Global Models From Sparse Data: A Robust Estimate of Earth's Residual Topography Spectrum

Abstract

A significant component of Earth's surface topography is maintained by stresses induced by underlying mantle flow. This “dynamic” topography cannot be directly observed, but it can be approximated—particularly at longer wavelengths—from measurements of residual topography, which are obtained by removing isostatic effects from the observed topography. However, as these measurements are made at discrete, unevenly distributed locations on Earth's surface, inferences about global properties can be challenging. In this paper, we present and apply a new approach to transforming pointwise measurements into a continuous global representation. The approach, based upon the statistical theory of Gaussian processes, is markedly more stable than existing approaches—especially for small data sets. We are therefore able to infer the spatial pattern, wavelength, and amplitude of residual topography using only the highest quality oceanic spot measurements within the database of Hoggard et al. (2017, https://doi.org/10.1002/2016JB013457). Our results indicate that the associated spherical harmonic power spectrum peaks at l = 2, with power likely in the range 0.46–0.76 km2. This decreases by over an order of magnitude to around 0.02 km2 at l = 30. Around 85% of the total power is concentrated in degrees 1–3. Our results therefore confirm previous findings: Earth's residual topography expression is principally driven by deep mantle flow, but shallow processes are also crucial in explaining the general form of the power spectrum. Finally, our approach allows us to determine the locations where collection of new data would most impact our knowledge of the spectrum.

Densely Sampled Global Dynamic Topographic Observations and Their Significance

“…This spectrum and the accompanying decrease of 𝐴𝐴 𝐴𝐴 2 𝑣𝑣 as a function of l are practically identical to what is presented in Figures 17a and 17b. Previously reported differences between the Hoggard et al (2016) and Valentine and Davies (2020) spectra are evidently caused by a combination of sparse and uneven coverage of spot measurements across the oceanic realm together with an absence of continental control. Analysis of the revised and extensively augmented database presented here has ameliorated these differences.…”

“…Hoggard et al (2016) gauged spectral uncertainty by analyzing a suite of spherical harmonic representations with different gradient regularizations and amplitude damping coefficients. Valentine and Davies (2020) used sampling of the a posteriori distribution obtained by computing 10 5 randomly generated models where each model is a realization of global residual topography that is compatible with observations, uncertainties, and the Bayesian prior. Spectra are computed for each model as a function of spherical harmonic degree.…”

“…The software packages have been previously published by Hoggard et al. (2016) and Valentine and Davies (2020). For access to these software packages please contact those authors.…”

Geodynamic, geodetic, and seismic constraints favour deflated and dense-cored LLVPs

“…Three key results emerge from this analysis. First, we find that acceptable fits to both the geoid and dynamic surface topography can be obtained for thermal and thermochemical density models (Table S1; Figures 2 and S1- Dark and light gray envelope = 99% and 50% confidence intervals for power spectrum of optimal spherical harmonic coefficients for oceanic residual depth measurements as constructed by Davies et al 32 using Automatic Relevance Determination algorithm (intervals derived from 100,000 random samples of inverted spherical harmonic coefficient probability distributions); solid gray line = power spectrum of mean spherical harmonic coefficients determined for oceanic residual depth measurements; dark and light red envelope = 99% and 50% confidence intervals for power spectrum of thermal model constructed by sampling predicted dynamic topography at locations of shiptrack and point-wise oceanic residual depth measurements and determining optimal spherical harmonic coefficients using Gaussian Process algorithm of Valentine & Davies 37 ; solid red line = power spectrum of mean spherical harmonic coefficients determined for thermal model. (c) Total geodynamic misfit, χ G , of best-fit thermochemical models for each combination of viscosity and seismic tomographic input.…”

Emerging Directions in Geophysical Inversion

“…This approach has formed the basis for a variety of geophysical studies (e.g. Tarantola & Nercessian, 1984;Montagner & Tanimoto, 1990, 1991Valentine & Davies, 2020) and has the attractive property that the inference problem is posed directly in a function space, avoiding some of the difficulties associated with discretization (such as spectral leakage).…”