Deblending of seismic data

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PROEFSCHRIFT

ter verkrijging van de graad van doctor

aan de Technische Universiteit Delft;

op gezag van de Rector Magnificus Prof. ir. K.C.A.M. Luyben;

voorzitter van het College voor Promoties

in het openbaar te verdedigen op dinsdag 13 november 2012 om

10.00 uur

door

Araz MAHDAD

Master of Science in Applied Geophysics

geboren te Esfahan, Iran.

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Rector Magniﬁcus voorzitter

Prof. dr. W.A. Mulder Technische Universiteit Delft, promotor Dr. ir. G. Blacqui`ere Technische Universiteit Delft, copromotor Prof. dr. ir. A. Gisolf Technische Universiteit Delft

Prof. dr. ir. E.C. Slob Technische Universiteit Delft Prof. dr. D.G. Simons Technische Universiteit Delft Dr. ir. P. Vermeer WesternGeco

Dr. ir. X.H. Campman Shell Global Solutions International B.V. Prof. dr. S.M. Luthi Technische Universiteit Delft, reservelid

support

The research for this thesis was ﬁnancially supported by the delphi Consortium

isbn 978-90-8891508-6

Copyright c 2012 by A. Mahdad

All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without the prior permission of the author.

Typeset by the author with the LA_{TEX2e Documentation System.}

Published by Uitgeverij BOXPress, Oisterwijk, The Netherlands Printed by Proefschriftmaken.nl

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1 Introduction

1

1.1 Introduction to hydrocarbon exploration . . . 1

1.2 Conventional and blended seismic acquisition . . . 1

1.3 Problem statement . . . 2

1.4 Literature . . . 3

1.4.1 Blended acquisition techniques . . . 3

1.4.2 Deblending methods . . . 5

1.5 Thesis outline . . . 6

2 Conventional and blended seismic data

9 2.1 Introduction . . . 9

2.2 Matrix representation of seismic data . . . 9

2.3 The concept of source blending . . . 13

2.4 Blended acquisition design . . . 28

2.4.1 Source encoding . . . 28

2.4.2 Lateral source conﬁguration . . . 28

2.4.3 Blending factor . . . 29

2.4.4 Survey condition . . . 29

2.5 Final remarks . . . 30

3 Deblending of blended seismic data

31 3.1 Introduction . . . 31 3.2 Deblending problem . . . 32 3.3 Pseudodeblending . . . 32 3.4 Deblending method . . . 36 3.4.1 Problem statement . . . 36 3.4.2 The iteration . . . 37 3.5 Coherence-pass ﬁlters . . . 42 i

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3.6.3 Common midpoint domain . . . 48

3.6.4 Common source domain . . . 49

3.7 Algorithmic aspects . . . 49

3.7.1 Convergence and stopping criterion . . . 49

3.7.2 Filter eﬀect . . . 50

3.7.3 Signal estimation errors . . . 52

3.8 Practical considerations . . . 53

3.8.1 Spatial aliasing . . . 54

3.8.2 Ground roll . . . 54

3.8.3 Seismic noise . . . 57

3.8.4 Near surface complexities and topographic variations . . . . 59

3.8.5 Irregularities in the source positions . . . 60

4 Field data examples

63 4.1 Introduction . . . 63

4.2 Application to 2D marine data . . . 64

4.3 Application to 2D land data . . . 67

4.3.1 General information . . . 67

4.3.2 Blended acquisition design . . . 67

4.3.3 Deblending strategy . . . 68

4.3.4 Results . . . 70

4.4 Application to 3D land data . . . 72

4.4.1 General information . . . 72

4.4.2 Blended acquisition design and deblending strategy . . . 76

4.4.3 Results . . . 77

5 Conclusions and recommendations

81 5.1 Conclusions . . . 81

5.2 Recommendations . . . 84

Bibliography

85

Summary

89

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Chapter

1

Introduction

1.1 Introduction to hydrocarbon exploration

The world in the 21st century is thirsty for energy and does not show any sign of saturation. The industrial and life expectancy growth in developing countries and rising economic powers such as China, Brazil, and India generated a new ascending trend in world energy consumption leading to 5.6 percent increase in 2010 – the largest increase since 1973 (BP statistical review of world energy, 2011). Although recently tremendous eﬀorts have been made to generate more energy from renewable sources, fossil fuels (mainly hydrocarbons) are still providing more than 80 percent of the world’s energy.

In the oil and gas industry, exploration is crucial to guarantee the energy supply for the coming decades. The success of the hydrocarbon exploration pro-jects is the key factor in identifying potential hydrocarbon resources and to turn potential resources into producing reservoirs. Hydrocarbon exploration mainly relies on highly sophisticated exploration techniques. The main goal of explora-tion geophysics is to extract structural informaexplora-tion and physical properties of the Earth’s subsurface from measurements that have been carried out on the basis of physical phenomena like acoustics, electro-magnetics, gravity, etc. Seismic reﬂec-tion surveying is one of the most common geophysical methods that are used in hydrocarbon exploration because of its depth of penetration and resolution.

1.2 Conventional and blended seismic acquisition

Seismic reﬂection surveying starts by acquisition design. There is a large num-ber of parameters that have to be taken into account during acquisition design. Types of sources and receivers, source and receiver sampling and geometry, and survey coverage are some of those parameters. These parameters may vary

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seismic data acquisition, sources are fired with large time intervals in order to avoid interference between successively firing source responses measured by the receivers. This leads to time-consuming and expensive surveys. Theoretically the waiting time between two successively firing sources has to be infinite, since the wavefield never vanishes completely. However, in practice this waiting time varies from a few seconds up to 30 s. This means that the source responses are negligible after the waiting time. As an example, within the time interval of 100 s, 20 source locations can be fired with 5 s waiting time, or 10 source locations can be fired with 10 s waiting time. Since decision making at the business level are usually based on minimizing acquisition cost, the source domain is usually poorly sampled to limit the survey duration, causing spatial aliasing. On the other hand, as it was shown in the previous example, modifying the waiting times brings flexibility in the source sampling and the survey time. The concept of simultaneous or blended acquisition has been introduced by Beasley et al. (1998) and Berkhout (2008) to address the aforementioned issues by either reducing the waiting time between firing sources, leading to reduced acquisition costs, or by increasing the number of sources within the same survey time, leading to a higher data quality. Note that a combination of the two approaches combines these benefits. This relation is shown schematically in Figure 1.1 by comparing the value of the blended acquisition with respect to the conventional acquisition in terms of cost and quality.

1.3 Problem statement

The price paid for achieving higher data quality at lower acquisition cost is dealing with the interfering sources in the data, which are acquired in blended acquisition. These data are called blended data. They can either be processed by specially designed least-squares migration algorithms, see Dai et al. (2010) and Verschuur and Berkhout (2011), or be first deblended and then further processed with the standard processing flows. Deblending is the process of retrieving the data as if they were acquired in a conventional, unblended way. This is an essential step in the case that standard processing flows are applied. Deblending of the blended seismic data is the major focus of this thesis.

As will become clear, deblending is an ill-posed problem, meaning that there are inﬁnite number of solutions for the deblending problem. Therefore, con-straints are necessary to solve the deblending problem. One of the most common approaches for solving ill-posed problems is to ﬁnd the least-squares solution. In

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Survey cost (survey duration) Quality (number of utilized sources) Conventional survey Blended survey focus on quality focus on economy balance economy - quality

Figure 1.1: Schematic comparison of the conventional and the blended seismic

acquisi-tion in terms of survey cost (survey duraacquisi-tion) with respect to quality (number of sources used).

most practical blending situations, the least-squares solution perfectly recovers the desired data. However, the least-squares solution suﬀers from the interference noise related to the interfering sources in the blended data, the so called blend-ing noise. Therefore, in order to achieve perfect deblendblend-ing, I propose that the blending noise must be estimated and subtracted from the least-squares solution.

1.4 Literature

1.4.1 Blended acquisition techniques

Several authors have discussed the concept of simultaneous shooting for impuls-ive and vibroseis type sources together with their particular advantages. Methods that utilize more than one source simultaneously in the ﬁeld have been considered for a long time in land surveys. The use of simultaneous vibrators transmitting the same or diﬀerent reference signals was proposed by Silverman (1979). The High Fidelity Vibratory Seismic (HFVS) method was developed by Sallas et al. (1998). In this method the sources are encoded with unique codes, designed such that the source responses can be separated through deterministic deconvolution. This method requires accurate ground force estimation for each vibrator location

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very low level of correlation noise. The Independent Simultaneous Sweeping (ISS) technique was introduced by Howe et al. (2008). In this technique, multiple vibro-seises are sweeping independently (spatial and temporal randomization) and the source responses are separated during processing. The Distance Separated Sim-ultaneous Sweeping (DSSS) technique by Bouska (2010) uses distance separation such that the source responses overlap only at travel times corresponding to the area below the zone of interest. As a result, the source responses can be separ-ated by simple muting. The Dithered Slip-Sweep (DSS) acquisition was proposed by Bagaini and Ji (2010); this technique combines the slip-sweep acquisition by Rozemond (1996) with a random time dithering scheme. The advantage of ﬁring with random time delays will be pointed out in the next subsection and will be discussed further in detail in this thesis. Pecholcs et al. (2010) acquired data from almost 45000 vibrator points per day taking advantage of distance separation as well as incoherent sweeping. These methods typically aim at reducing the acquis-ition time while utilizing more sources. The latter translates into better source sampling, reducing the threat of aliasing.

Such a paradigm of productivity increase has not been realized for the marine case yet. The use of simultaneous sources in marine surveys was introduced by Beasley et al. (1998). In this early study, two sources are firing simultaneously at the ends of a 2D marine cable and their responses are separated by dip filtering. Acquiring marine seismic data with random, quasi-random, or systematic delay times between firing sources was proposed by Vaage (2002). A near-simultaneous shooting technique with small random time delays between impulsive sources was presented by Hampson et al. (2008). The simultaneous shooting concept was extended to incoherent shooting and blending by Berkhout (2008) in order to achieve better illumination of the subsurface, see also Berkhout et al. (2010). Berkhout et al. (2009) extended the concept of source blending to the detector side by combining incoherent shooting with incoherent sensing and introducing the concept of double blending. Mansour et al. (2012) proposed an approach for quantifying the effectiveness of blended marine acquisition designs based on compressive sensing measures. Parkes and Hegna (2011) developed a new blended acquisition scheme for marine data that, in combination with dual sensor streamer technology, provides better source sampling and more efficient ghost suppression. In this scheme, the sources fire at different times and depths.

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The deblending process was addressed as a blind signal separation problem by Ikelle (2007), using Independent Component Analysis (ICA) to distinguish between the different blended sources. The similarities between the deblending problem and the well-known cocktail party effect1, see Cherry (1953), are pointed out in this work. Deblending can also be considered a denoising problem, treating the interference due to blending as noise. It has been reported by various au-thors (Doulgeris et al., 2010b, e.g.) that, when sources are fired with random time delays, the blending noise appears as random spikes after sorting the ac-quired blended data into a different domain than the common source domain, for instance, the common receiver, the common offset, and the common midpoint domain. Then, the separation process turns into a typical random noise removal procedure. Based on this property, Huo et al. (2009) use a vector median filter after resorting the data into common midpoint gathers. Kim et al. (2009) build a noise model from the data itself and then adaptively subtract the modelled noise from the acquired data. This algorithm is implemented in the common offset domain.

Although these methods have in certain cases produced promising results, many authors have proposed a formulation of deblending as an inverse problem that estimates the unknown unblended data (Bagaini et al., 2012, e.g.). Since this is an ill-posed problem, regularization is required. Moore et al. (2008), Akerberg et al. (2008), and Moore (2010) use a sparsity constraint in the Radon domain to regularize the inversion. A sparsity constraint is also utilized by Abma et al. (2010) in order to minimize the energy of the incoherent events present in the blended data. Lin and Herrmann (2009) and Herrmann et al. (2009), use an in-version approach requiring the separated data to be sparse in the curvelet domain. They make a link between their sparse inversion algorithm and the growing ﬁeld of compressive sensing, see e.g., Candes and Wakin (2008). It is common practice in such an approach to use sophisticated source codes (e.g., sweeps or random phase or/and amplitude encoding). Neelamani et al. (2008) used simultaneous sources and Compressive Sensing in a similar way to speed up forward modelling. Bagaini et al. (2012) studied two separation techniques for the DSS data, using the sparse inversion method by Moore (2010) and the f-x deconvolution denoising method by Canales (1984). The results from this study clearly demonstrate the advantage of the inversion methods over the random noise removal techniques. Wapen-aar et al. (2012) proposed a MultiDimensional Deconvolution (MDD) approach for deblending, which is closely linked to seismic interferometry.Van Borselen et

1_{The cocktail party eﬀect refers to a phenomenon related to the human’s selective attention}

ability. Consider a group of people talking simultaneously in a room. The individuals in the room are able to communicate with each other while others are talking at the same time. The ability to distinguish the voices related to the individuals in the case that two microphones are placed in the room, is referred to as the cocktail party problem. The relation between the deblending problem and the cocktail party problem is coming from the fact that the voices and the microphones can be interpreted as the sources and the receivers, respectively.

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is claimed to be less sensitive to aliasing than the other approaches due to its abil-ity to choose corrective projections to exploit data characteristics. An inversion approach that is based on coherence in some domain was proposed by Mahdad and Blacqui`ere (2010), and Doulgeris et al. (2010b). This work was later extended to a general framework that could integrate any multi-dimensional coherence-pass ﬁlter, see Mahdad et al. (2011), and Doulgeris et al. (2010a). The convergence properties and the algorithmic aspects of this method have been further discussed by Doulgeris et al. (2012) and Mahdad et al. (2012), respectively.

1.5 Thesis outline

The main focus of this thesis is deblending of blended seismic data. As already discussed, blended acquisition has the potential to change the future of seismic acquisition. Although direct migration is considered to be the most eﬃcient way to process blended data, deblending is the most practical approach with current industry standards. Figure 1.2 shows the general structure of the thesis, which consists of following chapters:

chapter 2 Conventional and blended seismic data. In this chapter the forward model describing conventional and blended seismic data is given by using a matrix notation. This is an essential step in solving inverse problems. Furthermore, the importance of incoherence in blended acquisition will be discussed in detail. The parameters that have to be taken into account in blended acquisition design also form an important part of this chapter. chapter 3 Deblending of blended seismic data. This chapter represents the core

of this thesis. A deblending method, which is an iterative inversion approach based on estimation and subtraction of blending noise is proposed and dis-cussed. Coherence of the signal and incoherence of the blending noise are the main ingredients for solving the ill-posed deblending problem. Imple-mentation of the method in diﬀerent data domains and ﬁltering options will also be discussed. Discussion on the optimization and algorithmic aspects of the deblending approach, addressed here, will lead to a better fundamental understanding of this deblending technique. Finally, the major issues that may be faced in practice regarding deblending will be explained. These practical considerations have to be also taken into account in blended ac-quisition design.

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until it is applied to field data. Deblending results related to field data are shown in this chapter. The blended data have been generated by numer-ical blending of three conventionally acquired datasets. Different blending schemes have been implemented in these examples. Although these data-sets are numerically blended, they provide a realistic representation, i.e., as if they were blended in the field. Furthermore, the obtained deblending results are numerically evaluated due to the fact that the unblended data are available in this particular case.

chapter 5 Conclusions and recommendations. The thesis ends by listing the con-clusions and providing recommendations for future work. A brief overview of the thesis will be given in this chapter, followed by remarks related to the key ﬁndings. Finally, the related areas of research that are worthwhile to investigate in the near future will be pointed out.

Further details about the chapter contents can be found at the beginning of each chapter. Some of the material in Chapters 3, 4 and 5 has already been published in the literature, see Mahdad et al. (2011) and Mahdad et al. (2012).

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Conventional vs. blended data Deblending Literature Forward model to describe conventional seismic data Forward model to describe blended seismic data Blending Incoherence Optimal solution: Deblended data Coherence Practical considerations

Chapter 1: Introduction

Chapter 2: Theory

Chapter 3: Method

Chapter 4: Field data examples

Deblending

Chapter 5: Conclusions and recommendations

3D land data 2D land data

2D marine data

General solution: Pseudodeblended data

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Chapter

2

Conventional and blended seismic

data

2.1 Introduction

As described in the previous chapter, source blending has tremendous capabilities in reducing seismic survey time, leading to lower acquisition costs, or by acquiring data from more source locations within the same survey time leading to higher data quality. The forward models describing the conventional and blended seismic data for stationary geometries (e.g., land seismic acquisition or ocean bottom systems) are discussed in this chapter. The system representation describing the conventionally acquired seismic data will be discussed ﬁrst in section 2.2. The main focus of this section will be on the data, source, and detector matrices. It will be followed by source blending and incoherent shooting in section 2.3, where the properties of the blended seismic data and the mathematical relations describing the blended seismic data will be discussed. Finally, the blended acquisition design parameters will be discussed in section 2.4. Some of these parameters have a direct impact on deblending, which will be discussed in the next chapter.

2.2 Matrix representation of seismic data

Berkhout (1982) showed that seismic data (2D or 3D) can be arranged in the so-called data matrix P. In the temporal frequency domain, each element of P corresponds to a complex-valued frequency component of a recorded trace. A system representation of seismic data is given by the following monochromatic expression:

P(zd, zs) =D(zd)X(zd, zs)S(zs). (2.1)

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d s

Figure 2.1: System representation of seismic data.

Here zsand zd correspond to the source and detector depth levels, respectively.

Figure 2.1 schematically demonstrates the forward model. The matrix operators in equation 2.1 are described as follows:

• S(zs): represents the source matrix. Each column of the source matrix

cor-responds to the downgoing source waveﬁeld atzsdue to one source (array)

and each row corresponds to a certain lateral position along the acquisi-tion surface. Therefore, each element of the source matrix shows the source wavefield due to a certain source array at a particular lateral position. Each source is represented by one element of the source matrix with columns representing the source (array) numbers and rows representing the lateral positions of the sources. Figure 2.2 demonstrates the source matrix forma-tion based on a particular source acquisiforma-tion geometry. As shown in Figure 2.2a, three elements of each column of the source matrix are filled with nonzero entries due to the fact that each array consists of three sources. If source signatures are identical, the values of the elements in one column are the same. This corresponds to the conventional field patterns, which are de-signed for suppressing ground-roll and improving the Signal to Noise Ratio (SNR) of the data. Figure 2.2b shows the source matrix for a single-source configuration. Due to the fact that multiple sources can be included in a source array, including blending is a straight-forward step. Such an array is called a blended source array and will be discussed in the next section. • D(zd): represents the detector matrix. Each row corresponds to one

de-tector (array) and each column corresponds to a dede-tector lateral position. Therefore, each element of the detector matrix shows the detector signa-ture at a particular location. Assuming that detectors are measuring the waveﬁeld exactly as it arrives at the acquisition surface, the elements of the detector matrix are ﬁlled with ones.

• X(zd, zs): represents the multidimensional transfer function of the Earth.

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mul-1 2 3 Lateral position 1 2 3 La te ra l position

Source geometry

Source matrix

1 2 3 Lateral position Array number 1 2 3 La te ra l position

Source geometry

Source matrix

(a)

(b)

Figure 2.2: Source matrix formation based on the source ﬁeld geometry. Conventional

ﬁeld array (a), single source conﬁguration (b).

tiples, surface waves, wave conversion, etc. Each element ofX corresponds to the entire subsurface impulse response from a certain lateral position along the source’s acquisition surface, indexed by columns, to a diﬀerent lateral location along the detector’s acquisition surface, indexed by rows.

• P(zd, zs): represents the data matrix. Each column of the data matrix

corresponds to a shot record, each row corresponds to a detector gather, each diagonal corresponds to a common oﬀset gather, and each anti-diagonal corresponds to a common-midpoint gather. The structure of the data matrix and the entire data cube are illustrated in Figure 2.3. An example of the diﬀerent gathers extracted from the data matrix and their corresponding f-k spectra are given in Figures 2.5 and 2.6, respectively, for a 2D numerical

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of theD and S matrices with the X matrix, which contains the subsurface’s reﬂectivity. If the source and detector side of the acquisition are sparsely sampled or poorly designed, X can not be well represented by P. Since the objective of seismic imaging is to retrieve the subsurface’s reﬂectivity, poor representation of theX matrix will ultimately result in a poor seismic image.

Common midpoint gather Common offset gather

De te c tor numbe r Source number Common detector gather Common source gather

(a) (b)

Figure 2.3: Monochromatic data matrix (a), and the data cube (b). Each column of the

data matrix corresponds to a shot record, each row corresponds to a detector gather, each diagonal corresponds to a common oﬀset gather, and each anti-diagonal corresponds to a common midpoint gather.

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Lateral position (m) Dept h ( m ) 0 1000 2000 3000 4000 5000 6000 0 1000 2000 3000 4000 5000 6000 1500 2000 2500 3000 3500 4000 4500

Figure 2.4: Synthetic velocity model used for the numerical data generation. The

sources and receivers are covering the 2D line of 5 km with 10 m sampling intervals.

2.3 The concept of source blending

Blended acquisition and incoherent shooting stand for the continuous recording of seismic responses from incoherent source arrays (Berkhout, 2008). As mentioned in the first chapter, blended acquisition reduces the survey time by reducing the temporal interval between sources, leading to a lower acquisition cost, or increases the number of sources within the same survey time, leading to a higher data quality. By incoherent shooting we aim at preserving the full temporal and spatial bandwidth, see also Lin and Herrmann (2009). Source blending is theoretically consistent with plane-wave synthesis (e.g. traditional field patterns) in the sense that multiple sources are activated within shorter time intervals with respect to the conventional data acquisition. However, they differ in the sense that the latter method generates continuous (coherent) wave fronts. In the case of source blending, it is important that such wave fronts are not generated. This is because of the sparsity in the data bandwidth introduced in this way. Instead, we know that the energy is distributed over the whole temporal and spatial bandwidth in the case that an incoherent signal is arriving at every subsurface location.

The importance of the incoherence in the blended acquisition will be shown by the following examples. Figure 2.7 shows four different configurations in which all sources are included in a source array. Each asterisk corresponds to a certain source that is firing at certain time and location. Note that all the sources are firing within 4 seconds time interval. Although it is not realistic to include all the sources in an array, these examples are aimed to demonstrate the coherence effect

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Lateral detector position (m) Time (s) 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 1.5 2 2.5 3 3.5 4

Lateral source position (m)

Time (s) 0 500 1000 1500 2000 2500 3000 3500 4000 4500 1.5 2 2.5 3 3.5 4

Time (s) 0 500 1000 1500 2000 2500 3000 3500 4000 4500 0 0.5 1 1.5 2 2.5 3 3.5 4

Time (s) 0 500 1000 1500 2000 2500 3000 3500 4000 4500 0 0.5 1 1.5 2 2.5 3 3.5 4 (a) (b) (c) (d)

Figure 2.5: Diﬀerent sections of the data matrix for a 2-D numerical data set. One

column of the data matrix (common source gather) (a), one row of the data matrix (common detector gather) (b), one diagonal of the data matrix (common oﬀset gather) (c), and one anti-diagonal of the data matrix (common midpoint gather) (d).

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Lateral wave number (m−1) Frequency (Hz) −0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0 10 20 30 40 50 60 70 80

Lateral wave number (m−1)

Frequency (Hz) −0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0 10 20 30 40 50 60 70 80

Frequency (Hz) −0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0 10 20 30 40 50 60 70 80 (a) (b) (c) (d)

Figure 2.6: The f-k spectra of the gathers shown in Figure 2.5. Common source gather

(a), common receiver gather (b), common oﬀset gather (c), and common midpoint gather (d).

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which corresponds to horizontal plane waves, See Figure 2.9a. As a consequence, the information related to large number of angles is destroyed. Note also the similarities between this configuration and the common offset gather of Figure 2.5c. This is due to the fact that the wavefield arrives to the receivers over the same range of angles for every source-receiver combination with a fixed offset. In the second configuration, which is shown in Figures 2.7b and 2.8b the sources are firing within regular time intervals. Although the sources are delayed, a dipping plane wave is generated by this configuration, which makes this case very similar to the first. The imprint of the dipping plane wave and missing angles can be clearly seen in Figure 2.9b. In the third case, the sources are still firing in a sequential order but with certain random time delays, see Figures 2.7c and 2.8c. As shown in Figure 2.9c, many more angles are present with respect to the afore-mentioned configurations. Finally, moving a step further, the sequential source ordering is replaced by random source ordering, which is shown in Figures 2.7d and 2.8d. The f-k spectrum in this case shows that the spatial bandwidth is fully covered, see Figure 2.9d.

As already mentioned, in reality the number of sources that are included in a source array is considerably less than the previous example. Figure 2.10 compares the same four array configurations, in the case that five sources are included in a source array and their corresponding f-k spectra are shown in Figure 2.11. The first configuration is similar to the conventional field arrays in the sense that sources are firing simultaneously, see Figure 2.10a. However, in conventional field arrays the lateral spacing between simultaneously firing sources is much smaller. Note the vertical striping in the f-k spectrum of this shot record. This effect is also known as the array effect, see e.g. Al-Ali (2007). The similar effect in a form of dipping stripes can be seen in the case that sources are firing within regular time intervals. In the case that the sources are still firing in a sequential order but with certain random time delays, the f-k spectrum is considerably less sparse compared to the previous configurations. Finally, in the fourth configuration in which the sequential source ordering is replaced by random source ordering, the optimal coverage of the spatial bandwidth is achieved.

Generally speaking, the incoherence can be also introduced in the lateral dir-ection. In the last example the sources are regularly spaced within the arrays. In the following example the regular source spacing is replaced by random lat-eral source spacing. Figure 2.12 shows the situation that five randomly spaced sources are firing simultaneously. The effect of lateral incoherence can be real-ized by comparing the f-k spectra in Figures 2.12b and 2.11a. Although in both cases the sources are firing simultaneously, the angle coverage is improved by

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0 1000 2000 3000 4000 5000 0.5 1 1.5 2 2.5 3 3.5 4

Lateral source location (m)

Source firing time (s)

0 1000 2000 3000 4000 5000 0.5 1 1.5 2 2.5 3 3.5 4

0 1000 2000 3000 4000 5000 0 0.5 1 1.5 2 2.5 3 3.5 4

(a) (b)

(c) (d)

Figure 2.7: Four diﬀerent source array conﬁgurations. All sources are included in the

source array. A source array with simultaneously firing sources (a), sequential sources firing within regular time intervals (b), blended source array with random firing times in a sequential order (c), and blended source array with random firing times in a random order(d).

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Lateral detector position (m) Time (s) 0 1000 2000 3000 4000 5000 3 4 5 6 7 8

Lateral detector position (m)

Time (s) 0 1000 2000 3000 4000 5000 3 4 5 6 7 8

Time (s) 0 1000 2000 3000 4000 5000 0 1 2 3 4 5 6 7 8

Time (s) 0 1000 2000 3000 4000 5000 0 1 2 3 4 5 6 7 8 (a) (b) (c) (d)

Figure 2.8: The resulting shot records of the diﬀerent source array conﬁgurations of

Figure 2.7. Amplitudes are normalized. Result of simultaneously firing sources (a), sequential sources firing within regular time intervals (b), blended sources with random firing times in a sequential order (c), and blended sources with random firing times in a random order (d).

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Lateral wave number (m−1) Frequency (Hz) −0.04 −0.02 0 0.02 0.04 0 10 20 30 40 50 60 70 80

Lateral wave number (m−1)

Frequency (Hz) −0.04 −0.02 0 0.02 0.04 0 10 20 30 40 50 60 70 80

Frequency (Hz) −0.04 −0.02 0 0.02 0.04 0 10 20 30 40 50 60 70 80 (a) (b) (c) (d)

Figure 2.9: The f-k spectra of diﬀerent source array conﬁgurations shown in Figure

2.8. Amplitudes are normalized. The f-k spectrum of Figure 2.10a (a), Figure 2.10b (b), Figure 2.10c (c), and Figure 2.10d (d).

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Time (s) 0 1000 2000 3000 4000 5000 3 4 5 6 7 8

Time (s) 0 1000 2000 3000 4000 5000 0 1 2 3 4 5 6 7 8

Time (s) 0 1000 2000 3000 4000 5000 0 1 2 3 4 5 6 7 8 (a) (b) (c) (d)

Figure 2.10: Four diﬀerent source array conﬁgurations. Conventional source array with

simultaneously firing sources (a), sequential sources firing within regular time intervals (b), blended source array with random firing times in a sequential order (c), and blended source array with random firing times in a random order (d).

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Lateral wave number (m−1) Frequency (Hz) −0.04 −0.02 0 0.02 0.04 0 10 20 30 40 50 60 70 80

Frequency (Hz) −0.04 −0.02 0 0.02 0.04 0 10 20 30 40 50 60 70 80

Frequency (Hz) −0.04 −0.02 0 0.02 0.04 0 10 20 30 40 50 60 70 80 (a) (b) (c) (d)

Figure 2.11: The f-k spectra of diﬀerent source array conﬁgurations shown in Figure

2.10. The f-k spectrum of Figure 2.10a (a), Figure 2.10b (b), Figure 2.10c (c), and Figure 2.10d (d).

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Frequency (Hz) −0.04 −0.02 0 0.02 0.04 30 40 50 60 70 80 (a) (b)

Figure 2.12: The eﬀect of lateral incoherence on the data bandwidth. Five irregularly

spaced sources ﬁring simultaneously (a), and the corresponding f-k spectrum (b).

randomizing the source locations. These examples clearly illustrate the eﬀect of incoherent shooting on preserving the angles present in the data bandwidth. In the next chapter, it will be discussed that incoherence is playing an essential role in deblending.

In general, source blending can be formulated as follows: S₍_z

s) =S(zs)Γ(zs), (2.2)

P₍_z

d, zs) =D(zd)X(zd, zs)S(zs) =P(zd, zs)Γ(zs), (2.3)

where S and P are the blended source matrix and the blended data matrix, respectively. Blending matrixΓ contains the blending parameters. Each column Γl is related to a blended source array and its elementsΓkl are the source codes

given to the individual sources that can be phase and/or amplitude terms. For example, in the simple case of random ﬁring times, Γkl = e−jωτkl is a linear

phase term that expresses the time delayτkl given to source k in blended source

arrayl. Similarly, in the case of vibrating sources transmitting a linear sweep, Γkl=e−jβklω

2

is a quadratic phase term describing the source code. The matrix multiplication of equation 2.3 can be interpreted as encoding of sources followed by summation (see Figure 2.13). If time delay is considered as source code, this procedure can be simulated by time shifting and summing shot records, see Figure 2.10d. An example of the diﬀerent gathers extracted from a blended data matrix

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( , )

z z

_d _s

P

………

=

( )

z

_s

ī

( , )

z z

_d _s

c

P

( , )

l d s k d s kl k

P z z

G

c

¦

P z z

G

*

……

………

*

De te c tor numbe r number De te c tor numbe r S our c e numbe r array

Figure 2.13: Matrix representation of source blending. The matrix multiplication can

be interpreted as encoding of sources followed by summation. A blended shot record Plis

simulated by summing over unblended shot records Pk that are encoded with incoherent

source codes Γkl.

and their f-k spectra is given in Figures 2.14 and 2.15. Here, five shot records from the dataset used in Figure 2.5 were blended by random time delays. Figure 2.16 shows the same gathers for a similar blending configuration utilizing random sweep lengthes as source codes instead of random time delays. The related f-k spectra are shown in Figure 2.17. Note that the diagonal and the anti-diagonal sections of the blended data do not correspond to the common offset and the common midpoint gathers. This is due to the fact the offsets related to multiple sources are included in a blended shot record. Consequently, common offset and common midpoint gathers do not exist in the blended data. The advantage of using more sophisticated source codes such as random sweep lengths over simple source codes such as random time delays will be discussed in the next chapter.

Note the incoherent structure of the blended data in diﬀerent domains and the data compression due to the blending. The data compression is coming from the fact that in all of these gathers, except common source gather, each trace represents the response of one blended source array in which ﬁve sources are blended together. Due to the fact that each source is encoded with a random time delay, in all domains of the blended data that each trace represents one blended source array, the incoherent structure is observed.

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Lateral detector position (m) Time (s) 0 1000 2000 3000 4000 5000 3 4 5 6 7 8 Source number Time (s) 20 40 60 80 100 3 4 5 6 7 8 Source number Time (s) 20 40 60 80 100 0 1 2 3 4 5 6 7 8 Source number Time (s) 20 40 60 80 100 0 1 2 3 4 5 6 7 8 (a) (b) (c) (d)

Figure 2.14: Diﬀerent sections of the blended data matrix (see Figure 2.5 for the

corresponding unblended data). Five sources are blended together utilizing random time delays as source codes. One column of the blended data matrix (blended shot record) (a), one row of the blended data matrix (blended detector gather) (b), one diagonal of the blended data matrix (c), and one anti-diagonal of the blended data matrix (d).

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Lateral wave number (m−1) Frequency (Hz) −0.05 0 0.05 0 10 20 30 40 50 60 70 80

Frequency (Hz) −0.05 0 0.05 0 10 20 30 40 50 60 70 80

Frequency (Hz) −0.05 0 0.05 0 10 20 30 40 50 60 70 80 (a) (b) (c) (d)

Figure 2.15: The f-k spectra of the sections shown in Figure 2.14. The f-k spectrum of

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Lateral detector position (m) Time (s) 0 1000 2000 3000 4000 5000 4 6 8 10 Source number Time (s) 20 40 60 80 100 4 5 6 7 8 9 10 Source number Time (s) 20 40 60 80 100 0 1 2 3 4 5 6 7 8 9 10 Source number Time (s) 20 40 60 80 100 0 1 2 3 4 5 6 7 8 9 10 (a) (b) (c) (d)

Figure 2.16: Diﬀerent sections of the blended data matrix (see Figure 2.5 for the

corresponding unblended data). Five sources are blended together utilizing random sweep lengths as source codes. One column of the blended data matrix (blended shot record) (a), one row of the blended data matrix (blended detector gather) (b), one diagonal of the blended data matrix (c), and one anti-diagonal of the blended data matrix (d).

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Lateral wave number (m−1) Frequency (Hz) −0.05 0 0.05 0 10 20 30 40 50 60 70 80

Frequency (Hz) −0.05 0 0.05 0 10 20 30 40 50 60 70 80

Frequency (Hz) −0.05 0 0.05 0 10 20 30 40 50 60 70 80 (a) (b) (c) (d)

Figure 2.17: The f-k spectra of the sections shown in Figure 2.14. The f-k spectrum of

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discussed here. The most important blended acquisition design parameters are brieﬂy discussed in this section.

2.4.1 Source encoding

As already mentioned, in blended acquisition the sources are encoded with inco-herent codes. The source codes can be very sophisticated like incoinco-herent sweeps in the case of vibratory sources or simple linear phase codes (random time delays) in the case of impulsive sources. The use of the sophisticated source codes is dir-ectly related to the deblending eﬀort, which will be discussed in the next chapter. In general, large delay times can be helpful in minimizing the deblending eﬀort due to time separation. However, one of the main objectives of the blended ac-quisition is to minimize the survey duration, which means that the delay times between blended sources have to be optimized. Further discussion on the source code optimization is beyond the focus of this thesis.

2.4.2 Lateral source configuration

As source encoding, the lateral spacing between blended sources plays an essen-tial role in blended acquisition. This is due to the fact that incoherence can be introduced by randomizing the lateral spacing between the blended sources, see Figure 2.12. In general, the larger the lateral separation between sources, the lower the effect of interference will be. Therefore, at a certain large lateral dis-tance between sources, they do not interfere anymore or the effect of interference is negligible as in the DSSS technique (Bouska, 2010). Therefore, large lateral separation between sources helps in minimizing the deblending effort. Special at-tention must be given to land seismic acquisition in which sources are shooting in different ground conditions, which may cause the interference of source responses that exhibit completely different dynamic ranges. On the other hand, the inter-ference of near offsets with each other can be problematic due to the existence of strong near-offset vibration noise. A better incoherence can be achieved by combining randomization in both spatial and temporal domains as in the ISS technique (Howe et al., 2008). Nevertheless, randomizing the source locations over the whole survey area might be infeasible due to the practical constraints.

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Berkhout (2008) defined two performance criteria for blended acquisition. Each criterion is directly related to either quality or economical aspects of the blended acquisition. The first performance measure, namely the Source Density Ratio (SDR), is defined as follows:

SDR = number of source in the blended survey

number of sources in the unblended survey. (2.4) As an example, for a blended acquisition in which five sources are blended together in every blended source array, the SDR is five in comparison to a conventional survey with single source configuration and the same number of experiments. This measure is directly related to the data quality improvement due to blending for a given survey time. The second performance measure is the Survey Time Ratio:

ST R = number of acquisition days in the unblended survey

number of acquisition days in the blended survey . (2.5) This measure is directly related to the survey time saving due to blending. As as example if the survey time is reduced by half in a blended acquisition, the STR is two. By multiplying both measures the overall eﬀectiveness of a certain blended acquisition is obtained which is called the “blending factor”. It is worth mentioning that these measures are irrespective of deblending quality. These choices depend to a great extent on several parameters such as acquisition area, expected cost quality ratio, utilized number of sources, and lateral source con-ﬁguration. Therefore, they demonstrate the necessity of an appraisal stage for strategic decision making prior to blended acquisition design.

2.4.4 Survey condition

Another important parameter is the survey condition. In general, the conven-tional marine data acquisition suffers from coarse source sampling (emphasis on improving the data quality), while the conventional land data acquisition suffers from very time consuming surveys (emphasis on minimizing the survey duration). Therefore, the major objectives of the blended acquisition are affected by the sur-vey condition. Furthermore, the raw marine data are less noisy and more coherent than the raw land data because of less incoherent noise, no groundroll, no near offset vibration noise, no source energy variation due to the acquisition surface, no complex near surface, and no topographic variations. Therefore, direct deblend-ing of blended marine data can be much more straightforward without any need for preprocessing. On the other hand, land seismic acquisition is blessed by the possibility of utilizing larger number of sources (customizable lateral source con-figuration), denser source sampling (less aliasing), and more sophisticated source encoding (better temporal incoherence).

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to step forward. Nevertheless, acquiring blended seismic data is not the end of the story. Blended data have to be processed in order to obtain the subsurface’s image. In order to be able to use standard processing ﬂows, the blended data have to be separated ﬁrst. This is the subject of the next chapter and the core of this thesis.

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Chapter

3

Deblending of blended seismic data

3.1 Introduction

In chapter 2 the forward models describing the conventional and blended seismic data were explained in matrix notation. Based on the forward model a method for deblending will be proposed in this chapter. The formulation of an inverse problem will be discussed in section 3.2. Due to the fact that we are dealing with an under-determined inverse problem, the full matrix inversion is not possible. The least-squares solution to the deblending problem (pseudodeblending) will be discussed in section 3.3. In most practical blending situations, the least-squares solution to the deblending problem generates interference (blending) noise. The blending noise amplitude level strongly depends on the source codes. By deblending, which will be discussed in section 3.4, the blending noise is estimated and subtracted from pseudodeblended data in an iterative inversion process. The deblending eﬃciency is directly related to the blended acquisition design.

This method takes advantage of the coherence of the signal and the incoher-ence of the blending noise by applying coherincoher-ence-pass filters. Different types of coherence-pass filters will be discussed in section 3.5. These coherence-pass filters can be implemented in different data domains, as discussed in section 3.6. The algorithmic aspects and practical considerations have to be taken into account re-garding deblending and the success of deblending rely on them. Optimization and algorithmic aspects of the deblending approach, which will be addressed in sec-tion 3.7 of this chapter, will lead to a better fundamental understanding of this deblending technique. Generally speaking, land seismic data raise major prac-tical problems and blended data are no exception. Ground roll, strong ambient noise, irregularities in the source positions, and topographic variations are some of these problems. Some of these practical issues compromise the coherence of the signal, while others can not be estimated by the coherence-pass filters, violating

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Blended data can be processed in two ways. The first option is separating the individual source responses prior to imaging (deblending). In this way, the ex-isting processing flows can be used after deblending. The second option is direct migration of blended data, see Verschuur and Berkhout (2011). In this thesis, we only focus on the first option, which is deblending of the acquired blended data. Deblending aims at retrieving the data as if it was acquired by conventional data acquisition. In other words, deblending aims at retrievingP from the acquired P_{. The optimization problem that will be solve is defined as:}

P₍_z

d, zs)− P(zd, zs)Γ(zs) =0. (3.1)

In order to solve this problem, a matrix inversion has to be performed. In general, the deblending problem is ill-posed (i.e., P has fewer columns than P and Γ is non-square matrix), which means that the blending matrix is not invertible and the solution is not unique. Therefore, the deblending problem needs to be constrained.

3.3 Pseudodeblending

Due to the fact that the blending matrix can not be inverted, a least-squares solu-tion of the deblending problem may be used. Minimizing the following objective function provides a least-squares solution of the deblending problem:

J = P₍_z

d, zs)− P(zd, zs)Γ(zs)22. (3.2)

From now on, the source and receiver depth indices,zs andzd, will be omitted

for brevity. The general solution of equation 3.2 is given by:

P = P_Γ−1_, _(3.3)

Γ−1₌_ΓH_{Γ + αI}−1_ΓH_, _(3.4)

where ΓH _{is the conjugate transpose (Hermitian) of} _{Γ, P is the least-squares}

solution, andΓ−1is the least-squares inverse of the blending matrix. The term αI in equation 3.4 is the regularization term which can be neglected in the absence of noise in the model. By omitting theαI term from the equation 3.4, Γ−1_is

referred to as the pseudo-inverse of the blending matrix. In the case that each source position is excited only once by a phase-encoded source,Γ has mutually orthogonal columns and contains only phase terms. If b sources are blended

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b

factor 1_b can be skipped for simplicity. This leads to the following expression for pseudodeblending:

P = P_ΓH_. _(3.5)

This process is also referred to as passive separation. From a practical point of view, pseudodeblending is a simple decoding procedure. E.g., ifb sources were blended using time delay as code, the blended shot record is copiedb times. Then, each of these copies is time-shifted (decoded) to undo the delays introduced in the field. Due to the fact that the responses of multiple sources are included in a single blended shot record and the source codes are not orthogonal, the pseudodeblend-ing process generates interference. This interference is known as blendpseudodeblend-ing noise. Figure 3.1 contains an example of different sections of pseudodeblended data re-covered from the blended data of Figure 2.14; their corresponding f-k spectra are shown in Figure 3.2. Note the existence of incoherent blending noise in the pseudodeblended common receiver, common offset and common mid-point gath-ers, in contrast with the signal, which is coherent in all gathers. This is because in these domains each trace belongs to a different source, which is decoded with a different source code. During the decoding process, the signal is phase corrected but the blending noise gains residual phase which is different from one trace to the other in all of the three aforementioned domains. However, the signal is coherent in all domains.

The blending noise amplitude level depends to a great extent on the choice of the source codes. In the case of random time delays (e−jωτk_{), the blending}

noise has the same level of amplitude as the signal. This is due the fact that pseudodeblending is a time correction process in this case, which is shown in Fig-ure 3.3. FigFig-ures 3.3a and 3.3b are two shifted delta functions representing two time delay codes. The autocorrelation and the crosscorrelation (e−jω(τ1−τ2)_{) of}

these codes are shown in Figure 3.3c, which represent the signal and the blend-ing noise, respectively. Note that they both have the same level of amplitude. However, in the case that random linear sweep lengths (e−jβkω2_{) are utilized as}

source codes, the blending noise (e−j(β1−β2)ω2_{) has a lower amplitude level than}

the signal, see Figure 3.4. This is due the fact that the blending noise energy spreads over the length of the sweep characterized by the diﬀerence between the length and the phase of the linear sweep codes (β1− β2). Therefore, in the

case that the linear upsweeps (e−jβkω2_{) and downsweeps (}_e+jβkω2_{) are utilized as}

source codes, the maximum diﬀerence and, as a result, the maximum blending noise sweep length with minimum amplitude level will be achieved, see Figure 3.5. However, in practice the use of downsweeps is often avoided because of the presence of strong harmonics at positive times, see Abd El-Aal (2010). These examples demonstrate the value of sophisticated source encoding in blended land seismic acquisition. Figure 3.6 shows diﬀerent sections of pseudodeblended data recovered from the blended data of Figure 2.16, in which the sources have been blended with random sweep lengths. The related f-k spectra are shown in

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Fig-Lateral detector position (m) Time (s) 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 2 2.5 3 3.5 4

Time (s) 0 500 1000 1500 2000 2500 3000 3500 4000 4500 2 2.5 3 3.5 4

Time (s) 0 500 1000 1500 2000 2500 3000 3500 4000 4500 0 0.5 1 1.5 2 2.5 3 3.5 4

Time (s) 0 500 1000 1500 2000 2500 3000 3500 4000 4500 0 0.5 1 1.5 2 2.5 3 3.5 4 (a) (b) (c) (d)

Figure 3.1: Diﬀerent sections of pseudodeblended data generated by expanding and

restoring the phase applied to the blended data of Figure 2.14. One column of the pseudodeblended data (common source gather) (a), one row of the pseudodeblended data (common receiver gather) (b), one diagonal of the pseudodeblended data (common oﬀ-set gather) (c), and one anti-diagonal of the pseudodeblended data (common midpoint gather) (d).

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Lateral wave number (m−1) Frequency (Hz) −0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0 10 20 30 40 50 60 70 80

Frequency (Hz) −0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0 10 20 30 40 50 60 70 80

Figure 3.2: The corresponding f-k spectra of diﬀerent gathers (Figure 3.1) extracted

from the pseudodeblended data. Common source gather (a), common receiver gather (b), common oﬀset gather (c), and common midpoint gather (d).

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to be discussed in detail in the next section. 0 2 4 6 8 10 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 Time (s) Amplitude 0 2 4 6 8 10 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 Time (s) Amplitude −10 −5 0 5 10 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 Time (s) Amplitude Autocorrelation Crosscorrelation (a) (b) (c)

Figure 3.3: The blending noise amplitude level with respect to the signal utilizing

ran-dom time delays as source codes. The time delay codes are represented by shifted delta functions. The autocorrelation corrects the phase shift and the crosscorrelation is repres-ented by the phase diﬀerence between the source codes. 3.56 s delay (a), 6.48 s delay (b), the autocorrelation (signal) and the crosscorrelation (blending noise) of the time delay codes (c).

3.4 Deblending method

3.4.1 Problem statement

In order to retrieve the unblended dataP in equation 3.1, an inversion procedure has to be carried out. The problem that we wish to solve can be stated as follows minimize P− PΓ2₂ subject to the condition P ∈ R(C), (3.6) whereC represents a ﬁlter function and R(C) is its range. The ﬁlter function is used in the deblending process in order to suppress the incoherent blending noise

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0 2 4 6 8 10 −1 −0.5 0 0.5 Time (s) Amplitude 0 2 4 6 8 10 −1 −0.5 0 0.5 Time (s) Amplitude −10 −5 0 5 10 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Time (s) Amplitude Autocorrelation Crosscorrelation (a) (b) (c)

Figure 3.4: The blending noise amplitude level with respect to the signal utilizing

lin-ear sweeps of different lengths. The sweep codes are represented by linlin-ear chirp signals of different lengths. The autocorrelation generates a band limited delta function and the crosscorrelation is characterized by the difference in sweep lengths. 6 s linear up-sweep (a), 10 s linear upup-sweep (b), the autocorrelation (signal) and the crosscorrelation (blending noise) of the linear sweep codes (c).

while leaving the coherent signal untouched. When the ﬁlter function is combined with a thresholding function, only the coherent part passes through. Therefore, the combination of both is called a coherence-pass ﬁlter, which will be discussed later.

3.4.2 The iteration

The blending information contained inΓ is known. This means that ΓH_{is known}

as well and therefore, if the unblended dataP were known, the blending noise that is present in the pseudodeblended dataP could be computed as the diﬀerence between the pseudodeblended and the unblended data. Using equation 3.5 we obtain

N = P − P = PΓΓH_{− I}_, _(3.7)

whereN represents the blending noise. However, the initial unblended data are not available and obviously, if they were there would be no need for a deblend-ing method. Suppose, though, that part of P could be extracted from the pseudodeblended data P. Then, an iterative estimation-and-subtraction pro-cess could be initiated where more of the blending noise could be computed and removed at each iteration. Such a method is depicted in Figure 3.8 and can be

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0 2 4 6 8 10 −1 Time (s) 0 2 4 6 8 10 −1 Time (s) −10 −5 0 5 10 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Time (s) Amplitude Autocorrelation Crosscorrelation (a) (b) (c)

Figure 3.5: The blending noise amplitude level with respect to the signal utilizing linear

upsweeps and downsweeps. 6 s linear upsweep (a), 6 s linear downsweep (b), the auto-correlation (signal) and the crossauto-correlation (blending noise) of the linear sweep codes (c).

formulated as follows:

Pi+1₌_P_ΓH_{− P}i_(ΓΓH_{− I),} _(3.8)

Pi=F{Ti{F−1{C{Pi}}}}, (3.9)

where Pi _{is the deblended estimate on iteration} _{i. The coherence-pass ﬁltered}

estimatePi is obtained by thresholding the ﬁltered estimate C{Pi_{} in the time}

domain (F and F−1 denote forward and inverse temporal Fourier transforms, respectively). Due to the fact that the forward and inverse Fourier transforms are performed in the temporal direction, the 3D multi-frequency data cube Pi

is used instead of the 2D mono-frequency data matrixPi_{. After coherence-pass}

ﬁltering, the mono-frequency components ofPi, i.e.,Pi, are used in equation 3.8. The thresholding process is described by a threshold functionTi _{which is deﬁned}

as follows:

Ti_{{x} =}

x if |x| ≥ ni_,

0 if |x| < ni_, (3.10)

where x and Ti_{(x) are the input and the thresholded output, respectively, and}

ni _{is the threshold level on iteration} _{i. The threshold level decreases at every}

iteration. In general, the filter can also be updated in every iteration. A typical implementation would involve a narrow-band filter during the first iterations that

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Lateral detector position (m) Time (s) 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0.5 1 1.5 2 2.5 3 3.5 4

Time (s) 0 500 1000 1500 2000 2500 3000 3500 4000 4500 0.5 1 1.5 2 2.5 3 3.5 4

Time (s) 0 500 1000 1500 2000 2500 3000 3500 4000 4500 0 0.5 1 1.5 2 2.5 3 3.5 4

Time (s) 0 500 1000 1500 2000 2500 3000 3500 4000 4500 0 0.5 1 1.5 2 2.5 3 3.5 4 (a) (b) (c) (d)

Figure 3.6: Diﬀerent sections of pseudodeblended data generated from the blended

data of Figure 2.16, in which the sources have been encoded with random sweep lengths. One column of the pseudodeblended data (common source gather) (a), one row of the pseudodeblended data (common receiver gather) (b), one diagonal of the pseudodeblen-ded data (common oﬀset gather) (c), and one anti-diagonal of the pseudodeblenpseudodeblen-ded data (common midpoint gather) (d).

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Lateral wave number (m−1) Frequency (Hz) −0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 30 40 50 60 70 80

Frequency (Hz) −0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 30 40 50 60 70 80

Frequency (Hz) −0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0 10 20 30 40 50 60 70 80

Figure 3.7: The corresponding f-k spectra of diﬀerent gathers (Figure 3.6) extracted

from the pseudodeblended data. Common source gather (a), common receiver gather (b), common oﬀset gather (c), and common midpoint gather (d).