3D Bivariate Spatial Modelling of Argo Ocean Temperature and Salinity Proles Mary Lai O. Salva na and Mikyoung Jun1

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3D Bivariate Spatial Modelling of Argo Ocean

Temperature and Salinity Proﬁles

Mary Lai O. Salva˜na and Mikyoung Jun1

October 24, 2022

Abstract

Variables contained within the global oceans can detect and reveal the eﬀects of the warm-

ing 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 multivari-

ate 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 vari-

ables while incorporating the underlying vertical structure of the ocean. We demonstrate that

proposed cross-covariance models describe the complex vertical cross-covariance structure well,

while existing cross-covariance models including bivariate Mat´ern models poorly ﬁt empirical

cross-covariance structure. Furthermore, the results show that using one more variable signiﬁ-

cantly enhances the prediction of the other variable and that the estimated spatial dependence

structures are consistent with the ocean stratiﬁcation.

Some key words: 3D covariance functions; Argo; cross-covariance function; nonstationary;

salinity; spatial; temperature.

1Department of Mathematics, University of Houston, 4800 Calhoun Rd, Houston, TX 77004, USA

E-mail: msalvana@central.uh.edu mjun@central.uh.edu.

Mikyoung Jun acknowledges support by NSF DMS-1925119 and DMS-2123247. The authors also acknowledge

helpful discussions with Mikael Kuusela on Argo data.

arXiv:2210.11611v1 [stat.AP] 20 Oct 2022

1 Introduction

The international scientiﬁc program named Array for Real-time Geostrophic Oceanography

(ARGO) was launched in the early 2000s as a response to the call of global observation net-

works to monitor the climate system (Argo, 2000; Johnson et al., 2022). Since then, the program

has launched a global network of 4000 free-drifting Argo proﬁling ﬂoats that measure ocean

variables in the upper 2000 meters of the world’s oceans. Each ﬂoat performs a ten day dura-

tion “park-and-proﬁle” mission. From the surface, the ﬂoat descends to the drift depth at 1000

meters where it will park for 9 days. At the 10th day, the ﬂoat descends to 2000 meters and

collects temperature (in degree Celsius or ◦C), salinity (in practical salinity unit or PSU), and

pressure (in decibars or dbar) measurements as it ascends to the surface. Once at the surface,

the data collected are transmitted via satellite. The full “park-and-proﬁle” mission is illustrated

in Figure 1. The data harvested by the ﬂoats are made available within 24 hours of its collection

as Argo data products (https://argo.ucsd.edu). By making the data publicly accessible, the

Argo program oﬀers the research community the opportunity to analyze ocean processes and

challenges modelers to improve existing data-driven scientiﬁc methods.

The core Argo data products are temperature and salinity measurements, the two most im-

portant variables through which other oceanographic variables, such as freezing point, electrical

conductivity, viscosity (Pawlowicz, 2013), ocean heat content and potential density (Yarger et al.,

2022), and tropical cyclone-induced ocean thermal response (Hu et al., 2020), can be derived. The

temperature and salinity variables have been key pieces in understanding the physical properties

and dynamics of the ocean. Their distributions were shown to drive ocean circulation (Chen

et al., 2022; Gangopadhyay, 2022), aﬀect climate processes (Olson et al., 2022), and change

biogeochemistry (Gal´an et al., 2021; Ding et al., 2022). They were also used to understand

underwater sound propagation for hydroacoustics research (Jana et al., 2022), design oﬀshore

wind farms (Escobar et al., 2016), and identify stressors to organisms in seawater and freshwater

(Walker et al., 2020).

Figure 1: Illustration of the standard Argo “park-and-proﬁle” mission. Source: (Wong et al.,

2020)

1.1 Spatial Interpolation of Argo Data

Spatially continuous maps of temperature and salinity measurements are essential to multidisci-

plinary scientiﬁc research. However, despite the planetary scale and subsurface reach of the Argo

proﬁling network, not all locations are sampled. Since other ocean variables depend heavily on

temperature and salinity, eﬀorts are focused on obtaining the best interpolated maps of these

two variables. Several institutions have produced high-resolution gridded global temperature and

salinity datasets from the sparse measurements recorded by the Argo proﬁling ﬂoats using vari-

ous interpolation techniques (Liu et al., 2020). The list includes the EN4 dataset from the U.K.

Met Oﬃce (Good et al., 2013) and the grid point value of the monthly objective analysis using

Argo data (MOAA) of the Japan Agency for Marine-Earth Science and Technology (JAMSTEC)

(Hosoda et al., 2008), both of which use various optimal interpolation methods and covariance

functions. Another example is the Barnes objective analysis Argo (BOA-Argo) gridded dataset

that employs objective interpolation based on Barnes successive correction method (Li et al.,

2017).

One of the most highly used Argo data product is the Roemmich–Gilson Argo climatology

from the Scripps Institution of Oceanography (Roemmich and Gilson, 2009). They used weighted

local regression in its high-resolution global mapping. Suppose s= (L, l, p)>is the spatial

location vector and tis time in yeardays. For s0= (L0, l0, p0)>, a reference spatial location

vector, and t0, the reference time index, they ﬁt the mean function:

µs0,t0(s, t) = β0+β1(L−L0) + β2(l−l0) + β3(L−L0)2+β4(l−l0)2+β5(p−p0) + β6(p−p0)2

k=1

γksin 2πk t

365.25+

k=1

δkcos 2πk t

365.25(1)

to measurements from the sampled location (L, l, p), such that (L, l) is one of the 100 nearest

neighbor from (L0, l0), for twithin 12 calendar months from t0, and for pwhich is one pressure

level above and below p0. Here β0and βk,γk, and δk,k= 1,...,6, are scalar coeﬃcients.

Furthermore, they modelled the residuals using the spatial covariance function:

C(L1, L2;l1, l2)∝0.77 exp "−hRG(L1, L2;l1, l2)

140 km 2#+ 0.23 exp −hRG(L1, L2;l1, l2)

1111 km ,(2)

where hRG(L1, L2;l1, l2) is a distance function of the form:

hRG(L1, L2;l1, l2) = p(L1−L2)2+ (l1−l2)2+ penalty(L1, L2;l1, l2)2.

Kuusela and Stein (2018) improved the modelling of the residuals by using an anisotropic

exponential space-time covariance function, i.e.,

C(L1, L2;l1, l2;t1, t2) = σ2exp {−hKS(L1, L2;l1, l2;t1, t2)},(3)

where σ2>0 is a variance parameter and hKS(L1, L2;l1, l2;t1, t2) is a new distance function that

captures anisotropy, i.e.,

hKS(L1, L2;l1, l2;t1, t2) = sL1−L2

θlat 2

+l1−l2

θlon 2

+t1−t2

θt2

Here θlat >0, θlon >0, and θt>0 are range parameters for the latitude, longitude, and time

dimensions, respectively.

Previous studies begin with discretizing the vertical domain of the Argo data into distinct

pressure levels and performing the high resolution spatial interpolation of each layer separately.

Furthermore, prior to ﬁtting their proposed means and covariance functions, given the irregular

sampling of Argo proﬁles along pressure, a linear interpolation is done to obtain measurements

at the ﬁxed pressure levels. This extra step can signiﬁcantly introduce errors to the model. To

address this problem, Yarger et al. (2022) proposed treating the Argo proﬁle data as a functional

data that changes with pressure and introduced a functional model for the mean of the form:

µL0,l0,t0(L, l, p, t, y) =

2016

˜y=2007

β0,˜y(p)(y=˜y)+β1(p)(L−L0) + β2(p)(l−l0)

+β3(p)(L−L0)2+β4(p)(l−l0)2+β5(p−p0) + β6(p−p0)2

k=1

γksin 2πk t

365.25+

k=1

γkcos 2πk t

365.25.(4)

Here ydenotes the year the measurement was obtained, (·)is an indicator function, and β0,y(p)

and βk(p), k= 1,...,6, are functions that direct how the coeﬃcients change along pressure.

Similar to the mean function in (1), the functional mean (4) is ﬁtted to the measurements from

the nearest neighbors. This functional data approach reports lower prediction errors than the

pressure-by-pressure approach. The residuals from Yarger et al. (2022) were then modelled using

functional principal components.

The aforementioned studies and other results in the Argo literature typically model temper-

ature separately from salinity rather than leverage joint information that may be contained by

the two variables. In this work, we extend the spatial modelling of temperature and salinity

residuals to include their joint spatial dependence structures. Furthermore, we propose bivariate

spatial models in 3D space that can incorporate special oceanographic structures. The ocean,

as a ﬂuid, has a density which is a continuous function of depth or pressure. This density dic-

tates the layering of the ocean waters or ocean stratiﬁcation which is a phenomenon that occurs

when diﬀerent types of water meet and mix, e.g. low-temperature and low-salinity subarctic

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3DBivariateSpatialModellingofArgoOceanTemperatureandSalinityProlesMaryLaiO.Salva~naandMikyoungJun1October24,2022AbstractVariablescontainedwithintheglobaloceanscandetectandrevealtheeectsofthewarm-ingclimateastheoceansabsorbhugeamountsofsolarenergy.Hence,informationregardingthejointspatialdistributi...

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