For each station, covariate information such as location, elevation and land use is available. The analysis is accompanied by a derivation of the resulting uncertainty of the estimates. Though different in details, the gridding procedures are algorithmically based, i. The ultimate goal is to reconstruct the entire spatiotemporal yearly or even monthly average temperature field, with appropriate measures of uncertainty, taking the model parameter uncertainty into account.

Since most of the spatial variation is linked to the rotational nature of the globe in relation to the sun, we shall here restrict ourselves to a rotationally invariant covariance model, which reduces the computational burden. However, we shall allow for regional deviations from rotational symmetry in the expectations. To avoid complications due to temporal dependence between monthly values, we aggregate the measurements into yearly means and model only the yearly average temperature at each location. Here, we include only the effect of elevation.

We also ignore any remaining residual dependences between consecutive years, analysing only the marginal distribution properties of each year. One of the most important steps is how to determine the conditional distribution for the weather given observations and model parameters, where is the conditional precision, and the expectation is the kriging estimator of x t.

The computational cost of the kriging estimates is in the number of observations, and approximately in the number of basis functions. This shows that when constructing computationally efficient models it is not enough to consider the theoretical properties of the prior model, but instead the whole sequence of computations needs to be taken into account. We implemented the model by using Rinla. Owing to the large size of the data set, this initial analysis is based on data only from the period —, requiring nodes in a joint model for the yearly temperature fields, measurements and linear covariate parameters, with nodes in each field, and the number of observations in each year ranging between approximately and , for each year including all stations with no missing monthly values.

## An innovation approach to random fields : application of white noise theory

The spatial covariance parameters are more difficult to interpret individually, but we instead show the resulting spatially varying field standard deviations and correlation ranges in Fig. Both curves show a clear dependence on latitude, with both larger variance and correlation range near the poles, compared with the equator.

The standard deviations range between 1. There is an asymmetric north—south pole effect for the variances, but a symmetric curve is admissible in the credible intervals. Also, the spherical harmonics that were used for the climate model are not of sufficiently high order to capture all regional effects.

To alleviate these problems, we base the presentation on what can reasonably be called the empirical climate and weather anomalies for the period —, in effect using the period average as reference. In Figs 9 a and 9 b , the posterior expectation of the empirical climate, , is shown including the estimated effect of elevation , together with the posterior expectation of the temperature anomaly for ,. The corresponding standard deviations are shown in Figs 9 c and 9 d.

As expected, the temperatures are low near the poles and high near the equator, and some of the relative warming effect of the thermohaline circulation on the Alaska and northern European climates can also be seen.

There is a clear effect of regional topography, showing cold areas for high elevations such as in the Himalayas, Andes and Rocky Mountains, as indicated by an estimated cooling effect of 5. Posterior means for a the empirical — climate and b the empirical mean anomaly with c and d the corresponding posterior standard deviations respectively: the climate includes the estimated effect of elevation; an area preserving cylindrical projection is used.

With this in mind, we might expect that the period of analysis and data coverage are too restricted to allow detection of global trends, especially since the simple model that we use a priori assumes a constant climate. The posterior standard deviation for each global average temperature anomaly was calculated to about 0. Comparing the values with the corresponding estimates in the GISS series, which has an observed trend of 1. Thus, the results here are similar to the GISS results, even without the use of ocean data.

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Although this result is not generally applicable for all covariance functions, the subclass of models where this result is applicable is substantial, and we expect to find additional versions and extensions in the future; see for example Bolin and Lindgren The explicit link makes these GFs much more practically applicable, as we might model and interpret the model by using covariance functions while doing the computations by using the GMRF representation which allows for sparse matrix numerical linear algebra.

Furthermore, the simplicity of the SPDE parameter specifications provides a new modelling approach that is not dependent on the theory for constructing positive definite covariance functions.

## Table of contents for Lectures on white noise functionals

The extension to manifolds is also useful, with fields on the globe as the main example. A third issue, which has not yet been discussed, is that the SPDE approach might help to interpret external covariates e. On the negative side, the approach comes with an implementation and preprocessing cost for setting up the models, as it involves the SPDE, triangulations and GMRF representations, but we firmly believe that such costs are unavoidable when efficient computations are required.

This paper is dedicated to the memory of Julian E. The authors thank the Research Section Committee and reviewers for their very helpful comments and suggestions. We are also grateful to Peter Guttorp for encouraging us to address the global temperature problem, Daniel Simpson for providing the convergence rate result 11 and the key reference in Appendix C. This appendix includes some explicit expressions and results that are not included in the main text.

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In this section, we derive explicit expressions for the building blocks for the precision matrices, for general triangulated domains with piecewise linear basis functions. To obtain explicit expressions for equation 19 , we need to introduce some notation for geometry of an arbitrary triangle. For the anisotropic version, parameterized as in Appendix A. As shown in Appendix C. The effects on the covariance functions resulting from using Neumann boundary conditions can be explicitly expressed as a folding effect. Theorem 1. Theorem 1, which extends naturally to arbitrary generalized rectangles in , is proved in Appendix D.

In practice, when the effective range is small compared with L , only the three main terms need to be included for a very close approximation: 24 The covariances for the oscillating model can be calculated explicitly for and , from the spectrum.

## An Innovation Approach to Random Fields: Application of White Noise Theory

To state concisely the theory needed for constructing solutions to SPDEs on more general spaces than , we need to introduce some concepts from differential geometry and manifolds. Here, we give a brief overview of the concepts that are used in the subsequent appendices.

For more details on manifolds, differential calculus and geometric measure theory see for example Auslander and MacKenzie , Federer and Krantz and Parks We also require the manifolds to be metric manifolds, so that distances between points and angles between vectors are well defined.

A bounded manifold has a finite maximal distance between points. If such a manifold is complete in the set sense, it is called compact. Finally, if the manifold is compact but has no boundary, it is closed. The most common metric manifolds are subsets of equipped with the Euclidean metric. The prime example of a closed manifold is the unit sphere embedded in. In Fourier analysis for images, the flat torus commonly appears, when considering periodic continuations of a rectangular region. From the metric that is associated with the manifold it is possible to define differential operators.

In Euclidean metric on , we can write. An alternative to defining integration on general manifolds through mapping subsets into is to replace Lebesgue integration with integrals defined through normalized Hausdorff measures Federer, , , here denoted. This leads to a natural generalization of Lebesgue measure and integration that coincides with the regular theory on. We also need to define Fourier transforms on general manifolds, where the usual cosine and sine functions do not exist.

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Definition 1 generalized Fourier representation. The Fourier transform pair for functions is given by Here, we briefly abuse our notation by including complex functions in the inner products. Definition 2. We restrict ourselves to GFs that are at most as irregular as white noise. Definition 3 GF. The complicating issue in dealing with the fractional SPDEs that are considered in this paper is that, for some parameter values, the solutions themselves are discontinuous everywhere, although still more regular than white noise.

Inspired by Adler and Taylor , we solve this by using a generalized definition based on generalized functions. Definition 4 generalized function. The definition allows many of the regular integration rules to be used for generalized functions, without any need to introduce heavy theoretical notational machinery, and provides a straightforward way of generalizing definition 3 to the kind of entities that we need for the subsequent analysis.

Definition 5 generalized GF.

### Introduction

Of particular importance is the fact that white noise can be defined directly as a generalized GF. Definition 6 Gaussian white noise. We note that the popular approach to defining white noise on via a Brownian sheet is not applicable for general manifolds, since the notion of globally orthogonal directions is not present.