the Cauchy Integral Analog.- 6.2.1 Gravitational Field Equations.- 6.2.2 Representation of the Gravitational Field of a Three-Dimensional Homogeneous Body in Terms of the Cauchy-Type Integral.- 6.2.3 Gravitational Field of a Body with an Arbitrary Density Distribution.- 6.2.4 Case of Vertical or One-Dimensional Horizontal Variations in the Density.- 6.2.5 Some Special Cases of Density Distribution.- 6.2.6 Calculation of the Gravitational Field of a Three-Dimensional Infinitely Extended Homogeneous Domain.- 6.2.7 Field of an Infinitely Extended Domain Filled with Masses of a Z-Variable Density.- 6.3 Representation of a Fixed Magnetic Field in Terms of the Cauchy Integral Analog.- 6.3.1 Intensity and Potential of a Fixed Magnetic Field.- 6.3.2 Representation of a Magnetic Field with an Arbitrary Distribution of Magnetized Masses.- 6.3.3 Potential Distribution of Magnetization.- 6.3.4 Laplace Distribution of Magnetization.- 6.3.5 Magnetic Field of a Uniformly Magnetized Body.- 6.4 Generalized Kertz-Siebert Technique for Separation of Three-Dimensional Geopotential Fields.- 6.4.1 Statement of the Problem of Separation of a Three-Dimensional Field.- 6.4.2 Separation of Fields at Ordinary Points of the Surface.- 6.4.3 Separation of Fields at Singular Points of the Surface.- 6.4.4 Generalized Kertz-Siebert Formulas.- 7 Analytical Continuation of a Three-Dimensional Geopotential Field.- 7.1 Fundamentals of Analytical Continuation of the Laplace Field.- 7.1.1 Analytical Nature of Laplace Vector Fields.- 7.1.2 Uniqueness of Laplace Vector Fields and Harmonic Functions.- 7.1.3 Concept of Analytical Continuation of a Vector Field and Its Riemann Space.- 7.1.4 Continuation of the Laplace Field Using the Taylor Series.- 7.1.5 Stal Theorem (Principle of Continuity for the Laplace Field).- 7.2 Analytical Continuation of the Three-Dimensional Cauchy Integral Analog Through the Integration Surface.- 7.2.1 Concept of an Analytical Part of the Surface; Surface Equations in a Harmonic Form.- 7.2.2 Relationship Between the Surface Equation in a Harmonic Form and the Plane Curve Equation in the Herglotz-Tsirulsky Form.- 7.2.3 Continuation of the Cauchy-Type Integral Through the Integration Surface.- 7.3 Analytical Continuation of a Three-Dimensional Gravitational Field into a Homogeneous Material Body.- 7.3.1 Properties of the Gravitational Field of a Body Bounded by an Analytical Surface.- 7.3.2 Relationship Between the Shape of the Surface of a Three-Dimensional Homogeneous Material Body and the Location of Singularities of the Gravitational Field Continued Analytically into the Body.- 7.3.3 Definition of the Shape of the Surface of Three-Dimensional Material Bodies by Analytical Continuation of the Gravitational Field.- 7.4 Continuation of the Gravitational and Magnetic Fields into a Domain with an Arbitrary Analytical Distribution of Field Sources.- 7.4.1 Analytical Representations of Fields Continued into Masses.- 7.4.2 Case of a Domain Bounded by an Analytical Surface.- 7.4.3 Case of a Domain Bounded by a Piecewise Analytical Surface.- 7.5 Integral Techniques for Analytical Continuation of Three-Dimensional Laplace Fields.- 7.5.1 Analytical Continuation of the Laplace Field into the Upper Half-Space.- 7.5.2 Analytical Continuation of the Laplace Field into the Lower Half-Space.- III Stratton-Chu Type Integrals in the Theory of Electromagnetic Fields.- 8 Stratton-Chu Type Integrals.- 8.1 Electromagnetic Field Equations.- 8.1.1 Maxwell Equations.- 8.1.2 Field in Homogeneous Domains of a Medium.- 8.1.3 Boundary Conditions.- 8.1.4 Monochromatic Field Equations.- 8.1.5 Quasi-Stationary Electromagnetic Field.- 8.1.6 Field Wave Equations.- 8.1.7 Field Equations Allowing for Magnetic Currents and Charges.- 8.1.8 Stationary Electromagnetic Field.- 8.2 Integration of Equations for an Arbitrary Vector Field.- 8.2.1 Auxiliary Integral Identities.- 8.2.2 Vector Analogs of the Pompei Formulas.- 8.3 Stratton-Chu Integral Formulas.- 8.3.1 Stratton-Chu Formulas for a Transient Electromagnetic Field (General Case).- 8.3.2 Stratton-Chu Formulas for a Quasi-Stationary Field.- 8.3.3 Wave Model of the Field.- 8.3.4 Case of a Stationary Field.- 8.3.5 Stratton-Chu Formulas for a Monochromatic Field (General Case).- 8.3.6 Modified Stratton-Chu Formulas for a Monochromatic Field.- 8.3.7 Two-Dimensional Stratton-Chu Formulas.- 8.3.8 Stratton-Chu Formulas as a Cauchy Formula Analog.- 8.4 Stratton-Chu Type Integrals.- 8.4.1 Concept of the Stratton-Chu Type Integral for a Monochromatic Field.- 8.4.2 Properties of the Stratton-Chu Type Integrals.- 8.4.3 Modified Stratton-Chu Type Integrals.- 8.4.4 Matrix Representation.- 8.4.5 Stratton-Chu Type Integrals for a Quasi-Stationary Field.- 8.5 Extension of the Stratton-Chu Formulas to Inhomogeneous Media.- 8.5.1 Green Electromagnetic Tensors and Their Properties.- 8.5.2 Stratton-Chu Formulas for an Inhomogeneous Medium.- 8.5.3 Transition to the Model of a Homogeneous Medium.- 8.5.4 Stratton-Chu Type Integrals in an Inhomogeneous Medium and Their Properties.- 8.6 Integral Transforms of Electromagnetic Fields.- 8.6.1 Integral Boundary Conditions for the Electromagnetic Field on the Boundary of a Homogeneous Domain.- 8.6.2 Integral Boundary Conditions for the Electromagnetic Field on the Boundary of an Inhomogeneous Domain.- 8.6.3 Determination of the Electromagnetic Field from a Specified Discontinuity.- 8.6.4 Inversion Formulas for the Stratton-Chu Type Integrals.- 8.6.5 Stratton-Chu Integral Transforms on a Plane.- 8.7 Techniques for Separation of the Earth's Electromagnetic Fields.- 8.7.1 Separation of the Electromagnetic Field into External and Internal Parts.- 8.7.2 Separation of the Electromagnetic Field into Normal and Anomalous Parts.- 9 Analytical Continuation of the Electromagnetic Field.- 9.1 General Principles.- 9.1.1 Analytical Nature of the Electromagnetic Field.- 9.1.2 Concept of Analytical Continuation of the Electromagnetic Field.- 9.1.3 Equations of Complete Analytical Functions.- 9.1.4 Principle of Continuity for the Electromagnetic Field.- 9.1.5 Electromagnetic Field in the Riemann Space.- 9.2 Analytical Continuation of the Electromagnetic Field into Geoelectrical Inhomogeneities.- 9.2.1 Analytical Continuation of the Stratton-Chu Type Integral Through the Integration Surface.- 9.2.2 Analytical Continuation of the Electromagnetic Field into a Homogeneous Domain Bounded by an Analytical and Piecewise Analytical Surfaces.- 9.3 Techniques for Analytical Continuation of the Electromagnetic Field.- 9.3.1 Forms of Analytical Continuation of the Electromagnetic Field in Geoelectrical Problems.- 9.3.2 Problem Statement.- 9.3.3 Continuation of the Field into a Layer.- 9.3.4 Continuation of a Two-Dimensional Electromagnetic Field.- 10 Migration of the Electromagnetic Field.- 10.1 Definition of the Concept of Migration.- 10.1.1 Definition of a Migration Field.- 10.1.2 System of Migration Transforms of Nonstationary Electromagnetic Fields.- 10.2 Properties of Migration Fields.- 10.2.1 Equation for a Migration Field in Direct Time.- 10.2.2 One-, Two-, and Three-Dimensional Migrations of Electromagnetic Source Fields.- 10.2.3 Extreme Values of Migration Fields.- 10.2.4 Migration of Theoretical and Model Electromagnetic Fields.- IV Kirchhoff-Type Integrals in the Elastic Wave Theory.- 11 Kirchhoff-Type Integrals.- 11.1 Elastic Waves in an Isotropic Medium.- 11.1.1 Stresses and Strains in Elastic Bodies.- 11.1.2 Equations of Motion of a Homogeneous Isotropic Elastic Medium.- 11.1.3 Longitudinal and Transverse Waves in a Homogeneous Isotropic Elastic Medium.- 11.2 Generalized Kirchhoff Integral Formula.- 11.2.1 Green Tensor and Vector Formulas.- 11.2.2 Kirchhoff Integral Formulas.- 11.2.3 Kirchhoff Integral Formulas for a Scalar Wave Field.- 11.2.4 Kirchhoff Integral Formulas in Matrix Notation.- 11.3 Kirchhoff-Type Integrals.- 11.3.1 Kirchhoff-Type Integrals for Wave Fields.- 11.3.2 Kirchhoff-Type Integrals for Elastic Displacement Fields.- 12 Continuation and Migration of Elastic Wave Fields.- 12.1 Analytical Continuation of Elastic Wave Fields.- 12.1.1 Continuation of the Elastic Displacement Field into the Upper and Lower Half-Spaces in a Homogeneous Isotropic Medium.- 12.1.2 Integral Formulas for Continuation of the Elastic Displacement Field into the Lower Half-Space for a Two-Layer Medium.- 12.1.3 Continuation of Elastic Displacement Fields Specified in a Horizontal Plane.- 12.1.4 Elaboration of Regularizing Algorithms for Wave Field Continuation.- 12.2 Migration of Wave Fields on the Basis of Analytical Continuation.- Appendix A Space Analogs of the Cauchy-Type Integrals and the Quaternion Theory.- A.1 Quaternions and Operations Thereon.- A.2 Monogenic Functions.- A.3 Quaternion Notation of Space Analogs of the Cauchy-Type Integral.- Appendix B Green Electromagnetic Functions for Inhomogeneous Media and Their Properties.- B.1 Field Equations.- B.2 Lorentz Lemma for an Inhomogeneous Medium.- B.3 Reciprocal Relations.- B.4 Integral Representations of the Electric and Magnetic Fields.- B.5 Some Formulas and Rules of Operations on Dyadic Tensor Functions.- B.6 Tensor Statements of the Ostrogradsky-Gauss Theorem.- References.