A technique for stabilizing interval velocities from the Dix equation

Geophysics ◽  
1988 ◽  
Vol 53 (9) ◽  
pp. 1241-1243 ◽  
Author(s):  
John B. Dubose
Keyword(s):  
Root Mean Square ◽  
Mean Square ◽  
Average Interval ◽  
Interval Velocity ◽  
The Earth ◽  
Interval Velocities

Interval velocities, the velocities at which sounds travel in the earth, can be computed from stacking or root‐mean‐square (rms) velocities by applying the Dix equation (Dix, 1955): [Formula: see text] where [Formula: see text] are the stacking velocity picks, [Formula: see text] are the associated times, and [Formula: see text] is the average interval velocity between [Formula: see text] and [Formula: see text].

Generalized form of the Dix equation for the calculation of interval velocities and layer thicknesses

Geophysics ◽  
1989 ◽  
Vol 54 (5) ◽  
pp. 659-661 ◽  
Author(s):  
Ali A. Nowroozi
Keyword(s):  
Root Mean Square ◽  
Seismic Data ◽  
Approximate Equation ◽  
Mean Square ◽  
Interval Velocity ◽  
Reflection Seismic ◽  
Interval Velocities ◽  
Horizontal Layers

Over three decades ago, Dix (1955) derived an approximate equation for the determination of interval velocity from observed reflection seismic data. Assuming a stack of m horizontal layers, with interval velocities [Formula: see text], layer thicknesses [Formula: see text], j = 1, m, and near‐vertical raypaths, Dix (1955) showed that [Formula: see text]where [Formula: see text] and [Formula: see text] are the two‐way vertical times and [Formula: see text] and [Formula: see text] are the root‐mean‐square (rms) velocities to interfaces j + 1 and j, respectively.

Travel-time errors in the laterally inhomogeneous earth

1971 ◽  
Vol 61 (6) ◽  
pp. 1639-1654 ◽  
Author(s):  
Cinna Lomnitz
Keyword(s):  
Travel Time ◽  
Root Mean Square ◽  
Time Estimation ◽  
Systematic Errors ◽  
Least Square ◽  
Travel Time Estimation ◽  
Mean Square ◽  
Travel Times ◽  
The Earth ◽  
Lateral Inhomogeneity

abstract Travel times from earthquakes or explosions contain both positive and negative systematic errors. Positive skews in travel-time residuals due to epicenter mislocation, and negative skews due to lateral inhomogeneity in the Earth, are analyzed. Methods for travel-time estimation are critically reviewed. Recent travel-time tables, including the J-B tables, are within the range of root-mean-square travel-time fluctuations; the J-B tables are systematically late but cannot be reliably improved by least-square methods. Effects of lateral inhomogeneity at teleseismic distances can be estimated by chronoidal methods independently of standard tables, but the available explosion data are insufficiently well-distributed in azimuth and distance for this purpose.

scholarly journals HYBRID INVERSION OF INTERVAL VELOCITIES IN MULTISCALE APPROACH

2017 ◽  
Vol 35 (4) ◽  
pp. 237
Author(s):  
Rodrigo de S. Santos ◽  
Milton J. Porsani
Keyword(s):  
Simulated Annealing ◽  
Hybrid Algorithm ◽  
Seismic Waves ◽  
Optimization Method ◽  
Multiscale Approach ◽  
Velocity Inversion ◽  
Interval Velocity ◽  
The Earth ◽  
Very Fast Simulated Annealing ◽  
Interval Velocities

ABSTRACT. The understanding of the interior of the planet by using the seismic method of reflection requires knowledge of the velocities with which the seismic waves propagate in the subsurface of the Earth. This work presents strategies to obtain the velocity intervals using RMS velocity inversion. Using a hybrid algorithm that combines the Very Fast Simulated Annealing (VFSA) global optimization method and the Fletcher-Reeves local search method, we have sought to reduce the dependence between the accuracy of the results and the model by which the optimization process begins. The main innovative contribution of this study was the development and presentation of the named inversion strategy of multiscale parameters. This technique allows the use of the VFSA method in inversion problems in which the number of variables is significantly large. The hybrid algorithm with multiscale approach was used to solve 1D and 2D problems, estimating models with high degrees of accuracy, which allowed to confirm the efficiency of the proposed method.Keywords: inversion, parameter multiscale, interval velocity, Very Fast Simulated Annealing, Fletcher-Reeves, hybrid.RESUMO. O entendimento do interior do planeta por meio do método sísmico de reflexão requer o conhecimento das velocidades com que as ondas sísmicas se propagam na subsuperfície da Terra. Este trabalho apresenta estratégias para a obtenção das velocidade intervalares por uso inversão de velocidades RMS. Utilizando um algoritmo híbrido, que combina o método de otimização global Very Fast Simulated Annealing (VFSA), e o método de busca local Fletcher-Reeves, buscou-se reduzir a dependência entre a acurácia dos resultados e o modelo pelo qual o processo de otimização se inicia. A principal contribuição inovadora deste estudo foi o desenvolvimento e apresentação da estratégia de inversão nomeada de multiescala de parâmetros. Esta técnica possibilita o uso do método VFSA em problemas de inversão em que o número de variáveis é significativamente grande. O algoritmo híbrido com abordagem multiescala foi usado para solucionar problemas 1D e 2D, estimando modelos com elevado grau de acurácia, o que permitiu confirmar a eficiência do método proposto.Palavras-chave: inversão, multiescala de parâmetros, velocidade intervalar, Very Fast Simulated Annealing, Fletcher-Reeves, híbrido.

A limitation of well velocity surveys in highly deviated wells drilled parallel to bedding

Geophysics ◽  
1996 ◽  
Vol 61 (3) ◽  
pp. 627-630
Author(s):  
Jesper M. Smidt
Keyword(s):  
Simple Formula ◽  
Seismic Energy ◽  
Depth Interval ◽  
Average Interval ◽  
Interval Velocity ◽  
Layer Interface ◽  
The Difference ◽  
Interval Velocities ◽  
Deviated Wells

Well velocity surveys (or check‐shot surveys) in vertical or moderately deviated wells provide average velocities and formation interval velocities by assuming a horizontally layered subsurface. The interval velocity, [Formula: see text], is calculated using the simple formula, [Formula: see text]where DZ is the vertical depth interval between two consecutive well geophone locations, [Formula: see text] and [Formula: see text], as shown in Figure 1a, and DT, the difference in the seismic traveltime from a source S at the wellhead to these two locations (Telford et al., 1976, 347). This quantity is the average interval velocity of Al‐Chalabi (1974) (see also Sheriff, 1991, 317). Vertical incidence of the seismic energy at the geophone is assumed, i.e., the source is located vertically above the geophone location. This assumption is made throughout this note; it would only serve to confuse the issue to correct for the usually small horizontal offset between source and receiver locations. I also assume throughout this note that no refraction takes place at any layer interface. Since only two‐layer cases are dealt with, this assumption is hardly a serious limitation.

Maximum uncertainty of interval velocity estimates

Geophysics ◽  
1981 ◽  
Vol 46 (11) ◽  
pp. 1543-1547 ◽  
Author(s):  
Z. Hajnal ◽  
I. T. Sereda
Keyword(s):  
Numerical Experiments ◽  
Normal Incidence ◽  
Mean Square ◽  
Order Error ◽  
First Order ◽  
Interval Velocity ◽  
Number Of Factors ◽  
Time Variables ◽  
Interval Velocities ◽  
Time Information

The Dix equation (Dix, 1955) is commonly used to estimate interval velocities from stacking velocity and travel‐time information. The errors in these estimates can result from a number of factors, including indiscriminate substitution of stacking velocities for root‐mean‐square (rms] velocities without compensating for the effects of spread‐length or dipping reflectors. A first‐order error equation has been developed which estimates the maximum uncertainty for Dix‐derived interval velocities when the accuracy of the input rms velocity and normal incidence time information is considered. Some simple numerical experiments using this equation indicate that the uncertainty in the calculated interval velocity increases with depth and is inversely proportional to layer thickness, even when the errors in the input velocity and time variables remain constant.

HYBRID INVERSION OF INTERVAL VELOCITIES IN MULTISCALE APPROACH

2018 ◽  
Vol 35 (4) ◽  
pp. 237
Author(s):  
Rodrigo De Santana Santos ◽  
Milton José Porsani
Keyword(s):  
Simulated Annealing ◽  
Hybrid Algorithm ◽  
Seismic Waves ◽  
Optimization Method ◽  
Multiscale Approach ◽  
Velocity Inversion ◽  
Interval Velocity ◽  
The Earth ◽  
Very Fast Simulated Annealing ◽  
Interval Velocities

RESUMO. O entendimento do interior do planeta por meio do método sísmico de reflexão, requer o conhecimento das velocidades com que as ondas sísmicas se propagam na subsuperfície da terra. Este trabalho apresenta estratégias para a obtenção das velocidades intervalares por uso inversão de velocidades RMS. Utilizando um algoritmo híbrido, que combina o poderoso método de otimização global Very Fast Simulated Annealing, e o método de busca local Fletcher-Revees, nós buscamos reduzir a dependência entre a acuracia dos resultados e o modelo pelo qual o processo de otimização se inicia. A principal contribuição inovadora deste estudo foi o desenvolvimento e apresentação da estratégia de inversão nomeada de multiescala de parâmetros. Esta técnica possibilita o uso do método VFSA em problemas de inversão em que o número de variáveis e signicativamente grande. O algoritmo hbrido com abordagem multiescala, foi usado para solucionar problemas 1D e 2D, estimando modelos com elevado grau de acurácia, o que permitiu confirmar a eci^encia do método proposto. Palavras-chave: inversão, multiescala de parâmetros, velocidade intervalar, Very Fast Simulated Annealing, Fletcher-Revees, híbrido. ABSTRACT. The understanding of the interior of the planet by using the seismic method of reflection requires knowledge of the velocities with which the seismic waves propagate in the subsurface of the earth. This work presents strategies to obtain the velocity intervals using RMS velocity inversion. Using a hybrid algorithm that combines the powerful Very Fast Simulated Annealing global optimization method and the Fletcher-Revees local search method, we have sought to reduce the dependence between the accuracy of the results and the model by which the optimization process begins. The main innovative contribution of this study was the development and presentation of the named inversion strategy of multiscale parameters. This technique allows the use of the VFSA method in inversion problems in which the number of variables is significantly large. The hybrid algorithm with multiscale approach was used to solve 1D and 2D problems, estimating models with high degrees of accuracy, which allowed to confirm the efficiency of the proposed method Keywords: inversion, parameter multiscale, interval velocity, Very Fast Simulated Annealing, Fletcher-Revees, hybrid.

The Relationship of Root Mean Square Value of Electromyography and Isometric Torque of Quadriceps in Normal Subjects

2016 ◽  
Vol 26 (1) ◽  
pp. 58
Author(s):  
Qiurong XIE ◽  
Zheng JIANG ◽  
Qinglu LUO ◽  
Jie LIANG ◽  
Xiaoling WANG ◽  
...  
Keyword(s):  
Root Mean Square ◽  
Normal Subjects ◽  
Mean Square ◽  
Isometric Torque ◽  
Relationship Of ◽  
The Relationship
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