Maximum-likelihood estimation of lithospheric flexural rigidity,
initial-loading fraction, and load correlation, under isotropy

Frederik J Simons¹ and Sofia C. Olhede²

1 Geosciences Department
Princeton University
Princeton NJ 08544, USA

2 Department of Statistical Science
University College London
London WC1E 6BT

Abstract

Topography and gravity are geophysical fields whose joint statistical structure derives from interface-loading processes modulated by the underlying mechanics of isostatic and flexural compensation in the shallow lithosphere. Under this dual statistical-mechanistic viewpoint an estimation problem can be formulated where the knowns are topography and gravity and the principal unknown the elastic flexural rigidity of the lithosphere. In the guise of an equivalent ``effective elastic thickness'', this important, geographically varying, structural parameter has been the subject of many interpretative studies, but precisely how well it is known or how best it can be found from the data, abundant nonetheless, has remained contentious and unresolved throughout the last few decades of dedicated study. The popular methods whereby admittance or coherence, both spectral measures of the relation between gravity and topography, are inverted for the flexural rigidity, have revealed themselves to have insufficient power to independently constrain both it and the additional unknown initial-loading fraction and load-correlation factors, respectively. Solving this extremely ill-posed inversion problem leads to non-uniqueness and is further complicated by practical considerations such as the choice of regularizing data tapers to render the analysis sufficiently selective both in the spatial and spectral domains. Here, we rewrite the problem in a form amenable to maximum-likelihood estimation theory, which we show yields unbiased, minimum-variance estimates of flexural rigidity, initial-loading fraction and load correlation, each of those separably resolved with little a posteriori correlation between their estimates. We are also able to separately characterize the isotropic spectral shape of the initial loading processes. Our procedure is well-posed and computationally tractable for the two-interface case. The resulting algorithm is validated by extensive simulations whose behavior is well matched by an analytical theory with numerous tests for its applicability to real-world data examples.

Figures

1. Figure 01 Synthetic data representing the standard model without initial-load correlation
   
2. Figure 02 Expected values of the admittance and coherence between Bouguer gravity anomalies and topography
   
3. Figure 03 Synthetic "topographies" generated from the Matérnrn spectral class
   
4. Figure 04 Synthetic data representing the standard model with initial-load correlation
   
5. Figure 05 The behavior of the quadratic residuals in a recovery simulation for correlated loading
   
6. Figure 06 Recovery statistics of simulations under uncorrelated loading on a 64x64 grid with 20 km spacing
   
7. Figure 07 Recovery statistics of simulations under uncorrelated loading on a 128x128 grid with 20 km spacing
   
8. Figure 08 Recovery statistics of simulations under correlated loading on a 32x32 grid with 20 km spacing
   
9. Figure 09 Recovery statistics of simulations under uncorrelated loading on a 64x64 grid with 20 km spacing
   
10. Figure 10a Correlation (normalized covariance) matrices for the uncorrelated-loading experiments previously reported
    
11. Figure 10b Correlation (normalized covariance) matrices for the uncorrelated-loading experiments previously reported
    
12. Figure 11a Correlation (normalized covariance) matrices for the correlated-loading experiments previously reported
    
13. Figure 11b Correlation (normalized covariance) matrices for the correlated-loading experiments previously reported
    
14. Figure 12a Admittance and coherence curves for the uncorrelated-loading experiments previously reported
    
15. Figure 12b Admittance and coherence curves for the uncorrelated-loading experiments previously reported
    
16. Figure 13a Admittance and coherence curves for the correlated-loading experiments previously reported
    
17. Figure 13b Admittance and coherence curves for the correlated-loading experiments previously reported
    

Frederik Simons