Reflectors in wave equation imaging

doi:10.1016/j.wavemoti.2008.01.001

Wave Motion

Volume 45, Issue 6, June 2008, Pages 776-784

https://doi.org/10.1016/j.wavemoti.2008.01.001 Get rights and content

Abstract

We consider the problem of imaging with the wave equation from backscatter. It is well known that the velocity can be recovered from multi-static measurements provided the source signature has zero frequencies. We show that in the presence of a plane reflector zero frequencies are not needed. The analysis is done within the Born approximation. We show by numerical simulations that this also holds in the fully nonlinear case, even for arbitrarily shaped and unknown reflectors.

Introduction

In this paper we consider the following imaging geometry: an object is placed in a slab in $R^{n}$ . Sources and receivers are placed on one of the surfaces of the slab. The field is measured for all sources at all receivers in that surface, i.e. we measure only the backscatter, not the transmitted waves (We use the word backscatter here in a wider sense than usual, i.e. we do not require zero offset geometry.). The problem is to recover the object from these measurements.

To be more specific we assume that the propagation obeys the wave equation $\frac{\partial^{2} u}{\partial t^{2}} = c^{2} (Δ u + q (t) δ (x - s))$ Here, $c = c (x), x \in R^{n}$ is the local speed of propagation. For $x \in R^{n}$ we write $x = (x^{'}, x_{n}), x^{'} \in R^{n - 1}$ . We assume that c² = c₀(x)²/(1 + f) where f vanishes outside the slab 0 ⩽ x_n ⩽ D and c₀ is the known – not necessarily constant – background velocity. This form of c is convenient for the computations to follow. s is the source location, s_n = D, and q is the source wavelet which is assumed to vanish for t < 0. The receivers r are sitting on the source plane x_n = D as well. We always assume u = 0 for t < 0.

It is well known that this problem can be solved fairly easily if the source wavelet q is δ-like, meaning that it has low frequencies [2]. Adjoint methods are the reconstruction methods of choice [8], [9]. Unfortunately, in practice low frequencies are not available. We note in passing that the problem with the missing low frequencies disappears if transmission measurements can be taken, i.e. if sources and receivers are on either side of the slab; see [8].

The absence of low frequencies has two consequences: first, one can show by Fourier analysis, that $\hat{f}$ is determined only outside the balls of radius k_min around (0, ±k_min), k_min being the wave number corresponding to the smallest frequency contained in q [8]. Thus, if the object has significant energy inside these balls, which is typically the case for objects without reflectors, there is no way to determine it. Second, even if the object has most of its energy outside these balls, its determination is difficult due to the presence of local minima in the cost functional [13].

In this paper we study the effect of reflectors. It belongs to the folklore of seismic imaging that the presence of reflectors greatly facilitates the reconstruction of the subsurface; see [3] for the basics and [4], Section 1A, for a state-of-the-art exposition. In the simplest case we assume that the reflector is the plane x_n = 0. We model this reflector by stipulating the boundary condition $\frac{\partial u}{\partial x_{n}} = 0 on x_{n} = 0 .$

We show by Fourier analysis that in the presence of this reflecting plane whose position is known $\hat{f}$ is uniquely determined by the backscatter. This is done within the Born approximation. For the fully nonlinear case we show by numerical simulations and heuristics that adjoint methods easily recover f from the backscatter, provided that the initial approximation f₀ satisfies the condition $|\int (f - f_{0}) d s| ⩽ λ$ where the integral is along geodesics of the background and λ is the largest wavelength that is contained with sufficient strength in the source wavelet q. A possible application is the CARI version of ultrasound mammography, where a metal plate is used as reflector [12]. If the position and shape of the reflectors is not known, as is the case e.g. in seismic imaging, we are not able to prove uniqueness, but adjoint methods still work.

Summing up, the presence of reflectors (be they known or unknown) greatly improves the imaging from backscatter. Adjoint methods work well provided the initial approximation is good enough so as to satisfy (1.3).

We note that the role of a plane reflector was studied in [10] for the monostatic case and limited aperture in the context of synthetic aperture radar (SAR). Our analysis does not apply to this case, nor can adjoint methods be expected to work.

Section snippets

Fourier analysis

We carry out the Fourier analysis for a plane reflector that is identified with the plane x_n = 0. This is modeled by the boundary condition ∂u/∂x_n = 0 on x_n = 0. For the Fourier transform in Rⁿ we use the notation $\hat{f} (ξ) = (2 π)^{- n / 2} \int_{R^{n}} e^{- i x \cdot ξ} f (x) d x$ and correspondingly for the inverse Fourier transform $\tilde{f}$ . Inverse Fourier transforming (1.1) with respect to t we obtain $Δ \tilde{u} + k^{2} (1 + f) \tilde{u} = \tilde{q} (ω) δ (x - s)$ where k = ω/c₀ is the wave number. The initial condition u = 0 for t < 0 translates into the Sommerfeld radiation condition

Uniqueness and resolution

Now we want to find out to which extent f is determined by (2.9) for |ρ′|, |σ′| ⩽ k. We restrict ourselves to the 2D case. Fix ξ′ ∈ [0, 2k] and consider ρ′, σ′ for which ρ′ + σ′ = ξ′. Put $a_{k, ξ^{'}}^{\pm} (ρ^{'}) = a (ρ^{'}) \pm a (σ^{'}) .$ ρ′ can assume all the values in the interval [ξ′ − k, k]. Putting $h_{ξ^{'}} = C \hat{f} (ξ^{'}, \cdot)$ we can rewrite (2.9) as $G_{k, ξ^{'}} (ρ^{'}) = h_{ξ^{'}} (a_{k, ξ^{'}}^{+} (ρ^{'})) + h_{ξ^{'}} (a_{k, ξ^{'}}^{-} (ρ^{'})),$ where $G_{k, ξ^{'}} (ρ^{'}) = (2 π)^{(1 - n) / 2} \frac{2}{A} \hat{g} (ρ^{'}, σ^{'})$ is determined by the data. Thus the question is whether (3.1) for ξ′ − k ⩽ ρ′ ⩽ k determines $h_{ξ^{'}}$ uniquely.

Fig. 1 shows the functions

Kaczmarz’ method

In our view Kaczmarz’ method is the method of choice for solving the inverse problem for (1.1) numerically. It is widely used in X-ray computed tomography and, in its nonlinear version, in inverse scattering; see Chap. 7 of [9]

Kaczmarz’s method is an iterative method of the adjoint type. Let $g_{s^{'}} (r^{'}, t) = (R_{s^{'}} (f)) (r^{'}, t) = u (r, t)$ where r = (r′, D) and u is the solution of (1.1). Starting out from an initial approximation for f the update is described by $f \leftarrow f - α {(R_{s^{'}}^{' (f)})}^{*} C_{s^{'}} (R_{s^{'}} (f) - g_{s^{'}})$ where α is a relaxation

Numerical experiments

In this section we demonstrate that Kaczmar’s method works for imaging with known or unknown reflectors, even if the source wavelet does not have zero frequencies. This is to be expected from our analysis in Section 2.

As a first example we choose the lens phantom from [13]: $c (x_{1}, x_{2}) = c_{0} - a \exp \{(- (x_{1}^{2} + {(x_{2} - 0.5)}^{2})/ 0.09)\}$ where a = 0.4 in the first example and c₀ = 2 is the background velocity. In Fig. 3 we display the corresponding function $f = c_{0}^{2} / c^{2} - 1$ in a rectangle of size 8 × 2. The bottom boundary x₂ = 2 serves as

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Reflectors in wave equation imaging

Abstract

Introduction

Section snippets

Fourier analysis

Uniqueness and resolution

Kaczmarz’ method

Numerical experiments

Ultrasonic Imaging

Multiscale seismic waveform inversion

Geophysics

Fundamentals of Geophysical Data Processing

Microlocal analysis of seismic inverse scattering