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3D Bivariate Spatial Modelling of Argo Ocean Temperature and Salinity

Mary Lai Salvana, Jian Cao, Mikyoung Jun

Abstract

Variables contained within the global oceans can detect and reveal the effects of the warming climate as the oceans absorb huge amounts of solar energy. Hence, information regarding the joint spatial distribution of ocean variables is critical for climate monitoring. In this paper, we investigate the spatial correlation structure between ocean temperature and salinity using data harvested from the Argo program and construct a model to capture their bivariate spatial dependence from the surface to the ocean's interior. We develop a flexible class of multivariate nonstationary covariance models defined in 3-dimensional (3D) space (longitude $\times$ latitude $\times$ depth) that allows for the variances and correlation to change along the vertical pressure dimension. These models are able to describe the joint spatial distribution of the two variables while incorporating the underlying vertical structure of the ocean. We demonstrate that the proposed cross-covariance models describe the complex vertical cross-covariance structure well, while existing cross-covariance models including bivariate Matérn models poorly fit empirical cross-covariance structure. Furthermore, the results show that using one more variable significantly enhances the prediction of the other variable and that the estimated spatial dependence structures are consistent with the ocean stratification.

3D Bivariate Spatial Modelling of Argo Ocean Temperature and Salinity

Abstract

Variables contained within the global oceans can detect and reveal the effects of the warming climate as the oceans absorb huge amounts of solar energy. Hence, information regarding the joint spatial distribution of ocean variables is critical for climate monitoring. In this paper, we investigate the spatial correlation structure between ocean temperature and salinity using data harvested from the Argo program and construct a model to capture their bivariate spatial dependence from the surface to the ocean's interior. We develop a flexible class of multivariate nonstationary covariance models defined in 3-dimensional (3D) space (longitude latitude depth) that allows for the variances and correlation to change along the vertical pressure dimension. These models are able to describe the joint spatial distribution of the two variables while incorporating the underlying vertical structure of the ocean. We demonstrate that the proposed cross-covariance models describe the complex vertical cross-covariance structure well, while existing cross-covariance models including bivariate Matérn models poorly fit empirical cross-covariance structure. Furthermore, the results show that using one more variable significantly enhances the prediction of the other variable and that the estimated spatial dependence structures are consistent with the ocean stratification.
Paper Structure (17 sections, 1 theorem, 28 equations, 13 figures, 4 tables)

This paper contains 17 sections, 1 theorem, 28 equations, 13 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Spatial covariance models in 2D and 3D space
  3. Background
  4. Spatial Interpolation of Argo Data
  5. Differential Operators Approach in 2D
  6. Methodology
  7. Multivariate Nonstationary Spatial Models in 3D Domain
  8. Modelling vertical coefficients for the differential operator
  9. Likelihood Approximation
  10. Application
  11. Data
  12. Covariance models
  13. Fitted results
  14. Kriging results
  15. Model diagnostics on differenced processes
  16. ...and 2 more sections

Key Result

Proposition 1

Let $\mathbf{X}_k (L, l, p)$, $k = 0, \ldots, K$, be the latent multivariate stationary processes defined above, each with cross-covariance $C_{ij}^X$. For $i = 1,\ldots,q$, let $a_{i,k}(L)$, $b_{i,k}(L)$, $c_{i,k}(L,p)$, and $d_i(L)$ denote deterministic coefficient functions that weight the differ Furthermore, suppose the cross-covariance function $C_{ij}^X$ is chosen to be the parsimonious Maté

Figures (13)

  • Figure 1: Standard Argo "park-and-profile" mission (Source: wong2020argo)
  • Figure 2: Empirical colocated correlation of temperature and salinity residuals from January to March in 2016.
  • Figure 3: Temperature residuals from January to March in 2016 obtained by yarger2022functional. (Top) Six reference locations are marked with their coordinates. Residuals are shown in their natural units ($^\circ C$) with the color scale truncated to $\pm 1$$^\circ C$ for visual clarity; a small number of more extreme values are omitted. No normalization or standardization was applied. (Bottom) Empirical standard deviations (in log-scale) at every 50-meter depth interval along two selected longitude transects: 150° W and 25° W. These transects were chosen because they intersect the most number of reference locations while avoiding land crossings, allowing for a continuous vertical slice of the ocean to be examined. The latitudes of the closest reference locations along each transect are delineated with dashed lines and labeled accordingly.
  • Figure 4: Same as Figure \ref{['fig:yarger_residuals_temp']} but for salinity residuals. Residuals are in PSU, with the color scale truncated to $\pm 0.5$ PSU for legibility. No normalization or standardization was applied.
  • Figure 5: Vertical shapes of $c_i(p)$ for three sets of coefficients (top row) and resulting colocated correlation curves (bottom row). Colocated correlations are shown for $\rho_{12} = -0.9, \ldots, 0.9$.
  • ...and 8 more figures

Theorems & Definitions (1)

  • Proposition 1