Techniques for Reconstructing a Riemannian Metric Via the Boundary Control Method

In this dissertation, we consider some new techniques related to the solution of the inverse boundary value problem for the wave equation with partial boundary data. Most results are formulated in a geometric setting, where waves propagate in the interior of a smooth manifold with smooth boundary M, and the wave speed is modelled by an unknown Riemannian metric g. For data, we focus mostly on using the Neumann-to-Dirichlet (N-to-D) map with sources and receivers restricted to a measurement set Γ ⊂ ∂M. The goal of the inverse problem, in this setting, is to use these wave boundary measurements to recover the geometry of (M, g) near the measurement set. We note that this geometric perspective accomodates, as special cases, both the scalar acoustic wave equation and elliptically anisotropic wave speeds. We consider three problems. In the first problem, we provide a technique to use the N-to-D map to construct the travel times between interior points with known semi-geodesic coordinates and boundary points belonging to Γ. Such travel times can be used to reconstruct the metric in semi-geodesic coordinates using one of several existing techniques, so this procedure can be viewed as providing a data processing step for a metric reconstruction procedure. In the second problem, we consider a redatuming procedure, where we use data on the boundary and known near-boundary geometry to synthesize wave measurements in this known near-boundary region. This allows us to construct a map which plays a similar role to the N-to-D map, but for interior sources and interior measurements. Our motivation for this procedure is that it can serve as a data propagation step for a layer stripping reconstruction method, in which one first reconstructs the metric near the boundary and then propagates data into this region to serve as data for an interior reconstruction step. In the third problem, we restrict attention to the case where M is a domain in Rn, and consider two related procedures to use the N-to-D map or Dirichlet-to-Neumann (D-to-N) map to directly reconstruct the metric. In the anisotropic case, we construct the metric in semi-geodesic coordinates via reconstruction of the wave field in the interior of the domain. In the isotropic case, we can go further and construct the wave speed in the Euclidean coordinates via reconstruction of the coordinate transformation from the boundary normal coordinates to the Euclidean coordinates. In addition to providing constructive procedures, we analyze the stability of some steps from these procedures. In particular we consider the stability of the redatuming procedure and the stability of the metric reconstruction procedure from internal data (for the third problem). Moreover, we provide computational experiments to demonstrate our three main procedures

Recovery of a Smooth Metric via Wave Field and Coordinate Transformation Reconstruction

In this paper, we study the inverse boundary value problem for the wave equation with a view towards an explicit reconstruction procedure. We consider both the anisotropic problem where the unknown is a general Riemannian metric smoothly varying in a domain, and the isotropic problem where the metric is conformal to the Euclidean metric. Our objective in both cases is to construct the metric, using either the Neumann-to-Dirichlet (N-to-D) map or Dirichlet-to-Neumann (D-to-N) map as the data. In the anisotropic case we construct the metric in the boundary normal (or semi-geodesic) coordinates via reconstruction of the wave field in the interior of the domain. In the isotropic case we can go further and construct the wave speed in the Euclidean coordinates via reconstruction of the coordinate transformation from the boundary normal coordinates to the Euclidean coordinates. Both cases utilize a variant of the Boundary Control method, and work by probing the interior using special boundary sources. We provide a computational experiment to demonstrate our procedure in the isotropic case with N-to-D data.Comment: 24 pages, 6 figure

Full-waveform redatuming via a TRAC approach: a first step towards target oriented inverse problem

In inverse problems, redatuming data consists in virtually moving thesensors from the original acquisition location to an arbitrary position. Thisis an essential tool for target oriented inversion. An exact redatumingmethod which has the peculiarity to be robust with respect to noise isproposed. Our iterative method is based on the Time Reversal AbsorbingConditions (TRAC) approach and avoids the need for a regularization strategy. Numerical results and comparisons with other redatumingapproaches illustrate the robustness of our method

A numerically exact local solver applied to salt boundary inversion in seismic full-waveform inversion

In a set of problems ranging from 4-D seismic to salt boundary estimation, updates to the velocity model often have a highly localized nature. Numerical techniques for these applications such as full-waveform inversion (FWI) require an estimate of the wavefield to compute the model updates. When dealing with localized problems, it is wasteful to compute these updates in the global domain, when we only need them in our region of interest. This paper introduces a local solver that generates forward and adjoint wavefields which are, to machine precision, identical to those generated by a full-domain solver evaluated within the region of interest. This means that the local solver computes all interactions between model updates within the region of interest and the inhomogeneities in the background model outside. Because no approximations are made in the calculation of the forward and adjoint wavefields, the local solver can compute the identical gradient in the region of interest as would be computed by the more expensive full-domain solver. In this paper, the local solver is used to efficiently generate the FWI gradient at the boundary of a salt body. This gradient is then used in a level set method to automatically update the salt boundary

Reciprocity-based imaging using multiply scattered waves

In exploration seismology, seismic waves are emitted into the structurally complex Earth. Its response, consisting of a mixture of arrivals including primary reflections, conversions, multiples, and transmissions, is used to infer the internal structure and properties. Waves that interact multiple times with the inhomogeneities in the medium probe areas of the subsurface that are sometimes inaccessible to singly scattered waves. However, these contributions are notoriously difficult to use for imaging because multiple scattering turns out to be a highly nonlinear process. Conventionally, imaging algorithms assume singly scattered energy dominates data. Hence these require that energy that scatters more than once is attenuated. The principal focus of this thesis is to incorporate the effect of complex nonlinear scattering in the construction of subsurface elastic images. Reciprocity theory is used to establish an exact relation between the full recorded data and the local (zero-offset, zero-time) scattering response in the subsurface which constitutes our image. Fully nonlinear, elastic imaging conditions are shown to lead to better illumination, higher resolution and improved amplitudes in pure-mode imaging. Strikingly it is also observed that when multiple scattering is correctly handled, no converted-wave energy is mapped to any image point. I explain this result by noting that conversions require finite time and space to manifest. The construction of wavefield propagators (Green’s functions) that are used to extrapolate recorded data from the surface to points in the Earth’s interior is a crucial component of any imaging technique. Classical approaches are based on strong assumptions about the propagation direction of recorded data, and their polarization; preliminary steps of wavefield decomposition (directional and modal) are required to extract upward propagating waves at the recording surface and separate different wave modes. These algorithms also generally fail to explain the trajectories of multiply scattered and converted waves, representing a major problem when constructing nonlinear images as we do not know where such energy interacted with the scatterers to be imaged. A secondary aim of this thesis is to improve on the practice of wavefield extrapolation or redatuming by taking advantage of the different nature of multi-component data compared with single-mode acoustic data. Two-way representation theorems are used to define novel formulations in elastic media which allow both up- and downward propagating fields to be back-propagated correctly without ambiguity in the direction, and such that no cross-talk between wave modes is generated. As an application of directional extrapolation, the acoustic counterpart of the new approach is tested on an ocean-bottom cable field dataset acquired over the Volve field, North Sea. Interestingly, the process of redatuming sources to locations beneath a complex overburden by means of multi-dimensional deconvolution also requires preliminary wavefield separation to be successful: I propose to use the two-way convolution-type representation as a way to combine full pressure and particle velocity recordings. Accurate redatumed wavefields can then be obtained directly from multi-component data without separation. Another major challenge in seismic imaging is to construct detailed velocity models through which recorded data will be extrapolated. Nowadays the information contained in the extension of subsurface images along either the time or space axis is commonly exploited by velocity model building techniques acting in the image domain. Recent research has shown that when both extensions are taken into account, it is possible to estimate the data that would have been recorded if a small, local seismic survey was conducted around any image point in the subsurface. I elaborate on the use of nonlinear elastic imaging conditions to construct such so-called extended image gathers: missing events, incorrect amplitudes, and spurious energy generated from the use of only primary arrivals are shown to be mitigated when multiple scattering is included in the migration process. Finally, having access to virtual recordings in the subsurface is also very useful for target-oriented imaging applications. In the context of one-way representation, I apply the novel methodology of Marchenko redatuming to the Volve field dataset as a way to unravel propagation effects in the overburden structure. Constructed wavefields are then used to synthesize local, subsurface reflection responses that are only sensitive to local heterogeneities, and detailed images of target areas of the subsurface are ultimately produced. Overall the findings of this thesis demonstrate that, while incorporating multiply scattered waves as well as multi-component data in imaging may be not a trivial task, such information is vital for achieving high-resolution and true-amplitude seismic imaging

CRS-stack-based seismic reflection imaging for land data in time and depth domains

Land data acquisition often suffers from rough top-surface topography and complicated near-surface conditions. The resulting poor data quality makes conventional data processing very difficult. Under such circumstances, where simple model assumptions may fail, it is of particular importance to extract as much information as possible directly from the measured data. Fortunately, the ongoing increase in available computing power makes advanced data-driven imaging approaches feasible; thus, these have increasingly gained in relevance during the past few years. The common-reflection-surface (CRS) stack, a generalized high-density velocity analysis and stacking process, is one of these promising methods. It is applied in a non-interactive manner and provides, besides an improved zero-offset simulation, an entire set of physically interpretable stacking parameters that include and complement the conventional stacking velocity. For every zero-offset sample, these so-called kinematic wavefield attributes are obtained as a by-product of the data-driven stacking process. As will be shown, they can be applied both to improve the stack itself and to support subsequent processing steps...thesi