Laplacian Eigenfunctions for Climate Analysis

Timothy DelSole George Mason University, Fairfax, Virginia, and Center for Ocean–Land–Atmosphere Studies, Calverton, Maryland

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Michael K. Tippett Department of Applied Physics and Applied Mathematics, Columbia University, New York, New York, and Center of Excellence for Climate Change Research, Department of Meteorology, King Abdulaziz University, Jeddah, Saudi Arabia

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Abstract

This paper proposes a new method for representing data in a general domain on a sphere. The method is based on the eigenfunctions of the Laplace operator, which form an orthogonal basis set that can be ordered by a measure of length scale. Representing data with Laplacian eigenfunctions is attractive if one wants to reduce the dimension of a dataset by filtering out small-scale variability. Although Laplacian eigenfunctions are ubiquitous in climate modeling, their use in arbitrary domains, such as over continents, is not common because of the numerical difficulties associated with irregular boundaries. Recent advances in machine learning and computational sciences are exploited to derive eigenfunctions of the Laplace operator over an arbitrary domain on a sphere. The eigenfunctions depend only on the geometry of the domain and hence require no training data from models or observations, a feature that is especially useful in small sample sizes. Another novel feature is that the method produces reasonable eigenfunctions even if the domain is disconnected, such as a land domain comprising isolated continents and islands. The eigenfunctions are illustrated by quantifying variability of monthly mean temperature and precipitation in climate models and observations. This analysis extends previous studies by showing that climate models have significant biases not only in global-scale spatial averages but also in global-scale dipoles and other physically important structures. MATLAB and R codes for deriving Laplacian eigenfunctions are available upon request.

Corresponding author address: Timothy DelSole, 112 Research Hall, MS 2B3, George Mason University, 4400 University Drive, Fairfax, VA 22030. E-mail: delsole@cola.iges.org

Abstract

This paper proposes a new method for representing data in a general domain on a sphere. The method is based on the eigenfunctions of the Laplace operator, which form an orthogonal basis set that can be ordered by a measure of length scale. Representing data with Laplacian eigenfunctions is attractive if one wants to reduce the dimension of a dataset by filtering out small-scale variability. Although Laplacian eigenfunctions are ubiquitous in climate modeling, their use in arbitrary domains, such as over continents, is not common because of the numerical difficulties associated with irregular boundaries. Recent advances in machine learning and computational sciences are exploited to derive eigenfunctions of the Laplace operator over an arbitrary domain on a sphere. The eigenfunctions depend only on the geometry of the domain and hence require no training data from models or observations, a feature that is especially useful in small sample sizes. Another novel feature is that the method produces reasonable eigenfunctions even if the domain is disconnected, such as a land domain comprising isolated continents and islands. The eigenfunctions are illustrated by quantifying variability of monthly mean temperature and precipitation in climate models and observations. This analysis extends previous studies by showing that climate models have significant biases not only in global-scale spatial averages but also in global-scale dipoles and other physically important structures. MATLAB and R codes for deriving Laplacian eigenfunctions are available upon request.

Corresponding author address: Timothy DelSole, 112 Research Hall, MS 2B3, George Mason University, 4400 University Drive, Fairfax, VA 22030. E-mail: delsole@cola.iges.org

1. Introduction

This paper proposes a new method for representing climate data in a general domain on a sphere. The method is based on eigenfunctions of the Laplace operator, which provide a natural basis for defining orthogonal spatial patterns that can be ordered by a measure of length scale. Such representation is useful for dimension reduction. Laplacian eigenfunctions are ubiquitous in climate modeling—for example, spherical harmonics provide the basis of all working spectral models—but the use of Laplacian eigenfunctions in arbitrary domains, such as over continents or ocean separately, is not common. One reason for this neglect is that Laplacian eigenfunctions in general domains under traditional Dirichlet or Neumann boundary conditions often are difficult to obtain. In particular, an irregular boundary typically cuts between grid points on a regular grid and therefore requires special differencing schemes that depend on the geometry of the boundary relative to the grid points. The development of schemes that have the same order accuracy as the scheme in the interior and yield stable numerics is challenging (Fogelson and Keener 2001). We exploit new techniques that have been proposed in computational science and machine learning literature. The basis set we propose depends only on the shape of the domain and thus is independent of the statistics of the data and is relatively easy to compute using standard eigenvalue solvers.

A major application of the proposed basis set is dimension reduction. Climate studies often involve massive datasets. For example, a field represented on a typical 2.5° × 2.5° longitude–latitude grid comprises over 10 000 numbers per field, and if the field evolves in time then the total number of numbers representing the data easily exceeds one million, which is too many for human comprehension. Accordingly, most climate studies effectively project data onto a reduced subspace whose dimension rarely exceeds 20. Indeed, the vast majority of “climate indices” are nothing more than projections of data onto prespecified spatial structures (e.g., the global average temperature is the projection of the data onto a spatially uniform pattern). The single most common method for representing climate data on a reduced subspace is principal component analysis (PCA), which is optimal in the sense that it captures the most variance for the specified dimension number (Mardia et al. 1979). Despite this optimality, there are at least three reasons why PCA is not ideal for climate studies. First, the components depend on the data and therefore change as the time period and data source change. This dependence makes comparisons across datasets and time periods difficult. Second, PCA tends to overfit variance, which leads to biases (Lawley 1956). These biases can be important in some statistical analyses (e.g., detection and attribution; see Allen and Tett 1999). Third, components that explain relatively little variance may be important to include in certain contexts (Farrell and Ioannou 2001; Giannakis and Majda 2012).

Another attractive property of Laplacian eigenfunctions is that they can be ordered by a measure of length scale. Many important components of climate variability are characterized by large-scale patterns. For instance, the response to changing greenhouse gas concentrations has a strong projection on the spatially uniform pattern (which is the largest possible scale on a globe). Seasonal variability caused by ENSO tends to be characterized by patterns that span continental scales (Ropelewski and Halpert 1987). In addition, identifying “signals” in data often requires spatial smoothing to filter out random weather noise on short spatial scales. Many spatial smoothers can be interpreted as projecting data on large-scale spatial structures. Laplacian eigenfunctions provide an objective, intuitive basis for filtering out small-scale variability. More generally, Laplacian eigenfunctions can be used to construct “low pass,” “band pass,” and “high pass” spatial filters.

The approach we propose is based on the work of Saito (2008). This approach derives Laplacian eigenfunctions from a Green’s function. In the case of a restricted domain, the Green’s function is generally unknown and difficult to compute. Nevertheless, one can still set up an integral equation for Laplacian eigenfunctions using the known Green’s function on a sphere. Saito (2008) proves that the eigenfunctions of the resulting operator are identical to those of the Laplace operator with a nonlocal boundary condition. Although the new boundary condition is more complicated than the traditional Dirichlet or Neumann boundary conditions, it is implicit in the solution method and does not appear explicitly. Moreover, our purpose is to define an orthogonal basis set to represent large-scale fields, not to solve Laplace’s equation, so the particular boundary condition imposed on the eigenfunctions is not critical. The main differences between our approach and that of Saito (2008) are that we use the exact Green’s function on the surface of a sphere, account for singularities more systematically, and enforce the first eigenfunction to be the spatially uniform pattern.

The proposed approach is related to recent work at elucidating the geometrical structure of low-dimensional nonlinear manifolds embedded in a high-dimensional space (Belkin and Niyogi 2003; Giannakis and Majda 2012). Belkin and Niyogi (2003) proposed representing the low-dimensional manifold using a graph composed of one node for each point and an edge connecting two nodes, with an edge weight that decays with the distance between the two nodes. The Laplacian of this graph was shown to be intimately related to the Laplace–Beltrami operator on manifolds whose solution involves the Green’s function of the heat equation. For short time intervals and short distances, the Green’s function effectively corresponds to an approximately Gaussian weight decay function. The resulting eigenfunctions are discrete approximations of the eigenfunctions of the Laplace–Beltrami operator on the manifold. Coifman and Lafon (2006) proposed the concept of diffusion maps based on eigenfunctions of a Markov matrix describing a random walk on the data. Diffusion maps unified many previous methods and clarified their relation to spectral clustering and the Fokker–Planck equation (Nadler et al. 2006). Giannakis and Majda (2012) applied these concepts to time-lagged embedded state vectors to identify features characteristic of nonlinear dynamics, such as intermittency and rare events, which are not accessible by classical techniques. While these methods are important to the overall problem of data representation, they depend on the data. Here, we are interested in basis sets that do not depend on data. Accordingly, we determine Laplacian eigenfunctions of the physical domain, which depend only on the geometry of the domain and hence can be precomputed independently of the data. Importantly, the resulting eigenfunctions can be applied to small sample sizes without encountering statistical complexities that arise from data-dependent basis vectors. Since the geometry of a sphere is known, we use the exact Green’s function for a sphere rather than a Gaussian approximation. In separate analysis (not shown), similar results were found using the Gaussian approximation. However, the Gaussian approximation requires specifying an arbitrary parameter, which effectively corresponds to the time scale of the diffusion process, whereas the method proposed here does not involve any tunable parameters. Also, since we use the exact Green’s function, our eigenfunctions converge to familiar spherical harmonics when the domain is the whole globe (as will be shown).

The next section describes our methodology in detail. Following that, section 3 compares the Laplacian eigenfunctions computed from our method to those computed from the discretized Laplace operator on a globe. It is shown that the resulting eigenvectors and eigenvalues are very similar, which validates the numerical method, at least on a globe, despite being derived from two very different calculations. Section 4 shows that the leading Laplacian eigenfunctions of a high-resolution grid can be obtained quite accurately from a coarsened grid. Section 5 discusses certain mathematical questions that arise from the geometry of the boundary conditions. Sections 6 and 7 show the results of applying Laplacian eigenfunctions to describe the variability of surface temperature and precipitation. We conclude with a summary of our results.

2. Laplacian eigenfunctions on the surface of a sphere

The Laplace operator on a sphere is
e1
where θ and ϕ and are latitude and longitude, respectively. The eigenfunctions of the Laplace operator satisfy
e2
where λ is an eigenvalue. There are at least two numerical methods for finding eigenfunctions of the Laplace operator: one based on the differential operator (1) and one based on the integral operator. We use the integral operator, which requires knowing the Green’s function. The Green’s function gives the response at the point arising from a point impulse at the source point . The Green’s function also satisfies certain regularity conditions everywhere except at the point (), where the nature of the singularity is determined in a standard way by Green’s formula. If the Green’s function is known, then the solution to the inhomogeneous equation
e3
where f is a suitable forcing function, is given by
e4
Substituting in the above equation leads to the homogeneous integral equation
e5
Eigenfunctions are unique up to a multiplicative constant. We choose the constant such that the area average square of the eigenfunction is unity:
e6
Courant and Hilbert (1962) show that the Green’s function of the Laplace operator on a sphere is
e7
where ρ is the great circle distance between the points and . The great circle distance is given by the so-called haversine formula
e8
where and . Substituting the haversine function into the Green’s function yields
e9
The Green’s function satisfies the reciprocity property: it is invariant to swapping the forcing and response points. Thus, G is a symmetric operator. Unfortunately, the kernel in (5) is not symmetric. It is advantageous to use symmetric kernels, so we define the new function
e10
where β is a constant to be determined later, in which case the integral (5) becomes
e11
where
e12
Since K is symmetric, the theorems on symmetric kernels can be used; in particular, the kernel has a complete orthonormal basis and can be represented by an eigenfunction expansion.
Our goal is to compute Laplacian eigenfunctions on a discretized domain of arbitrary shape on a spherical surface. Accordingly, we discretize the integral equation by representing variables on a regular grid in which the longitudes are equally spaced with increment and the latitudes are equally spaced with increment . Our procedure also could be applied to nonuniform grids, but only regular grids are considered here because of their prevalence. Also, the grid is assumed to exclude the pole points , to avoid indeterminate ratios. Suppose the grid cells are ordered as , where S is the total number of grid cells. Let the value of Y at the center of the ith grid cell be , and let the Green’s function corresponding to a forcing at j and response at i be
e13
Using the midpoint rule to approximate integrals, the integral (11) becomes
e14
where
e15
Unfortunately, the Green’s function (7) is singular at , implying that the diagonal elements of are infinite. To handle this singularity, we use a different discretization at elements corresponding to . Specifically, at the ith grid cell, the integral (5) is
e16
where W is the antiderivative of :
e17
For diagonal elements corresponding to , the small-angle approximation can be invoked to write the Green’s function (7) as
e18
This function should be integrated over the rectangular grid cell or perhaps an ellipse covering the same area. The resulting integrals cannot be evaluated in closed form but can be reduced to one-dimensional quadrature, which in turn can be evaluated numerically. Alternatively, the integration can be performed over a circle with the same area as the grid cell, which is attractive because the integral can be evaluated analytically as
e19
where is the radius of a circle whose area is approximately that of the ith grid cell:
e20
We explored these different approaches but could not find any significant difference between using an ellipse, rectangle, or circle to approximate the integral equation at the source point. Accordingly, we use the circle since it is much simpler and can be expressed in closed form.
Consolidating the above results, the integral (14) can be written as the eigenvalue problem
e21
where γ is an eigenvalue of and
e22
Note that the eigenvalues of correspond to inverse eigenvalues of the Laplace operator. The matrix elements depend only on the coordinates inside the domain and hence are trivial to evaluate numerically for arbitrary domains. The eigenvector problem (21) yields S eigenvectors , which can be collected as separate column vectors to form the matrix . The eigenvectors of (21) are related to the Laplacian eigenvectors according to (10), which in matrix form is
e23
where is a diagonal matrix whose ith diagonal element is , and the columns of are the Laplacian eigenvectors. In this case, the Laplacian eigenfunctions are obtained as . The matrix is never singular because the pole points are excluded from the state vector.
As a result of the discretization, the constant function is no longer an eigenfunction. However, in climate studies, we often are interested in the spatial average over a domain. For instance, the global average temperature is of great interest in climate change studies, and standard climate indices such as Niño-3.4 and the Atlantic multidecadal oscillation (AMO) are derived from spatial averages. Therefore, a basis set that includes the spatially uniform pattern as a distinct member would be attractive. This can be done as follows: Let the spatially uniform pattern be denoted by the vector j, which is an S dimensional vector of all ones. Let the transformed eigenvector be , where α is a normalization constant chosen to give . Then we project onto the orthogonal complement of :
e24
Note that ; hence, is an eigenvalue of with zero eigenvalue. Inverting the variable transformation implies that is a Laplacian eigenvector (α is ignored because eigenvectors are unique up to a multiplicative factor). Therefore, we compute the eigenvectors of with nonzero eigenvalues and join these vectors with to form a complete, orthogonal basis set.
The matrix is symmetric and hence has real eigenvalues. On a spherical domain, these eigenvalues correspond to the inverse total wavenumber. Analytically, the eigenvalues of the Laplace operator on a sphere are for , where each eigenvalue has the multiplicity . Numerically, the eigenvalues of are not always of the same sign. For instance, on a 5° × 5° grid the largest and smallest eigenvalues are about 0.50 and −0.002. We have found that the eigenvalues usually can be rendered of the same sign by slightly increasing the radius of (20). Since the circle used to derive (20) is only an approximation to the grid cell, this adjustment seems allowable. However, this adjustment mostly shifts the eigenvalues upward by the same amount. Therefore, choosing the adjustment to render the last eigenvalue zero is approximately equivalent to simply shifting all the eigenvalues by a constant, in which case the total wavenumber would be defined as
e25
where is the ith largest eigenvalue of . This definition breaks down for high wavenumbers (i.e., for , but high wavenumber solutions (corresponding to small length scales) are inaccurate because of discretization errors anyway and are usually not of interest.
Owing to symmetry, the eigenvectors of are orthogonal with respect to the L2 norm. It is natural to normalize each eigenvector to unit length; that is, . This normalization, plus orthogonality, implies that is an orthogonal matrix:
e26
The transformation (23) and normalization (26) further imply
e27
Finally, we choose the normalization constant β to satisfy (6), namely,
e28
This choice implies that the column vectors of are orthogonal with respect to an area-weighted inner product, consistent with Laplacian eigenfunctions. To find the amplitudes of the Laplacian eigenfunctions, we need the inverse of . It is clear from (27) that the inverse of is
e29
This relation allows the inverse to be computed directly from the eigenvectors without numerically inverting a matrix. Using this inverse, a given S-dimensional state vector can be written as a sum of eigenvectors with amplitudes:
e30
More generally, if there exist N state vectors, say , then the corresponding amplitudes are
e31
and the data can be expressed as a linear combination of eigenvectors as
e32
The total area-weighted sum square is then
e33
where (27) has been used, and is a N-dimensional vector denoting the kth row of . The above equation shows that the total sum square can be expressed as a sum of “variances” for each component with no cross terms. Therefore, is the “variance” explained by the jth Laplacian eigenfunction, and the fraction of variance explained by the jth Laplacian eigenfunction is
e34

As an illustration, we computed the Laplacian eigenfunctions over the complete globe on a 5° × 5° regular grid. The leading Laplacian eigenfunctions, shown in Fig. 1, are virtually identical to the exact spherical harmonics sampled on the grid.

Fig. 1.
Fig. 1.

The first nine Laplacian eigenfunctions derived from the Green’s function method over the complete globe. The patterns are orthogonal with respect to an area-weighted inner product and normalized such that the area-averaged square = 1.

Citation: Journal of Climate 28, 18; 10.1175/JCLI-D-15-0049.1

Now consider Laplacian eigenfunctions over a restricted domain. In this case, the Green’s function is generally unknown and difficult to compute. Nevertheless, the eigenvalue problem (21) can still be set up by including only those grid points inside the domain. Saito (2008) showed that the continuous version of this new operator commutes with the Laplace operator, provided the solution satisfies certain nonlocal boundary conditions. A standard theorem in functional analysis states that if two operators commute, then they share the same eigenfunctions. Thus, one can find the eigenfunctions of whichever operator is easier to solve. The Green’s function approach is especially simple because it requires only the location of the grid points and pairwise distances. In particular, the nonlocal boundary conditions are implicit and are not used explicitly to obtain the eigenfunctions. Although the resulting eigenfunctions satisfy neither the Dirichlet nor the Neumann boundary conditions, they still define an orthogonal basis set that can be ordered by length scale. Since our purpose is to define an orthogonal basis set to represent large-scale fields, not to solve Laplace’s equation, the particular boundary condition imposed on the eigenfunctions is not critical, as long as it does not eliminate important modes of variability. In the above method, the derived basis set is complete, so all modes of variability are captured. The main differences between our approach and that of Saito (2008) are that we use the exact Green’s function on the surface of a sphere, account for singularities along the diagonal more systematically, and enforce the first eigenfunction to be the spatially uniform pattern.

Some mathematical subtleties related to the smoothness and connectedness of the boundary conditions are worth discussing. We defer this discussion to section 5 so that we can complete our description of the method for readers interested in practical applications.

3. Comparison with eigenvectors of the discretized Laplace operator

We now assess the accuracy of the above method. To do this, one might calculate Laplacian eigenfunctions on a sphere and compare them to the exact spherical harmonics. However, the continuous spherical harmonics are not orthogonal when sampled on a grid with equally spaced latitudes, whereas the eigenfunctions derived above are. Clearly, then, the two sets of vectors cannot be equal. Moreover, we do not want to relax the orthogonality constraint of the discretized eigenfunctions. Instead, a more appropriate comparison is with eigenfunctions of the discretized Laplace operator, which are orthogonal on the discretized domain. As in the previous section, we discretize the sphere on a regular grid. Accordingly, let denote the value of Y at the ith longitude and jth latitude. Let I and J denote the total number of longitudes and latitudes, respectively. A second-order differencing scheme for the Laplace operator (1) is
e35
where the subscript denotes the latitude halfway in between and j. The grid is defined such that the poles lie at and and is periodic in i in the sense that . The Laplacian eigenfunctions are then derived by solving the eigenvalue problem
e36
where is with all grid points stacked into a single vector, and is the corresponding operator (35). It should be recognized that defining the matrix for arbitrary domains is difficult because of the imposition of the boundary conditions. Note that the Laplacian eigenfunctions obtained from the discretized operator will not be exactly the same as those from the Green’s function method, owing to degeneracy, irrelevant phase shifts in the zonal direction, and numerical differences. To account for different orderings, the eigenfunctions from the Green’s function method are matched in one-to-one correspondence to the eigenfunctions from the discretized operator, based on the pair that maximizes the pattern correlation after all possible cyclic permutations in the zonal direction have been considered. The resulting pattern correlations for the first 100 eigenfunctions for a 5° × 5° grid are shown in Fig. 2. The pattern correlations are all above 0.96, implying that the eigenfunctions derived from the two methods are practically the same for the leading eigenfunctions.
Fig. 2.
Fig. 2.

Pattern correlation between Laplacian eigenfunctions computed from the Green’s function method and from the discretized Laplace operator. The eigenfunctions are computed from a 5° × 5° grid on the globe. The latter Laplacian eigenfunctions are resorted to maximize the pairwise correlation with the eigenfunction in the former set, to account for changes in order arising from numerical differences. The maximization also is performed for all possible cyclic permutations in the longitudinal direction, to account for irrelevant phase shifts in the zonal direction. The pattern correlation is defined based on an area-weighted inner product.

Citation: Journal of Climate 28, 18; 10.1175/JCLI-D-15-0049.1

4. Numerical considerations

Perhaps the most important limitation of the above methodology is that it requires solving the eigenvectors of a matrix whose dimension equals the number of grid points. For a global domain on a 2.5° × 2.5° regular grid, the number of grid points exceeds 10 000. Thus, even moderate resolution requires solving a high-dimensional eigenvalue problem. However, in practice, we typically are interested only in the leading eigenvectors characterized by large length scales. Fortunately, the leading Laplacian eigenfunctions are well approximated on coarser grids. To demonstrate this, we compare Laplacian eigenfunctions computed on a 5° × 5° grid with Laplacian eigenfunctions computed on a 10° × 10° grid and then bilinearly interpolated onto a 5° × 5° grid. The pattern correlation between the first 50 Laplacian eigenfunctions in the two sets, after reordering, as discussed in the previous section, is shown in Fig. 3. The pattern correlations exceed 0.96, with most exceeding 0.99. These high pattern correlations indicate that the leading Laplacian eigenfunctions computed on coarser grids and then interpolated onto finer grids are practically the same as the leading Laplacian eigenfunctions computed on the finer grid.

Fig. 3.
Fig. 3.

Pattern correlation between Laplacian eigenfunctions computed on a 5° × 5° and computed on a 10° × 10° grid and then linearly interpolated onto the 5° × 5° grid. The latter Laplacian eigenfunctions are resorted to maximize the pairwise correlation with the eigenfunction in the former set, to account for changes in order arising from numerical differences. The maximization also is performed for all possible cyclic permutations in the longitudinal direction, to account for irrelevant phase shifts in the zonal direction. The pattern correlation is defined based on an area-weighted inner product.

Citation: Journal of Climate 28, 18; 10.1175/JCLI-D-15-0049.1

If Laplacian eigenfunctions on a very high-resolution grid are desired, we recommend coarsening the grid and then computing Laplacian eigenfunctions on the coarsened grid. To quantify the error involved in this approach, let the leading M Laplacian eigenfunctions computed on the fine grid be the column vectors of the matrix , where the dot indicates that the matrix contains a truncated set of Laplacian eigenfunctions. If is a field on the fine grid, the representation of using M leading Laplacian eigenfunctions is
e37
where contains the first M columns of . Alternatively, we interpolate onto a coarse grid, perform the projection based on Laplacian eigenfunctions in the coarse grid, and then interpolate back to the fine grid. Let be an operator that bilinearly interpolates from low to high resolution. Then the representation based on the alternative approach is
e38
where is the pseudoinverse of . The mean square error of the latter representation is
e39
where denotes the Euclidean length of the vector . If is “white noise,” that is, , then
e40
where denotes the Frobenius norm of the matrix . The mean relative error
e41
shown in Fig. 4, reveals that even at 50 Laplacian eigenfunctions the mean error is less than 3%. This result shows that interpolating high-resolution data onto a coarse grid and using leading Laplacian eigenfunctions from the coarse grid to represent the data results in a very small error relative to using the leading Laplacian eigenfunctions on the high-resolution grid.
Fig. 4.
Fig. 4.

Mean percentage error Γ of representing data on a 5° × 5° grid using Laplacian eigenfunctions computed on a 10° × 10° grid. The data are first interpolated onto the coarse grid, where they are represented by Laplacian eigenfunctions computed on the coarse grid, and then interpolated back to the fine grid.

Citation: Journal of Climate 28, 18; 10.1175/JCLI-D-15-0049.1

Another potential concern is that numerical problems may be encountered when the grid is high resolution, especially if the grid includes points close to the poles (recall that pole points themselves are excluded from the grid). In particular, the rows and columns of the kernel matrix (15) associated with points close to the poles become arbitrarily small as the resolution increases. However, our experiments with high-resolution grids localized near the poles reveal that the eigenfunctions are remarkably insensitive to resolution, suggesting that this ill conditioning is not a problem in practice. This robustness is perhaps a consequence of the fact that when a row and column of a matrix approaches zero, the resulting eigenvalues also approach zero. Importantly, eigenvalues near zero correspond to high wavenumbers because of the inverse relation between eigenvalues of (15) and wavenumber. Thus, this singularity probably influences mostly the high wavenumber eigenfunctions, which typically are ignored. Technically, the matrix in (23) also becomes ill conditioned for points near the poles (because the ratio of the largest to smallest eigenvalues is large), but it is a diagonal matrix and hence its inverse can be computed accurately.

An alternative approach to finding the leading Laplacian eigenfunctions on a fine grid is to apply iterative techniques like Rayleigh quotient iteration or inverse iteration. However, these techniques work best for sparse matrices, which is not. In any case, iterative techniques require a first guess, and the above results suggest that very good first guesses can be obtained from coarsened grids. Saito (2008) also discusses approaches based on the fast multipole method. Another possible approach is to take advantage of the methods proposed by Belkin and Niyogi (2003) and Giannakis and Majda (2012). These methods invoke Gaussian approximations for Laplacian eigenfunctions, which can be represented by sparse matrices (because of the decay of the Gaussian function), which in turn have favorable scaling of the memory and computational costs.

Finally, the above Laplacian eigenfunctions depend on the discretization. This dependence is a necessary part of orthogonality; vectors that are orthogonal on one grid are not orthogonal when interpolated onto another grid. The results shown in Figs. 3 and 4, which involve dramatic changes in grid, suggest that the leading Laplacian eigenfunctions are not sensitive to discretization.

5. Issues related to the geometry of the boundary conditions

The properties of Laplacian eigenfunctions in bounded domains have received considerable attention in various disciplines, including mathematics, physics, and computational sciences [see Grebenkov and Nguyen (2013) for an extensive review]. In this section, we discuss additional aspects of the proposed method, some of which raise difficult mathematical questions.

a. Connectedness

Standard Laplacian eigenvalue problems assume that the domain geometry is connected; that is, the domain cannot be broken into disconnected spaces. In contrast, climate studies often involve disconnected domains, such as a land domain comprising isolated continents and islands. If the domain comprises disconnected spaces, then eigenfunctions satisfying Dirichlet or Neumann boundary conditions are uncoupled and can be solved separately for each domain. In contrast, applying the above Green’s function method to a set of disconnected spaces results in eigenfunctions in which the spaces are coupled. That is, each eigenfunction has nonzero amplitude in each geographically disconnected domain. The reason for this coupling is that the implicit boundary conditions are nonlocal and impose long-range connections between disconnected spaces. If desired, these long-range connections can be broken simply by setting to zero those matrix elements in the Green’s function corresponding to impulses and responses in disconnected spaces. Whether the desired eigenfunctions should in fact be coupled or uncoupled is a choice that depends on application. In the examples shown in section 7, we derive coupled eigenfunctions for the land domain, which itself is disconnected because of islands and continents.

b. Appropriateness of the boundary condition

Laplacian eigenfunctions depend on the boundary conditions. For instance, if Neumann boundary conditions are imposed, then the resulting eigenfunctions have vanishing normal gradients at the boundary and therefore cannot describe functions with nonvanishing normal gradients at the boundary. The question arises as to whether the utility of the eigenfunctions is diminished if the climate variable under study does satisfy the assumed boundary conditions?

First, note that the eigenfunctions produced by the discrete method are complete. This is clear from the fact that they are the eigenvectors of a square, symmetric matrix. When the eigenfunctions are used to represent a function that does not satisfy the boundary conditions, spurious oscillations tend to occur near the boundaries (e.g., Gibbs phenomena). In the continuous problem, these oscillations become increasingly concentrated near the boundaries as more eigenfunctions are added together but always exist in any finite series representation. In the discrete problem, these oscillations similarly become concentrated near the boundaries but eventually become concentrated into a space smaller than the grid size, so that they eventually disappear in the finite dimensional problem when the complete set of eigenfunctions are used.

Whether the boundary conditions imposed by the method are realistic is unknown. This question can be answered only empirically in the context of specific datasets. Our experience suggests that the implied boundary conditions tend to produce physically appealing eigenfunctions in a wide variety of geometries. Note that incompatible boundary conditions is not problematic in data-driven methods like principal component analysis because the basis vectors are obtained by linear projection and hence inherit whatever (linear) boundary conditions are satisfied by the data.

c. Smoothness

If the boundary is irregular or fractal, then a new phenomenon known as localization can occur (Grebenkov and Nguyen 2013). Localization refers to the tendency for the amplitude of the eigenfunction to be concentrated in a small subset of the domain, often near a boundary. Thus, as the domain boundary becomes increasingly irregular, especially at fractal-like coastlines, the Laplacian eigenfunctions may exhibit sensitive and complicated behavior. In particular, if the eigenfunctions are sensitive to the structure of a particular domain boundary, then the eigenfunctions may have limited usefulness as a basis set for representing data in that domain. Unfortunately, a rigorous theory of such phenomena is lacking. Fortunately, we have not found this behavior to be problematic, as will be shown shortly, but it is a possibility of which to be aware.

6. Data

We illustrate the usefulness of Laplacian eigenfunctions by using them to study variability of surface temperature and precipitation. For climate models, we use historical simulations from phase 5 of the Coupled Model Intercomparison Project (CMIP5) (Taylor et al. 2012). Historical simulations refer to climate model runs that simulate the period 1880–2000 forced by anthropogenic and natural forcings. We project Laplacian eigenfunctions on monthly mean atmospheric 2-m temperature and precipitation (the variables “tas” and “pr” in the CMIP5 protocol) and sea surface temperatures (“tos”). All available models were included in this analysis.

To validate model simulations, simulated 2-m temperatures over land are compared to the observational estimates of Fan and van den Dool (2008). These estimates are monthly means starting from 1948 and derived from land station data from the Global Historical Climatology Network, version 2 (GHCNv2), and the Climate Anomaly Monitoring System (CAMS). Over the ocean, sea surface temperature is compared to observational estimates from the Extended Reconstructed Sea Surface Temperature (ERSST), version 3b (Smith et al. 2008). These estimates start from 1854 and are derived from the International Comprehensive Ocean–Atmosphere Data Set. Simulated precipitation is compared with observational estimates derived from NOAA’s Precipitation Reconstruction over Land (PREC/L) (Chen et al. 2002). These estimates are monthly means starting from 1948 and derived from gauge observations over land from GHCNv2 and CAMS.

All datasets were interpolated onto a common 5° × 5° regular grid and then projected onto the Laplacian eigenfunctions calculated from the Green’s function method.

7. Comparison between observations and climate models

a. Land

The first nine Laplacian eigenfunctions over global land are shown in Fig. 5. The first Laplacian eigenfunction obviously corresponds to the land average temperature. The next two correspond mostly to a Eurasian–South American gradient and to a North American–African gradient. Note that the eigenfunctions are smooth and exhibit no localization near jagged coastlines. Moreover, the patterns are global even though some continents are disconnected.

Fig. 5.
Fig. 5.

As in Fig. 1, but for derived over land on a 5° × 5° regular grid.

Citation: Journal of Climate 28, 18; 10.1175/JCLI-D-15-0049.1

The projection coefficients are computed for the period 1950–2000 without removing the mean climatology. The mean and standard error (i.e., mean value plus and minus one standard deviation) of the amplitudes of the first three Laplacian eigenfunctions for the CMIP5 models for four selected months are shown in Fig. 6. The gray lines show the mean plus or minus the standard error of observations. The figure shows that the variability of average land temperature is inconsistent with observations for about a third of the CMIP5 models (as indicated by the fact that the standard error of these models do not overlap with the standard error of the observations). Also, HadGEM2 falls several standard deviations from the observed mean in some months but not in other months, indicating a significant error in the annual cycle that cannot be corrected by a single bias correction. Interestingly, the dispersion about the mean is fairly consistent across models and fairly consistent with observations. The trend of the projection coefficients during the 26-yr period 1975–2000 are shown in Fig. 7. The trend for the full 51-yr period 1950–2000 turns out to be insignificant owing to the cooling during 1950–70 that offset the warming after 1970; model trends are consistent with each other and with observations for the 1950–2000 period for the first three Laplacian eigenfunctions. For the first eigenfunction, observed 2-m temperature has a significant trend in all four months, and model trends are consistent with observed trends (as indicated by the fact that the confidence intervals for models overlap with those for observations). For the second eigenfunction, significant positive trends are found in observations only for July and October, but a few models give significant negative trends. For the third eigenfunction, no significant trends are found for observations, and all but one model is consistent with observed trends. In general, the vast majority of model trends of 2-m land temperature are consistent with observed trends for all three Laplacian eigenfunctions.

Fig. 6.
Fig. 6.

The mean and standard error of the projection coefficients for (top)–(bottom) the first three Laplacian eigenfunctions of 2-m land temperature in CMIP5 model simulations (color bars) and observations (gray horizontal lines). The colored bars are centered at the 1950–2000 mean value and have a width equal to twice the standard deviation of the monthly mean values. The gray lines are similarly constructed for observations. For clarity, the CMIP5 models are ordered based on the mean value in January.

Citation: Journal of Climate 28, 18; 10.1175/JCLI-D-15-0049.1

Fig. 7.
Fig. 7.

The 1975–2000 linear trend of the projection coefficients for (top)–(bottom) the first three Laplacian eigenfunctions of 2-m land temperature in CMIP5 model simulations (color bars) and observations (gray horizontal lines). The colored bars show the 95% confidence interval of the trend, and the gray lines show the 95% confidence interval for observations. For clarity, the CMIP5 models are ordered based on the mean value in January.

Citation: Journal of Climate 28, 18; 10.1175/JCLI-D-15-0049.1

Amplitudes also were computed for land precipitation. The mean value plus and minus one standard deviation of the amplitudes of the Laplacian eigenfunctions for the CMIP5 models for four selected months are shown in Fig. 8. More than half the models are seen to be inconsistent with observations for all three Laplacian eigenfunctions. Models disagree significantly even with the sign of the mean for the second and third eigenfunctions. Note also that the dispersion about the mean is similar between models and fairly consistent with observations for most models. Trends also were computed, but the trends are insignificant and model trends are consistent with observations.

Fig. 8.
Fig. 8.

The mean and standard error of the projection coefficients for (top)–(bottom) the first three Laplacian eigenfunctions of land precipitation in CMIP5 model simulations (color bars) and observations (gray horizontal lines). The colored bars are centered at the 1950–2000 mean value and have a width equal to twice the standard deviation of the monthly mean values. The gray lines are similarly constructed for observations. For clarity, the CMIP5 models are ordered based on the mean value in January.

Citation: Journal of Climate 28, 18; 10.1175/JCLI-D-15-0049.1

b. Ocean

The first nine Laplacian eigenfunctions over the ocean are shown in Fig. 9. As expected, the first eigenfunction corresponds to the ocean mean. The next two Laplacian eigenfunctions correspond to mostly an Atlantic–Pacific gradient and to an Indian–American gradient. The mean and standard error of the amplitudes of Laplacian eigenfunctions 2–4 are shown in Fig. 10. Results for the first eigenfunction are not shown because they essentially reproduce the well-known fact that models have large biases in global average temperature (e.g., Flato et al. 2013). The figure reveals considerable disagreement between models, and between model and observations, for all three Laplacian eigenfunctions. Some models are several standard deviations from observations. No model is consistent with observations in all 4 months for the second eigenfunction. Some models disagree even with the sign of the third eigenfunction. Most models are consistent with observed trends for all three Laplacian eigenfunctions (not shown). The fourth eigenfunction over the ocean is of interest because it corresponds to the north–south hemispheric gradient. The mean and standard error of the amplitudes, shown in the bottom panel of Fig. 10, reveals major inconsistencies between models and between model and observations. Surprisingly, some models disagree even with the sign of the pattern in some months.

Fig. 9.
Fig. 9.

As in Fig. 5, but for over the ocean. The domain is chosen to avoid grid points with sea ice.

Citation: Journal of Climate 28, 18; 10.1175/JCLI-D-15-0049.1

Fig. 10.
Fig. 10.

The mean and standard error of the projection coefficients for (top)–(bottom) Laplacian eigenfunctions 2–4 of sea surface temperature in CMIP5 model simulations (color bars) and observations (gray horizontal lines). The colored bars are centered at the 1950–2000 mean value and have a width equal to twice the standard deviation of the monthly mean values. The gray lines are similarly constructed for observations. For clarity, the CMIP5 models are ordered based on the mean value in January.

Citation: Journal of Climate 28, 18; 10.1175/JCLI-D-15-0049.1

c. Variance

The percent of variance explained by the Laplacian eigenfunctions is shown in Fig. 11. The thin gray lines show variances in CMIP5 historical simulations, while the line with dots shows variances in observations. Only results for the month of June are shown, but results for other months support the same general conclusion: climate models tend to accumulate variance faster than observations for land temperature, and climate models are most consistent with observations for land precipitation. The variance spectrum for land precipitation is approximately constant (“flat”), so the explained variance increases approximately linearly.

Fig. 11.
Fig. 11.

The percent of variance explained by Laplacian eigenvectors in CMIP5 historical simulations (thin gray lines) and observations (line with dots) for (top) June land temperature, (middle) June SST, and (bottom) June land precipitation. The variance is calculated over the 1950–2000 period for the month of June. Note the change in scale for the y axis.

Citation: Journal of Climate 28, 18; 10.1175/JCLI-D-15-0049.1

8. Summary and discussion

This paper proposed a new method for representing climate data based on eigenfunctions of the Laplace operator over an arbitrary domain on a sphere. The eigenfunctions were computed using an approach suggested by Saito (2008) but specialized to domains on a sphere and modified to include the spatially uniform pattern and to account for singularities more systematically. The resulting eigenfunctions satisfy a nonlocal boundary condition that differs from the traditional Dirichlet and Neumann boundary conditions of mathematical physics, but this difference is secondary for studies that use these functions to define an orthogonal basis set for representing data rather than to solve Laplace’s equation directly. Happily, the nonlocal boundary condition is implicit in the method and does not appear explicitly in the method. The resulting eigenfunctions are orthogonal with respect to an area-weighted inner product and can be ordered by a measure of length scale. As a result, these patterns provide an objective, intuitive basis for filtering small-scale variability. These patterns depend only on the geometry of the domain and hence do not require training data from models or observations. Potentially, this method may have significant advantages in studies with small sample sizes compared to data-based pattern methods (e.g., principal component analysis or singular value decomposition).

Computation of the above Laplacian eigenfunctions is relatively straightforward. First, one defines the spatial domain of interest. The domain can be multiply connected, defined on an irregular grid, or interrupted by “missing grid points.” In particular, the domain can be defined for just land, just ocean, or selected subsets of each. Second, one constructs the matrix , defined in (22), which is easy because each matrix element is an explicit function of the coordinates of the points in the domain. Certainly this step is trivial compared to numerical methods based on finite differences (Fogelson and Keener 2001) or finite elements. Third, one removes the spatially uniform component from to construct [see (24)]. Then the eigenvectors of are computed and transformed according to (23) to obtain the Laplacian eigenfunctions. The eigenvalues, after a slight shift, give the inverse total wavenumber. The Laplacian eigenfunctions computed this way were compared to the Laplacian eigenfunctions computed by discretizing the Laplace operator directly and found to be very similar, especially at low wavenumbers (corresponding to large length scale). For data on a high-resolution grid, we show that interpolating the data onto a coarse grid and representing the data using the leading Laplacian eigenfunctions, computed on the coarse grid, results in very little error relative to performing all computations on the high-resolution grid.

An interesting property of the eigenfunctions can be seen when they are derived for a domain comprising disconnected domains. In the case of traditional Dirichlet or Neumann boundary conditions, Laplacian eigenfunctions of disconnected domains are uncoupled and can be solved separately. In contrast, the procedure in this paper produces Laplacian eigenfunctions that have nonzero amplitudes in each of the disconnected domains; that is, the domains are coupled together. This coupling arises from the (implicit) nonlocal boundary conditions imposed in the procedure but can be “turned off,” if desired, by zeroing out appropriate elements of the matrix . We argue, however, that this coupling can be very attractive in some studies, especially those that attempt to characterize variability holistically across disconnected domains (e.g., quantifying land average quantities and land variability about that average). Whether the nonlocal boundary condition is appropriate for a particular study is a question that can be answered only empirically.

The above methodology was applied to monthly mean temperature and precipitation in CMIP5 simulations and in observations. The first eigenfunction is always the spatially uniform pattern. The amplitude of this pattern is obtained by performing an area average of the data over the domain. Such averages are common for defining climate indices; for example, the global average temperature is a key indicator of climate change. The secondary Laplacian eigenfunctions receive much less attention but seem no less important to understand and simulate correctly. As has been found in previous studies, the comparison revealed significant model biases in the uniform pattern for temperature and precipitation. The present study extends this result by showing significant model biases in the secondary Laplacian eigenfunctions as well, with some models constituting “outliers” and disagreeing even with the sign of the pattern. Models are generally consistent with observed trends, although some models are clear outliers in certain cases. The fourth Laplacian eigenfunction over the ocean, which measures the north–south hemispheric gradient, is not consistent across models nor consistent between models and observations, with some models disagreeing even with the sign of this pattern.

MATLAB and R codes for deriving Laplacian eigenfunctions are available upon request.

Acknowledgments

We thank Dimitris Giannakis for insightful comments that led to significant clarifications in this paper. Discussions with Naoki Saito, Dimitris Giannakis, Robert Sachs, and Tim Sauer were extremely helpful in the development of this work. We also thank two anonymous reviewers for helpful comments. This research was supported primarily by the National Oceanic and Atmospheric Administration, under the Climate Test Bed program (NA10OAR4310264). Additional support was provided by the National Science Foundation (NSF 1338427), National Aeronautics and Space Administration (NNX14AM19G), and the National Oceanic and Atmospheric Administration (NA14OAR4310160). The views expressed herein are those of the authors and do not necessarily reflect the views of these agencies.

REFERENCES

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    • Search Google Scholar
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    • Search Google Scholar
    • Export Citation
  • Grebenkov, D. S., and B.-T. Nguyen, 2013: Geometrical structure of Laplacian eigenfunctions. SIAM Rev., 55, 601667, doi:10.1137/120880173.

    • Search Google Scholar
    • Export Citation
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    • Search Google Scholar
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  • Nadler, B., S. Lafon, R. R. Coifman, and I. G. Kevrekidis, 2006: Diffusion maps, spectral clustering and reaction coordinates of dynamical systems. Appl. Comput. Harmonic Anal., 21, 113127, doi:10.1016/j.acha.2005.07.004.

    • Search Google Scholar
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  • Ropelewski, C., and M. Halpert, 1987: Global and regional scale precipitation patterns associated with the El Niño/Southern Oscillation. Mon. Wea. Rev., 115, 16061626, doi:10.1175/1520-0493(1987)115<1606:GARSPP>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Saito, N., 2008: Data analysis and representation on a general domain using eigenfunctions of Laplacian. Appl. Comput. Harmonic Anal., 25, 6897, doi:10.1016/j.acha.2007.09.005.

    • Search Google Scholar
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  • Smith, T. M., R. W. Reynolds, T. C. Peterson, and J. Lawrimore, 2008: Improvements to NOAA’s historical merged land–ocean surface temperature analysis (1880–2006). J. Climate, 21, 22832296, doi:10.1175/2007JCLI2100.1.

    • Search Google Scholar
    • Export Citation
  • Taylor, K. E., R. J. Stouffer, and G. A. Meehl, 2012: An overview of CMIP5 and the experimental design. Bull. Amer. Meteor. Soc., 93, 485498, doi:10.1175/BAMS-D-11-00094.1.

    • Search Google Scholar
    • Export Citation
Save
  • Allen, M. R., and S. F. B. Tett, 1999: Checking for model consistency in optimal fingerprinting. Climate Dyn., 15, 419434, doi:10.1007/s003820050291.

    • Search Google Scholar
    • Export Citation
  • Belkin, M., and P. Niyogi, 2003: Laplacian eigenmaps for dimensionality reduction and data representation. Neural Comput., 15, 13731396, doi:10.1162/089976603321780317.

    • Search Google Scholar
    • Export Citation
  • Chen, M., P. Xie, J. E. Janowiak, and P. A. Arkin, 2002: Global land precipitation: A 50-year monthly analysis based on gauge observations. J. Hydrometeor., 3, 249266, doi:10.1175/1525-7541(2002)003<0249:GLPAYM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Coifman, R. R., and S. Lafon, 2006: Diffusion maps. Appl. Comput. Harmonic Anal., 21, 530, doi:10.1016/j.acha.2006.04.006.

  • Courant, R., and D. Hilbert, 1962: Methods of Mathematical Physics. Vol. 1. John Wiley and Sons, 560 pp.

  • Fan, Y., and H. van den Dool, 2008: A global monthly land surface air temperature analysis for 1948–present. J. Geophys. Res., 113, D01103, doi:10.1029/2007JD008470.

    • Search Google Scholar
    • Export Citation
  • Farrell, B. F., and P. J. Ioannou, 2001: Accurate low-dimensional approximation of the linear dynamics of fluid flow. J. Atmos. Sci., 58, 27712789, doi:10.1175/1520-0469(2001)058<2771:ALDAOT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Flato, G., and Coauthors, 2013: Evaluation of climate models. Climate Change 2013: The Physical Science Basis, T. F. Stocker et al., Eds., Cambridge University Press, 741–866.

  • Fogelson, A., and J. Keener, 2001: Immersed interface methods for Neumann and related problems in two and three dimensions. SIAM J. Sci. Comput., 22, 16301654, doi:10.1137/S1064827597327541.

    • Search Google Scholar
    • Export Citation
  • Giannakis, D., and A. J. Majda, 2012: Nonlinear Laplacian spectral analysis for time series with intermittency and low-frequency variability. Proc. Natl. Acad. Sci. USA, 109, 22222227, doi:10.1073/pnas.1118984109.

    • Search Google Scholar
    • Export Citation
  • Grebenkov, D. S., and B.-T. Nguyen, 2013: Geometrical structure of Laplacian eigenfunctions. SIAM Rev., 55, 601667, doi:10.1137/120880173.

    • Search Google Scholar
    • Export Citation
  • Lawley, D. N., 1956: Tests of significance for the latent roots of covariance and correlation matrices. Biometrika, 43, 128136, doi:10.1093/biomet/43.1-2.128.

    • Search Google Scholar
    • Export Citation
  • Mardia, K. V., J. T. Kent, and J. M. Bibby, 1979: Multivariate Analysis. Academic Press, 518 pp.

  • Nadler, B., S. Lafon, R. R. Coifman, and I. G. Kevrekidis, 2006: Diffusion maps, spectral clustering and reaction coordinates of dynamical systems. Appl. Comput. Harmonic Anal., 21, 113127, doi:10.1016/j.acha.2005.07.004.

    • Search Google Scholar
    • Export Citation
  • Ropelewski, C., and M. Halpert, 1987: Global and regional scale precipitation patterns associated with the El Niño/Southern Oscillation. Mon. Wea. Rev., 115, 16061626, doi:10.1175/1520-0493(1987)115<1606:GARSPP>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Saito, N., 2008: Data analysis and representation on a general domain using eigenfunctions of Laplacian. Appl. Comput. Harmonic Anal., 25, 6897, doi:10.1016/j.acha.2007.09.005.

    • Search Google Scholar
    • Export Citation
  • Smith, T. M., R. W. Reynolds, T. C. Peterson, and J. Lawrimore, 2008: Improvements to NOAA’s historical merged land–ocean surface temperature analysis (1880–2006). J. Climate, 21, 22832296, doi:10.1175/2007JCLI2100.1.

    • Search Google Scholar
    • Export Citation
  • Taylor, K. E., R. J. Stouffer, and G. A. Meehl, 2012: An overview of CMIP5 and the experimental design. Bull. Amer. Meteor. Soc., 93, 485498, doi:10.1175/BAMS-D-11-00094.1.

    • Search Google Scholar
    • Export Citation
  • Fig. 1.

    The first nine Laplacian eigenfunctions derived from the Green’s function method over the complete globe. The patterns are orthogonal with respect to an area-weighted inner product and normalized such that the area-averaged square = 1.

  • Fig. 2.

    Pattern correlation between Laplacian eigenfunctions computed from the Green’s function method and from the discretized Laplace operator. The eigenfunctions are computed from a 5° × 5° grid on the globe. The latter Laplacian eigenfunctions are resorted to maximize the pairwise correlation with the eigenfunction in the former set, to account for changes in order arising from numerical differences. The maximization also is performed for all possible cyclic permutations in the longitudinal direction, to account for irrelevant phase shifts in the zonal direction. The pattern correlation is defined based on an area-weighted inner product.

  • Fig. 3.

    Pattern correlation between Laplacian eigenfunctions computed on a 5° × 5° and computed on a 10° × 10° grid and then linearly interpolated onto the 5° × 5° grid. The latter Laplacian eigenfunctions are resorted to maximize the pairwise correlation with the eigenfunction in the former set, to account for changes in order arising from numerical differences. The maximization also is performed for all possible cyclic permutations in the longitudinal direction, to account for irrelevant phase shifts in the zonal direction. The pattern correlation is defined based on an area-weighted inner product.

  • Fig. 4.

    Mean percentage error Γ of representing data on a 5° × 5° grid using Laplacian eigenfunctions computed on a 10° × 10° grid. The data are first interpolated onto the coarse grid, where they are represented by Laplacian eigenfunctions computed on the coarse grid, and then interpolated back to the fine grid.

  • Fig. 5.

    As in Fig. 1, but for derived over land on a 5° × 5° regular grid.

  • Fig. 6.

    The mean and standard error of the projection coefficients for (top)–(bottom) the first three Laplacian eigenfunctions of 2-m land temperature in CMIP5 model simulations (color bars) and observations (gray horizontal lines). The colored bars are centered at the 1950–2000 mean value and have a width equal to twice the standard deviation of the monthly mean values. The gray lines are similarly constructed for observations. For clarity, the CMIP5 models are ordered based on the mean value in January.

  • Fig. 7.

    The 1975–2000 linear trend of the projection coefficients for (top)–(bottom) the first three Laplacian eigenfunctions of 2-m land temperature in CMIP5 model simulations (color bars) and observations (gray horizontal lines). The colored bars show the 95% confidence interval of the trend, and the gray lines show the 95% confidence interval for observations. For clarity, the CMIP5 models are ordered based on the mean value in January.

  • Fig. 8.

    The mean and standard error of the projection coefficients for (top)–(bottom) the first three Laplacian eigenfunctions of land precipitation in CMIP5 model simulations (color bars) and observations (gray horizontal lines). The colored bars are centered at the 1950–2000 mean value and have a width equal to twice the standard deviation of the monthly mean values. The gray lines are similarly constructed for observations. For clarity, the CMIP5 models are ordered based on the mean value in January.

  • Fig. 9.

    As in Fig. 5, but for over the ocean. The domain is chosen to avoid grid points with sea ice.

  • Fig. 10.

    The mean and standard error of the projection coefficients for (top)–(bottom) Laplacian eigenfunctions 2–4 of sea surface temperature in CMIP5 model simulations (color bars) and observations (gray horizontal lines). The colored bars are centered at the 1950–2000 mean value and have a width equal to twice the standard deviation of the monthly mean values. The gray lines are similarly constructed for observations. For clarity, the CMIP5 models are ordered based on the mean value in January.

  • Fig. 11.

    The percent of variance explained by Laplacian eigenvectors in CMIP5 historical simulations (thin gray lines) and observations (line with dots) for (top) June land temperature, (middle) June SST, and (bottom) June land precipitation. The variance is calculated over the 1950–2000 period for the month of June. Note the change in scale for the y axis.

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