# Problems in Exploration Seismology and their Solutions

Series | Geophysical References Series |
---|---|

Title | Problems in Exploration Seismology and their Solutions |

Author | Lloyd P. Geldart and Robert E. Sheriff |

DOI | http://dx.doi.org/10.1190/1.9781560801733 |

ISBN | ISBN 9781560801153 |

Store | SEG Online Store |

Focusing on the basic theory required to solve practical problems, this book provides 212 problems, and solutions, which cover a wide range of issues, including least-squares methods, choosing velocities for various situations, z-transforms, determining 2D and 3D field geometries, and solving processing and interpretation problems.

This book is designed for students and for geophysicists who need a refresher on the basic theory required to solve practical problems. Geophysical texts often provide problems, but this book is unique in that it provides solutions also. The authors include a summary of the basic theory required to solve each problem. The 212 problems cover a wide range, including least-squares methods, choosing velocities for various situations, z-transforms, determining 2D and 3D field geometries, and solving processing and interpretation problems.

## Contents

- 1 Chapter 1: Introduction
- 2 Chapter 2: Theory of seismic waves
- 3 Chapter 3: Partitioning at an interface
- 4 Chapter 4: Geometry of seismic waves
- 5 Chapter 5: Seismic velocity
- 6 Chapter 6: Characteristics of seismic events
- 7 Chapter 7: Seismic equipment
- 8 Chapter 8: Reflection field methods
- 9 Chapter 9: Data processing
- 10 Chapter 10: Geologic interpretation of reflection data
- 11 Chapter 11: Refraction methods
- 12 Chapter 12: 3D methods
- 13 Chapter 13: Specialized techniques
- 14 Chapter 14: Specialized applications

## Chapter 1: Introduction

## Chapter 2: Theory of seismic waves

- The basic elastic constants
- Interrelationships among elastic constants
- Magnitude of disturbance from a seismic source
- Magnitudes of elastic constants
- General solutions of the wave equation
- Wave equation in cylindrical and spherical coordinates
- Sum of waves of different frequencies and group velocity
- Magnitudes of seismic wave parameters
- Potential functions used to solve wave equations
- Boundary conditions at different types of interfaces
- Boundary conditions in terms of potential functions
- Disturbance produced by a point source
- Far- and near-field effects for a point source
- Rayleigh-wave relationships
- Directional geophone responses to different waves
- Tube-wave relationships
- Relation between nepers and decibels
- Attenuation calculations
- Diffraction from a half-plane

## Chapter 3: Partitioning at an interface

- General form of Snell’s law
- Reflection/refraction at a solid/solid interface and displacement of a free surface
- Reflection/refraction at a liquid/solid interface
- Zoeppritz’s equations for incident SV- and SH-waves
- Reinforcement depth in marine recording
- Complex coefficient of reflection
- Reflection and transmission coefficients
- Amplitude/energy of reflections and multiples
- Reflection/transmission coefficients at small angles and magnitude
- Magnitude
- AVO versus AVA and effect of velocity gradient
- Variation of reflectivity with angle (AVA)

## Chapter 4: Geometry of seismic waves

- Accuracy of normal-moveout calculations
- Dip, cross-dip, and angle of approach
- Relationship for a dipping bed
- Reflector dip in terms of traveltimes squared
- Second approximation for dip moveout
- Calculation of reflector depths and dips
- Plotting raypaths for primary and multiple reflections
- Effect of migration on plotted reflector locations
- Resolution of cross-dip
- Cross-dip
- Variation of reflection point with offset
- Functional fits for velocity-depth data
- Relation between average and rms velocities
- Vertical depth calculations using velocity functions
- Depth and dip calculations using velocity functions
- Weathering corrections and dip/depth calculations
- Using a velocity function linear with depth
- Head waves (refractions) and effect of hidden layer
- Interpretation of sonobuoy data
- Diving waves
- Linear increase in velocity above a refractor
- Time-distance curves for various situations
- Locating the bottom of a borehole
- Two-layer refraction problem

## Chapter 5: Seismic velocity

- Maximum porosity versus depth
- Relation between lithology and seismic velocities
- Porosities, velocities, and densities of rocks
- Velocities in limestone and sandstone
- Dependence of velocity-depth curves on geology
- Effect of burial history on velocity
- Determining lithology from well-velocity surveys
- Reflectivity versus water saturation
- Effect of overpressure
- Effects of weathered layer (LVL) and permafrost
- Horizontal component of head waves
- Stacking velocity versus rms and average velocities
- Quick-look velocity analysis and effects of errors
- Well-velocity survey
- Interval velocities
- Finding velocity
- Effect of timing errors on stacking velocity, depth, and dip
- Estimating lithology from stacking velocity
- Velocity versus depth from sonobuoy data
- Influence of direction on velocity analyses
- Effect of time picks, NMO stretch, and datum choice on stacking velocity

## Chapter 6: Characteristics of seismic events

- Characteristics of different types of events and noise
- Horizontal resolution
- Reflection and refraction laws and Fermat’s principle
- Effect of reflector curvature on a plane wave
- Diffraction traveltime curves
- Amplitude variation with offset for seafloor multiples
- Ghost amplitude and energy
- Directivity of a source plus its ghost
- Directivity of a harmonic source plus ghost
- Differential moveout between primary and multiple
- Suppressing multiples by NMO differences
- Distinguishing horizontal/vertical discontinuities
- Identification of events
- Traveltime curves for various events
- Reflections/diffractions from refractor terminations
- Refractions and refraction multiples
- Destructive and constructive interference for a wedge
- Dependence of resolvable limit on frequency
- Vertical resolution
- Causes of high-frequency losses
- Ricker wavelet relations
- Improvement of signal/noise ratio by stacking

## Chapter 7: Seismic equipment

- Radiolocation errors because of velocity variations
- Effect of station angle on location errors
- Transit satellite navigation
- Effective penetration of profiler sources
- Directivity of linear sources
- Sosie method
- Energy from an air-gun array
- Dominant frequencies of marine sources
- Effect of coil inductance on geophone equation
- Streamer feathering due to cross-currents
- Filtering effect of geophones and amplifiers
- Filter effects on waveshape
- Effect of filtering on event picking
- Binary numbers

## Chapter 8: Reflection field methods

- Effect of too many groups connected to the cable
- Reflection-point smear for dipping reflectors
- Stacking charts
- Attenuation of air waves
- Maximum array length for given apparent velocity
- Response of a linear array
- Directivities of linear arrays and linear sources
- Tapered arrays
- Directivity of marine arrays
- Response of a triangular array
- Noise tests
- Selecting optimum field methods
- Optimizing field layouts
- Determining vibroseis parameters
- Selecting survey parameters
- Effect of signal/noise ratio on event picking
- Interpreting uphole surveys
- Weathering and elevation (near-surface) corrections
- Determining static corrections from first breaks
- Determining reflector location
- Blondeau weathering corrections

## Chapter 9: Data processing

- Fourier series
- Space-domain convolution and vibroseis acquisition
- Fourier transforms of the unit impulse and boxcar
- Extension of the sampling theorem
- Alias filters
- The convolutional model
- Water reverberation filter
- Calculating crosscorrelation and autocorrelation
- Digital calculations
- Semblance
- Convolution and correlation calculations
- Properties of minimum-phase wavelets
- Phase of composite wavelets
- Tuning and waveshape
- Making a wavelet minimum-phase
- Zero-phase filtering of a minimum-phase wavelet
- Deconvolution methods
- Calculation of inverse filters
- Inverse filter to remove ghosting and recursive filtering
- Ghosting as a notch filter
- Autocorrelation
- Wiener (least-squares) inverse filters
- Interpreting stacking velocity
- Effect of local high-velocity body
- Apparent-velocity filtering
- Complex-trace analysis
- Kirchhoff migration
- Using an upward-traveling coordinate system
- Finite-difference migration
- Effect of migration on fault interpretation
- Derivative and integral operators
- Effects of normal-moveout (NMO) removal
- Weighted least-squares

## Chapter 10: Geologic interpretation of reflection data

- Improvement due to amplitude preservation
- Deducing fault geometry from well data
- Structural style
- Faulting
- Mapping faults using a grid of lines
- Fault and stratigraphic interpretation
- Interpretation of salt uplift
- Determining the nature of flow structures
- Mapping irregularly spaced data
- Evidences of thickening and thinning
- Recognition of a reef
- Seismic sequence boundaries
- Unconformities
- Effect of horizontal velocity gradient
- Stratigraphic interpretation
- Interpretation of a depth-migrated section
- Hydrocarbon indicators
- Waveshapes as hydrocarbon accumulation thickens

## Chapter 11: Refraction methods

- Salt lead time as a function of depth
- Effect of assumptions on refraction interpretation
- Effect of a hidden layer
- Proof of the ABC refraction equation
- Adachi’s method
- Refraction interpretation by stripping
- Proof of a generalized reciprocal method relation
- Delay time
- Barry’s delay-time refraction interpretation method
- Parallelism of half-intercept and delay-time curves
- Wyrobek’s refraction interpretation method
- Properties of a coincident-time curve
- Interpretation by the plus-minus method
- Comparison of refraction interpretation methods
- Feasibility of mapping a horizon using head waves
- Refraction blind spot
- Interpreting marine refraction data

## Chapter 12: 3D methods

- Spatial sampling restrictions
- Bin size in marine work
- Effect of crosscurrents
- Number of seismic sources
- Circle shooting
- Ocean-bottom cable surveys
- Vibroseis land survey
- Loop layout for a 3D survey
- Fault interpretation using time slices
- Acquisition direction for marine 3D surveys

## Chapter 13: Specialized techniques

- S-wave conversion in marine surveys
- Equally inclined orthogonal geophones
- Guided (channel) waves and normal-mode propagation
- Vertical seismic profiling
- Effect of velocity change on VSP traveltime
- Mapping the vertical flank of a salt dome
- Poission’s ratio from P- and S-wave traveltimes