Research Proposal for ITT3: Spatio-Temporal Reconstruction of OceanTemperature and Salinity from Historical Data

Robbie Peck, University of Bath. r.peck@bath.ac.uk

February 2016

SummaryA statistical reconstruction of global ocean temperatures and salinity at different depths since 1950 isdesirable for climate modelling research. Bayesian Hierarchical Models are proposed for this reconstructionfitted using recently developed computational methods. A relative lack of data in the 20th Centrury,at large depths, and the different instruments used require that the model can quantify uncertaintysufficiently.

1 Introduction

The transport of heat and fresh water in the oceans drives the climate in different parts of the globe.Monitoring and understanding temperature and salinity are therefore scientifically important. An under-standing of the temperature and salinity flows in the past will help make predictions about the behaviourof the ocean makeup which will be useful for predictions of climate and weather. It is possible to measuretemperature at the ocean surface from space. However it is not possible to measure temperature belowthe surface in this way. In addition, salinity can only be measured by in water instruments.

The instruments used to measure temperature and salinity differ and some may offer more accuratereadings. In addition, the makeup of the types of instruments recording temperature and salinity varyover the decades from 1950. Until recently, temperature and salinity measurements have only come fromexpensive research ships producing relatively few readings. However recently, technology such as Argofloats have enabled more measurements around the globe at a finer spatial scale.

Spatial-temporal modelling characterises the distribution of underlying processes over space and timethat we assume data we observe has come from. It greatly increases our power to assess the relationshipsbetween modelled variables and the space and time at which they are observed.

The Met Office have available global observations of temperature and salinity since 1950. The data sethas a short period recently of good sampling of the full globe, but sparse early observations. In addition,there are few observations at large depths.

Figure 1: Plot of observations available in April 1965 (far left), April 1985 (mid left), April 1995 (mid right), April 2010(far right)

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2 Motivation

There are many aspects of this project that are challenging, and will require significant time and thoughtto overcome. Over the last few years, advances made in computational methods for spatio-statisticalmodelling have made the process of fitting complex models with large amounts of data quicker. This hasenabled researchers to fit models to larger data sets which until recently, would have been computationallyinfeasible. Central to this project is the ’Big N’ problem; that is the data has periods in time in whichit has large numbers of observations. Whilst previous approaches such as [11] have used Markov ChainMonte Carlo (MCMC) methods to fit similar types of models, this is not feasible here.

The sparse early observations in the 20th century will lead to great uncertainty in the posterior distribu-tions of the processes in this time period. Communicating the results of the model and comparisons toother available data sources from these time periods will help to build a picture of underlying processes inthe oceans. The data output from this project may be used for a data assimilation approach from anothermodel that models similar variables overlapping in in time with the time period we consider here.

3 National Importance

The Met Office is a world leader in climate modelling and weather prediction. Continuous novel researchin partnership with the Met Office will keep the UK at the forefront of climate research. The Met Officemust innovate and develop new methods in order to remain competitive compared to other weatherprediction services based around the world.

Of growing importance in the UK is the availability of ’big data’ and the development of complex modelsand novel computational methods that can cope with this. The Alan Turing Institute for Data Sciencehas recently been funded by UK government and aims to be at the forefront of novel techniques for dealingwith ’big data’. Hence this research would link with this initiative. Investment in this field of statistics isconsidered important as it would allow the publicising of models and computational methods that maybenefit reconstruction and forecasting models in applications different to this one.

The University of Bath has a large pool of expertise in using complex statistical models with ’big data’and in particular the development and application of INLA in order to fit these models. Given the existinglinks with the Met Office through academics and the Institute for Mathematical Innovation, investmentin this area may be considered an efficient use of a research grant. Given the recruitment of a motivatedand competent doctoral student, this project would be feasible with a main academic supervisor such asFinn Lindgren combined with an industrial external secondary supervisor from the Met Office.

4 Research Hypothesis and Objectives

To infer meaningful results from the model, simplifications of the full model will be fitted to subsets ofthe data. An examination of the changes in the fitted model parameters in the analysis of these simplermodels would allow an assessment of the effect of the model assumptions. A subset of the data includesthe EN4 data set, which spans an area in the North Pacific Ocean with measurements of temperatureand salinity and has been studied before in [13]. In particular, the following simplifications of the modelare proposed:

• EN4 subset of the data, time homogeneous, temperature only: The model is fitted to the EN4subset of the data with variable temperature only. This consists of constructing a 3D mesh andusing finite element methods to solve an Stochastic Partial Differential Equation within this space.This will examine the fit of the model with regards to whether the spatial component explains the

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variability in the data. The separation of depth from longitude and latitude will be investigatedincluding whether depth should be a co-variate or part of the spatial field.

• EN4 subset of the data, time heterogeneous, temperature only: The time component will be addedin. The objective of this is to examine any assumptions made about time effects, in particularthe separability of time and spatial correlation structures in the model. A random walk and auto-regressive time structure will be investigated.

• EN4 subset of the data, time homogeneous temperature-salinity joint modelling: Given the physicallink between temperature and salinity, one can model the two processes jointly. The two variableswill be modelled independently at first, and then jointly. The exact link specified between the twocan be examined in terms of its physical meaning and to what extent it explains the variability inthe data.

• EN4 subset of the data, time heterogeneous, temperature-salinity joint modelling: The time com-ponent of the model will then be added in. One will examine the validity about the time correlationstructures and temperature-salinity joint modelling together.

The above four models are then fitted to the global data set. This will examine the scalability of themethod. Once the model is fitted, inference can be made on trends, such as how temperature changeswith time. Temperature and Salinity measurements should be available at any point by possibly usingsome interpolation technique (depending on the exact methodology used) in space globally from 1950.

The main challenge of the reconstruction is the sparse data set. This can be overcome by the combinationof spatial covariance structure assumptions, temporal processes, and dependence characteristics enablingus to ’fill in the gaps’ and obtain a posterior distribution for credible observations at all points in spaceand time. This is done in the context of a Bayesian Hierarchical Model.

5 Methodologies

5.1 Bayesian Hierarchical Model

The models proposed are of a hierarchical nature, which is often assumed to be the inherent structureof spatial and spatio-temporal data. A Bayesian hierarchical model is presented in three stages and isa flexible framework for statistical modelling. The use of this model allows us to perform inference ondifferent levels in the model such as the underlying latent process. The reader is referred to Shaddick andZidek (2015) [12].

The first level of the Bayesian Hierarchical model is the observational level; where the data is assumedto come from an underlying process with measurement error. The second level is the underlying process,which is assumed to vary spatially and temporally. The third level is the hyperprior, which describesprior knowledge about the model parameters.

Measurements or other variables that may help us explain the variability in temperature and salinity maybe available in the form of covariates which contribute to the underlying process in the second level.

Let ys,t and zs,t represent the temperature and salinity respectively at location s in terms of latitude,longitude, and depth, and time t. The location s varies over S, the locations on the globe that are madeup of oceans, and t varies over T = {1, ..., 792} representing each month from Jan 1950 to Dec 2015. Aparticular type of model of temperature only may be described as follows:

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Level 1: Observations

ys,t = us,t + εs,t (1)

• εs,t represents measurement error, which is independent and and identically distributed N(0, σ2ε )

random variables.

Level 2: Underlying Process

us,t = Xs,tβ +ms + θt (2)

• Xs,t is a 1× q-dimensional vector of covariate data that may change in space and time.

• β is a q × 1-dimensional vector of regression coefficients

• ms represents the spatial effect at location s. This may or may not include depth.

• θt represents the temporal process at time t

Level 3: Hyperprior

All the covariates in β, the parameters of the spatial correlation structure, the time correlation structure,and the variance of the measurement error are assigned prior distributions.

5.1.1 The Underlying Spatial Effect Model ms and Computational Methods for fitting theModel

Let an unidirected graph be G = (V, E), where V are the set of nodes and E the set of edges. Supposexi is the realisation of a node i for any i ∈ V. Then x = {xi : i ∈ V} is a Gaussian Random Field ifx ∼ MVN(0, σ2Σ). The entry (i, j) of the correlation matrix Σ represent the correlation between xi andxj on nodes i and j. The Gaussian Random field is stationary if the correlation is a function of sites onlybetween two nodes.

A Gaussian Markov Random Field is a Guassian Random Field if ∀(i, j) ∈ V such that (i, j) /∈ E , xi isconditionally independent of zj given all other nodes z{−i,j}. [2]

The covariance of a Gaussian Markov Random Field can be governed by a precision matrix Q = Σ−1

where Q is of the form:

Qij ={

1 if i ∈ Nj0 if i /∈ Nj

(3)

where Nj is the set of neighbours of node i. The precision matrix Q must be positive definite in orderthat the Gaussian Markov Random Field is well defined. [2, 12]

The idea is that the structure of the precision matrix allows for more efficient computations. HenceGaussian Markov Random Fields are convenient models from a theoretical and computational point ofview, which has led to them being widespread in spatio-temporal modelling. In the methodology forfitting the model, we aim to exploit this computational efficiency.

Let the vector x := {xs | s ∈ S2 ⊂ S} be some finite element representation of the spatial field at a finitenumber of points S2 ⊂ S (For example, a basis expansion of the spatial field or if the model is defined ona finite number of points on space). We stipulate that:

m ∼ MVN(0, σ2xΣx) (4)

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The spatial structure is controlled in the correlation matrix. A common and intuitive form of the covari-ance structure is the Matérn Covariance structure [3].

One possible way of fitting the model is by MCMC which is asymptotically exact and is very wellknown [9, 14]. Software like WinBUGS [4] enable a quick specification of the model, but MCMC iscomputationally heavy and difficult to implement. Laplace approximation is an approximate Bayesiantechnique to find posterior distributions which is less computationally heavy. Integrated Nested LaplaceApproximation (INLA) is an approximate Bayesian method proposed in [6]. The method performs cal-culations to obtain the posterior distribution of a latent Gaussian model (which the model above is)using Laplace approximations. The aim of the approach is to find posterior densities for the underlingprocess distributions us,t, and the parameters of the model. Given our model is guassian, the laplaceapproximation is exact.

It is proposed that the INLA-type approach is undertaken in conjunction with the approach developed byLindgren [5]. In this approach, the spatial effect formulated by a Stochastic Partial Differential Equation(SPDE) approach which exploits the sparse precision matrix as described above [8]. The package R-INLAallows [7] provides a user interface which is able to handle a multitude of different models with fixed effects,non-linear terms, and random effects. The functions in the package return results from fitted models suchas the marginal distributions of the latent process and model parameters.

In this SPDE approach, ones finds a Gaussian Markov Random Field with a neighbourhood structurethat produces a sparse precision matrix Q that best represents the Matérn field. To do this, one startswith the Gaussian Random Field with Matérn covariance function and a Markov structure is induced onit to make it a Gaussian Markov Random Field. In particular, assume that

x ∼ MVN(µ, Q−1) (5)

is a Gaussian Random Field in domain D.

Then we stipulate that {xs|s ∈ D} is a solution of:

(κ2 −∆)α/2xs =Ws s ∈ Rd, α = ν + d/2, κ > 0, ν > 0 (6)

where (κ2 −∆)α/2 is a pseudo-differential operator, ∆ is the Laplacian and W(s) is spatial white noisewith unit variance. The marginal variance of the process is given by

σ2 = Γ(ν)Γ(ν + d/2)(4π)d/2κ2ν (7)

This representation provides the link from a Gaussian Random Field to a Gaussian Markov RandomField, using an approximate solution to the SPDE. As in [5], one builds an approximate solution using afinite element approach with a Delauney triangulation over the domain of interest D. This triangulisationis adaptive and places a finer mesh at points where there is more data available. By defining a set of basisfunctions, one can give a finite element representation of the solution xs. Details about implementationcan be found in [5]. Given that depth is included in the spatial field, one may find it suitable to specifylarger basis functions at larger depths where the variability is less extreme.

One possible approach is to fit this model at local areas on the globe, each with fixed covariates and modelparameters. Then one could use techniques such as Bayesian Melding to combine the results togetherinto a global model.

6 Case Studies and Intepretation

As outlined in the objectives, case studies using particular subsets of the data will be performed in thisproject. This will examine the validity of the model in a more local area. The data provided in the form

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of the E14 data set covers an area of the North-West Pacific Ocean and has been studied before [13]. Thiswill show the practicalities and limitations of the proposed framework of the model. The case studies willbe examined whilst working closely with researchers at the Met Office.

Figure 2: A 2D reconstruction ofthe temperature at depth -10m. Thered points are the observations withthe mean of the posterior distributionof the spatial field ranging from blue(colder) to red (red) in the background.

Currently the Met Office have the methodology for a 2D reconstructionof subsets such as E14 using INLA as shown in Figure 2. Working withthem to bring in the depth as in our full model would be desirable.

Academic papers such as [11] have attempted fitting similar modelsusing the time period in which large data coverage is available. Thismostly comes from Argo floats. Using the INLA approach in contrastto the MCMC approach taken in [11] will make a valid case study foruse in model checking.

Over the 3 years, 3 Workshops will be held so that the best practiseswith regards to fitting spatio-temporal models in this context can beshared. This will open up discussion regarding the methods used andallow for any criticisms of the method. It is hoped researchers involvedin spatio-temporal statistics and industrialists working on problemsrequiring these types of models will attend and be able to contribute.New developments and results of case studies will naturally be releasedas articles in journals that are relevant to the project. Examples ofjournals of interest include Spatial Statistics and the Journal of Cli-mate.

7 Impact

Figure 3: Globally averaged oceantemperature anomalies as a functionof depth and time relative to a 1971-2010 mean reveal warming at all depthsin the upper 700m over the relativelywell-sampled 40-year period considered(top), and the mean over all depths(bottom), from [10]

Research into climate is a globally important objective. Specifically,the change in the statistical distribution of weather patterns when thischange is over a long period of time has received much attention andfunding in recent years. National and pan-national policy is influencedand shaped by research in this area in attempts to mitigate the effectsof the changing climate.

Ocean analyses in particular are useful for a range of purposes includ-ing the study of ocean circulation, climate variability, monitoring theaccumulation of heat in the oceans under climate change and validationof ocean and coupled ocean-atmosphere climate models. [13] [15]

Changes in global ocean heat content (that is, heat integrated acrossthe ocean globally) is a primary indicator of global climate change.Approaches have been made to measure this change in global oceanheat content, such as in Levitus et al. [10] using interpolation of pastrecords. The mean global average temperature is shown as a functionof depth in Figure 3.

Hence a full reconstruction of temperatures and salinity will enable further analyses of the global oceanheat content with respect to time to enable us to contrast the results against those of studies such as [10].Further, reconstructions can be used as input to ocean reanalyses [1].

Global oceans are the largest store of heat accumulated by the Earth under climate change, owing to thelarge mass and high heat capacity compared to the atmosphere. Ocean temperature reconstructions can

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herefore also be used to study and monitor major patterns of variability in the climate, such as the ElNiño Southern Oscillation (ENSO).

In summary, the main impact of this research would be the availability of a huge amount of reconstructeddata available to climate scientists. Due to the nature of the problem, this would benefit both academicand industrial scientists involved in the problem. The reconstructed data would enable the building ofmore accurate models to measure global temperature trends and validate weather prediction approachesthat use coupled ocean-atmosphere climate models.

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[5] Havard Rue Finn Lindgre and Johan Lindstrom. An explicit link between gaussian fields and guassianmarkov random fields: the stochastic partial differential equation approach. Journal of the RoyalStatistical Society: Series B (Statistical Methodology), pages 73(4):423–498, 2011.

[6] Sara Martino Havard Rue and Nicolas Chopin. Approximate bayesian inference for latent gaussianmodels by using integrated laplace approximations. Journal of the Royal Statistical Society: SeriesB (Statistical Methodology), pages 71(2):319–392, 2009.

[7] Sara Martino Havard Rue and Finn Lindgren. The r-inla project. http://www.r-inla.org. Accessed:2016-01-01.

[8] Gianluca Baio Marta Blangiardo Michela Cameletti and Havard Rue. Spatial and spatio-temporalmodels with r-inla. Spatial and Spatio-Temporal Epidemiology, pages 7:39–55, 2013.

[9] Marshall N Rosenbluth Augusta H Teller Nicholas Metropolis, Arianna W Rosenbluth and EdwardTeller. Equation of state calculations by fast computing machines. The Journal of Chmical Physics,pages 1087–1092, 1993.

[10] T. P. Boyer R. A. Locarnini H. E. Garcia S. Levitus, J. I. Antonov and A. V. Mishonov. Global oceanheat content 19552008 in light of recently revealed instrumentation problems. Geophysical ResearchLetters, 2009.

[11] S. K. Sahu and P Challenor. A space-time model for joint modeling of ocean temperature and salinitylevels as measured by argo floats. Environmetrics, pages 19: 509–528, 2008.

[12] G. Shaddick and J. Zidek. Spatio-Temporal Methods in Environmental Epidemiology. Texts inStatistical Science. CRC Press, Florida, 2016.

[13] Nick A. Rayner Simon A. Good, Matthew J. Martin. En4: Quality controlled ocean temperature andsalinity profiles and monthly objective analyses with uncertainty estimates. Journal of GeophysicalResearch, 2013.

[14] Adrian FM Smith, Gareth O Roberts. Bayesian Computation via the Gibbs Sampler, and RelatedMarkov Chain Monte Carlo Methods. The matrn function as a general model for soil variograms.Journal of Royal Statistical Society. Series B (Methodological), pages 3–23, 1993.

[15] S. Cusack A. W. Colman C. K. Folland G. R. Harris Smith, D. M. and J. M. Murphy. Improvedsurface temperature prediction for the coming decade from a global climate model,. Journal ofGeophysical Research, pages 317, 796799, 2007.

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