Description: The ocean load in glacial isostatic adjustment (GIA) modelling is represented by the so-called sea level equation (SLE). The SLE describes the mass redistribution of water between ice sheets and oceans on a deforming Earth. Despite various teams independently investigating GIA, there has been no systematic intercomparison among the numerical solvers of the SLE through which the methods may be validated. The goal of this paper is to present a series of synthetic examples designed for testing and comparing the numerical implementations of the SLE in GIA modelling. The 10 numerical codes tested combine various temporal and spatial parametrizations. The time-domain or Laplace-domain discretizations are used to solve the SLE through time, while spherical harmonics, finite differences or finite elements parametrize the GIA-related field variables spatially. The surface ice-water load and solid Earth’s topography are represented spatially either on an equiangular grid, a Gauss–Legendre or an equiarea grid with icosahedron-shaped spherical pixels. Comparisons are made in a series of five benchmark examples with an increasing degree of complexity. Due to the complexity of the SLE, there is no analytical solution to it. The accuracy of the numerical implementations is therefore assessed by the differences of the individual solutions with respect to a reference solution. While the benchmark study does not result in GIA predictions for a realistic loading scenario, we establish a set of agreed-upon results that can be extended in the future by including more complex case studies, such as solutions with realistic loading scenarios, the rotational feedback in the linear-momentum equation, and by considering a 3-D viscosity structure of the Earth’s mantle. The test computations performed so far show very good agreement between the individual results and their ability to capture the main features of sea-surface variation and the surface vertical displacement. The differences found can often be attributed to the different approximations inherent in the various algorithms. This shows the accuracy that can be expected from different implementations of the SLE, which helps to assess differences noted in the literature between predictions for realistic loading cases.

Description: The identification of a source model for the catastrophic 1908 December 28 Messina earthquake ( M w  = 7.2) has been the subject of many papers in the last decades. Several authors proposed different models on the basis of seismological, macroseismic and geodetic data sets; among these models, remarkable differences exist with regard to almost all parameters. We selected a subset of six models among those most cited in literature and used them to model the post-seismic sea level variation recorded at the tide gauge station of Messina (until 1923), to attempt an independent discrimination among them. For each model, we assumed a simple rheological structure and carried out a direct-search inversion of upper crust thickness and lower crust viscosity to fit the post-seismic sea level signal. This approach enabled us to identify a class of fault geometries which is consistent with the post-seismic signal at the Messina tide gauge and with the known structural and rheological features of the Messina strait.

Description: We propose a procedure for uncertainty quantification in Probabilistic Tsunami Hazard Analysis (PTHA), with a special emphasis on the uncertainty related to statistical modelling of the earthquake source in Seismic PTHA (SPTHA), and on the separate treatment of subduction and crustal earthquakes (treated as background seismicity). An event tree approach and ensemble modelling are used in spite of more classical approaches, such as the hazard integral and the logic tree. This procedure consists of four steps: (1) exploration of aleatory uncertainty through an event tree, with alternative implementations for exploring epistemic uncertainty; (2) numerical computation of tsunami generation and propagation up to a given offshore isobath; (3) (optional) site-specific quantification of inundation; (4) simultaneous quantification of aleatory and epistemic uncertainty through ensemble modelling. The proposed procedure is general and independent of the kind of tsunami source considered; however, we implement step 1, the event tree, specifically for SPTHA, focusing on seismic source uncertainty. To exemplify the procedure, we develop a case study considering seismic sources in the Ionian Sea (central-eastern Mediterranean Sea), using the coasts of Southern Italy as a target zone. The results show that an efficient and complete quantification of all the uncertainties is feasible even when treating a large number of potential sources and a large set of alternative model formulations. We also find that (i) treating separately subduction and background (crustal) earthquakes allows for optimal use of available information and for avoiding significant biases; (ii) both subduction interface and crustal faults contribute to the SPTHA, with different proportions that depend on source-target position and tsunami intensity; (iii) the proposed framework allows sensitivity and deaggregation analyses, demonstrating the applicability of the method for operational assessments.

Description: We review the theory of the Earth's elastic and gravitational response to a surface disk load. The solutions for displacement of the surface and the geoid are developed using expansions of Legendre polynomials, their derivatives and the load Love numbers. We provide a matlab  function called diskload that computes the solutions for both uncompensated and compensated disk loads. In order to numerically implement the Legendre expansions, it is necessary to choose a harmonic degree, n max , at which to truncate the series used to construct the solutions. We present a rule of thumb (ROT) for choosing an appropriate value of n max , describe the consequences of truncating the expansions prematurely and provide a means to judiciously violate the ROT when that becomes a practical necessity.

Description: The present work reports the analysis of a possible relationship due to stress transfer between the two earthquakes that hit the province of Van, Eastern Turkey, on October 23, 2011 (Mw = 7.2) and on November 9, 2011 (Mw = 5.6). The surface displacement field of the mainshock has been obtained through a combined data set made up of differential interferograms from COSMO-SkyMed and ENVISAT satellites, integrated with continuous GPS recordings from the Turkish TUSAGA-AKTIF network. This allowed us to retrieve the geometry and the slip distribution of the seismic source and to compute the Coulomb Failure Function (CFF) variation on the aftershock plane, in order to assess a possible causal relationship between the two events. Our results show that the November 9 earthquake could have been triggered by the October 23 shock, with transferred stress values largely exceeding 1 bar.

Description: The sea-level load in glacial isostatic adjustment (GIA) is described by the so called sea-level equation (SLE), which represents the mass redistribution between ice sheets and oceans on a deforming earth. Various levels of complexity of SLE have been proposed in the past, ranging from a simple mean global sea level (the so-called eu- static sea level) to the load with a deforming ocean bottom, migrating coastlines and a changing shape of the geoid. Several approaches to solve the SLE have been derived, from purely analytical formulations to fully numerical methods. Despite various teams independently investigating GIA, there has been no systematic intercomparison amongst the solvers through which the methods may be validated. The goal of this paper is to present a series of benchmark experiments designed for testing and comparing numerical implementations of the SLE. Our approach starts with simple load cases even though the benchmark will not result in GIA predictions for a realistic loading scenario. In the longer term we aim for a benchmark with a realistic loading scenario, and also for benchmark solutions with rotational feedback. The current benchmark uses an earth model for which Love numbers have been computed and benchmarked in Spada et al (2011). In spite of the significant differences in the numerical methods employed, the test computations performed so far show a satisfactory agreement between the results provided by the participants. The differences found can often be attributed to the different approximations inherent to the various algorithms.

Description: The ocean load in glacial isostatic adjustment (GIA) modelling is represented by the so-called sea level equation (SLE). The SLE describes the mass redistribution of water between ice sheets and oceans on a deforming Earth. Despite various teams independently investigating GIA, there has been no systematic intercomparison among the numerical solvers of the SLE through which the methods may be validated. The goal of this paper is to present a series of synthetic examples designed for testing and comparing the numerical implementations of the SLE in GIA modelling. The 10 numerical codes tested combine various temporal and spatial parametrizations. The time-domain or Laplace-domain discretizations are used to solve the SLE through time, while spherical harmonics, finite differences or finite elements parametrize the GIA-related field variables spatially. The surface ice-water load and solid Earth’s topography are represented spatially either on an equiangular grid, a Gauss–Legendre or an equiarea grid with icosahedron-shaped spherical pixels. Comparisons are made in a series of five benchmark examples with an increasing degree of complexity. Due to the complexity of the SLE, there is no analytical solution to it. The accuracy of the numerical implementations is therefore assessed by the differences of the individual solutions with respect to a reference solution. While the benchmark study does not result in GIA predictions for a realistic loading scenario, we establish a set of agreed-upon results that can be extended in the future by including more complex case studies, such as solutions with realistic loading scenarios, the rotational feedback in the linear-momentum equation, and by considering a 3-D viscosity structure of the Earth’s mantle. The test computations performed so far show very good agreement between the individual results and their ability to capture the main features of sea-surface variation and the surface vertical displacement. The differences found can often be attributed to the different approximations inherent in the various algorithms. This shows the accuracy that can be expected from different implementations of the SLE, which helps to assess differences noted in the literature between predictions for realistic loading cases.

Description: The sea-level load in glacial isostatic adjustment (GIA) is described by the so called sea-level equation (SLE), which represents the mass redistribution between ice sheets and oceans on a deforming earth. Despite various teams independently investigating GIA, there has been no systematic intercomparison amongst the solvers through which the methods may be validated. The goal of this paper is to present a series of synthetic examples designed for testing and comparing numerical implementations of SLE in GIA modelling. Ten numerical codes tested combine various time and spatial parameterizations. The explicit time-domain discretization or the Laplace-domain transformation are applied to evolve the SLE in time, while spherical harmonics, finite differences or finite elements parameterize the field variables spatially. The surface load distribution and solid Earth’s topography are represented spatially either on the grid of the Gauss-Legendre longitudinal nodes and equal-angle latitudinal nodes, or equalarea, icosahedron-shaped, spherical pixels. Comparisons are made in a series of five benchmark test cases with an increasing degree of complexity. Due to the complexity of SLE, there is no analytical solution to it. The accuracy of the numerical implementations is therefore assessed by comparison of the differences of individual solutions with respect to a chosen solution. The benchmark study does not result in GIA predictions for a realistic loading scenario, but we establish a set of agreed results that can be extended in future by including more complex case studies such as the solution with a realistic loading scenario, the rotational feedback in the linear momentum equation and three-dimensional viscosity structure of the Earth’s mantle. The current benchmark study is a follow-on to the Spada et al. (2011) benchmark study of various GIA modelling codes and mathematical formulations in which here the comparison is made for the case of surface mass loading by the SLE. In spite of the significant differences in the implementations of the SLE, the test computations performed so far show a very good agreement between the results and their ability to capture the main features of sea-level change and viscoelastic behaviour of the Earth. Literature G. Spada, V. R. Barletta, V. Klemann, R. E. M. Riva, Z. Martinec, P. Gasperini, B. Lund, D. Wolf, L. L. A. Vermeersen, and M. A. King, 2011. A benchmark study for glacial isostatic adjustment codes. Geophys. J. Int. 185: 106-132 doi:10.1111/j.1365-246X.2011.04952.x