VELOCITY FUNCTIONS IN SEISMIC PROSPECTING

Geophysics ◽  
1953 ◽  
Vol 18 (2) ◽  
pp. 289-297 ◽  
Author(s):  
H. Kaufman
Keyword(s):  
Travel Time ◽  
Average Velocity ◽  
Time Function ◽  
Instantaneous Velocity ◽  
Geophysical Prospecting ◽  
Velocity Function ◽  
Mathematical Properties ◽  
Basic Equations ◽  
Vertical Travel ◽  
Quantities Of Interest

An account of the mathematical properties of velocity functions used in the seismic method of geophysical prospecting is presented. Basic equations are given for deriving various quantities of interest associated with a particular velocity function, such as average velocity as a function of depth, average velocity as a function of vertical travel time, parametric equations for displacement, depth, travel time, etc. In Part I the relations are based on a given instantaneous‐velocity—depth function. Part II contains a similar analysis based on a given instantaneous‐velocity—vertical‐time function. The results are incorporated in Tables I and II.

A VELOCITY FUNCTION INCLUDING LITHOLOGIC VARIATION

Geophysics ◽  
1953 ◽  
Vol 18 (2) ◽  
pp. 271-288 ◽  
Author(s):  
L. Y. Faust
Keyword(s):  
Travel Time ◽  
Direct Measurement ◽  
General Knowledge ◽  
Velocity Function ◽  
Geologic Time ◽  
True Resistivity ◽  
Resistivity Log ◽  
Vertical Travel ◽  
Water Resistivity

Assuming velocity (V) a function of depth (Z), geologic time (T), and lithology (L) the resistivity log is an approach to the determination of L. Since general knowledge of water resistivity values [Formula: see text] is lacking, the values of true resistivity [Formula: see text] against [Formula: see text] were compared for 670,000 feet of section widely distributed geographically. Variations in [Formula: see text] were presumably averaged out thereby, and the results indicate that statistically [Formula: see text] and [Formula: see text] This formula was applied to an additional 270,000 feet of section more localized geographically to observe its accuracy in predicting vertical travel time. If a correction map for [Formula: see text] variations is applied the results are encouraging but less accurate than good velocity surveys. Examination of an inconclusively small amount of data with more careful measurements of [Formula: see text] suggests that accuracy comparable to direct measurement may be attainable. The cooperation of other investigators and of the electric‐logging specialists is desired.

Intensities and travel times of seismic rays in a sphere with a linear velocity function

1977 ◽  
Vol 67 (1) ◽  
pp. 33-42
Author(s):  
Mark E. Odegard ◽  
Gerard J. Fryer
Keyword(s):  
Velocity Profile ◽  
Travel Time ◽  
Upper Mantle ◽  
Intensity Ratio ◽  
Computation Time ◽  
Spherical Body ◽  
Travel Times ◽  
Velocity Function ◽  
Intensity Ratios ◽  
Distance Travel

Abstract Equations are presented which permit the calculation of distances, travel times and intensity ratios of seismic rays propagating through a spherical body with concentric layers having velocities which vary linearly with radius. In addition, a method is described which removes the infinite singularities in amplitude generated by second-order discontinuities in the velocity profile. Numerical calculations involving a reasonable upper mantle model show that the standard deviations of the errors for distance, travel time and intensity ratio are 0.0046°, 0.057 sec, and 0.04 dB, respectively. Computation time is short.

Simulated Estimation of Davidson's travel time function

1992 ◽  
Vol 16 (4) ◽  
pp. 251-259 ◽  
Author(s):  
Geoffrey Rose ◽  
Mark Raymond
Keyword(s):  
Travel Time ◽  
Time Function

scholarly journals INVESTIGATIONS INTO VERTICAL TRAVEL TIME OF THE FRONT OF BUOYANT FIRE PLUMES

1997 ◽  
Vol 62 (502) ◽  
pp. 1-8 ◽  
Author(s):  
Takashi FUJITA ◽  
Jun-ichi YAMAGUCHI ◽  
Takeyoshi TANAKA ◽  
Takao WAKAMATSU
Keyword(s):  
Travel Time ◽  
Fire Plumes ◽  
Vertical Travel

Discrete closed one-particle chain of contours

2021 ◽  
pp. 114-125
Author(s):  
P. A. Myshkis ◽  
◽  
A. G. Tatashev ◽  
M. V. Yashina ◽  
◽  
...  
Keyword(s):  
Limit Cycle ◽  
Average Velocity ◽  
Discrete Dynamical System ◽  
Time Function ◽  
Particle Chain ◽  
A Value ◽  
Closed Chain ◽  
A Cell ◽  
Left And Right ◽  
Increasing Function

A discrete dynamical system called a closed chain of contours is considered. This system belongs to the class of the contour networks introduced by A. P. Buslaev. The closed chain contains N contours. There are 2m cells and a particle at each contour. There are two points on any contour called a node such that each of these points is common for this contour and one of two adjacent contours located on the left and right. The nodes divide each contour into equal parts. At any time t = 0,1, 2,... any particle moves onto a cell forward in the prescribed direction. If two particles simultaneously try to cross the same node, then only the particle of the left contour moves. The time function is introduced, that is equal to 0 or 1. This function is called the potential delay of the particle. For t ≥ m, the equality of this function to 1 implies that the time before the delay of the particle is not greater than m. The sum of all particles potential delays is called the potential of delays. From a certain moment, the states of the system are periodically repeated (limit cycles). Suppose the number of transitions of a particle on the limit cycle is equal to S(T) and the period is equal to T. The ratio S(T) to T is called the average velocity of the particle. The following theorem have been proved. 1) The delay potential is a non-increasing function of time, and the delay potential does not change in any limit cycle, and the value of the delay potential is equal to a non-negative integer and does not exceed 2N/3. 2) If the average velocity of particles is less than 1 for a limit cycle, then the period of the cycle (this period may not be minimal) is equal to (m + 1)N. 3) The average velocity of particles is equal to v = 1 - H/((m + 1)N), where H is the potential of delays on the limit cycle. 4) For any m, there exists a value N such that there exists a limit cycle with H > 0 and, therefore, v < 1.

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