Curie depth offers a valuable constraint on the thermal structure of the lithosphere, based on its interpretation as the depth to 580

Magnetic data are, along with gravity anomalies, the most commonly available geophysical observables for subsurface imaging of the Earth. Surveys of the magnetic anomaly capture, among other features, the contribution of various magnetic minerals to the surface field recorded by magnetometers. Ferromagnetic minerals retain their magnetism until they reach their Curie temperature. Magnetite is the most prevalent magnetic mineral, in terms of susceptibility and quantity, and has a Curie temperature of 580

Various methods have been proposed to estimate Curie depth from a spectral analysis of the magnetic anomaly, which have been successfully applied to many regions of the Earth, but often under-represent the degree of uncertainty within each Curie depth estimate. We propose a Bayesian approach where Curie depth is expressed in probabilistic terms. We apply this methodology to the British Isles to quantify the degree of uncertainty in Curie depth and use this as a boundary condition to compute the temperature distribution in the crust, integrating a stratified geological model constrained by heat production, seismic, and surface heat flow data. We validate our prediction of geothermal heat flow across the British Isles against surface heat flow data to examine the variations and controls on crustal heat flow.

The British Isles comprise Laurentian and Avalonian terranes that were brought together at the closure of the Iapetus Ocean at 420 Ma

The

Igneous outcrop and major geological structures across the British Isles. Surface heat flow data are compiled from

We apply probabilistic techniques to quantify the uncertainty in Curie depth (interpreted as the 580

Most methods to estimate Curie depth relate the spectrum of magnetic anomalies and the depth of magnetic sources by transforming the data into the Fourier domain, which is computed over a square window of the magnetic anomaly. Depth to the bottom of magnetic sources is estimated from the slope of the radial power spectrum,

Alternatively,

The relationship between window size and the uncertainty in Curie depth estimates (

Here we consider the analytic expression from Eq. (

The Bayesian framework we have described produces an ensemble of models that sample the posterior density function. Here we design a synthetic sensitivity test to explore the inversion parameters. As the inversion operates in the spectral domain a crucial parameter is the size of the spatial window used to transform the spatially distributed magnetic data to the Fourier domain. A synthetic magnetic anomaly was generated from random fractal noise in three dimensions with known model parameters (

In addition to window size, retrieval of accurate Curie depth values is also complicated by a strong correlation between

Ensemble of radial power spectra from a MCMC (Markov chain Monte Carlo) simulation.

Geothermal heat flow can be determined by modelling a geotherm from the surface of the crust to the Curie depth. A geotherm is modelled by solving the steady-state heat equation,

Maps of Curie depth in the study region were taken from the magnetic anomaly of the British Isles that was extracted from the EMAG2 global compilation (Fig.

The implemented map-generation algorithm starts at the MAP estimate for the largest window size (800 km) at every point in the model domain and iteratively reduces at 25 km increments until the standard deviation of the posterior exceeds that of the maximum window size. All other Curie depth estimates using smaller or larger window sizes for a given point are rejected. As a consequence, the inverted Curie depth map contains a combination of window sizes: smaller windows are used where the magnetic base is shallow to resolve small-scale features and larger windows are used where the magnetic base is deep in order to improve precision (Fig.

A broad linear increase in the MAP estimate of Curie depth is observed from north to south of the British Isles (Fig.

Curie depth is

For the most part, the Curie depth we determine is concordant with previous studies. The global reference model of

We find that quantifying Curie depth within a Bayesian framework adds significant insight into the thermal structure of the crust. The ensemble of model simulations reveals that uncertainty correlates with Curie depth (Fig.

Magnetic anomaly map of the British Isles, extracted from the EMAG2 version 3 global compilation

The ensemble of Curie depth estimates was computed across centroids spaced

MAP estimate of Curie depth in the British Isles and its uncertainty;

Analysis of Curie depth estimates.

The heat flow distribution over the British Isles is determined by modelling a geotherm from the surface of the crust to the Curie depth. The top of the model is the surface topography from ETOPO2

Our geothermal heat flow map offers very high resolution (approximately 4 km) compared to the spatial density of borehole measurements, and provides insights into the variation in thermal regimes across the British Isles (Fig.

Thermal properties assigned to each layer in the British Isles, summarized from

Maps of layer geometries extracted from the model of

Schematic of each layer in the 3-D thermal model of the British Isles, subdivided from surfaces compiled in

Surface heat flow in the British Isles and its uncertainty,

Heat flow data are clustered mainly within coastlines and in some localized areas offshore. The simulated heat flow is mostly concordant with heat flow data except for locations above felsic bodies or mafic volcanics. These are not included within the 3-D model due to the unconstrained nature of their geometry. Granites can have very high rates of heat production which, when integrated across their thickness, contribute significantly to surface heat flow. This is most prevalent in the southwest England and northern England batholiths that contain rates of heat production between 3 and 5

The uncertainty of surface heat flow was estimated from spatially varying the depth of the lower boundary condition within the posterior distribution assembled in Sect.

Thinning of the lithosphere towards the north of the ISZ has been associated with a branch of the Icelandic mantle plume

The Curie depth, often interpreted as the depth to 580

The uncertainty in Curie depth increases rapidly with depth but can be tempered with larger window sizes. Windows of the magnetic anomaly need to be 15–30 times larger than the deepest possible magnetic base in the study area to resolve precise Curie depth estimates. In the British Isles, the Curie depth is broadly delineated by the SW–NE Iapetus Suture, which separates Laurentian and Avalonian continental blocks. The uncertainty exponentially increases with Curie depth, which is due to fewer points in the low wave number range of the radial power spectrum. Curie depths of

Surface heat flow is estimated by simulating the temperature distribution across a stratified model of the crust from the surface to the Curie depth. This 3-D model incorporates sedimentary thickness and vertically partitioned heat-producing elements. The simulated heat flow matches heat flow data to a reasonable degree of uncertainty, except where igneous bodies outcrop. Granitic intrusions can contribute significantly to surface heat flow due to their high rates of heat production, and heat refracts around mafic volcanics because of their low thermal conductivity. These results lend further support to lithospheric thinning in northern Britain and Ireland due to the presence of the proto-Icelandic mantle plume.

All data presented in this article may be obtained in the Supplement from the online version, and may be reproduced from algorithms implemented within PyCurious (

The commonly used Metropolis–Hastings algorithm samples the posterior distribution,

Generate a proposal

Calculate the acceptance ratio between each sample of the posterior

Generate a random number,

accept if

reject if

Numerous chains were initiated to ensure they converged to a similar solution.

The supplement related to this article is available online at:

BM developed the Curie depth calculation software, interpreted the results across the British Isles, and wrote most of the paper; JF interpreted the Curie depth variation and provided the geological context.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Understanding the unknowns: the impact of uncertainty in the geosciences”. It is not associated with a conference.

This work was made possible by the G.O.THERM.3D project, supported by an Irish Research Council Research for Policy & Society grant (RfPS/2016/50) co-funded by Geological Survey Ireland and by the Sustainable Energy Authority of Ireland. Javier Fullea was supported by a Science Foundation Ireland grant iTHERC (16/ERCD/4303).

This research has been supported by the Irish Research Council (grant no. RfPS/2016/50) and the H2020 Marie Skłodowska-Curie Actions (grant WINTERC-3D (657357)).

This paper was edited by Juan Alcalde and reviewed by Jian Wang and one anonymous referee.