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

Posterior  ~  likelihood x prior

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

-norm objective function

l_2
$l_2$
• 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).

Efficiently invert a large number of parameters

Entrapment by local minima

Number of evaluations increase exponentially with dimensions

Copes better with highly nonlinear problems

Posterior distribution

MCMC

+

+

+

-

-

-

Thermal solver

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

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}
$S(\mathbf{m}) = \sum_i \frac{\lvert g^i(\mathbf{m}) - \mathrm{d}^i \lvert^2}{(2\sigma_{\mathrm{d}}^i)^2} + \sum_{j} \frac{\vert \mathrm{m}^j - \mathrm{m}^j_p \vert^2}{(2\sigma^j_p)^2}$
\mathbf{x}: [k_1, k_2,\ldots, k_n, \\ H_1, H_2,\ldots, H_n ]
$\mathbf{x}: [k_1, k_2,\ldots, k_n, \\ H_1, H_2,\ldots, H_n ]$

Input parameters

Forward

Model

Model

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

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

A

A'

## 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

## Uncertainty reduction

Coupling gradient inversion with local sampling builds an approximation of the posterior

1. Perturb observation within PDF
2. Perturb priors within PDF
3. Build the ensemble
\delta\sigma = 1 - \frac{\sigma_{\mathrm{posterior}}}{\sigma_{\mathrm{prior}}}
$\delta\sigma = 1 - \frac{\sigma_{\mathrm{posterior}}}{\sigma_{\mathrm{prior}}}$

Uncertainty reduction

MAP estimate

Standard deviation

thermal

conductivity

heat production

## Thermal model of Ireland

• Significant heat refraction in Northern Ireland.
• High temperature related to lateral branch of Iceland plume(?)

Temperature model with surface heat flow

Temperature, thermal conductivity, heat production

at various depths

# Thank you

Dr. Ben Mather

The University of Sydney, NSW 2006

https://benmather.info

By Ben Mather

# BGH meeting Feb 2019

11th February 2019

• 994
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