arXiv subjects

David Al-Attar

Publications and source records attributed to David Al-Attar.

9 recordsLinked to original sources

arXiv2026

A scalable Bayesian framework for modern sea-level inference

This paper presents a scalable Bayesian framework for the inference of modern-day sea-level change and surface mass redistribution. To motivate this approach, we first review and quantify the performance of some standard methods for the analysis of satellite gravity and ocean altimetry observations. We find that these methods substantially underestimate uncertainties and are subject to systematic biases, deficiencies that stem from their incomplete treatment of sea-level physics and from uncertainty estimates based solely on the propagation of observational noise. To address these limitations, our approach combines three advances. First, we use recent developments in adjoint sea-level theory to embed the full physics into the forward and inverse modelling. Second, we formulate the Bayesian inverse problem in an infinite-dimensional setting, thereby avoiding discretisation artefacts and the underestimation of uncertainties inherent in truncated model spaces. Finally, our computational methods render such inversions tractable at full observational resolution while supporting joint model spaces and multiple data types. By employing a matrix-free approach with iterative solvers and randomised low-rank decompositions -- implemented in the linked open-source libraries pygeoinf and pyslfp -- basic calculations are possible on a single laptop, with the most intensive tasks parallelising trivially across available cores. We demonstrate the methodology through a series of synthetic experiments, culminating in a joint inversion of satellite gravity and ocean altimetry data that decomposes regionally averaged sea-level change into steric and manometric components with quantified uncertainties, including the degeneracies that remain.

physics.geo-ph↗

arXiv2026

The direct spectral element method for the calculation of synthetic seismograms in self-gravitating, spherically symmetric planets

This paper describes the implementation of the direct solution method (DSM) using radial spectral elements for the calculation of synthetic seismograms in self-gravitating, spherically symmetric, non-rotating, anelastic, and transversely isotropic Earth models. In contrast to previous implementations of the DSM that used a potential formulation within fluid regions, we use a displacement formulation throughout. It is this feature that allows us to extend the DSM to account fully for self-gravitation along with arbitrary fluid stratification. Our code, $\texttt{DSpecM1D}$, is benchmarked against the normal mode summation code $\texttt{MINEOS}$ as well as the direct radial integration code $\texttt{YSpec}$. Agreement between the codes is excellent for both elastic and anelastic models.

physics.geo-ph↗

arXiv2026

Efficient parallel finite-element methods for planetary gravitation: DtN and multipole expansions

The Poisson equation governing a planet's gravitational field is posed on the unbounded domain, $\mathbb{R}^3$, whereas finite-element computations require bounded meshes. We implement and compare three strategies for handling the infinite exterior in the finite-element method: (i) naive domain truncation; (ii) Dirichlet-to-Neumann (DtN) map on a truncated boundary; (iii) multipole expansion on a truncated boundary. While all these methods are known within the geophysical literature, we discuss their parallel implementations within modern open-source finite-element codes, focusing specifically on the widely-used MFEM package. We consider both calculating the gravitational potential for a static density structure and computing the linearised perturbation to the potential caused by a displacement field - a necessary step for coupling self-gravitation into planetary dynamics. In contrast to some earlier studies, we find that the domain truncation method can provide accurate solutions at an acceptable cost, with suitable coarsening of the mesh within the exterior domain. Nevertheless, the DtN and multipole methods provide superior accuracy at a lower cost within large-scale parallel geophysical simulations despite their need for non-local communication associated with spherical harmonic expansions. The DtN method, in particular, admits an efficient parallel implementation based on an MPI-communicator limited to processors that contain part of the mesh's outer boundary. A series of further illustrative calculations are provided to show the potential of the DtN and multipole methods within realistic geophysical modelling.

astro-ph.EP↗

arXiv2025

Forward and adjoint calculations of gravitational potential in heterogeneous, aspherical planets

We have developed a computational package for the calculation of numerically exact internal and external gravitational potential, its functional derivatives and sensitivity kernels, in an aspherical, heterogeneous planet. We detail our implementation, utilizing a transformation of the Poisson equation into a reference domain, as well as a pseudospectral/spectral element discretisation. The use of the forward solver within the package is demonstrated by calculating the gravitational potential of Phobos with homogeneous and heterogeneous density models. Equations for the first-order perturbation expansion of potential in the referential formulation are found, and the magnitude of the error is quantified based on the exact method. The adjoint Poisson equation is derived, and from it the sensitivity kernels for objective functionals of the potential, which are calculated for Phobos. The expression for perturbations to an objective functional is reduced to a single body and surface integral. Finally, a relation is obtained between the sensitivity kernels which must be satisfied when using a computational domain different to the physical domain.

physics.geo-ph↗

arXiv2024

Application of first- and second-order adjoint methods to glacial isostatic adjustment incorporating rotational feedbacks

This paper revisits and extends the adjoint theory for glacial isostatic adjustment (GIA) of Crawford et al. (2018). Rotational feedbacks are now incorporated, and the application of the second-order adjoint method is described for the first time. The first-order adjoint method provides an efficient means for computing sensitivity kernels for a chosen objective functional, while the second-order adjoint method provides second-derivative information in the form of Hessian kernels. These latter kernels are required by efficient Newton-type optimisation schemes and within methods for quantifying uncertainty for non-linear inverse problems. Most importantly, the entire theory has been reformulated so as to simplify its implementation by others within the GIA community. In particular, the rate-formulation for the GIA forward problem introduced by Crawford et al. (2018) has been replaced with the conventional equations for modelling GIA in laterally heterogeneous earth models. The implementation of the first- and second-order adjoint problems should be relatively easy within both existing and new GIA codes, with only the inclusions of more general force terms being required.

physics.geo-ph↗

arXiv2023

On the elastodynamics of rotating planets

Equations of motion are derived for (visco)elastic, self-gravitating, and variably-rotating planets. The equations are written using a decomposition of the elastic motion that separates the body's elastic deformation from its net translational and rotational motion as far as possible. This separation is achieved by introducing degrees of freedom that represent the body's rigid motions; it is made precise by imposing constraints that are physically motivated and should be practically useful. In essence, a Tisserand frame is introduced exactly into the equations of solid mechanics. The necessary concepts are first introduced in the context of a solid body, motivated by symmetries and conservation laws, and the corresponding equations of motion are derived. Next, it is shown how those ideas and equations of motion can readily be extended to describe a layered fluid--solid body. A possibly new conservation law concerning inviscid fluids is then stated. Thereafter the equilibria and linearisation of the fluid--solid equations of motion are discussed, along with new equations for use within normal-mode coupling calculations and other Galerkin methods. Finally, the extension of these ideas to the description of multiple, interacting fluid--solid planets is qualitatively discussed.

physics.class-ph↗

arXiv2023

Reciprocity and sensitivity kernels for sea level fingerprints

Reciprocity theorems are established for the elastic sea level fingerprint problem including rotational feedbacks. In their simplest form, these results show that the sea level change at a location x due to melting a unit point mass of ice at x' is equal to the sea level change at x' due to melting a unit point mass of ice at x. This identity holds irrespective of the shoreline geometry or of lateral variations in elastic Earth structure. Using the reciprocity theorems, sensitivity kernels for sea level and related observables with respect to the ice load can be readily derived. It is notable that calculation of the sensitivity kernels is possible using standard fingerprint codes, though for some types of observable a slight generalisation to the fingerprint problem must be considered. These results are of use within coastal hazard assessment and have a range of potential applications within studies of modern-day sea level change.

physics.geo-ph↗

arXiv2021

Linear inference problems with deterministic constraints

Methods are described for the solution of linear inference problems subject to deterministic constraints. The approach builds on work by Backus (1970a,b,c) and Parker (1977), but a range useful advances are suggested to address both conceptual and practical issues. The theory is motivated by, and illustrated with, the estimation of a finite number of a function's spherical harmonic coefficients from a finite set of its point values. Numerical examples are included to demonstrate that the methods can be efficiently applied to realistic problems.

physics.geo-ph↗

arXiv2020

On the stress dependence of the elastic tensor

The dependence of the elastic tensor on the equilibrium stress is investigated theoretically. Using ideas from finite-elasticity, it is first shown that both the equilibrium stress and elastic tensor are given uniquely in terms of the equilibrium deformation gradient relative to a fixed choice of reference body. Inversion of the relation between the deformation gradient and stress might, therefore, be expected to lead neatly to the desired expression for the elastic tensor. Unfortunately, the deformation gradient can only be recovered from the stress up to a choice of rotation matrix. Hence it is not possible in general to express the elastic tensor as a unique function of the equilibrium stress. By considering material symmetries, though, it is shown that the degree of non-uniqueness can sometimes be reduced, and in some cases even removed entirely. These results are illustrated through a range numerical calculations, and we also obtain linearised relations applicable to small perturbations in equilibrium stress. Finally, we make a comparison with previous studies, before considering implications for geophysical forward- and inverse-modelling.

physics.geo-ph↗