A inversão da forma de onda completa pode compensar a falta de iluminação na tomografia poço-a-poço?

Universidade Federal do Rio Grande do Norte Centro de Ciências Exatas e da Terra

Programa de Pós-Graduação em Geodinâmica e Geofísica

DISSERTAÇÃO DE MESTRADO

A inversão da forma de onda completa pode

compensar a falta de iluminação na tomografia

poço-a-poço?

Autor:

Alex Tito de Oliveira

Orientador:

Prof. Dr. Walter Eugênio de Medeiros (UFRN)

UNIVERSIDADE FEDERAL DO RIO GRANDE DO NORTE CENTRO DE CIÊNCIAS EXATAS E DA TERRA

PROGRAMA DE PÓS-GRADUAÇÃO EM GEODINÂMICA E GEOFÍSICA

DISSERTAÇÃO DE MESTRADO

A inversão da forma de onda completa pode

compensar a falta de iluminação na tomografia

poço-a-poço?

Autor:

Alex Tito de Oliveira

Dissertação apresentada em 17 de De-zembro de dois mil e dezoito, ao Pro-grama de Pós-Graduação em Geodinâ-mica e Geofísica – PPGG, da Universidade Federal do Rio Grande do Norte -UFRN como requisito à obtenção do Tí-tulo de Mestre em Geodinâmica e sica, com área de concentração em Geofí-sica.

Comissão Examinadora:

Prof. Dr. Walter Eugênio de Medeiros (UFRN) - Orientador

Prof. Dr. Amin Bassrei (UFBA) - Examinador externo

Oliveira, Alex Tito de.

A inversão da forma de onda completa pode compensar a falta de iluminação na tomografia poço-a-poço? / Alex Tito de

Oliveira. - 2018. 85f.: il.

Dissertação (Mestrado) - Universidade Federal do Rio Grande do Norte, Centro de Ciências Exatas e da Terra, Programa de Pós-Graduação em Geodinâmica e Geofísica. Natal, 2018.

Orientador: Walter Eugênio de Medeiros.

1. Geofísica Dissertação. 2. Inversão da forma de onda -Dissertação. 3. Arranjo poço-a-poço - -Dissertação. 4. Tomografia sísmica - Dissertação. I. Medeiros, Walter Eugênio de. II. Título.

RN/UF/CCET CDU 550.3

Universidade Federal do Rio Grande do Norte - UFRN Sistema de Bibliotecas - SISBI

Catalogação de Publicação na Fonte. UFRN - Biblioteca Setorial Prof. Ronaldo Xavier de Arruda - CCET

Resumo

A iluminação sísmica em cada ponto da região interpoços pode ser definida como o ângulo máximo entre os raios que passam por esse ponto. Interfaces completamente contidas nas aberturas angulares podem ser imageadas com a tomografia de tempo de trânsito da primeira chegada (first arrival travel time tomography, ou FATTT). Nós investigamos se a inversão de forma de onda (full waveform inversion, ou FWI) 2D acústica pode compensar a falta de iluminação. Nós usamos dados sintéticos gerados com fontes de forma Ricker com frequências de pico de 100 ou 500 Hz, resultando em superposição pequena das bandas de frequência, de tal forma que uma abordagem de FWI multiescala é aplicada, em que os resultados com o conjunto de dados de 100 Hz são usados como entrada para o conjunto de 500 Hz. Nós investigamos dois casos: no primeiro (FWI T), somente as ondas registradas no poço oposto são usadas enquanto, no segundo caso (FWI T+R), as ondas registradas em ambos os poços são usadas. Para uma única interface separando dois meios, a forma da onda transmitida varia significantemente apenas quando a interface está contida dentro das aberturas angulares. Portanto, famílias de tiro comum para modelos de camadas com interfaces fora das aberturas angulares podem ser aproximadamente reproduzidas com um meio homogêneo equivalente. Dessa forma, em comparação com FATTT, ambos os casos de FWI resultam em uma melhoria moderada para modelos com interfaces dentro da cobertura angular, mas não conseguem compensar a falta de iluminação. Nessa situação, pequenos aumentos de resolução são obtidos tanto com FWI T como com FWI T+R. Contudo, para modelos na condição mista em que camadas com interfaces contidas na abertura angular são cortadas por uma falha, a FWI oferece melhorias substanciais sobre a FATTT, mesmo se o plano de falha está fora da cobertura angular e a FWI T é aplicada. Nessa situação mista, a resolução também aumenta quando FWI T+R e fontes de maior conteúdo de frequência são usadas.

Abstract

Seismic illumination at each point of the interwell region can be defined as the maximum an-gle between the rays that pass through the point. Interfaces completely contained in the angular apertures can be imaged with first arrival travel time tomography (FATTT), but interfaces comple-tely outside cannot be imaged even under regularized FATTT. We investigate if 2D acoustic full waveform inversion (FWI) can compensate for the lack of illumination. We use synthetic data generated with Ricker source wavelets with peak frequencies at 100 or 500 Hz, resulting in a small overlapping in the frequency bandwidths, so that a multiscale FWI approach is employed where the results with the 100 Hz dataset are used as input for the 500 Hz dataset. We investigate two FWI cases: in the first (FWI T), just the waves recorded at the opposite borehole are used whilst, in the second case (FWI T+R), the waves recorded at the two boreholes are used. For a single interface separating two media, the shape of the transmitted waveform varies significantly only when the interface is contained in the angular apertures. Accordingly, shot gathers for layered mo-dels with interfaces outside the angular apertures can be approximately reproduced with equivalent homogeneous media. As a result, in comparison with FATTT, both FWI cases give a mild impro-vement for models with interfaces inside the angular coverage, but cannot compensate for the lack of illumination. In this situation, minor resolution increases are obtained either with FWI T or FWI T+R cases. However, for models in the mixed condition where layers with interfaces inside the angular coverage are cut by a fault, FWI offers substantial improvements over FATTT, even if the fault plane is outside the angular coverage and FWI T is employed. In this mixed situation, resolution also increases when FWI T+R and source wavelets with a higher frequency content are used.

Agradecimentos

Agradeço, primeiramente, a Deus, por todas as coisas em minha vida.

Agradeço ao PRH 229 pela bolsa de mestrado e pelo apoio financeiro para participar do Congresso Internacional de Geofísica (SBGf 2017).

Agradeço aos meus familiares, Maria Cristina Tito, Maria Lucia de Oliveira Bastos, Severina Vicente Ferreira e Iracema Leopoldo por todo apoio e dedicação que elas sempre mostraram. Agradeço a Fabiana Cirino dos Santos por todo apoio, dedicação e conselhos em grande parte dessa jornada.

Agradeço ao meu orientador, Prof. Dr. Walter Eugênio de Medeiros, por ter me orientado por dois anos e pelas valiosas contribuições humanas.

Agradeço a Renato Ramos pela participação essencial nessa pesquisa, por toda ajuda e compreen-são durante toda essa jornada.

Agradeço a Jessé C. Costa pelas contribuições para o desenvolvimento dessa pesquisa.

Agradeço aos alunos do PPGG Elizangela Amaral, Gilsijane Vieira, Rafaela Silva, Marcio Barboza e Renato Ramos, pelos momentos de descontração e pelas importantes discussões e contribuições.

Sumário

1.1 Importância da tomografia sísmica . . . 1

1.2 Resolução em tomografia sísmica poço-a-poço . . . 2

1.3 Modelagem sísmica acústica . . . 2

1.4 Inversão da forma de onda completa . . . 3

2 Manuscrito submetido: Can full waveform inversion compensate for the lack of

illu-mination on crosswell tomography?" 4

Referências bibliográficas 63

Apêndice A: Modelagem direta 70

Apêndice B: Modelagem inversa 73

Capítulo 1

Contextualização

1.1

Importância da tomografia sísmica

A tomografia sísmica poço-a-poço é uma das técnicas clássicas da inversão de dados sísmicos, frequentemente aplicada na exploração de recursos minerais, monitoramento de reservatórios de hidrocarbonetos e imageamento de estruturas geológicas complexas (e.g.Franklin, 2009, Ajo-Franklin et al., 2007, Byun et al., 2010, Plessix, 2006b). A tomografia é importante para o estudo de áreas de descarte de lixo radioativo (e.g. Peterson et al., 1985), e também tem grande atuação em projetos de injeção de CO2(e.g. Ajo-Franklin, 2009, Byun et al., 2010, Harris et al., 1995). Ainda

na questão exploratória, pode ser utilizada na exploração mineral em minas, mapeando tanto corpos mineralizados de alta densidade como possíveis zonas de fraqueza, auxiliando na lavra da mina (e.g. Gustavsson et al., 1986). A tomografia também tem relevância em problemas de geofísca rasa. De Iaco et al. (2003) e Lanz et al. (1998) mostram a utilidade e as limitações da tomografia na delimitação das bases de aterros. Liu and Guo (2005) imageiam a distribuição de velocidades da coluna de concreto de uma ponte, de modo a avaliar a competência do material e localizar possíveis zonas de fraqueza. Outra aplicação da tomografia em em problemas de geofísca rasa é na investigação de sítios arqueológicos (Metwaly et al., 2005, Polymenakos and Papamarinopoulos, 2005).

CAPÍTULO 1. CONTEXTUALIZAÇÃO 2

1.2

Resolução em tomografia sísmica poço-a-poço

Os estudos sobre a resolução tomográfica poço-a-poço definiram que o aspecto principal que limita a resolução é a iluminação. Rector III and Washbourne (1994) mostraram que, para um modelo homogêneo, a limitação da cobertura angular acarreta em uma variação espacial da reso-lução do experimento. Essa variação se deve ao fato de que a resoreso-lução associada a determinado ponto do espaço está diretamente relacionada à maior abertura angular dos raios disponíveis para esse ponto. Menke (1984) e Rector III and Washbourne (1994) concluíram que para uma arranjo padrão com fontes e receptores igualmente espaçados em cada poço e dentro do mesmo intervalo de profundidade, a resolução máxima disponível para um modelo homogêneo é obtida no centro da região coberta pelos raios. Dantas and Medeiros (2016) evidenciaram que não é possível para a tomografia de tempos de trânsito reconstruir interfaces de alto ângulo devido à ausência de raios quase verticais, mesmo com a utilização de vínculos mais sofisticados.

1.3

Modelagem sísmica acústica

A modelagem sísmica é uma simulação do campo de ondas sísmicas, na qual são estabelecidas as amplitudes sísmicas em todo o tempo de registro e para cada par fonte-receptor (Rego, 2014). A Terra é um meio heterogêneo, anisotrópico, inelástico e dispersivo. Contudo, realizar uma modelagem considerando todas essas características incorpora um alto nível de complexidade ao problema e pode demandar um alto custo computacional. Assim, em geral, utilizamos modelos que fazem uma boa aproximação da realidade e possibilitam resultados satisfatórios, como no caso da modelagem acústica.

As modelagem sísmicas se baseiam no fato de que grandezas mensuráveis, como esforço e deformação, estão relacionadas através das leis constitutivas. A maneira como essas grandezas se relacionam depende do meio e em geral é possível agrupar a modelagem dos materiais em modelos constitutivos que incluem um ou mais comportamentos como os que são citados nas modelagens de elasticidade, plasticidade, viscoelasticidade, viscoplasticidade, dentre outras.

CAPÍTULO 1. CONTEXTUALIZAÇÃO 3 estão relacionados através da relação constitutiva para fluidos perfeitamente elásticos, que é uma generalização da Lei de Hooke, e também se relacionam pela segunda lei de Newton (Di Bartolo, 2010).

1.4

Inversão da forma de onda completa

A inversão de forma de onda completa (Full Waveform Inversion, ou FWI) é um dos méto-dos utilizaméto-dos para superar as limitações da teoria do raio e conseguir uma melhor resolução de imageamento de um meio. Ela se baseia em resolver numericamente a equação da onda durante a etapa de modelagem. Desse modo, elimina-se a necessidade de filtragem de várias fases que não seriam utilizadas pelas abordagens padrões (múltiplas, por exemplo), aproveitando-se quase todo o conteúdo do traço sísmico como sinal e utilizando essa informação adicional para melhorar substancialmente a resolução da imagem reconstruída (Virieux and Operto, 2009).

Teoricamente, a técnica de FWI é capaz de fornecer modelos de velocidade com maior reso-lução do que a tomografia de tempos de trânsito. Espera-se esse resultado porque a FWI tenta ajustar simultaneamente as informações de fase e amplitude obtidas através da equação da onda. Por outro lado, estudos já mostraram que a FWI necessita de um modelo de velocidade inicial próximo do modelo real, de modo a garantir sua convergência, sendo a construção de um modelo inicial adequado um dos grandes desafios para o uso dessa técnica.

Capítulo 2

Manuscrito submetido: Can full waveform

inversion compensate for the lack of

illumination on crosswell tomography?"

Can full waveform inversion compensate for the lack of

illumination in crosswell tomography?

Alex T. Oliveiraa,d_{, Walter E. Medeiros}b,d,∗_{, Renato R. S. Dantas}a,d_{, Jess´e C.}

Costac,d

a_{Programa de P´os-gradua¸c˜ao em Geodinˆamica e Geof´ısica,}

Universidade Federal do Rio G. do Norte - UFRN, Natal/RN, Brazil

b_{Departamento de Geof´ısica, Universidade Federal do Rio G. do Norte - UFRN,}

Natal/RN, Brazil

c_{Faculdade de Geof´ısica, Universidade Federal do Par´a - UFPA, Bel´em/PA, Brazil} d_{INCT-GP/CNPq/CAPES - Instituto Nacional de Ciˆencias e Tecnologia em Geof´ısica do}

Petr´oleo - CNPq, Brazil

Abstract

Seismic illumination at each point of the interwell region can be de-fined as the maximum angle between the rays that pass through the point. Interfaces completely contained in the angular apertures can be imaged with first arrival travel time tomography (FATTT), but interfaces completely outside cannot be imaged even under regular-ized FATTT. We investigate if 2D acoustic full waveform inversion (FWI) can compensate for the lack of illumination. We use synthetic data generated with Ricker source wavelets with peak frequencies at 100 or 500 Hz, resulting in a small overlapping in the frequency bandwidths, so that a multiscale FWI approach is employed where the results with the 100 Hz dataset are used as input for the 500 Hz dataset. We investigate two FWI cases: in the first (FWI T), just the waves recorded at the opposite borehole are used whilst, in the

∗_{Corresponding author}

Email addresses: alextdo.geophysics@gmail.com (Alex T. Oliveira),

walter.ufrn@gmail.com(Walter E. Medeiros), rrsdantas@gmail.com (Renato R. S. Dantas), jesse.ufpa@gmail.com(Jess´e C. Costa)

second case (FWI T+R), the waves recorded at the two boreholes are used. For a single interface separating two media, the shape of the transmitted waveform varies significantly only when the inter-face is contained in the angular apertures. Accordingly, shot gathers for layered models with interfaces outside the angular apertures can be approximately reproduced with equivalent homogeneous media. As a result, in comparison with FATTT, both FWI cases give a mild improvement for models with interfaces inside the angular coverage, but cannot compensate for the lack of illumination. In this situa-tion, minor resolution increases are obtained either with FWI T or FWI T+R cases. However, for models in the mixed condition where layers with interfaces inside the angular coverage are cut by a fault, FWI offers substantial improvements over FATTT, even if the fault plane is outside the angular coverage and FWI T is employed. In this mixed situation, resolution also increases when FWI T+R and source wavelets with a higher frequency content are used.

Keywords: Full waveform inversion, crosswell tomography, seismic tomography

1. Introduction

Crosswell tomography is a classic technique of seismic inversion which, in its simplest form, is based on the inversion of first ar-rival travel times of the transmitted waves between two boreholes (e.g. Lo and Inderwiesen, 1994). Crosswell tomography might also

5

be formulated as a full waveform inversion (FWI) (e.g. Pratt and Goulty, 1991). In fact a substantial part of the effort to develop

and understand FWI was done in crosswell problems. Belina et al. (2009) compare the results of FWI and travel time inversion in crosswell tomography using synthetic horizontally-layered

stochas-10

tic models and highlight the advantages and limitations of each ap-proach. Pratt et al. (1996) point out that inverting the waveform offers tomograms with higher resolution than the ones obtained with first arrival travel time inversion. The better resolution of FWI is a result of the fact that travel time inversion resolution is limited

15

by the width of the first Fresnel zone (Williamson, 1991), while the resolution of waveform inversion is of the order of half the wave-length (Pratt et al., 1996). The solution of travel time tomogra-phy might be suitable as an input to waveform inversion, due to its low wavenumber content, which is important to avoid cycle-skipping

20

(Pratt and Goulty, 1991; Song et al., 1995; Pratt, 1999). Trying to obtain the better from the two inversion approaches, Zhou et al. (1995) jointly invert travel time and waveform in crosswell tomog-raphy. In addition, Zhou and Greenhalgh (2003) normalize the am-plitude in the FWI misfit functional to attenuate the influence of

25

the highest amplitudes.

Either as first arrival inversion or as FWI, crosswell tomography has been applied in many problems such as oil reservoir charac-terization and monitoring (e.g. Mathisen et al., 1995; Pratt and Sams, 1996; Watanabe et al., 2004; Plessix, 2006b; Zhang et al.,

30

2007; Asnaashari et al., 2012; Hicks et al., 2016), hydrogeology, environmental, and engineering geology problems (e.g. Hyndman et al. 1994; Yamamoto et al. 1994; Daily and Ramirez 1995; Daley

et al. 2004; Moret et al. 2006; Almalki et al. 2013; Emery and Parra 2013; Rumpf and Tronicke 2014; Gheymasi et al. 2016), monitoring

35

gas carbon sequestration (e.g. Li, 2003; Gasperikova and Hover-sten, 2006; Saito et al., 2006; Ajo-Franklin et al., 2007; Daley et al., 2007; Franklin, 2009; Onishi et al., 2009; Byun et al., 2010; Ajo-Franklin et al., 2013), mineral exploration (e.g. Greenhalgh et al., 2003; Xu and Greenhalgh, 2010; Perozzi et al., 2012), and civil

engi-40

neering and archaeology problems (e.g. Soupios et al., 2011; Cheng et al., 2016; Butchibabu et al., 2017).

Compared with the seismic reflection method based on surface measurements, crosswell tomography might offer higher resolution because it uses higher frequency wavelet sources. However,

cross-45

well tomography has severe limitations associated with illumination (Menke, 1984; Rector III and Washbourne, 1994). Seismic illumina-tion at each point of the interwell region can be defined as the maxi-mum angle between the rays that pass through the point. Crosswell tomography based on first arrival travel time inversion cannot

im-50

age interfaces dipping in angles which are not contained in the an-gular coverage. In this situation, Dantas and Medeiros (2016) show that estimated tomograms are unreliable because they might con-tain artefacts with no correspondence to actual structures. To make matters worse, even inversion approaches incorporating constraints

55

might not alleviate this problem (Dantas and Medeiros, 2016). We investigate now if 2D acoustic full waveform inversion (FWI) can compensate for the lack of illumination, allowing to image interfaces outside the angular coverage or in mixed condition.

2. Methodology summary

60

2.1. Validating the acoustic modeling for the crosswell case

In the inversion results to be presented we assume that the inter-well region might be represented by a 2D isotropic non homogeneous medium described by its P-wave distribution. After discretizing the P-wave velocity field in a regular mesh, the resulting acoustic wave

65

equation is solved using a finite-difference scheme of second order in time and 14th order in space (Silva Neto et al., 2005). In order to reduce numerical dispersion and numerical anisotropy we optimized the spatial operators according to Holberg (1987).

The finite-difference modeling of the wave equation might present

70

undesirable reflections caused by the boundaries that artificially sim-ulate infinitely distant interfaces (e.g. Cerjan et al., 1985). These undesirable reflections might be eliminated or at least, highly at-tenuated by using absorbing boundary conditions (Cerjan et al., 1985; Sochacki et al., 1987; Gao et al., 2015). Figure 1 shows three

75

snapshots of a wave front propagating in a homogeneous isotropic medium that was generated at position 64 m in the left borehole for non absorbing and absorbing boundaries associated with the limits of the interwell region. By comparing the three pairs of snapshots one can conclude that the undesirable artificial reflections were

sat-80

isfactorily attenuated.

2.2. Full waveform inversion

Full waveform inversion (FWI) consists in estimating model pa-rameter fields based on the reproduction of the complete waveform

of the observed seismic dataset (e.g. Virieux and Operto, 2009), ac-cordingly to a given wave propagation assumption that is compatible with the observed dataset. For the 2D acoustic case, defining c(r) as the P-wave velocity at point r, and ugs(rg, c, t; rs) and vgs(rg, t; rs)

as respectively the modeled and measured seismic traces at time t (t_{∈ [0, T ]) and at points r}g due to a source located at point rs, the

FWI solution can be described as the minimum in relation to c(r) of the functional ψ = 1 T NgNs Ng X g Ns X s Z T 0 F (ugs, vgs)dt , (1)

where Nsand Ngare the number of source and measurement points,

respectively, and F (ugs, vgs) is the function defining the misfit

be-tween measured and modeled seismic traces.

85

We use the classic FWI version where F (ugs, vgs) is given by the

least-squares misfit function F (ugs, vgs) =

1

2σ2[ugs(rg, t; rs; c)− vgs(rg, t; rs)]

2_{, (2)}

being σ2 _{an estimate of the variance of F (u}

gs, vgs).

After discretizing the P-wave velocity field in a 2D mesh, min-imizing ψ (equation 1) is often solved with local methods of opti-mization. In the synthetic examples to be presented, the mesh used for inversion and modeling is the same. In addition, because FWI

90

(even in this simple acoustic formulation) is computationally very expensive and time-consuming, the adjoint-state method is the most common approach to calculate efficiently the gradient of ψ (Chavent, 1974; Tarantola, 1984; Plessix, 2006a; Chavent, 2010).

We use the conjugate gradient method (e.g. Press et al., 2007) to

95

minimize ψ (equation 1) and, in order to obtain better convergence, the gradient of ψ is preconditioned using the pseudo-Hessian oper-ator proposed by Shin et al. (2001). In this approximation to the Hessian, only the diagonal elements of this matrix are taken into account, their values being estimated from the autocorrelation of

100

the incident wavefield at each mesh point.

As stopping criteria, we impose a maximum number of iterations (50) or a Cauchy-type convergence criterion (e.g. Bartle, 1964) given by:

kck+1− ckk

kckk

<  , (3)

where ck and ck+1 are the estimates of c(r) at iterations k and

k+1, respectively, and  is a small positive number (typically around 10−3).

We will obtain FWI solutions for the two arrays outlined in Figure

105

2, where the source is always positioned in the left borehole. In the first array, named as FWI T, the generated wavefield is recorded just in the right borehole whilst in the second case, named as FWI T+R, the wavefield is recorded in both boreholes (except at the source point). Although we name the first FWI case as FWI T

110

(T as a mnemonic for transmission), note that in this array events caused by internal reflections might also be recorded at the right borehole. The advantages of the FWI T+R case over the FWI T case were studied by Bube and Langan (1995), Van Schaack (1997), and Bube and Langan (2008) for the first arrival travel time tomography

115

Except when it is explicitly stated, the seismic array for the FWI T case contains 64 sources, spaced of 2 m and located in the left borehole, and 64 receivers, spaced of 2 m and located in the right borehole. For the FWI T+R case, 127 receivers are then used for

120

each shot. In addition, for the conceptual models we treat, the first arrival in each receiver located at the left borehole is the wave propagating along a subvertical trajectory joining the receiver and the source. This event has a relatively high amplitude and adds no information to the velocity profile along the borehole, which is

125

assumed to be known. Because of the unfavorable influence of the high amplitude events in the classic FWI functional (equations 1 and 2) (e.g. Zhou and Greenhalgh, 2003), the first arrival is silenced (or at least strongly attenuated) by applying a mute filter in the form of a Gaussian window. For each pair source-receiver located in the

130

same borehole, the peak time of the Gaussian filter is estimated from the velocity profile along the borehole and the filter width is estimated from the source wavelet width.

We employ two seismic datasets generated with Ricker wavelets in the two frequency bands shown in Figure 3. The source wavelets

135

have peak frequencies at 100 and 500 Hz, so that there is little overlapping in the frequency content and the usual criteria (Sirgue and Pratt, 2004; Boonyasiriwat et al., 2009) for separating frequency bandwidths in multiscale FWI approaches (Bunks et al., 1995; Ficht-ner, 2011) are satisfied. To model the wave propagation in the 100

140

and 500 Hz cases, we use square meshes with sizes 25 m and 5 m, respectively.

3. Results

3.1. Dependence of waveform on angular coverage

The concept of seismic illumination in crosswell tomography as

145

result of angular coverage (Rector III and Washbourne, 1994; Dan-tas and Medeiros, 2016) is illustrated in Figures 4a and 4b for an isotropic homogeneous medium. For each point of the interwell re-gion, Figure 4a shows the angular aperture defined as the maximum angle between the rays that pass through the point. Figure 4b is

150

a simplified version where the interwell region is divided into just nine sectors and, for each sector, the angular aperture in its center is shown. Note that angular aperture varies significantly, being higher around the center of the interwell region.

Dantas and Medeiros (2016) show that interfaces completely

con-155

tained in the angular apertures, as the interfaces shown in Model 1 (Figure 5a), can be imaged with crosswell first arrival travel time to-mography. On the other hand, interfaces outside the angular aper-tures, as in Model 2 (Figure 5b), can not be imaged even under regularized inversion (Dantas and Medeiros, 2016). So the question

160

we answer in this work is: can FWI image the interfaces completely outside the angular apertures, as in Model 2, or at least in mixed condition?

Certainly the possibility of imaging the interfaces of Model 2 relies on their influence on the shape of the recorded waves. For

165

Models 1 and 2, Figure 6 shows the common shot gathers of the waves recorded in the right borehole due to a source located in the left borehole at three different depths (the green stars in Figure 5).

The source is a Ricker wavelet with peak frequency equal to 100 Hz. Note that the common shot gathers of Model 1 (Figures 6a, 6c,

170

and 6e) has comparatively much more geometric details than the one generated with Model 2 (Figures 6b, 6d, and 6f). To explain the shot gathers of Model 1 an heterogeneous medium is necessary. However, taking as example Figure 6d generated with the source located in the center of the left borehole, except for the presence of a slight

175

asymmetry and of delayed events of weak amplitude, this shot gather can be approximately reproduced with an isotropic homogeneous medium. In fact Figure 7, which was generated with a uniform medium with velocity equal to 2300 m/s, reasonably reproduces Figure 6d. The value 2300 m/s for the equivalent velocity results

180

from an approximate visual reproduction by trial-and-error of Figure 6d.

Let us now investigate how the waveform itself changes its shape in relation to the incidence angle with a single interface separat-ing two isotropic homogeneous media (Model 3), includseparat-ing the two

185

group of cases where the interface is contained or not contained in the angular aperture. As shown in Figures 8a, 8c, 8e, and 8g, the locations of source and receiver are kept fixed but the interface is rotated around the center point of the interwell region, being the recorded traces shown in Figures 8b, 8d, 8f, and 8h, respectively.

190

Note that the first arrival travel time varies significantly (Figure 9). The waveform variation every 10 degrees is shown in Figure 10. The cases where the interface is not contained in the angular aperture are shown in Figures 10a and 10c whilst the cases where the interface

is contained in the angular aperture are shown in Figures 10b and

195

10d. Comparatively, the variation of waveform shape is small for the cases where the interface is not contained in the angular aperture.

Both results described above sinalize that imaging interfaces out-side the angular aperture with crosswell tomography is a hard task even with FWI. In the following sections, we investigate this

ques-200

tion using the FWI T and the FWI T+R cases for the two source wavelets with peak frequencies equal to 100 and 500 Hz (Figure 3). For all shown results, the velocity distribution referred as the initial model is the starting model for the FWI 100 Hz T and FWI 100 Hz T+R cases. On the other hand, the final results of the FWI 100 Hz

205

T and FWI 100 Hz T+R cases are the starting models for the FWI 500 Hz T and FWI 500 Hz T+R cases, respectively.

3.2. Vertical interface

In Figure 11a one of the cases of vertical interfaces of Model 3 is shown. This is the worst situation for crosswell tomography

imag-210

ing (Rector III and Washbourne, 1994; Dantas and Medeiros, 2016). Figure 11b is the initial model for FWI, which is a linear interpola-tion between the velocity values at the two boreholes. In addiinterpola-tion, Figures 11c, 11d, 11e, and 11f are the tomograms resulting from the FWI 100 Hz T, FWI 500 Hz T, FWI 100 Hz T+R, and FWI 500

215

Hz T+R cases, respectively. The true and initial models, besides all FWI results, are shown in Figure 12 as horizontal profiles pass-ing through the interwell center. None of the FWI results is even a reasonable reproduction of the true model. In fact, the changes in relation to the initial model are small and there are almost no

provements with the frequency increase of the source wavelet. Also, employing the waves reaching at the left borehole (T+R cases) did not add significant improvements and even some spurious oscilla-tions were introduced (marked by the arrow in Figure 12).

3.3. Layers completely inside or outside the angular coverage

225

We now apply FWI to Models 1 and 2 (Figures 5a and 5b, re-spectively). Figure 13 shows the gradients at the first iteration for both Model 1 (left column of Figure 13) and Model 2 (right column of Figure 13) for the cases FWI T (upper row of Figure 13) and FWI T+R (lower row of Figure 13), in all cases with 500 Hz. For Model

230

1, the gradient is sensitive to the velocity contrasts and interfaces for both FWI T (Figure 13a) and FWI T+R (Figure 13c) cases, with practically no improvement from the T to the T+R case. On the other hand, for Model 2 the striking features of the gradient do not conform with the interfaces for the FWI T case (Figure 13b)

235

or show spurious features of the same magnitude of those associ-ated with the interface for the FWI T+R case (for example, see the features inside the rectangle in Figure 13d).

The true and initial models, besides the FWI results, are shown in Figures 14 and 15 for Models 1 and 2, respectively. In addition,

240

Figure 16 shows vertical or diagonal (left column) and horizontal (right column) profiles along the tomograms. From now on, all shown initial models were obtained from a non linear first arrival travel time regularized tomography using ray tracing (e.g. Dantas and Medeiros, 2016). All FWI results reproduce satisfactorily the

245

1 and 2, respectively, the shot gathers for the source positioned at depth 64 m of the recorded, fitted, and residual wavefields in the 500 Hz cases.

For Model 1, because all interfaces are inside the angular coverage

250

(Figure 5a), the initial model (Figure 14b) is already a good esti-mate of the true model (Figures 14a and 14b). Nonetheless, quite good improvements of the velocity contrats were obtained with FWI (Figures 16a and 16b). In addition, some spurious artefacts were even reduced when the T+R array is employed or two frequencies

255

were used (see features near the right borehole in Figures 14c-14f). On the other hand, for Model 2 (Figure 5b) no significant improve-ments on the velocity contrasts were obtained, even for the T+R cases (Figure 15). Basically FWI introduced oscillations around the initial solution (Figures 16c and 16d). In some cases, it appear that

260

these oscillations are related with the corners of the velocity con-trasts, as the case marked by a arrow in Figure 16c; however, there are other oscillations that show no correlation with corners, as the cases marked by arrows in Figure 16d.

3.4. Horizontal layers cut by a vertical fault

265

Each model above treated falls into one of the two extreme cate-gories: the interfaces are completely inside or completely outside the angular coverage. Now we treat a mixed case (Model 4) shown in Figure 19a, where horizontal layers (whose interfaces are completely inside the angular coverage) are cut by a vertical fault (a plane

com-270

pletely outside the angular coverage). The initial model, besides the FWI results, are shown in Figures 19b to 19f. In addition, Figure

20 shows a vertical profile along the tomograms at position 15 m. The initial model (Figure 19b) allows the interpreter to infer the presence of vertical velocity contrasts. However it is not possible to

275

infer the fault because the tomogram features might be explained with curved deposition surfaces. As expected, this first arrival travel time tomogram is a very smoothed version of the true velocity dis-tribution (Figure 20). On the other hand, the fault presence can be readily inferred from any of the FWI results (Figures 19c to 19f),

280

particularly in the FWI T+R cases (Figures 19e and 19f), in spite of the presence of some oscillations in the estimated velocity profiles (see arrows in Figure 20).

The results of this model evidence that discontinuities, such as faults, cutting interfaces contained in the angular coverage might be

285

well imaged with FWI even when the discontinuity plane is outside the angular coverage.

3.5. A realistic layered sequence cut by a dipping fault

We investigate now in more detail using Model 5 (Figure 21) the possibility of imaging with FWI a complex layered sequence cut by

290

a subvertical fault. In this model, we use 80 sources in the left bore-hole and 80 receivers in the right borebore-hole, both spaced every 1.0 m. Model 5 was designed to represent a realist sedimentary case, where a curved erosional surface located around depth 20 m (Figure 21a) separates two major sedimentary sequences. The velocity

val-295

ues were attributed to the modeled lithologies according to Sch¨on (2015). Above the erosional surface, it was deposited a sandstone package and, below the erosional surface, there are three

sedimen-tary packages (Figure 21b) representing a sandstone sequence (the dark blue one in Figure 21a) intercalated between two shale

se-300

quences. Note that each sedimentary package is formed by thin layers showing velocity variation (Figure 21b), including a high ve-locity thin layer around depth 45 m. In addition, note that the layer package above the erosional surface is dipping (_{≈ 20}o_{) and}

that the upper part of the interwell region, where it is located, has

305

very poor angular coverage (Figure 4), so that the layer interfaces are in most cases outside the local angular aperture. Note also that a subvertical normal fault affects just the sedimentary package be-low the erosional surface (Figure 21a). This fault might be possibly inferred from the vertical shift in the velocity profiles of the two

310

boreholes (Figure 21b). However, this fault is syndepositional be-cause its offset varies with depth, a characteristic that is clear in the true model (Figure 22a) because the offset of the high velocity thin layer around depth 45 m is smaller than the offset of the thick sandstone sequence. This characteristic of the fault could hardly be

315

inferred from the velocity profiles in the boreholes (Figure 21a). The true and initial models, besides the FWI results, are shown in Figure 22 and vertical profiles along the tomograms at position 75 m are shown in Figure 23. As in the previous example, the initial model (Figure 22b) allows the interpreter to infer the presence

320

of the main vertical velocity contrasts, besides the lateral velocity variation above the erosional surface. However, inferring a fault from this tomogram is a hard task because their features might be explained with curved deposition surfaces. Note that this first

arrival travel time tomogram, besides being a very smoothed version

325

of the true velocity distribution (Figure 23), has spurious artefacts particularly around the fault region (Figure 22b). In addition, no velocity variations inside the sedimentary packages can be inferred (Figure 23) and the geometry of the erosional surface is wrongly imaged, possibly due to the superposition of the referred spurious

330

artefacts around the fault region. On the other hand, the FWI results, even for the 100 Hz T case, show clearly the fault presence, the velocity variation inside the sedimentary packages (Figure 23), and the correct geometry of the erosional surface. In particular, the FWI T+R 500 Hz result (Figure 22f) shows very good resolution

335

and images all relevant features of the model, including the sharp boundaries associated with the fault, even in the region where the high velocity thin layer is present (depth 45 m). Because of this good resolution, the fact that the fault is syndepositional can be inferred from Figure 22f, due to the clear imaged variation with depth of the

340

fault offset.

4. Conclusions

In comparison with the classic first arrival regularized tomo-grams, for the tested class of models FWI gives a mild improvement in the case where all interfaces are completely inside the angular

cov-345

erage, but FWI can not compensate for the lack of illumination in crosswell tomography when the interfaces are completely outside the angular coverage. In this extreme case, minor resolution increases are obtained with FWI, even when the waves recorded in the two

boreholes are taken into account. However, in the mixed and very

350

important case where discontinuities, such as faults, cut interfaces contained in the angular coverage, the FWI results offer substantial improvements over the first arrival tomograms, even when the dis-continuity plane is outside the angular coverage and only the waves regisitered at the opposite borehole are employed. In this case,

reso-355

lution also increases in the tomograms after taking into account the waves recorded in the two boreholes and employing source wavelets with a higher frequency content.

5. Acknowledgments

The Human Resources Training Program PRH-229

(PETRO-360

BRAS, UFRN, and ANP) is thanked for the MSc scholarship to ATO. The Brazilian agency CNPq is thanked for the PhD scholar-ship to RRSD and the research fellowscholar-ships and associated grants to WEM and JCC. The financial support to purchase the computa-tional infrastructure used in this study was given by the INCT-GP

365

(CNPq/CAPES).

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List of Figures

Figure 1. Snapshots in a homogeneous isotropic medium

570

with P-wave velocity equal to 3000 m/s at propaga-tion times equal to 21 ms (upper row), 32 ms (middle row), and 43 ms (bottom row). A 100 Hz Ricker wavelet was generated at depth 64 m in the left bore-hole. Left and right columns show results for non

ab-575

sorbing and absorbing boundary conditions, respec-tively. . . 36 Figure 2. Schematic figure showing the two seismic arrays

used in this study for Full Waveform Inversion (FWI). A source (the star) positioned in the left borehole B1

580

generates the incident wave (I), that propagates to a point P of an interface and generates transmitted (T) and reflected (R) waves, which are respectively recorded in the right (B2) and left (B1) boreholes (at the triangles). In this simplified figure, internal

585

reflections in the interwell region generating events that might also be recorded at the right borehole are not included. In the first FWI array, only the waves recorded at the opposite borehole B2 are used whilst, in the second array, both the waves recorded at

bore-590

holes B1 and B2 are used (except at the point co-incinding with the source). For the sake of simplicity, we refer to the first and second FWI cases as FWI T and FWI T+R, respectively. . . 37

Figure 3. Ricker wavelets used as source signatures for

595

FWI. The wavelets have peak frequencies at 100 Hz (in black) and 500 Hz (in red). Note that there is little overlapping in the frequency content. . . 38 Figure 4. Seismic illumination in crosswell tomography

as result of angular coverage for an isotropic

homoge-600

neous medium. For each point of the interwell region, the angular aperture defined as the maximum angle between the rays that pass through the point is shown in (a). A simplified version is given in (b), where the interwell region is divided into just nine sectors and,

605

for each sector, the angular aperture in its center is shown. Adapted from Dantas and Medeiros (2016). . 39 Figure 5. Models 1 (a) and 2 (b) which have interfaces

completely inside or completely outside, respectively, the available angular coverage. That is, in (a) the

610

dip of the interface at every point is contained in the angular aperture at the point whilst, in (b), it is not contained. The green stars show the source positions that generate the shot gathers shown in Figure 6. Adapted from Dantas and Medeiros (2016). . . 40

Figure 6. Shot gathers for Model 1 (left column) and Model 2 (right column) formed with the wavefield recorded at the opposite borehole for sources located at depths 12 m (upper row), 64 m (middle row), and 116 m (bottom row). The source positions are shown

620

as green stars in Figure 5. The source wavelet is a Ricker pulse with peak frequency at 100 Hz (Figure 3). 41 Figure 7. Shot gather formed with the transmitted

wave-field in an isotropic homogeneous medium with veloc-ity equal to 2300 m/s for a source located at depth

625

64 m, which is at the center of the left borehole. The source wavelet is a Ricker pulse with peak frequency at 100 Hz. This shot gather reasonably reproduces Figure 6d, except for the slight asymmetry and de-layed events of weak amplitude in the latter figure. . 42

Figure 8. Model 3 - Synthetic experiment showing how time arrival and shape of a Ricker pulse (peak fre-quency at 100 Hz) change in relation to the incidence angle with a plane interface separating two isotropic homogeneous media with velocities equal to 2000 m/s

635

(white region) and 3000 m/s (black region). Four cases of the interface angle are shown in the left col-umn and, for each case, the resulting trace is shown at the right in the same row. The source (green star) and receiver (red triangle) positions are kept fixed but

640

the interface is rotated around the center of the inter-well region. The blue lines show the angular aperture at the center. The trace amplitudes are normalized by the maximum value of the four traces. The first arrival travel time varies significantly as shown in

Fig-645

ure 9. . . 43 Figure 9. Model 3 - First arrival travel times (red curve)

for the synthetic experiment outlined in Figure 8. The black line shows the travel time for the approxi-mate straight ray trajectory. . . 44

Figure 10. Model 3 - Waveform variation every 10 de-grees of the synthetic experiment outlined in Figure 8. The two groups of cases where the interface angle is not contained in the angular aperture at the center of the interwell region (the rotating point of the

in-655

terface) are shown in (a) and (c). On the other hand, the two groups of cases where the interface angle is contained in the angular aperture are shown in (b) and (d). The trace amplitudes are normalized by the maximum value of all traces. The source wavelet is a

660

Ricker pulse with peak frequency at 100 Hz. . . 45 Figure 11. Model 3 - True model (a), initial model (b),

and FWI results for the 100 Hz T (c), 500 Hz T (d), 100 Hz T+R (e), and 500 Hz T+R (f) cases. The initial model in (b) is a linear interpolation between

665

the velocity values at the two boreholes. . . 46 Figure 12. Model 3 - Horizontal profiles at depth 64

m along the velocity distributions shown in Figure 11. None of the FWI results is even a reasonable reproduction of the true model. . . 47

Figure 13. Models 1 and 2 - Gradients at the first iter-ation for Model 1 (left column) and Model 2 (right column) for the cases FWI T (upper row) and FWI T+R (lower row). In all cases the source wavelet is a Ricker pulse with peak frequency at 500 Hz. The true

675

interfaces are shown in black (left column) or white (right column) lines. The rectangle in (d) contains spurious features of the same magnitude of those as-sociated with the interfaces. . . 48 Figure 14. Model 1 - True model (a), initial model (b),

680

and FWI results for the 100 Hz T (c), 500 Hz T (d), 100 Hz T+R (e), and 500 Hz T+R (f) cases. The initial model in (b) was obtained from a non linear first arrival travel time regularized tomography using ray tracing (e.g. Dantas and Medeiros, 2016). The

685

true interfaces are shown in black lines. . . 49 Figure 15. Model 2 - True model (a), initial model (b),

and FWI results for the 100 Hz T (c), 500 Hz T (d), 100 Hz T+R (e), and 500 Hz T+R (f) cases. The initial model in (b) was obtained from a non linear

690

first arrival travel time regularized tomography using ray tracing (e.g. Dantas and Medeiros, 2016). The true interfaces are shown in white lines. . . 50

Figure 16. Models 1 and 2 - Profiles of the FWI results shown in Figures 14 and 15, respectively. For Model

695

1, vertical profiles at position 64 m (a) and horizon-tal profiles at depth 64 m (b); for Model 2, profiles along the diagonal direction that is perpendicular to the interfaces (b) and horizontal profiles at depth 64 m (d). The arrow in (c) marks oscillations in the

700

FWI results that are possibly related with a corner of the velocity contrast whilst the arrows in (d) mark oscillations that apparently show no correlation with corners. . . 51 Figure 17. Model 1 - Shot gathers for the source

po-705

sitioned at depth 64 m of the observed (upper row), modeled (middle row), and residual (lower row) wave-fields for the FWI T (left column) and FWI T+R (right column) cases. In all cases, the source wavelet is a Ricker pulse with peak frequency at 500 Hz. The

710

shot gathers for the T+R cases (right column) are in fact the superposition of the shot gathers observed in the two boreholes; in these cases, the channel identi-fies the two receivers which are at the same depth in the two boreholes. . . 52

Figure 18. Model 2 - Shot gathers for the source po-sitioned at depth 64 m of the observed (upper row), modeled (middle row), and residual (lower row) wave-fields for the FWI T (left column) and FWI T+R (right column) cases. In all cases, the source wavelet

720

is a Ricker pulse with peak frequency at 500 Hz. The shot gathers for the T+R cases (right column) are in fact the superposition of the shot gathers observed in the two boreholes; in these case, the channel identi-fies the two receivers which are at the same depth in

725

the two boreholes. . . 53 Figure 19. Model 4 - True model (a), initial model (b),

and FWI results for the 100 Hz T (c), 500 Hz T (d), 100 Hz T+R (e), and 500 Hz T+R (f) cases. The initial model in (b) was obtained from a non linear

730

first arrival travel time regularized tomography using ray tracing (e.g. Dantas and Medeiros, 2016). The fault plane is shown in dotted white line. . . 54 Figure 20. Model 4 - Vertical profiles at position 15 m

along the velocity distributions shown in Figure 19.

735

The black arrows mark oscillations in the FWI re-sults. . . 55

Figure 21. Model 5 - Realistic layered sequence cut by a syndepositional subvertical fault (a) and velocity profiles at the two boreholes (b). The black and red

740

lines in (b) identify the velocity profiles in the left and right boreholes, respectively. . . 56 Figure 22. Model 5 - True model (a), initial model (b),

and FWI results for the 100 Hz T (c), 500 Hz T (d), 100 Hz T+R (e), and 500 Hz T+R (f) cases. The

745

initial model in (b) was obtained from a non linear first arrival travel time regularized tomography using ray tracing (e.g. Dantas and Medeiros, 2016). The fault plane is shown in black line. . . 57 Figure 23. Model 5 - Vertical profiles at position 75 m

750

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