Reducing the uncertainty of lithospheric thermal structure

integrated modelling from geothermal, magnetic, and seismic data

Dr. Ben Mather

EarthByte Group & Sydney Informatics Hub

Motivation

Understanding temperature in the crust can help to interpret:

hydrocarbon maturity
geothermal potential
seismic velocity

Quantifying its uncertainty is important to estimate the risk associated with a resource

Bayesian Inference

P(\mathbf{m}|\mathbf{d}) \propto P(\mathbf{d}|\mathbf{m}) \cdot P(\mathbf{m})

Posterior ~ likelihood x prior

P(\mathbf{m}|\mathbf{d}) = A \exp (-S(\mathbf{m}))

S(\mathbf{m}) = \sum_i \frac{\lvert g^i(\mathbf{m}) - \mathrm{d}^i \lvert^2}{(2\sigma_{\mathrm{d}}^i)^2} + \sum_{j} \frac{\vert \mathrm{m}^j - \mathrm{m}^j_p \vert^2}{(2\sigma^j_p)^2}

-norm objective function

l_2

Bayes theorem describes the posterior as the probability of a model given the data

The posterior probability can be evaluated with an objective function

We seek the maximum a posteriori (MAP) model, which can be obtained by minimising S(m).

Adjoint inversion

Efficiently invert a large number of parameters

Added complexity to compute forward problem AND gradient vectors

Entrapment by local minima

Number of evaluations increase exponentially with dimensions

Copes better with highly nonlinear problems

Posterior distribution

MCMC

Adjoint

Thermal solver

\nabla(\mathbf{k} \nabla \mathbf{T}) = - \mathbf{H}

Objective function

S(\mathbf{m}) = \sum_i \frac{\lvert g^i(\mathbf{m}) - \mathrm{d}^i \lvert^2}{(2\sigma_{\mathrm{d}}^i)^2} + \sum_{j} \frac{\vert \mathrm{m}^j - \mathrm{m}^j_p \vert^2}{(2\sigma^j_p)^2}

\mathbf{x}: [k_1, k_2,\ldots, k_n, \\ H_1, H_2,\ldots, H_n ]

Input parameters

Forward

Model

Adjoint

Model

Gradient descent

The gradient descent method for finding successibely better approximations to the minimum of S(m) is:

\mathbf{m}_{i+1} = \mathbf{m}_i - \mu \nabla S(\mathbf{m}_i)

However, we use a quasi-Newton nonlinear solver to approximate the Hessian (2nd derivatives)

Useful because the Hessian is difficult to compute

Regional context

SE Australia is an accretionary terrane combining Proterozoic and Phanerozoic crust

Data assimilation

On their own, any of these datasets result in non-unique solutions of temperature.
Together, they constrain temperature in the lithosphere to a reasonable degree of uncertainty.

HEAT FLOW

CURIE DEPTH

SHEAR VELOCITY