Recovering the past history of natural recording media by Bayesian inversion

Spatial growth patterns are natural recording media (NRMs) that preserve important historical information, which can be accessed and analyzed to reconstruct past environmental conditions and events. Here, we propose the Bayesian inversion method, which can reconstruct the evolution of target parameters from purely spatial data by extending data assimilation (DA), a method for integrating numerical simulations with time-series observations. Our method converts discrete spatial observation data to time-series data with the help of a law representing the NRM's time-evolution dynamics and Gaussian process regression, enabling us to directly compare the observations with a numerical simulation based on the DA framework. The method's effectiveness is demonstrated using a synthetic inversion problem, namely reconstructing the $\mathit{pressure}–\mathit{temperature}–\mathit{time}$ ($P–T–t$) path of a metamorphic rock from chemical composition profiles of its zoned minerals. The proposed method is broadly applicable to a wide variety of NRMs.

Figure 1

(a) X-ray Mg intensity map of a garnet grain in the Sambagawa metamorphic rocks, Japan. (b) Schematic image of mineral zones. The mineral grows according to a mineral growth law, and the outer rim's mineral composition is controlled by thermodynamic variables, in this case pressure and temperature.

Figure 2

Conceptual state-space models. (a) Conventional DA. (b) History-tracking inversion, as proposed in this study.

Figure 3

Overview of predictive, filter, and smoothed PDFs in DA. Here, $p\left({\mathit{z}}_j\right|{\mathit{y}}_{1:k})$ is the conditional probability of $\mathit{z}$ at time $j$, given the observations $\mathit{y}$ between times 1–$k$.

Figure 4

Problem setup for the numerical experiments. (a) True $T$–$t$ path. (b) True $P$–$t$ path. (c) True $P$–$T$ path. (d) Composition of $X_{\mathrm{Fe}}$ in a synthetic garnet grain. The scale bar represents 5.0 mm. (e) Observed composition profile for $X_{\mathrm{Fe}}$. (f) Observed composition profile for $X_{\mathrm{Ca}}$.

Figure 5

Results of interpolation using Gaussian process regression for the observed compositions, $X_{\mathrm{Fe}}$ and $X_{\mathrm{Ca}}$, showing the means and $\pm 2\sigma$ limits.

Figure 6

Results for the numerical experiments. (a) Estimated $T$–$t$ path. (b) Estimated $P$–$t$ path. (c) Estimated $P$–$T$ path. (d) Radius of the growing grain. (e) Composition profile for $X_{\mathrm{Fe}}$. (f) Composition profile for $X_{\mathrm{Ca}}$. In (a)–(d), the estimated and true paths shown as red and black lines, respectively. In (a), (b), and (d), the estimated $\pm 2\sigma$ uncertainties are also shown. In (e) and (f), the small colored points correspond to the compositions calculated using Monte Carlo particles for the smoothed PDFs, while the black circles are the observed compositions. To show the overall distributions, only 100 points are shown, randomly selected from the 4 000 Monte-Carlo particles used.

Figure 7

Relationship between RMSE and measurement noise. (a) $T$–$t$ path. (b) $P$–$t$ path. The blue, red, and orange circles represent 5, 20, and 50 measurement points, respectively.

Figure 8

Relationship between RMSE and the number of measurement points. (a) $T$–$t$ path. (b) $P$–$t$ path. The blue, red, and orange circles represent noise standard deviations of 0.000, 0.005, and 0.015, respectively.