Waveﬁeld reconstruction inversion modelling of Marchenko focusing functions Ruhul F. Hajjaj12 Sjoerd A.L. de Ridder1 Philip W. Livermore1

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Waveﬁeld reconstruction inversion modelling of

Marchenko focusing functions

Ruhul F. Hajjaj∗1,2, Sjoerd A.L. de Ridder1, Philip W. Livermore1,

and Matteo Ravasi3

1University of Leeds, Leeds, United Kingdom.

2Institut Teknologi Sumatera, Lampung, Indonesia.

3King Abdullah University of Science and Technology, Thuwal, Saudi Arabia.

October 2022

Abstract

Marchenko focusing functions are in their essence waveﬁelds that sat-

isfy the wave equation subject to a set of boundary, initial, and focusing

conditions. Here, we show how Marchenko focusing functions can be

modeled by ﬁnding the solution to a waveﬁeld reconstruction inversion

problem. Our solution yields all elements of the focusing function includ-

ing evanescent, refracted, and trapped waves (tunneling). Our examples

indicate that focusing function solutions in higher dimensions are how-

ever challenging to compute by numerical means due to the appearance

of strong evanescent waves.

1 Introduction

Focusing functions are a new concept in waveﬁeld propagation that lies at the

foundations of various Marchenko imaging schemes. They enable Green’s func-

tion retrieval inside a medium from a single-sided (surface based) recording of

seismic reﬂection data. The Marchenko equations provide a relationship be-

tween the Green’s function in the interior of a medium (GD) and the reﬂection

data at the surface (R) via the focusing functions (f) (Burridge, 1980; Brog-

gini and Snieder, 2012; K. Wapenaar et al., 2014; K. Wapenaar, Snieder, et al.,

2021). Conventionally, these kinds of focusing functions are estimated from the

reﬂection data in a data-driven manner by means of Neumann series of direct

inversion, e.g. (Broggini et al., 2014; van der Neut et al., 2015; Vargas et al.,

2021). Even though focusing functions are solutions of the wave equation them-

selves, they are traditionally thought of more as a mathematical than a physical

∗Email: eerﬁ@leeds.ac.uk. This document constitutes a summary of results in the PhD

Transfer Report of R.F. Hajjaj, successfully defended on October 19th, 2022.

arXiv:2210.14570v1 [physics.geo-ph] 26 Oct 2022

entity (C. A. et al., 2018). To our knowledge, there are no theoretical proofs

available for model-based validations in dimension higher than 1D.

Marchenko focusing function modelling can be done using full-waveﬁeld prop-

agation methods when a detailed subsurface model is available. Elison et al.,

2021 use the two-way waveﬁeld extrapolation method to model focusing func-

tions (Kosloﬀ and Baysal, 1983; C. P. Wapenaar and Berkhout, 1986). Focus-

ing state is achieved by incorporating the focusing boundary condition that is

deﬁned for the pressure (a delta function, or its spatio-temporal bandlimited

version) and the particle velocity ﬁeld (i.e. the vertical derivative of the delta

function) (C. P. A. Wapenaar, 1993). The method eliminates evanescent and

downward propagating waves at each integration step e.g. using spectral projec-

tors (Sandberg and Beylkin, 2009). This approach had earlier been indicated by

Becker et al., 2016 and K. Wapenaar et al., 2017. Meanwhile, focusing functions

will not be able to perfectly focus the waveﬁeld as long as evanescent waves are

not compensated (K. Wapenaar, Brackenhoﬀ, et al., 2021).

In this study, we follow a model-driven approach to compute focusing functions.

We deﬁne focusing function modelling as a full-waveﬁeld reconstruction inverse

problem in actual (non-truncated) media. In this way, functions can be gen-

eralised to accommodate full-waveﬁelds propagation that can ultimately lead

to more general Marchenko schemes, with the ability to accurately image steep

ﬂanks and to account for evanescent and refracted waves.

2 Algorithm

2.1 Deﬁning the focusing functions

A focusing waveﬁeld is a solution to the wave equation that forms a focus

in space on a horizontal plane at a given focal depth (i.e. the focal plane).

We deﬁne a focusing waveﬁeld as the solution to a partial diﬀerential equation

(PDE) and a set of boundary, initial, ﬁnal, and focusing conditions, in and

on a domain. The spatial boundaries of the domain are open (i.e. we impose

no boundary condition) and the waveﬁeld has support throughout the domain,

with the exception of the focal plane, where the support is conﬁned to the focal

point. The focusing waveﬁelds satisfy the following conditions:

ρ(x)∇T1

ρ(x)∇ − 1

c2(x)

∂2

∂t2F±(x,xf, t) = 0,(1)

F±(x,xf, t) = 0,for t↓ −∞,(2)

F±(x,xf, t) = 0,for t↑+∞,(3)

and

F±(x,xf, t)z=zf

=δ(t)δ(xH−xH,f ),(4)

"∂

∂z ∓s1

c2(z)

∂2

∂t2−∂2

∂x2−∂2

∂y2#F±(x,xf, t)z=zf

= 0,(5)

where ∇T=∂

∂x ,∂

∂y ,∂

∂z , and the subscript ±of Fdenotes respectively an

down- or upgoing focusing solution at z=zf. These conditions are illustrated

in Figure 1. The initial and ﬁnal conditions are required to eliminate a null-

space containing waves that enter or leave the domain boundaries, without

passing through the focal level (and therefore avoid constraint by the focusing

conditions).

Figure 1: Conceptual illustration of a focusing waveﬁeld (here a down-going

focusing waveﬁeld). (a) Focusing conditions at z=zf(the focal plane) consist

of the condition that the waveﬁeld forms an impulse at x=xH,f and obeys the

one-way wave equation (for down-going waves). Ergo, there is no propagation

on the focal plane. Above and below the focal plane there are super-positions

of both up- and down-going waves. (b) Illustration of the domain’s boundary,

initial, ﬁnal, and focusing conditions on the waveﬁeld.

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摘要：

WaveeldreconstructioninversionmodellingofMarchenkofocusingfunctionsRuhulF.Hajjaj*1,2,SjoerdA.L.deRidder1,PhilipW.Livermore1,andMatteoRavasi31UniversityofLeeds,Leeds,UnitedKingdom.2InstitutTeknologiSumatera,Lampung,Indonesia.3KingAbdullahUniversityofScienceandTechnology,Thuwal,SaudiArabia.October202...

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Waveﬁeld reconstruction inversion modelling of Marchenko focusing functions Ruhul F. Hajjaj12 Sjoerd A.L. de Ridder1 Philip W. Livermore1

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