quality factor inversion applied to 3d seismic data in

Quality factor inversion applied to 3D

seismic data in Colombia

Jorge Luis Ñustes Andrade

201424022

Department of Geosciences

Universidad de los Andes

This dissertation is submitted for the degree of

BSc in Geosciences

December 2017

For my loving parents.

Declaration

I declare that except where specific reference is made to the work of others, this thesis is an

original report of my research and has not been submitted for any previous degree in this, or

any other university.

Jorge Luis Ñustes Andrade

201424022

December 2017

Acknowledgements

I would like to acknowledge Dr Jean Baptiste Tary for helping me through all the process of

my research, and giving me important reviews of my thesis. I would also like to acknowledge

HOCOL S.A and especially to MSc Jaime Checa for believing on my research, and giving

me the seismic data used in this work. Special thanks to Halliburton for allowing me to

use their ProMax seismic proccessing software. I wish to thank MSc Camilo Gonzalez from

Halliburton for his invaluable help with the preprocessing of the data and for writing a

Matlab code to split the segy files. Finally, thanks to Dr Carl Reine for helping me with all the

questions I had through my investigation, he is the real inspiration behind this work.

Abstract

Seismic attenuation, usually quantified by the dimensionless parameter Q, contains valu-

able information of the petrophysical parameters of the subsurface, and presents a valuable

tool for reservoir characterization. Yet, due to the number of issues that must be addressed

to obtain reliable measurements, it is rarely calculated.

In this work I present a robust method to calculate attenuation on the τ−p domain from

prestack CMP gathers. The PSQI method (Reine, 2009), uses a variable-window time-

frequency transform, a simultaneous inversion scheme, and a τ− p domain transform

to get an accurate attenuation measurement in the form of 1/Q.

To test the robustness of the method, I apply the PSQI to a 3D dataset from Colombia.

Before doing the inversion, I describe a step by step preproccesing that must be done to

improve the attenuation measurements, and reduce the uncertainty of the inversion. The

final result proves the reliability of the PSQI method.

Abstract

La atenuación sísmica, generalmente cuantificada por medio del parámetro adimensional

Q, contiene información valiosa de los parámetros petrofísicos del subsuelo y representa

una herramienta valiosa para la caracterización de reservorios. Sin embargo, debido a la

cantidad de problemas que se deben considerar para obtener mediaciones confiables, rara

vez se calcula.

En este trabajo presento un método robusto para calcular la atenuación en el dominio

τ− p a partir de un arreglo CMP. El método PSQI (Reine, 2009), usa una transformada

tiempo-frequencia de ventana variable, un esquema de inversión simultánea, y una trans-

formación al dominio τ−p para obtener una mediación de atenuación precisa en la forma

de 1/Q.

Para probar la solidez del método, apliqué el procedimiento PSQI a un conjunto de datos

sísmicos 3D en Colombia. Antes de llevar a acabo la inversión, describo el paso a paso del

preprocesamiento que se debe realizar para mejorar las mediciones de atenuación y reducir

la incertidumbre de la inversión. El resultado final demuestra la fiabilidad del método PSQI.

Contents

List of Figures xvi

List of Tables xix

1 Introduction 1

1.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Seismic attenuation 4

2.1 Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.1 Attenuation and Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.2 Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.3 Effective Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Types of Attenuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Intrinsic Attenuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.2 Apparent Attenuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Methods to Calculate Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3.1 Time domain methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3.2 Frequency domain methods . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.4 The Spectral Ratio Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.5 Surface Seismic Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.6 Pre-stack Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Spectral Analysis 12

3.1 Non stationary signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2 Heisenberg-Gabor Uncertainty Principle . . . . . . . . . . . . . . . . . . . . . . 13

3.3 Time-frequency Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3.1 Short-time Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.3.2 Gabor transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.3.3 S transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

xiv Contents

3.4 Synthetic examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.4.1 Analysis of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.4.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4 Inversion Scheme 22

4.1 Simultaneous inversion scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.2 Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.2.1 Weighting functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.2.2 Weighted inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5 τ−p Domain 28

5.1 The τ−p transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5.1.2 Artifacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.1.3 Moveout and anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

6 Multimensional Interpolation 33

6.1 Seismic data interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

6.2 MWNI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

6.3 POSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

6.3.1 5D Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

7 The Prestack Q-Inversion Method 36

7.1 Preproccesing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

7.1.1 Static Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

7.1.2 Velocity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

7.1.3 Band-Pass Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

7.1.4 Amplitude Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

7.1.5 5D Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

7.1.6 τ−p Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

7.2 Calculate Moveout Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

7.2.1 Interval Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

7.2.2 Equivalent Traveltime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

7.3 Spectral Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

7.3.1 Data Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

7.3.2 Spectral information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

7.4 Inversion for 1/Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Contents xv

8 3D Data Example 44

8.1 Background Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

8.1.1 Geological Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

8.1.2 Survey Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

8.2 PSQI Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

8.2.1 Preprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

8.2.2 Boundary of Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

8.2.3 Calculate Moveout Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

8.2.4 Spectral Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

8.2.5 Inversion for 1/Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

8.3 PSQI Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

8.3.1 Weighted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

8.3.2 Unweighted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

9 Conclusions 60

9.1 Recommended future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Bibliography 63

Appendix A Other methods to measure Q 68

A.1 Risetime Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Appendix B Multimensional algorithms 69

B.1 MWNI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

List of Figures

2.1 Classification scheme for common attenuation processes.From (Liner, 2012). 6

2.2 Schematic diagram showing the natural log spectral ratio data as a function of

frequency, from (Reine, 2009). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.1 Sampling of the time-frequency plane. Different forms of sampling: Shannon,

Fourier, Gabor, Wavelet , modified from (Flandrin, 1998). . . . . . . . . . . . . 15

3.2 The Short time Fourier transform, from (Gröchenig, 2013). . . . . . . . . . . . 16

3.3 Impulse signal (a) with the time-frequency representations of STFT (b) and

S-transform (c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.4 Sine signal (a) with the time-frequency representations of STFT (b) and S-

transform (c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.5 Chirp signal (a) with the time-frequency representations of STFT (b) and S-

transform (c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.1 Schematic drawing of the linear relationship of the natural log spectral ratio

data for all traces. The blue osculations represent some random noise. . . . . 23

4.2 Another visualization of equation 4.2 but in the ∆tω space. Note that attenua-

tion causes the surface to descend as one coordinate increases relative to the

other. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.3 Solution to 4.6 with a variable intercept. Shows undulations in the data along

the ∆t coordinate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5.1 Mapping of hyperbolic reflections in the t −x domain shown as ellipses in the

τ−p domain, and linear events like refractions in the t −x domain shown as

points in the τ−p domain. The vertical axis of the right figure is the intercept

time τ. From (Reine, 2009). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

7.1 Flow diagram of the PSQI process. Modified from (Reine, 2009). . . . . . . . . 37

List of Figures xvii

7.2 The horizontal slowness of the data must be truncated at the slowness of

refraction or maximum offset for the lowest interface. Here, the truncation

shown in gray occurs at the refraction of the red horizon. From (Reine, 2009). 42

8.1 Location map of the Sinú-San Jacinto Basin. From (Sánchez and Permanyer,

2006). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

8.2 Foldmap of the 3D survey. The black-dotted rectangle shows the subset where

the inversion was done. See that most of the fold values are consistent within

the subset, averaging a fold of 30. . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

8.3 Stratigraphic column of the Sinú-San Jacinto Basin. From (Barrero et al., 2007). 46

8.4 Raw CMP with static corrections applied. Look at the high-amplitude surface

waves indicated in red. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

8.5 The average spectrum for the entire CMP (green), the surface waves (red), and

the reflections (blue). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

8.6 Band-pass filtered CMP gather. The surface waves, shown by the red triangle,

were effectively removed. The high frequency noise present on various traces

has also been removed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

8.7 The CMP gather from figure 8.4 after the multidimensional interpolation has

been applied. Noisy traces were removed, and anomalous amplitudes were

corrected. All this processes are necessary to reduce the artifacts during the

τ−p transform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

8.8 (a) shows the offset distribution of a raw CMP gather and (b) shows the in-

terpolated offset distribution of that same CMP. Offset spacing in the original

data is regularized after the 5D interpolation. Moreover, offsets lower than 600

m (Including offset 0 m) that were not present in the raw data are created with

the interpolation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

8.9 τ−p transform of the raw data. The horizontal slowness values are in ms/m 51

8.10 τ−p transform of band-pass filtered data. The horizontal slowness values are

in ms/m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

8.11 τ−p transform of the interpolated data. The horizontal slowness values are in

ms/m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

8.12 Pre-stack time migration amplitude map. The circled zone indicates the area

of interest where high amplitude values are observed. The large amplitudes

are often associated with gas. The two horizons used for the inversion are

labeled as A and B. Courtesy of Hocol SA & PetroSeis LTDA . . . . . . . . . . . 53

8.13 Two-way travel times for (a) horizon A, and (b) horizon B. Time is in seconds. 53

xviii List of Figures

8.14 Stacking velocity profile (a), and its interval stacking velocity profile (b) calcu-

lated using Dix’s equation. Values are shown in m/s. . . . . . . . . . . . . . . . 54

8.15 CMP gather in the τ−p domain as (a) amplitude data, and (b) instantaneous

amplitude data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

8.16 Maximum horizontal slowness trace for horizon B . . . . . . . . . . . . . . . . 55

8.17 From left to right, the stacking velocity, interval stacking velocty, and critical

angle for each time sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

8.18 Natural log spectral ratio surface between horizons A and B. . . . . . . . . . . 56

8.19 The weighted PSQI map of 1/Q between horizons A and B is shown on (a). The

frequency bandwidth used is 15 Hz to 75 Hz, and the colorbar is clipped below

0. The uncertainty map of the weighted inversion between horizons A and B is

given in (b). (c) shows the unweighted PSQI map of 1/Q between horizons A

and B with the same frequency bandwidth used in the weighed inversion. The

uncertainty map of the unweighted solution is shown in figure (d). . . . . . . . 58

8.20 The filtered PSQI calculations of weighted 1/Q for the interval between hori-

zons A and B is shown in (a). likewise, the filtered PSQI calculations of the

unweighted 1/Q solution is shown in (b). The 1/Q color bar is clipped below

zero for both results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

List of Tables

2.1 Spectral amplitude ratios for determination of attenuation, modified from

(Bath, 1974) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Chapter 1

Introduction

Reliable measurements of seismic attenuation, usually presented as 1/Q, are desirable

for improving resolution of signal and reservoir characterization. As higher frequencies

attenuate faster than lower frequencies, dominant signal wavelength and period lowers

as the seismic wave propagates. Attenuation can be measured either in the time domain

(amplitude decrease) and the frequency domain (changes in frequency content). In both

cases, it is accompanied with a decrease in resolution (Zhang and Ulrych, 2002).

The attenuation coefficient is linked closely to petrophysical parameters that can be used in

reservoir characterization. For example, Winkler and Nur (1982) established that attenuation

may serve as an indicator of permeability, mobility of fluids, and fluids saturation. Depend-

ing on the properties of the fluid, such as viscosity, compressibility, and on the properties

of the pores, like porosity and permeability, amplitude losses due to attenuation may be

affected in different ways. For hydrocarbon exploration, it has been shown that attenuation

is highly dependent on the gas content in saturated rocks (White, 1975). Furthermore, at-

tenuation is sensitive to fractures and mobility of fluids, and can be used to monitor fluid

dynamics in a reservoir (Macride and Kanasewich, 1987).

Knowledge of quality factor is very desirable, yet due to the number of issues that must be

addressed to obtain reliable measurements, it is rarely calculated. Post-stacked is appealing

for Q factor calculation because of the optimum Signal to Noise ratio and the apparent zero-

offset. However, in post-stack data, each of the individual stacked traces have a different

ray-path for every case except for zero offset (Dasgupta and Clark, 1998). Additionally, for a

given reflection event, raypath geometries, spectral distortions, and angle dependent effects,

corrupt the measurement of attenuation and make estimation of Q from surface seismic

stacked data potentially erroneous.

2 Introduction

One method was proposed by Dasgupta and Clark (1998) to improve the measurements

of attenuation using surface pre-stack seismic data. The Q versus offset method (QVO),

exhibits real enhancements in comparison to other pre-stack and post-stacked common

methods used to calculate attenuation. Moreover, other studies have shown the applications

of QVO to determine the azimuthal discrimination of fractures and time-lapse changes in

anisotropic media (Clark et al., 2009). Despite the enhancements, attenuation measure-

ments obtained by the QVO method are still corrupted by the effects of spectral interference,

directivity and, anisotropy, making this technique less reliable for a robust calculation of the

attenuation factor.

A novel method to calculate attenuation from surface pre-stacked data was introduced

by Reine et al. (2012a) and proved to be better than other previous methods like QVO.

The pre-stack Q inversion (PSQI), uses a variable-window time frequency transform and

a scheme to invert for 1/Q from natural log spectral-ratio data, to overcome the effects

induced by spectral interference. Additionally, authors take the data to the τ−p domain

to reduce the angle-dependent effects like directivity, anisotropy, and raypath differences.

Reine et al. (2012a)

Reine et al. (2012a), shown that the combination of these techniques provides accurate

calculation of the quality factor by combining Q estimates using synthetic data and real 3D

data with VSP measurements. Results show the robustness of this new method to address

the main complications to calculate a precise value for the Q factor.

1.1 Objective

The main purpose of this research is to apply the PSQI method on 3D seismic data from

Colombia. To do that, a series of processing steps must be followed to get the precise value of

quality factor from pre-stack CMP surface seismic data. Considering that one of the reasons

of calculating attenuation, in the form of 1/Q, is to predict the properties of subsurface,

there are three major components that must be executed to achieve the expected results:

1. Transform common midpoint (CMP) data into the τ−p domain.

1.1 Objective 3

2. Calculate time-frequency transforms of the data using a variable-window time-frequency

transform.

3. Perform simultaneous inversion of the natural log spectral ratio data with respect to

frequency and time difference to get attenuation in the form of 1/Q.

In addition to the application of the PSQI method to seismic data, this research also includes

the application of pre-proccessing techniques required to obtain reliable measurements of

the attenuation factor. Furthermore, this thesis aims to provide significant results that can

be use to characterize the potential of a hydrocarbon reservoir in the area of study.

Chapter 2

Seismic attenuation

2.1 Q

2.1.1 Attenuation and Q

Q stands for quality factor. It is a dimensionless parameter that measures how much me-

chanical energy is converted to heat and fluid flow as a seismic wave propagates away

from its source (Costain and Çoruh, 2004; Müller et al., 2010). When the restoring force is

proportional to the amplitude of vibration and the dissipative force is proportional to the

velocity, Q can be defined as:1

Q(ω)=− ∆E

2πE(2.1)

Where E is the elastic energy stored in the specimen when stress is maximum, and −∆E is

the energy dissipated in a specimen through a stress cycle (Aki and Richards, 2002).

In a medium with linear stress-strain relation, the measured amplitude of a wave A is pro-

portional to E12 . By replacing the energy terms in equation (2.1) and assuming that Q ≫ 1,

the relationship can be expressed in a way that we can obtain the amplitude fluctuations

due to attenuation. Hence:1

Q(ω)=− 1

π

∆A

A(2.2)

In order to observe the spatial decay of A we will assume that the direction of propagation

and maximum attenuation will occur along the x axis. Then, we can write (2.2) as:

d A

d x= ∆A

λ(2.3)

2.1 Q 5

Where λ is the wavelength given in terms of ω and phase velocity c by:

λ= 2πc

ω(2.4)

Replacing (2.4) on (2.3) we ended up with the exponential solution of A(x):

A(x) = A0e− ωx2cQ (2.5)

The exponential term in (2.5) is known as the attenuation coefficient α, a quantity which

measures energy absorption, and is related to 1Q by:

α(ω) = ω

2c(ω)Q(ω)(2.6)

Several attenuation measurements have been done in the laboratory over a wide range of

frequencies (Knopoff, 1964; Toksöz et al., 1979). The results showed that there exists a linear

relationship between attenuation coefficient and frequency. This leads to a quality factor Q

that is independent of frequency. Equation (2.6) can be simplified to:

α= ω

2cQ(2.7)

Hence (2.5) is simply:

A(x) = A0e−αx (2.8)

2.1.2 Dispersion

Body wave dispersion can be defined as the dependence of body wave velocity on frequency

(Costain and Çoruh, 2004). This dependence causes each frequency component ω to travel

at a different phase velocity c. As the higher frequencies attenuate faster than the lower

frequencies, the wavelet changes its shape. This change in shape must be such that the

wavelet remains causal. It has been proved that for a linear attenuation theory the existence

of absorption implies some dispersion (Aki and Richards, 2002). Moreover, experiments done

by Futterman (1962) proved that phase shift of the wave is a direct cause of the dispersion

that guarantees a causal arrival of the signal. Depending on the attenuation model used, the

dispersion equation varies. Futterman (1962) introduced a dispersion relation given by:

k = ω

c0

[1− 1

πQln

(ω

ω0

)](2.9)

Where c0 is the phase velocity at a reference frequency ω0.

6 Seismic attenuation

2.1.3 Effective Q

In order to compute attenuation from real seismic data, effects like source spectrum, geomet-

rical spreading, anisotropy, scattering, and noise must be separated in advance (Tonn, 1991).

However some of them are difficult to handle, and often they cause a distorted measurement

of Q. This corrupted measure is known as the effective Qe , and can be defined as the sum of:

1

Qe= 1

Qi+ 1

Qa(2.10)

Where 1Qi

is the intrinsic component and 1Qa

is the apparent attenuation (Spencer et al.,

1982).

2.2 Types of Attenuation

Attenuation can be caused by a variety of physical phenomena that can be divided broadly

into elastic and inelastic processes. Figure 2.1 shows a scheme with the most common

causes of attenuation.

Figure 2.1 Classification scheme for common attenuation processes.From (Liner, 2012).

2.2.1 Intrinsic Attenuation

Intrinsic or inelastic attenuation (1/Qi ) is the energy loss in waves from the conversion of

the wave’s energy into heat and fluid flow (Raji and Rietbrock, 2013). It is mainly caused by

the presence of fluids in the pore space of rocks by a mechanism known as wave-induced

fluid flow (Müller et al., 2010). As a compressional wave propagates through a porous media,

a fluid pressure gradient occurs between the peaks and troughs of the wave, causing the

fluid to move from the peaks to the troughs until reaching an equilibrium state (Pride, 2005).

2.3 Methods to Calculate Q 7

This macroscopic flow is the main source of intrinsic attenuation described by the equations

of porous media acoustics introduced by Biot (1956a,b).

2.2.2 Apparent Attenuation

Contrary to intrinsic attenuation, apparent or elastic attenuation does not involve any lost

of energy to the wave. It is caused by the layered structure of the earth, and mainly involves

a redistribution of the energy by processes such as reflection, transmission, multiples,

scattering, and mode conversion (Liner, 2012).

2.3 Methods to Calculate Q

Tonn (1991) purposed 10 different methods for the computation of the quality factor Q.

These methods are separated into those in the time domain, and those in the frequency

domain. In this section I will briefly explain some of those methods, and then make a detail

explanation of the Spectral Ratio method which is the one used in the PSQI calculations.

2.3.1 Time domain methods

One of the simplest methods to compute Q in the time domain is the Amplitude decay

method. This method is based on the decay of amplitudes between two different distances

or times. For this method Q is defined as:

Q = ω∆x

2c

(ln

(A(x1)

A(x2)

))−1

(2.11)

Where f = 2πω is the dominant frequency and c is the phase velocity (Tonn, 1991). One of

the limitations of this method is that it requires true amplitude recordings, usually difficult

to obtain on exploration seismic data. Other methods like the rise-time, analytical signal,

and pulse-amplitude also demand real amplitude measurements (REFERENCE) (See further

information of these methods on Appendix A).

2.3.2 Frequency domain methods

The matching technique is a method to calculate Q based on the transfer function computed

on the frequency domain. It was introduced by Raikes and White (1984) to determine the

operator that transforms the down going pulse recorded at one level into that recorded at a

deeper level. The matching of a signal at depth 1 to a signal at depth 2 results on the transfer


function H12(ω) and the inverse transfer function H21(ω). Tonn (1991) defines the ratio of

the transfer function as:

ln

(H21(ω)

H12(ω)

)= k −mω (2.12)

Where k is a constant and m is the slope:

m = ∆t

Q(2.13)

Note that this method yields the same result as the spectral ratio method shown on equation

2.18, apart for a factor of 2. However, the PSQI method uses the spectral ratio method not just

because it is one of the best known methods for computation of Q, but due to its simplicity

and flexibility in how it is parametrized (Reine, 2009).

2.4 The Spectral Ratio Method

As described in section 2.3, spectral analysis is the most used technique to calculate seismic

wave attenuation. We can write amplitude of a recorded seismic wave as:

|A(ω,r )| = |S(ω)||D(θ)||G(r )||P (ω)||I (ω)| (2.14)

Where S(ω) corresponds to the source spectrum, D(θ) is the source space function (that

depends on the direction θ from the source), G(r ) stands for the propagation effects (ge-

ometrical spreading, attenuation,and dispersion), P (ω) is the effect of the interfaces on

the spectrum and I (ω) is the instrument response. The spectral ratio method estimates

attenuation by comparing the amplitude spectra of two seismic arrivals. Bath (1974) defined

3 spectral equalization techniques to calculate attenuation from spectral amplitude ratios.

Method Number offrequencies compared

Number ofstations compared

Number ofwaves compared

Frequency ratio two or more one oneStation ratio one at a time two or more oneWave ratio one at a time one two or more

Table 2.1 Spectral amplitude ratios for determination of attenuation, modified from (Bath,1974)

2.4 The Spectral Ratio Method 9

For the PSQI method we will use the Frequency ratio method as the preferred spectral

equalization technique. Dasios et al. (2001) defines three conditions that must be set before

applying the Spectral Ratio Method:

1. Q must be independent of frequency.

2. The medium must be assumed to be non-dispersive.

3. Velocity is considered to be constant within the frequency range examined.

As stated in equation 2.8, the amplitude spectrum A(ω) of a spherical wave can be expressed

as an exponential decay process. However, 2.14 states that instrument response, geometrical

spreading, reflection-transmission interaction and directivity, must be considered for the

amplitude analysis. By now it will be only considered the amplitude loses due to geometric

spreading G and energy partitioning P . For an initial source A0(ω), the spectrum at a given

reflection will be:

A(ω) = (PG)A0(ω)e− ωt

2Q (2.15)

Where t is the travel time for that event and Q is the average quality factor. To measure 1/Q

between two reflection events, the ratio of the two spectra A1(ω) and A2(ω) may be taken,

eliminating the need to know the source spectra (Tonn, 1991). Then applying the natural

logarithm to the spectral ratio we get the equation:

ln

(A2(ω)

A1(ω)

)=−ω∆t

2Q+ ln(PG) (2.16)

Function 2.16 is a linear equation with respect to frequency ω. Figure 2.2 shows that the

slope contains 1/Q. We could define it as:

y(ω) = Eω+F (2.17)

E =−∆t

2Q(2.18)

F = ln(PG) (2.19)


Figure 2.2 Schematic diagram showing the natural log spectral ratio data as a function offrequency, from (Reine, 2009).

2.5 Surface Seismic Measurements

Surface seismic Q measurements differ a lot from the direct measurements done through

core-laboratory and vertical seismic profiling (VSP) methods (Toksöz et al., 1979). While

these methods have the advantage of direct observations of the boundaries of interest, in

surface seismic measurements the layers cannot be isolated. This leads to a wide variety of

errors associated with the seismic wave propagating through a layered medium. However,

direct measurements present some problems that made them less suitable for reservoir

characterization. Laboratory Q measurements showed that attenuation in a single rock

type is highly dependent on the pressure, porosity and pore fluid (Winkler and Nur, 1982).

Furthermore, these measurements are usually done at ultrasonic frequencies (500-9000 Hz)

that are different for the seismic band (Raikes and White, 1984) and fail to simulate seismic

conditions accurately. On the other hand, VSP or check-shot surveys depend on the sparse

location of the well, and can only give a local measurement of the Quality factor.

2.6 Pre-stack Measurements 11

Despite the common sources of error like noise, geometrical spreading, and energy partition-

ing, a common midpoint (CMP) gather provides information in the time and offset domains,

allowing the extraction of information concerning structure, lithology, and material proper-

ties such as velocity and Q-factor of a dense spatial coverage (Zhang and Ulrych, 2002). This

is why robust methods (PSQI) for computing Q based on surface seismic measurements are

required to better understand attenuation at a reservoir scale.

2.6 Pre-stack Measurements

As described in equation 2.14 there are a number of factors that make the calculation of

attenuation elusive. These effects, combined with the obvious influences of noise and mul-

tiples, make estimation of Q even more difficult. A stacked seismic section might be used

for the computation of Q due to its optimum signal to noise ratio (S/N) and its similarity

with zero-offset VSP (Dasgupta and Clark, 1998). However, it has been proved that seismic

attenuation is highly corrupted by the normal processing steps done in stacking. Ebrom

(2004) described the stack related mechanisms that could affect the frequency content

of traces, and hence corrupt attenuation measurements. Some of the problems include:

Mis-stacking due to coarse velocity picking, mis-stacking due to nonhyperbolic moveout,

mis-stacking of converted shear waves and, mis-stacking of high amplitude multiples. All

of this effects lead to a spectral amplitude of the stacked trace with a changed frequency

content which is corrupting any attenuation signature.

Dasgupta and Clark (1998), introduced a novel method to improve surface seismic measure-

ments of attenuation based on prestack data. The Q versus offset method (QVO), uses the

variations of attenuation with offset to extract a zero offset attenuation value from prestack

gathers. Despite the effectiveness of this method, authors like Reine et al. (2012b) have

proved that QVO exhibits some issues that increase the uncertainty of the 1/Q measure-

ments and makes it less robust than the PSQI approach presented in this work. Some of

the disadvantages of the QVO method over the PSQI method include: high sensibility to

the bandwidth choice, corrupting influences related to the isochron between the horizons

of interest and, occurrence of spikes and ridges in the data due to the two step inversion

scheme.

Chapter 3

Spectral Analysis

The spectrum of a finite time series is a representation of how the total power is distributed

over frequency (Stoica and Moses, 1997). Attenuation measurements require a careful

analysis of the relevant spectra to be used. Hence, it is necessary to compare different types

of time-frequency (t-f) transforms and decide which are better for attenuation analyses.

On this chapter, I will discuss the general concepts of spectral analysis, I will expose some

relevant types of t-f transforms, and finally I will show synthetic examples comparing the

behavior of some t-f transforms to decide which one is suitable for the PSQI method.

3.1 Non stationary signals

The time representation of a signal is usually the first approach that we can have to any

signal obtained by receivers recording variations with time. However, the spectral analysis

involves the Fourier analysis and specially the use of the Fourier transform (FT), which

decomposes continuous signals, with infinite length and stationary frequency behavior into

their component sinusoidal basis functions (Bracewell, 1986). Furthermore, the Fourier

analysis plays an important role in signal processing because it is suited to common trans-

form methods , such as linear filtering (Flandrin, 1998).

The frequency representation obtained by the Fourier transform is:

X ( f ) =∫ ∞

−∞x(t )e−i 2π f t d t (3.1)

Equation 3.1 shows that the Fourier analysis demands each frequency component to exist

with a constant amplitude over the entire time range, this is known as stationary, or time

invariant data (Bracewell, 1986). However, seismic wavelets are non-stationary, which is

3.2 Heisenberg-Gabor Uncertainty Principle 13

caused by the effects of wave front divergence, dispersion and attenuation (Zhou et al.,

2016).

3.2 Heisenberg-Gabor Uncertainty Principle

The deduction of this principle was well established by Flandrin (1998).

Defining the energy of a signal x(t ), as:

Ex =∫ ∞

−∞|x(t )|2d t <+∞ (3.2)

Assuming that the signal x(t ) and its Fourier transform X ( f ) have a center of gravity that is:∫ ∞

−∞t |x(t )|2d t = 0

∫ ∞

−∞f |X ( f )|2d f = 0 (3.3)

Introducing the respective moments of inertia:

∆t 2 = 1

Ex

∫ ∞

−∞t |x(t )|2d t ∆ f 2 = 1

Ex

∫ ∞

−∞f |X ( f )|2d f (3.4)

Defining the auxiliar quantuty I :

I =∫ ∞

−∞t∗x(t )

d x

d t(t )d t (3.5)

We can use the Parsevals identity and find that:

[Re{I }]2 ≤ |I |2 ≤∫ ∞

−∞t 2|x(t )|2d t ·

∫ ∞

−∞t 2

∣∣∣∣d x

d t(t )

∣∣∣∣2

d t = 4π2E 2x∆t 2∆ f 2 (3.6)

Intergration by parts show that I is:

I = t |x(t )|2∣∣∣−∞∞−Ex −

∫ ∞

−∞t x(t )

d∗x

d t(t )d t =−Ex −

∗I (3.7)

Then:

Re{I } =−Ex

2(3.8)

Replacing 3.8 into 3.6 and using the assumption that x(t ) decays so fast that t |x(t )|2 vanishes

at infinity, we get:

∆t∆ f ≥ 1

4π(3.9)

14 Spectral Analysis

Where ∆t is the duration time and ∆ f the frequency bandwidth (Flandrin, 1998), relate to

the resolution of adjacent signal components. If the standard deviations of ∆t and ∆ f are

measured, we can define the Heisenberg Gabor uncertainty principle as:

σtσ f ≥1

4π(3.10)

Where σt and σ f are the standard deviations of time and frequency estimates respectively

(Hall, 2006). The final consequence of the Heisenberg Gabor uncertainty principle is that

for windows with improved time resolution there will be poorer frequency resolution, and

vice versa. Usually time dimension is typically fixed for spectral analysis, however it is not

a requirement. It is possible to use a time-window that changes with the frequency being

analyzed and still follow the inequality (Reine, 2009).

3.3 Time-frequency Representations

A spectrogram is defined as a real-valued, and non-negative distribution of the spectral

energy density (Bracewell, 1986). For a one dimensional (1D) time series, the time-frequency

transform is a 2D spectrogram with coordinates of time and frequency. Depending on the

transform used, the resolution in time or frequency domain varies following the Heisenberg

Gabor uncertainty principle.

3.3 Time-frequency Representations 15

Figure 3.1 Sampling of the time-frequency plane. Different forms of sampling: Shannon,Fourier, Gabor, Wavelet , modified from (Flandrin, 1998).

3.3.1 Short-time Fourier transform

The short-time Fourier transform (STFT) is used to obtain local frequency spectrum by by

taking the Fourier transform in sliding windows over the signal. In it, the data trace s(t)

is gated by a sliding window function g (t), and the Fourier transform is applied. A sharp

cut-off may introduce artificial discontinuities and create unwanted problems, so a smooth


cut-off window function should be used (Gröchenig, 2013). The STFT is defined as:

SF (x, f ) =∫ ∞

−∞g (t −x)s(t )e−i 2πt f d t (3.11)

In equation 3.11 x represents the time lag to the center of the window function.

For signals sampled at a rate ∆t , this leads to a discrete STFT given by:

SFn,m =N−1∑n=0

gn−x sne− i 2πnmN (3.12)

Wheres x describes a discrete lag (Reine, 2009).

Figure 3.2 The Short time Fourier transform, from (Gröchenig, 2013).

3.3.2 Gabor transform

The STFT and the Gabor transform are known as fixed-window transforms. The STFT doesn’t

have an inherent window function. However, Gaussian functions are the only solutions that

3.4 Synthetic examples 17

minimize the duration-bandwidth product in the Heisenberg-Gabor sense (Flandrin, 1998).

Thus, the Gabor transform uses a window defined by a Gaussian function. This leads to a

transform defined by:

SGn,m =N−1∑n=0

gGn−x sne− i 2πnmN (3.13)

Where the Gaussian window defined by (Zhou et al., 2016) is:

g (t ) = 1

αp

2πe− t2

2α2 (3.14)

Note that this window is parametirized by α, which acts as a reciprocal of the standard

deviation, a measure of the width of the Fourier transform (Harris, 1978).

3.3.3 S transform

When the size of the time-window is a function of the frequency, we refer to variable-window

transforms. By replacing equation 3.14 into 3.1 we get the The S-transform (Stockwell et al.,

1996):

S(τ, f ) =∫ ∞

−∞s(t )

| f |p2π

e−(τ−t )2 f 2

2 e−i 2π f t d t (3.15)

Equation 3.15, where τ is the time lag. Note that unlike the normal Gaussian window, the S-

transform uses a Gaussian window that depends on frequency and time. Spectral-temporal

resolution trade off cannot be avoided since is a direct consequence of the Heisenberg

Gabor uncertainty principle. However, it can be optimized. It is desired to have good

frequency resolution at low frequencies (Resolve near signals at low frequencies) and, good

time resolution at high frequencies (Resolve near signals at high frequencies).

3.4 Synthetic examples

To better understand time-frequency transforms, 3 example signals (impulse, sine, chirp),

were generated and transformed using one fixed-window transform (STFT) and, one variable-

window transform (S-transform). All signals are 1000 ms long with a sample interval of 1 ms.

The fixed window transform uses a Hamming window of 128 points.


(a) Input signal

(b) STFT

(c) S-transform

Figure 3.3 Impulse signal (a) with the time-frequency representations of STFT (b) and S-transform (c).


(a) Input signal

(b) STFT

(c) S-transform

Figure 3.4 Sine signal (a) with the time-frequency representations of STFT (b) and S-transform (c).


(a) Input signal

(b) STFT

(c) S-transform

Figure 3.5 Chirp signal (a) with the time-frequency representations of STFT (b) and S-transform (c).


3.4.1 Analysis of results

Impulse

The impulse function is a useful signal to analyze because it contains all frequency compo-

nents at a single time. This impulse has an amplitude of 1 at 0.5s. There is a remarkable

difference between the fixed-window transform and the variable-window transform. While

the STFT exposes a fixed time resolution for all frequencies, the S-transform has a better

time resolution for higher frequencies than for lower ones.

Sine Wave

Another function useful for time-frequency analysis is the sine wave, since it has a specific

frequency for all times. Figures 3.4b and 3.4c shows the time-frequency transform of an

input sine wave with a frequency of 125H z. In this case, both transforms have constant

frequency resolution for all times.

Chirp

A chirp signal presents a gradual change of frequency with time. Figure 3.5a shows a linear

chirp varying linearly from 25H z to 250H z. While the STFT shows a fixed frequency resolu-

tion, the S-transform presents a high frequency resolution at lower frequencies.

3.4.2 Conclusion

The results clearly state the advantage of a variable-window time frequency transform

(S-transform) over a fixed-window transform (STFT). Figure 3.5c shows that a frequency

dependent window is desirable for the detection of high frequency contents. Furthermore,

figure 3.3c reveals that the S-transform has a better time resolution at higher frequencies

compared to low frequencies. This is useful when trying to resolve two near signals at high

frequencies. Finally, the inverse frequency dependence of the localizing Gaussian window is

an improvement over the fixed width window used in the STFT (Stockwell et al., 1996).

Chapter 4

Inversion Scheme

In this work, inversion refers to the determination of 1/Q from the calculated natural log

spectral ratio data. The purpose of this chapter is to show the advantage of applying a

simultaneous inversion scheme with respect to frequency and time difference. Moreover,

this aims to demonstrate the effects of weighting the inverse problem in the presence of

high-frequency noise and random velocity fluctuations.

4.1 Simultaneous inversion scheme

Equation 2.16 establishes the relation between the natural log spectral ratio and the inverse

quality factor 1/Q. On each trace, the spectrum of the windowed event A2 (ω) is divided by

that of the reference spectrum A1 (ω) giving the following equation:

ln

(A2 (∆t ,ω)

A1 (∆t ,ω)

)=−ω∆t

2Q+ ln(P (∆t ,ω)G (∆t ,ω)) (4.1)

Note that 4.1 introduces the dependence of A2 (∆t ,ω), A1 (∆t ,ω), P (∆t ,ω) and G (∆t ,ω)

on ∆t (the difference in travel time between the two measurements) and ω (the angular

frequency). In section 2.4 is presented a linear solution to equation 2.16, here it extends to:

y(∆tω) = E∆tω+F(∆tω) (4.2)

4.1 Simultaneous inversion scheme 23

Where

y(∆tω) = ln

(A2 (∆t ,ω)

A1 (∆t ,ω)

)

E =− 1

2Q

F(∆tω) = ln(P (∆t ,ω)G (∆t ,ω))

The inversion is considered as simultaneous because the product of ∆t and ω is treated as a

single variable (Reine, 2009).

As a first approximation, I will consider the term F(∆tω) in equation 4.2 to be constant

with respect to the variable ∆tω. However, this assumption introduces bias to the measure-

ment because both the reflectivity and the geometric spreading varies form trace to trace. At

a constant intercept F(∆tω) = F .

Then, equation 4.2 can be written as:

y(∆tω) = E∆tω+F (4.3)

Figure 4.1 Schematic drawing of the linear relationship of the natural log spectral ratio datafor all traces. The blue osculations represent some random noise.

24 Inversion Scheme

Figure 4.2 Another visualization of equation 4.2 but in the ∆tω space. Note that attenuationcauses the surface to descend as one coordinate increases relative to the other.

To reduce the bias induced by the assumption of a constant ln(P (∆t ,ω)G (∆t ,ω)), it

is recommended to apply a geometric spreading correction to compensate for wavefront

divergence early in processing (Yilmaz, 2001). The energy partitioning term P is not easily

corrected, but some approximations may by applied to reduce its effects (Reine, 2009).

As a second and better approach, I will treat the inversion of equation 4.2 with variable trace

intercepts. In this case I will only consider a ∆t dependence of the ln(PG) term. Equation

4.2 takes the form:

y(∆tω) = E∆tω+F(∆t ) (4.4)

Whit a matrix notation:

4.1 Simultaneous inversion scheme 25

y11

y21...

yN 1

y12...

yN 2...

yN M

=

∆t1ω1 1 0 . . . 0

∆t2ω1 0 1 . . . 0...

......

. . ....

∆tNω1 0 0 . . . 1

∆t1ω2 1 0 . . . 0...

......

. . ....

∆tNω2 0 . . . 1...

......

. . ....

∆tNωM 0 0 . . . 1

E

F1

F2...

BN

(4.5)

Equation 4.5 can be represented with the explicit linear equation:

d =Gm (4.6)

Equation 4.6 is considered as the foundation of the study of discrete inverse theory (Menke,

2012). In here,G stands for the data kernel, d is the data and, m refers to the model.

Since inverse theory is concerned with deducing knowledge from observational data that

has a discrete nature, d necessarily has to be discrete. Therefore, the discrete inverse theory

can be defined as:

d =M∑

j=1Gi j mi (4.7)

Equation 4.6 has no exact solution. However, a least square solution, is as a good approach

to estimate the solution based on the values of the model parameters that gave the best

approximate solution. Thus, Equation 4.6 has the least squares solution:

m = [GT G

]−1GT d (4.8)

26 Inversion Scheme

Figure 4.3 Solution to 4.6 with a variable intercept. Shows undulations in the data along the∆t coordinate.

The solution of the inverse problem is a smoothly varying surface. The undulating values

of the surface shown in figure 4.3 correspond to the F∆t term of equation 4.4.

4.2 Uncertainty

The two approaches for the inversion of equation 4.2 presented in this section (Variable

trace intercepts and constant trace intercepts), consider an equal degree of uncertainty over

all traces and frequencies. Nevertheless, this is not true for seismic data which is subjected

to frequency-dependent noise. As a result, each data point has its own uncertainty and can’t

be treated equally in the inversion. By applying a weighting function to the data, points

with low uncertainty will contribute more to the solution than points with high uncertainty

values (Taylor, 1997).

4.2.1 Weighting functions

If we assume that the measurements of the inversion solution obey a Gaussian distribution,

a good approach to provide an estimate of error is through the use of the standard deviation.

Taylor (1997) defines the weighting function wi as the reciprocal square of the corresponding

uncertainty σi .

wi = 1

σi2

(4.9)

4.2 Uncertainty 27

4.2.2 Weighted inversion

The inverse problem presented in equation 4.6 has a weighted least-squares solution defined

by Menke (2012) as:

m = [GT W G

]−1GT W d (4.10)

Chapter 5

τ−p Domain

5.1 The τ−p transform

5.1.1 Definition

Huygens’ principle of superposition allows the synthesis of a downward plane wave from the

spherical waveforms of the multiple sources (Schultz and Claerbout, 1978). If an appropriate

time lag is chosen, we can create a downward plane wave at an arbitrary angle θ. Time lag

∆t may be defined as:

∆t = sinθ

v∆x (5.1)

Where θ is the incidence angle of the plane wave, v is its velocity, and ∆x is the separation

between sources. Snell’s law states that for a ray path in a medium where velocity depends

only on depth z, (sinθ/v) is constant, and the ray is confined to a vertical plane (Aki and

Richards, 2002). The ray parameter p is defined as:

p = sinθ(z)

v(z)=Const ant (5.2)

Where p is usually known as the ray parameter or horizontal slowness. Replacing equation

5.2 into 5.1:

p = ∆t

∆x(5.3)

Equation 5.3 reveals that the downward plane wave created by the superposition of spherical

waveforms is a line with constant slope in the time-distance (t −x) domain. Hence, a linear

moveout can be defined as:

t = px +τ (5.4)

5.1 The τ−p transform 29

Where τ is the zero-offset intercept time. (Turner, 1990) resumes the mathematical definition

of the τ−p transform.

s(τ, p) =∫ ∞

−∞s(x,τ+px

)d x (5.5)

And for discrete data:

s(τ, p) =N∑

i=1s(xi ,τ+pxi

)(5.6)

In equation 5.6 N is the number of seismic traces used in the transform, and s(τ, p) is the

amplitude at (τ, p) in the τ− p domain. For multiple layers, the τ− p mappings for the

reflections are the sums of ellipses, while the direct arrival and head waves are mapped

as points (Diebold and Stoffa, 1981). Figure 5.1 shows synthetic traveltime curves in t − x

domain, and their τ− p mapping for a model with plane homogeneous layers. See that

reflections on the τ− p domain are mapped as ellipses because their apparent velocity

changes with offset, while for the head and direct waves the apparent velocity is constant so

they are appear as points in the τ−p domain.

Figure 5.1 Mapping of hyperbolic reflections in the t − x domain shown as ellipses in theτ−p domain, and linear events like refractions in the t −x domain shown as points in theτ−p domain. The vertical axis of the right figure is the intercept time τ. From (Reine, 2009).

30 τ−p Domain

5.1.2 Artifacts

To create a perfect downgoing composite plane wave by the superposition of many spherical

wavefronts, there must be an infinite lateral extent of the shot array and an infinitesimal shot

separation. However, normal seismic acquisitions are of finite spatial and temporal extent,

and discretely sampled in space and time. Hence, an introduction of offset truncation

artifacts and aliasing are unavoidable (Schultz and Claerbout, 1978).

A relation between τ and p can be derived from equation 5.4.

τ= t −px (5.7)

Equation 5.7 shows that the slope of a linear artifact has a value equal to the negative of the

offset were it was produced.

In this work we will refer to aliasing as the appearance of energy at multiple values of

the transformed horizontal slowness. Yilmaz (2001) establishes that spatial aliasing occurs

when the wavefront separation in time ∆t equals half the dominant period T . To avoid

spatial aliasing, we require that:

∆x ≤ v

2 fmax(5.8)

Where ∆x is the spatial sample interval, v is the lowest horizontal phase velocity in the data,

and fmax is the maximum frequency of interest (Stoffa et al., 1981). Using spatial sampling

principles, we can define a Nyquist ray parameter pN .

pN = 1

2∆x fmax(5.9)

Horizontal slownesses below pN are mapped correctly, while higher slowness values are

aliased. To reduce the effects of truncation and aliasing, Schultz and Claerbout (1978)

suggest to use a limited portion of the horizontal slowness line so that the windowed data

will be the only included into the τ−p transform.

5.1.3 Moveout and anisotropy

Whereas traces are stacked over offset after a conventional t −x moveout correction, they

are stacked over slowness after a τ−p moveout correction (van der Baan, 2004). Figure 5.1

shows that for a reflection in a homogeneous and isotropic media, the moveout follows an

ellipse in the τ−p domain. For a layered media, the τ−p moveout is just the sum of ellipses

5.1 The τ−p transform 31

(Diebold and Stoffa, 1981).

For a plane wave traveling in a homogeneous and isotopic media the total slowness u

can be defined as:

u = 1

v=

√p2 +q2 (5.10)

Where v is the the wave velocity in the medium, p is the horizontal component of the

slowness and q the vertical one. For multiple layers, the intercept time τ can be found by

summing contributions from each layer.

τi = 2N∑

i=1qi zi (5.11)

With N the number of layers and zi the thickness of layers down to the reflector. Replacing

q from equation 5.10 into 5.11 we get:

τi = 2zi(u2

i −p2) (5.12)

τi = τ0i

(1− v2

i p2) (5.13)

Where τ0i is the vertical incidence traveltime (Diebold and Stoffa, 1981).

Seismic anisotropy implies that the velocity of plane waves is direction-dependent (Hake,

1986). Moreover, it causes that the velocities of plane waves and the rays that carry the

energy to vary as a function of travel direction (Levin, 1990). Thus, ignoring the effects

of anisotropy may seriously affect the results of processing and interpretation steps, such

as normal moveout (NMO) correction (Thomsen, 1986). Tsvankin and Thomsen (1994)

describe the main distortions in reflection moveouts caused by the presence of anisotropy.

1. The short-spread moveout velocity is not equal to the root mean square (rms) velocity.

Hence, the difference between vertical rms and moveout velocities may lead to serious

errors in interval velocity, even for weak anisotropy.

2. Anisotropy leads to nonhyperbolic moveout, causing serious distortions in velocity

estimation.

The simplest anisotropic case of broad geophysical applicability is the transverse isotropy

(TI) with vertical symmetry axis. For this work, we will only study this particular case of

anisotropy, applied specially to weak anisotropy . Alkhalifah (1997) defines the η anisotropy

32 τ−p Domain

parameter to describe weak anisotropy.

η= ϵ−δ1+2δ

(5.14)

Where |ϵ|≪ 1 and |δ|≪ 1 are the weak anisotropic parameters defined by Thomsen (1986).

From equation 5.13, the anisotropic moveout in the τ−p domain within a given layer i is:

τi = τ0i

(1− p2v2

i

1−2ηi p2v2i

) 12

(5.15)

Where τ0 is the traveltime for a vertical incidence wave and vi is the interval stacking velocity

(van der Baan and Kendall, 2002). The parameter η for TI models with a vertical (VTI) axis of

symmetry can be obtained by inverting either NMO velocity from dip-moveout behavior of

P-wave surface seismic data, or nonhyperbolic moveout (Alkhalifah, 1997; Tsvankin et al.,

2010) For the isotropic case η= 0, equation 5.15 reduces to equation 5.13. To compute the

inversion of attenuation, the equivalent traveltime for a given reflection on the τ−p domain

is required.

t = τ

1+∑ p2[(1

v2i−p2

)(1

v2i− p2

1−2ηi p2v2i

)] 12 (

1−2ηi p2v2i

)2

(5.16)

(Reine, 2009). Equation 5.16 shows that for a given τ and p, the traveltime in the τ− p

domain can be calculated by knowing the interval values of v and η.

Chapter 6

Multimensional Interpolation

The most common preconditioning of seismic data is done to improve the signal-to-noise

(S/N) ratio by the removing of noise, the reduction of unwanted arrivals, and the regulariza-

tion of amplitude bursts. Although, this processing steps are sufficient to enhance seismic

images for interpretation (Brown, 2011), they usually ignore the effects of missing spatial

data. Missing offsets and azimuths, such as dead traces and lower-fold areas, may intro-

duce attribute artifacts and hence, have a negative impact in prestack inversion algorithms

(Chopra and Marfurt, 2013).

In this chapter I will discuss two interpolation techniques, the minimum weighted norm

interpolation (MWNI) method by Liu and Sacchi (2004), and the projection onto convex sets

(POSC) method by Abma and Kabir (2006). By the end of the chapter, I will choose the most

suitable algorithm to be used in a multidimensional reconstruction of seismic wavefields.

6.1 Seismic data interpolation

Seismic data reconstruction is crucial for any processing step requiring regular sampling.

Cary (1998) stated that the apparition of truncation artifacts and aliasing in the τ−p do-

main are mainly caused by gaps in the input data and missing data at near and far offsets.

Therefore, the proper infill of missing data prior to the transformation into τ−p domain,

can significantly reduce the presence of artifacts and aliasing.

Interpolation algorithms involving multiple spatial dimensions have many advantages

over one-dimensional methods (Trad, 2008). Particularly, interpolation in all five dimen-

sions (bin spacing along CMP X, bin spacing along CMP Y, bin spacing along azimuth, bin

34 Multimensional Interpolation

spacing along offset, time) is the most robust way to interpolate missing data that represents

the wavefield in azimuth and offset in a better way. For simplicity, I will expose the multidi-

mensional sampling functions for the 2D case, that can be generalized in a straightforward

manner to the N-D case (Naghizadeh and Sacchi, 2010). Consider the structure u with size

Nx ×Ny given by

u(nx ,ny

)=nx ∈ 0 : Nx −1

ny ∈ 0 : Ny −1(6.1)

The discrete Fourier transform (DFT) of u, is given by:

U (kx ,ky ) = 1

Ny

Ny−1∑ny=0

(1

Nx

Nx−1∑nx=0

u(nx ,ny

)e

−2iπkx nxNx

)e

−2iπky nyNy (6.2)

(Naghizadeh and Sacchi, 2010). Assuming that M samples of the data are 0 (Death traces), a

signal with missing samples uz may be defined. Therefore, the DFT of the data with missing

samples Uz , is obtained by

Uz =U ~Q (6.3)

Where ~ is the 2D convolution operator, and Q is the 2D Fourier response given by

Q(kx ,ky ) = 1

Nx Ny

M−1∑m=0

e−2iπkx hx (m)

Nx e−2iπky hy (m)

Ny (6.4)

Where, hx and hy are the locations of the available samples on the X and Y axis respectively

(Naghizadeh and Sacchi, 2010).

6.2 MWNI

Minimum weighted norm interpolation is a band-limited frequency-domain interpolation

algorithm, which uses a conjugate-gradient iteration to invert for an interpolation solution

with minimum error (Liu and Sacchi, 2004).

The method starts with a multi-dimensional matrix of the specified output sampling that is

filled with ones or zeros depending on the availability of the traces. All traces are transformed

to frequency domain using the discrete Fourier transform, and each band-limited frequency

slice is processed through a conjugate gradient inversion.

6.3 POSC 35

The band-limiting for MWNI occurs in the wavenumber k space, and is defined through a

weighted norm. The final result is a denser data set, with fewer gaps, suitable for a variety of

prestack algorithms including the τ−p transform. A step by step explanation of the MWNI

algorithm is given in Appendix B.

6.3 POSC

The projection onto convex sets (POCS) is a frequency-domain interpolation algorithm,

which applies multidimensional Fourier transform to interpolate irregularly populated grids

of seismic data with a simple iterative method (Abma and Kabir, 2006).

The method defines a threshold value that varies linearly from a large value in the first

iteration to a low value on the last iteration. Each threshold is applied to the data to keep

only the highest amplitudes. Due to the thresholding, some amplitudes will fill areas that

were previously zeros while keeping the orginal trace implitudes. The iterative process

continues until a proper convergence is reached.

6.3.1 5D Interpolation

Interpolation in five dimensions usually refers to one of the following cases:

1. 5 dimenions: Bin spacing along CMP X, bin spacing along CMP Y, bin spacing along

azimuth, bin spacing along offset, time or depth.

2. 5 dimenions: Bin spacing along CMP X, bin spacing along CMP Y, bin spacing along

offset X, bin spacing along offset Y, time or depth.

Since MWNI algorithm uses a spatial band-limiting of the data and the POCS algorithm does

not, results obtained with the MWNI are more robust and have less noise. Additionally, the

objective of applying the interpolation in this work is to reduce the gaps between offsets .

Therefore, the azimuth dimension is required to compensate for any azimuthal errors in

the shooting geometry. As a conclusion, I will use the MWNI algorithm applied to the five

dimensions including the bin spacing along offset.

Chapter 7

The Prestack Q-Inversion Method

In the previous chapters I described the main components that make PSQI a robust method

to calculate attenuation from surface pre-stack seismic data. These components involved

using a variable-window time-frequency transform (S-transform), inverting the natural log

spectral ratio surface in frequency and time difference simultaneously, and operating in the

τ−p domain.

In this chapter I discuss the main seismic processing steps that must be done to accomplish

the best results with the PSQI method. Furthermore, I show how the three components

discussed in chapters 3, 4, and 5 may be integrated to form the PSQI method.

Reine (2009) designed the PSQI method to remain constant to the nature of the surface

seismic data, no matter if it is land or marine data, 2D or 3D surveys. Figure 7.1 shows an

overview of the PSQI process.

7.1 Preproccesing

The primary objective in seismic processing is to obtain an earth model in time with an ac-

companying earth image in time. To achieve this, some common steps in seismic processing

include deconvolution, CMP stacking, and migration (Yilmaz, 2001). Although these steps

produce an optimal image for interpretation, they corrupt the seismic spectrum which is

needed to measure attenuation. For this reason, the processing steps needed for attenuation

analysis differ from the common processing steps used for seismic interpretation.

7.1 Preproccesing 37

Figure 7.1 Flow diagram of the PSQI process. Modified from (Reine, 2009).

38 The Prestack Q-Inversion Method

7.1.1 Static Corrections

Proper alignment of reflections with the expected traveltime curves is highly sensitive to

static shifts due to variable topography, the presence of high-velocity rocks near surface,

and complex subsurface velocity (Bevc, 1997). Therefore, corrections for static shifts are

necessary for attenuation analysis that requires well defined traveltime curves for a correct

location of the target spectra.

Elevation and static corrections shift the datum of a collection of seismic traces from one

surface of arbitrary shape to another reference horizontal datum based on a near-surface

velocity model (Berryhill, 1979; Cox, 1999). Similarly, surface-consistent residual statics

compensates for the effects of time delays in the highly variable near-surface weathering

zone (Wiggins et al., 1976). Residual statics may be run in conjunction with velocity analysis

to improve both processes.

7.1.2 Velocity Analysis

To calculate the traveltime curves, the moveout velocity for each CMP must be known. Yil-

maz (2001) established that velocity analysis can be done in selected CMP gathers or group

of gathers. For the purpose of attenuation measurements, the main goal of the velocity

analysis is to provide the best match between the moveout curves and the data.

For anisotropic data, the velocity analysis may be performed directly on the τ−p domain

with the advantage of using phase velocity, which is the natural velocity used in VTI and

orthorhombic media analysis (Tsvankin, 1997; van der Baan and Kendall, 2002). If the

maximum offset is large enough, Alkhalifah (1997) defined a way to extract the effective

anisotropy parameter η, described in chapter 5, directly from the velocity analysis. Equation

5.15 shows that the availability of this parameter can significantly increase the accuracy of

the calculated moveout curves in the τ−p domain.

7.1.3 Band-Pass Filter

Surface waves can be a serious issue for attenuation analysis in land data. While shallow

events may not be affected by surface waves, usually deeper events will. Since these waves

have a lower frequency content than the reflections, the difference in the combined spectra

distorts the process of measuring attenuation (Reine, 2009). Furthermore, the high ampli-

tudes of the surface waves can introduce artifacts in the τ−p transformed data (Stoffa et al.,

1981).

7.1 Preproccesing 39

To correct these problems, a single band-pass filter may be applied to the data. This solution

works well to remove the surface waves, although some of them will still be present after the

filtering process. Surface waves are different from reflections in both temporal and spatial

frequency. For this reason, they are usually removed by applying a frequency-wave number

f −k filter. Nevertheless, when the amplitude of the ground roll is bigger than those of

the reflection signals, f −k filters cause a severe distortion in the signal, and may corrupt

attenuation measurements (Liu, 1999).

7.1.4 Amplitude Regularization

The main importance of having a trace to trace consistency of amplitudes stems for the

τ−p transform. Anomalously high amplitude traces introduce artifacts during the τ−p

transform process, while weak frequency traces do not properly contribute to the plane wave

synthesis. Noisy traces that can’t be removed after the band-pass filter must be removed

from the data set through the use of a mute.

After the elimination of noisy traces and ground roll, the effect of geometric spreading

must be removed too. Ursin (1989) derived an offset dependent geometrical spreading

correction as a function of two-way zero-offset traveltime, and the Root Mean Square (RMS)

velocity estimated in the velocity analysis. Finally, with the proper amplitude and geometri-

cal spreading corrections, the surface consistent amplitudes are calculated for all of the data

simultaneously.

7.1.5 5D Interpolation

The PSQI code (Reine, 2009; Reine et al., 2012a,b), uses supergathers to fill up the gaps

between offsets. Although, this solution efficiently reduce the gaps, a multidimensional

interpolation is much more efficient (Trad, 2008). The 5D interpolation approach, defined in

chapter 6, uses supergathers as input for the interpolation. The algorithm predicts missing

traces with data from the surrounding CMPs, rather than just creating a huge supergather

with N CMPs inside.

7.1.6 τ−p Transform

The final stage in the preprocessing is to transform the data into the τ−p domain. A number

of parameters must be specified, and some may vary depending on the transform used. It


is important to make a proper choice of the parameters to avoid artifacts in the transform

process. For example, the maximum frequency used in the transform should be well above

the maximum frequency of the signal and the high-cut frequency of the band-pass filter

(Reine, 2009). A more detail explanation of the τ−p transform parameters used in this work

is given in section 8.

Additionally, the maximum horizontal slowness to be used in the transform should

exceed that of the reflection with the largest moveout. Equation 5.4 shows that this value

can be obtained by taking the derivative of time with respect to offset, this is equivalent to

differentiating the moveout equation with respect to offset. Alkhalifah and Tsvankin (1995)

derived a moveout equation for anisotropic VTI media.

t =(τ2

0 +x2

v2r ms

− 2ηx4

v2r ms

[t 2

0 v2r ms +

(1+2η

)x2

]) 12

(7.1)

Where vr ms is the stacking velocity. Differentiating this function with respect to offset gives:

p = x

t v2r ms

+ 2ηx3

t v2r ms

[t 2

0 v2r ms +

(1+2η

)x2

] ( (1+2η

)x2[

t 20 v2

r ms +(1+2η

)x2

] −2

)(7.2)

(Reine, 2009). Note that if the medium is isotropic η= 0, equation 7.2 reduces to:

p = x

t v2r ms

(7.3)

Evaluating equation 7.2 or 7.3 at the maximum offset, deepest horizon, and maximum

velocity in the zone of interest, gives the value of the maximum horizontal slowness present

in the data.

7.2 Calculate Moveout Curves

The moveout curves are required to track reflections in the τ−p domain across the gather.

Hence, moveout curves should match the data as accurately as possible to correctly extract

the relevant spectra for each trace.

7.3 Spectral Decomposition 41

7.2.1 Interval Velocity

Velocity analysis carried out in the t −x domain yields a stacking velocity or approximate

RMS velocity (Yilmaz, 2001). In section 7.1.4, I examined the importance of doing the velocity

analysis in the time-offset domain to address the problem of geometrical spreading. How-

ever, it is necessary to convert this RMS velocities to interval stacking velocities. Equation

7.4 shows how to obtain interval velocities from RMS velocities.

vi =√(

v2n tn

)− (v2

n−1tn−1)

tn − tn−1(7.4)

Where vn and vn−1 are the RMS velocities at the layer boundaries n and n −1 respectively,

and tn and tn−1 are the horizon times at these layer boundaries (Dix, 1955). Alternatively, if

the velocity analysis is done in the τ−p domain, interval stacking velocities can be obtained

directly in this domain (van der Baan, 2004).

7.2.2 Equivalent Traveltime

Equation 5.16 describes the equivalent traveltime for a given τ and p. This quantity is

necessary to calculate ∆t for the inversion of the natural log spectral ratio described in

chapter 4. The ∆t values are calculated by subtracting the equivalent traveltimes of the

relevant reflections.

7.3 Spectral Decomposition

In chapter 3, I mentioned the advantages of using a variable-window time-frequency

transform for the spectral analysis required in attenuation measurements. I picked the

S-transform because of its optimal time-frequency resolution, and the dependence of its

Gaussian window on time and frequency.

7.3.1 Data Range

The S-transform takes data with n time samples and m traces and produces an output of

n×n×m points. The result may be very huge and thus, computationally inefficient. For this

reason, the PSQI process requires a truncation of the data to avoid the transform of refracted

waves that are not properly recorded in the τ−p space. Therefore, the data is truncated

before the horizontal slowness of refraction at the lowest interface. To do so, a window

corresponding to approximately 1/4 of the dominant period is chosen. Then, the minimum


critical angle, for each interface within the window, is calculated using the interval velocities

and the Snell’s law. Next, the extracted critical angle is converted into a value of horizontal

slowness.

sin(θc ) = vn

vn+1(7.5)

pc = sin(θc )

vmax(7.6)

Where θc is the critical angle, pc is the critical slowness corresponding to the horizontal slow-

ness of refraction, and vmax is the maximum interval velocity within the selected window

(Reine, 2009).

Additionally, the horizontal slowness corresponding to the maximum offset in the data

must also be considered. Equations 7.2 and 7.3 may be used to define the maximum hori-

zontal slowness at which the data has to be truncated. Figure 7.2 shows the truncation of

the data at the slowness of refraction.

Figure 7.2 The horizontal slowness of the data must be truncated at the slowness of refractionor maximum offset for the lowest interface. Here, the truncation shown in gray occurs at therefraction of the red horizon. From (Reine, 2009).

7.4 Inversion for 1/Q 43

7.3.2 Spectral information

Once the data has been truncated and properly conditioned, the S-transform is applied.

It is important to mention that only the positive frequencies of the transform need to be

calculated. The spectra relevant to each reflection are matched to the appropriate moveout

curve, and the natural log of the ratio of each reflection combination is calculated. The final

product is a natural log spectral ratio surface for each pair of reflections.

7.4 Inversion for 1/Q

The final result of the PSQI method is the effective attenuation in the form of 1/Q, discussed

in section 2.1.3. This result is obtained by a simultaneous inversion of the natural log spec-

tral ratio data with respect to frequency and time difference. In chapter 4, I mentioned

the advantage of using a variable trace intercept for the inversion scheme, rather than a

constant trace intercept. Furthermore, I discussed the use of a weighting function to reduce

the uncertainty in the inversion of 1/Q.

The inversion is performed in the natural log spectral ratio surface, restricted in the ∆t

coordinate by the truncation of horizontal slownesses in the data. Additionally, the inversion

is also restricted in the frequency coordinate by the choice of an appropriate bandwidth over

which the attenuation is to be measured. Once the 1/Q value for a single CMP is calculated,

the entire process is repeated for the remaining CMPs. By the end, a map of 1/Q is obtained

for all the data, as well as a map with the corresponding uncertainty measurements.

Chapter 8

PSQI applied to Seismic Data in Colombia

The main objective of this work is to apply the PSQI method on real 3D seismic data from

Colombia. The data used in this work was provided by the colombian oil and gas company

Hocol S.A.. The prospecting reservoir for attenuation analysis is located approximately at

650 ms. I first give a short description of the area’s geology and then, review the seismic

acquisition parameters of the survey. Next, I show how each stage of the preprocessing

affects the quality of the data and make a detailed explanation of each of the PSQI steps

applied to real data. At the end, I present the results in the form of 1/Q maps and their

uncertainties.

8.1 Background Information

8.1.1 Geological Background

The potential reservoir investigated in this work is located at the Sinú-San Jacinto Basin in

Northwest Colombia. Figure 8.1 shows a general location map of the Sinú-San Jacinto Basin.

Hydrocarbon generation in this basin, is mainly attributed to the Cretaceous Cansona For-

mation. This formation is favorable for the generation of liquid hydrocarbons and consists

mainly of organic rich kerogen types I-II with a total organic carbon (TOC) of 2-11% (Olaya,

1994). Furthermore, Marín et al. (2010) suggest the presence of heavy oil with American

Petroleum Institute (API) gravities greater than 30◦.

The main zone of interest in this work is located somewhere between the Cienaga De

Oro Formation, a sandstone reservoir located at lower Miocene. Just above this formation,

lies the shale Carmen Formation that works as the regional seal rock of the hydrocarbon

8.1 Background Information 45

system in this zone Barrero et al. (2007). Figure 8.3 shows a stratigraphic column of the

Sinú-San Jacinto Basin.

Figure 8.1 Location map of the Sinú-San Jacinto Basin. From (Sánchez and Permanyer,2006).

8.1.2 Survey Parameters

The 3D survey acquisition parameters were designed to image up to 8s, with a sample

interval of 2ms. The main zone of interest is located before 1s. A surface grid of explosive

sources and receivers was deployed with a natural bin size of 20 m × 40 m, with a fold of 30.

The entire survey is made of a 557×312 grid of CMPs, resulting in 92868 CMPs. However,

the CMP data that I use in this work comprises a subset of 104×32 CMPs. Figure 8.2 shows

the foldmap of the survey and the subset that is used in this work.

Figure 8.2 Foldmap of the 3D survey. The black-dotted rectangle shows the subset wherethe inversion was done. See that most of the fold values are consistent within the subset,averaging a fold of 30.

46 3D Data Example

Figure 8.3 Stratigraphic column of the Sinú-San Jacinto Basin. From (Barrero et al., 2007).

8.2 PSQI Process

In this section I will apply the PSQI process to the real 3D seismic survey described in section

8.1.2. The steps of the PSQI are resumed in figure 7.1 and explained in detail in chapter 7. All

the examples used in this section correspond to the CMP located at inline 281 and xline 176.

8.2 PSQI Process 47

8.2.1 Preprocessing

Static Corrections and Velocity Analysis

CMP sorting, surface consistent amplitude corrections, static corrections, random noise

attenuation, and velocity analysis were done by PetroSeis LTDA using ProMax. Elevation

and refraction static corrections were applied (Using the Gauss Seidel algorithm in ProMax,

with a reference datum of 0m and a correction velocity of 2200 m/s), and two passes of

surface-consistent residual static corrections were iteratively applied (Using the Max Power

Autostatic algorithm in ProMax) in conjunction with the velocity analysis. The raw CMP

gather with static corrections is shown in figure 8.4.

Figure 8.4 Raw CMP with static corrections applied. Look at the high-amplitude surfacewaves indicated in red.

Band-Pass Filter

To apply a proper band-pass filter to the data, it is important to understand the different

components of the spectrum. The spectra of the surface waves, the reflections, and the

entire CMP gather are shown in figure 8.5. The peak of the surface wave spectrum is located

approximately at 5 Hz, with the majority of its energy ocurring between 4 Hz and 10 Hz. In

contrast, the spectrum of the reflections shows two peaks of frequency at 20 Hz and 30 Hz.

Likewise, the frequency peaks of the entire CMP occur at 20 Hz and 40 Hz approximately.

I design an Ormsby band-pass filter with corner frequencies 8 Hz, 12 Hz, 70 Hz, and 80

Hz. This values were picked to exclude the surface wave energy below its peak, and the

48 3D Data Example

low-amplitude high-frequency contents seen in the entire CMP spectrum. Figure 8.6 shows

the filtered CMP gather. It can be seen that much of the surface wave energy has been

eliminated.

Figure 8.5 The average spectrum for the entire CMP (green), the surface waves (red), and thereflections (blue).

Amplitude Corrections

As I mentioned before, the amplitude corrections were done by PetroSeis LTDA in a surface

consistent manner (Taner and Koehler, 1981). Yet, noisy amplitudes capable of producing

artifacts during the τ−p transform were still present in some traces. To account for this

problem, I used the TFD Noise Rejection tool from ProMax. This tool uses the STFT to

transform each record into t − f space and then, filters out isolated noise by replacing it with

the amplitudes from adjacent traces. While this is not a spectrally preserved operation, I

applied it sparingly to avoid the creation of strong artifacts in the τ−p domain.

8.2 PSQI Process 49

Figure 8.6 Band-pass filtered CMP gather. The surface waves, shown by the red triangle,were effectively removed. The high frequency noise present on various traces has also beenremoved.

5D Interpolation

In chapter 6, I discussed the multidimensional sampling functions and described two of

the most common multidimensional interpolation methods. To interpolate missing data, I

used the ProMax 4D/5D Interpolation tool with MWNI algorithm. As an input to the tool, I

created a 3×3 supergather arrangement and defined a maximum expected fold of 34. The

dimensionality of computation was set to 5 (Bin spacing along CMP X, bin spacing along

CMP Y, bin spacing along azimuth, bin spacing along offset, and time). To improve the result

of the interpolation, I just pick the center CMP from the 3×3 supergather. Figure 8.7 shows

the interpolated gather.

50 3D Data Example

Figure 8.7 The CMP gather from figure 8.4 after the multidimensional interpolation has beenapplied. Noisy traces were removed, and anomalous amplitudes were corrected. All thisprocesses are necessary to reduce the artifacts during the τ−p transform.

τ−p Transform

Figure 8.8 shows the offset distribution for the CMP data, as well as the distribution after

applying the 5D interpolation. It can be seen that with the multidimensional interpolation,

some offset gaps are filled, and the distribution is improved.

The τ−p transform that I use in this work is the ProMax forward linear Radon transform

described in chapter 5. I set a maximum transform slowness of such that it exceeds the

largest slowness of the shallow reflections. The number of horizontal slowness values is

the same as the maximum offset in the t − x space prior to the transform. The minimum

transform slowness is just the negative of the maximum slowness. Frequencies from 0 Hz to

200 Hz are considered, with a damping factor of 0.01, and a reference offset of 1800 m.

To demonstrate that various preprocessing steps are required, I compare the τ−p transform

of the data for each process. The τ−p transform of the raw CMP data is shown in Figure 8.9.

These data appears free from any near or far-offset truncation artifact. However, there are

some with very low amplitudes specifically below 300 ms.

The surface waves and high-frequency noise were removed by applying the band-pass

filter in the t − x domain. Figure 8.10 shows the result of the filtered τ− p transformed

data. The most significant change between figures 8.8 and 8.9 is the introduction of higher

8.2 PSQI Process 51

amplitudes at the early times.

To regularize the offset distribution, I applied a multidimensional interpolation to the

filtered data. Figure 8.11 shows the final τ− p transform for the interpolated data. The

amplitudes for the early times have significantly increased.

(a) (b)

Figure 8.8 (a) shows the offset distribution of a raw CMP gather and (b) shows the interpo-lated offset distribution of that same CMP. Offset spacing in the original data is regularizedafter the 5D interpolation. Moreover, offsets lower than 600 m (Including offset 0 m) thatwere not present in the raw data are created with the interpolation.

Figure 8.9 τ−p transform of the raw data. The horizontal slowness values are in ms/m

52 3D Data Example

Figure 8.10 τ−p transform of band-pass filtered data. The horizontal slowness values are inms/m

Figure 8.11 τ−p transform of the interpolated data. The horizontal slowness values are inms/m

8.2.2 Boundary of Interest

The zone of interest is located approximately at 650 ms on a high amplitude area within

the survey. Reflectors dip approximately 30◦. Two horizons were picked across the area of

interest (From now referred as horizons A and B). Figure 8.12 shows the seismic position of

8.2 PSQI Process 53

these horizons over a Pre-stack time migrated (PSTM) amplitude map. Likewise, figure 8.13

shows the two-way traveltimes for each event.

Figure 8.12 Pre-stack time migration amplitude map. The circled zone indicates the area ofinterest where high amplitude values are observed. The large amplitudes are often associatedwith gas. The two horizons used for the inversion are labeled as A and B. Courtesy of HocolSA & PetroSeis LTDA

(a) (b)

Figure 8.13 Two-way travel times for (a) horizon A, and (b) horizon B. Time is in seconds.

8.2.3 Calculate Moveout Curves

Because velocity field and zero-offset times are referenced to the original 20 m × 40 m bins

(the center CMP from the 3×3 supergather), the calculations are done directly on the τ−p

domain CMP gathers. Zero-offset traveltimes are calculated, and interval stacking velocities

are determined using Dix’s equation (7.4).

54 3D Data Example

With the interval velocity values and zero-offset travel times, I calculate the τ−p domain

moveout curves and the equivalent traveltimes (equations 5.15 and 5.16 respectively). Figure

8.14 shows RMS velocity profile and its converted interval stacking velocity section. Because

of the use of isotropic velocities, the calculated moveout curves might not align with the

actual data. To have a better alignment, moveout curves are snapped to the instantaneous

amplitude of the τ−p domain gathers. Figure 8.15 shows the CMP gather in the τ−p domain

as amplitude data, and as instantaneous amplitude data.

8.2.4 Spectral Decomposition

To reduce the total computational time, I limit the total number of traces to be transformed

into the time-frequency domain. Traces occurring at horizontal slownesses bigger than

that of the refraction of horizon B are first eliminated (See section 7.3.1). Additionally, I

determined the critical angle from the interval velocity data using search windows of length

10 ms. With the critical angle, the maximum horizontal slowness trace for horizon B is

calculated for each CMP using equation 7.6 (figure 8.16). Figure 8.17 shows the velocity data

and the critical angles calculated within a 10 ms search window.

Finally, I calculate the natural log spectral ratio for each of the reflection points between

horizons A and B. Each trace has its own traveltime difference calculated from the equivalent

traveltime curves. Figure 8.18 shows an example of the natural log spectral ratio surface

between horizons A and B. Each positive or negative notch in the surface has an associated

frequency position, which depends on the trace considered.

(a) (b)

Figure 8.14 Stacking velocity profile (a), and its interval stacking velocity profile (b) calculatedusing Dix’s equation. Values are shown in m/s.

8.2 PSQI Process 55

(a) (b)

Figure 8.15 CMP gather in the τ−p domain as (a) amplitude data, and (b) instantaneousamplitude data.

Figure 8.16 Maximum horizontal slowness trace for horizon B

56 3D Data Example

Figure 8.17 From left to right, the stacking velocity, interval stacking velocty, and criticalangle for each time sample.

Figure 8.18 Natural log spectral ratio surface between horizons A and B.

8.2.5 Inversion for 1/Q

The inversion for 1/Q is done with and without a weighting function. To compute the

weighted inversion, the natural log spectral ratios are calculated for each CMP individually,

and the mean and standard deviation are calculated for each CMP. The calculated mean is

8.3 PSQI Results 57

input into the inversion, which is weighted using equation 4.9.

The inversion of the natural log spectral ratio surface demands a bandwidth to be specified.

For this work I choose a bandwidth of 15 Hz to 75 Hz, which is coherent with the frequency

values of the reflections.

8.3 PSQI Results

Using the inversion solution, I calculate for each CMP the value of 1/Q and it´s uncertainty

(section 4). The final result is four maps of 1/Q representing the weighted and unweighted

inversion solutions, and its corresponding uncertainty maps.

8.3.1 Weighted

The maps of 1/Q and their uncertainty from a weighted inversion are shown in figure 8.19.

These maps show the attenuation measured between horizons A and B. To highlight the

results of the inversion, I apply a simple spatial filter to the mapped data. The filter smooths

the data using a mean filter over a rectangle of size (2Nr +1)× (2Nc +1), where Nr is the

number of points used to smooth rows, and Nc is the number of pints used to smooth

columns. The results of the spatial filtered map is shown in figure 8.20. The filtered map

shows various zones of high attenuation values, four high 1/Q areas are marked showing the

potential zones for hydrocarbon content within the reservoir.

8.3.2 Unweighted

Figure 8.19 shows the 1/Q map and its corresponding uncertainty using the unweighted

inversion. To understand better the results, I applied the same spatial filter used for the

weighted solution. The filtered map is displayed in figure 8.20. Although this result shows

many of the trends seen in figure filtered weighting solution, the attenuation values obtained

with the unweighed inversion are bigger. This may be due to interference effects that are not

corrected in the unweighted solution, allowing the presence of sparse high values.

58 3D Data Example

(a) (b)

(c) (d)

Figure 8.19 The weighted PSQI map of 1/Q between horizons A and B is shown on (a). Thefrequency bandwidth used is 15 Hz to 75 Hz, and the colorbar is clipped below 0. Theuncertainty map of the weighted inversion between horizons A and B is given in (b). (c)shows the unweighted PSQI map of 1/Q between horizons A and B with the same frequencybandwidth used in the weighed inversion. The uncertainty map of the unweighted solutionis shown in figure (d).

8.3 PSQI Results 59

(a) (b)

Figure 8.20 The filtered PSQI calculations of weighted 1/Q for the interval between horizonsA and B is shown in (a). likewise, the filtered PSQI calculations of the unweighted 1/Qsolution is shown in (b). The 1/Q color bar is clipped below zero for both results.

Chapter 9

Conclusions

Attenuation contains important information of the petrophysical parameters of the rock

and serves as an indicator of the saturation and mobility of fluids. Unlike other seismic

attributes like AVO, the use of attenuation for reservoir characterization is rare and usually

limited to laboratory experiments that fail to recreate the conditions of a real reservoir.

In this work, I presented a robust method to calculate attenuation, in the form of 1/Q,

from prestack CMP gathers. The PSQI method, developed by Reine (2009) uses three main

components to obtain a robust measurement of effective attenuation.

1. Use of a variable-window time-frequency transform to obtain better spectral estimates.

2. A simultaneous inversion scheme with respect to frequency and time difference.

3. Operate into the τ−p domain to reduce the effects of different raypaths.

I described each of these components and demonstrate the reliability of the PSQI method

by applying it to a real 3D seismic dataset from Colombia. The results obtained, proved the

robustness of the method and define an important tool to be used in the future characteriza-

tion of reservoirs.

9.1 Recommended future work

Although I have proved the effectiveness of the PSQI method to deliver reliable measure-

ments of attenuation, further investigation should be done to make the approach much

stronger. For example, the inversion could be updated to accommodate a non-least-square

solution. Furthermore, the the incorporation of well-log data into the analysis can make the

attenuation measurements more robust. Moreover, applying the PSQI method over different

9.1 Recommended future work 61

sets of seismic data with other acquisition geometries, may give important results to test the

effectiveness of the method in other types of datasets.

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Appendix A

Other methods to measure Q

A.1 Risetime Method

This is an empirical method based on the measurement of acoustic pulses propagating in

massive rocks. (Gladwin and Stacey, 1974) defines τ as the pulse rise time, and t as the time

of propagation of a pulse. This times are related to 1Q by:

τ= τ0 +C∫ t

0

1

Qd t (A.1)

Where C is a constant whose value is estimated experimentally. Equation A.1 can also be

expressed as the difference between two times, τN for a layer N and τ0. This is given by:

τN −τ0 =CN∑

i=1

T1

Qi(A.2)

Where Ti is the traveltime within layer i , and Qi is the quality factor or layer i (Jannsen

et al., 1985). From A.2 the effective quality factor Qe described on equation 2.10 for the layer

between N −1 and N is:

QeN =C

(TN −TN−1

τN −τN−1

)(A.3)

Appendix B

Multimensional algorithms

B.1 MWNI

This method begins with a vector x of length N sampled on a regular grid.

x = x1, x2, x3, ...., xN

With a vector y of observed data given by:

y = (xn(1), xn(2), xn(3), ...., xn(N )

)T

And a multi-dimensional matrix T of the specified output sampling defined by:

Ti , j = δn(i ), j

Where δ is the Kronecker operator. Liu and Sacchi (2004) defined a linear relationship

between the data and the observations given by the linear system:

y = T x (B.1)

Since this is a frequency-domain interpolation algorithm, the discrete Fourier transform

(DFT) is introduced.

Xk = 1pN

N∑n=1

xne−ı2π(m−1)(k−1)

N (B.2)

With inverse discrete Fourier transform:

xn = 1pN

N∑k=1

Kk e−ı2π(m−1)(k−1)

N (B.3)

70 Multimensional algorithms

(Bracewell, 1986). In chapter 4, I discussed the non-uniqueness nature of inverse linear

theory. To approximate the best solution, I defined a PSQI weighed inversion. Likewise,

MWNI uses a solution that minimizes a weighted model norm: ||x||2W . The wavenumber-

domain norm is defined as:

||x||2W = ∑k∈κ

X ∗k Xk

P 2k

(B.4)

Where Pk represents the spectral power at wavenumber k, and κ indicates the region of

spectral support of the signal. A diagonal matrix ∧ is introduced as:

∧k =P 2

k k ∈ κ0 k✚∈κ

(B.5)

Hence, the wavenumber-domain norm can be expressed as:

||x||2W = X H ∧s X (B.6)

Where X H is the the complex conjugate transform of X , and ∧s is the pseudoinverse of the

diagonal matrix ∧. Then, ||x||2W may be expressed as:

||x||2W = xHQ s x (B.7)

Where Q s = F H ∧s F . And F is the discrete Fourier transform. The minimum norm solution

is found by minimizing the folloing cost function:

J = bT (T x − y

)+||x||2W (B.8)

Where b is the vector of Lagrange multipliers (Liu and Sacchi, 2004). Finally, minimizing J

with respect to x leads to the MWNI solution:

x =QT T (T QT T )−1

y (B.9)

quality factor inversion applied to 3d seismic data in

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