A New Set of Orthogonal Patterns in Weather and Climate: Optimally Interpolated Patterns

A. Hannachi Department of Meteorology, King Abdulaziz University, Jeddah, Saudi Arabia

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Abstract

A new spectral-based approach is presented to find orthogonal patterns from gridded weather/climate data. The method is based on optimizing the interpolation error variance. The optimally interpolated patterns (OIP) are then given by the eigenvectors of the interpolation error covariance matrix, obtained using the cross-spectral matrix. The formulation of the approach is presented, and the application to low-dimension stochastic toy models and to various reanalyses datasets is performed. In particular, it is found that the lowest-frequency patterns correspond to largest eigenvalues, that is, variances, of the interpolation error matrix. The approach has been applied to the Northern Hemispheric (NH) and tropical sea level pressure (SLP) and to the Indian Ocean sea surface temperature (SST). Two main OIP patterns are found for the NH SLP representing respectively the North Atlantic Oscillation and the North Pacific pattern. The leading tropical SLP OIP represents the Southern Oscillation. For the Indian Ocean SST, the leading OIP pattern shows a tripole-like structure having one sign over the eastern and north- and southwestern parts and an opposite sign in the remaining parts of the basin. The pattern is also found to have a high lagged correlation with the Niño-3 index with 6-months lag.

Corresponding author address: A. Hannachi, Department of Meteorology, King Abdulaziz University, P.O. Box 80208, Jeddah 21589, Saudi Arabia. Email: ahannachi@kau.edu.sa

Abstract

A new spectral-based approach is presented to find orthogonal patterns from gridded weather/climate data. The method is based on optimizing the interpolation error variance. The optimally interpolated patterns (OIP) are then given by the eigenvectors of the interpolation error covariance matrix, obtained using the cross-spectral matrix. The formulation of the approach is presented, and the application to low-dimension stochastic toy models and to various reanalyses datasets is performed. In particular, it is found that the lowest-frequency patterns correspond to largest eigenvalues, that is, variances, of the interpolation error matrix. The approach has been applied to the Northern Hemispheric (NH) and tropical sea level pressure (SLP) and to the Indian Ocean sea surface temperature (SST). Two main OIP patterns are found for the NH SLP representing respectively the North Atlantic Oscillation and the North Pacific pattern. The leading tropical SLP OIP represents the Southern Oscillation. For the Indian Ocean SST, the leading OIP pattern shows a tripole-like structure having one sign over the eastern and north- and southwestern parts and an opposite sign in the remaining parts of the basin. The pattern is also found to have a high lagged correlation with the Niño-3 index with 6-months lag.

Corresponding author address: A. Hannachi, Department of Meteorology, King Abdulaziz University, P.O. Box 80208, Jeddah 21589, Saudi Arabia. Email: ahannachi@kau.edu.sa

Online correction: The supplemental file for this article was inadvertently left out. This online supplementary material is included by clicking on the following link: http://dx.doi.org/10.1175/2008JCLI2328.s1. A corrigendum will be published in a forthcoming issue.

1. Introduction

It is probably unquestionable that given the complexity of the climate system no single analysis tool can, alone, reveal all hidden structures embedded in it, and several methods are needed to serve that purpose. For example, stationary and propagating Rossby waves (Hoskins and Karoly 1981) and other propagating disturbances, for example, the Madden–Julian oscillation (Madden and Julian 1971, 1972, 1994), have different spatial and temporal characteristics and require therefore different identification tools. The empirical orthogonal functions (EOFs) method is one of the earliest tools used to analyze weather and climate (Obukhov 1947, 1960; Lorenz 1956) and is, undoubtedly, one of the most widely used methods in atmospheric science to identify and analyze the modes of variability and to reduce the system dimension for postprocessing, for example, for prediction purposes (see, e.g., von Storch and Zwiers 1999; Jolliffe 2002).

Because of variance maximization the desirable properties of EOFs and associated principal components (PCs), for example, space and time orthogonality, lead to some difficulties, such as physical interpretability (Jolliffe et al. 2003; Dommenget and Latif 2002; Hannachi et al. 2007; Dommenget 2007). For example, EOFs tend in general not to capture trends (e.g., Hannachi 2007). This has led weather–climate researchers to borrow tools developed in the social sciences, namely rotated EOFs (REOFs), to ease the interpretation of EOFs difficulties (Horel 1981; Richman 1981, 1986). Rotation of EOFs aims, in general, at achieving simplicity; see Hannachi et al. (2006) for an alternative to rotation. The atmosphere contains also other features such as propagating structures. EOFs form a complete basis and explain all the variance in the data. So in theory they can deal with all features including propagating ones. For example, a traveling feature in EOFs is associated with a pair of (nearly) equal but not degenerate eigenvalues. In practice, however, we do not know where, in the spectrum, this pair is, given that it is in general very difficult to distinguish between (nearly) equal and degenerate eigenvalues for high-ranked EOFs. Extended EOF (EEOF; Weare and Nasstrom 1982) and complex Hilbert EOF (HEOF; Rasmusson et al. 1981; Barnett 1983; Horel 1984) methods make it easier to extract such features in an automated and unambiguous manner. For more details on EOFs and related methods in atmospheric sciences see the recent review by Hannachi et al. (2007).

All the mentioned methods are examples of time domain approaches to analyze atmospheric fields. In the time domain it is in general difficult to control the pattern spectrum, see, for example, Dommenget (2007) for some examples, such as the leading pattern of variability of tropical sea level pressure (SLP), which is different from the low-frequency El Niño–Southern Oscillation pattern. Consider, for instance, the EOF method. It is known that the leading modes of variability have the largest variance, and many authors identify those with low-frequency modes. In reality, however, this need not be the case because variance is simply an integral or sum of the whole power spectrum. In fact, one can have different spectrum shapes with the same integral or total power. Another example was also given by Percival et al. (2001) regarding the variability of the North Pacific SLP. Percival et al. (2001) show that the winter-averaged SLP variability for the Aleutian low can be equally modeled as a red noise (short memory) and also as a fractionally differenced (long memory) model. The short-memory process is characterized by rapidly decaying autocovariance, that is, less power at low frequency, whereas the long-memory process has a slowly decaying autocovariance, that is, with more power at low frequency.

A natural and desirable alternative is then to use the spectral domain as, for example, Jenkins and Watts (1968), Koopmans (1974), and Parzen (1959, 1961). Now, to the best of my knowledge, there exist only very few spectral domain–based methods developed in the atmospheric science literature to analyze gridded weather–climate fields. One of these is the frequency domain EOFs (FDEOFs) method, which finds EOFs based on the cross-spectrum matrix averaged over a specific small frequency band (Wallace and Dickinson 1972; Wallace 1972; Brillinger 1981; Johnson and McPhaden 1993). I note here that FDEOFs generalize ordinary EOFs in the sense that the covariance matrix is only related to the real part of the cross-spectrum matrix and hence does not use the whole information from the cross-spectrum matrix. As pointed out by Horel (1984), however, FDEOFs may be cumbersome in practice. This is particularly the case if the power spectrum of an EOF is spread, for example, over a wide frequency band, requiring an averaging of the cross spectrum over this wide frequency range, where the theory behind FDEOFs is no longer applicable (Wallace and Dickinson 1972). This difficulty has resulted in the method being abandoned in climate research.

The other spectral method, not very well known in climate research, is predictive oscillation patterns (PrOPs) method (Kooperberg and O’Sullivan 1996). PrOPs are patterns based on minimizing the one-step-ahead prediction error using the famous Kolmogorov formula (Kolmogorov 1939; 1941) and are considered the most predictable patterns. Kooperberg and O’Sullivan (1996) have introduced PrOPs as an alternative to principal oscillation patterns (POPs; Hasselmann 1988; von Storch et al. 1988; von Storch and Zwiers 1999), which incorporate the first-order Markov model assumption to isolate patterns with strong temporal dependence. PrOPs are then regarded as being less reliant on this Markov assumption.

I note here that Horel (1984) regards HEOFs as frequency domain EOFs averaged over all frequency bands. Preisendorfer and Mobley (1988) also provide an account on computing EOFs of Fourier-transformed data. Outside the circle of atmospheric science literature the number of published works on spectral PC analysis is also very limited. The main reference to the subject is that of Brillinger (1981). Jolliffe (2002, section 12.4) provides a description of the approach along with few other related references. The point is that all these methods do not provide, in general, a systematic frequency ranking of the obtained patterns.

In this manuscript I present a new spectral method to identify orthogonal patterns from gridded weather–climate data that could benefit the climate researcher, and which I label optimally interpolated patterns (OIPs) or, similarly, interpolated EOFs (IEOFs). As the name indicates, OIPs are based on interpolation—unlike PrOPs, which are based on prediction—which is a form of extrapolation. The manuscript is organized as follows. Section 2 presents a brief background on spectral techniques applied in atmospheric science. The methodology to derive OIPs is then presented in section 3. Section 4 presents the estimation procedure. Application to a stochastic low-dimension toy model is presented in section 5 and to reanalyses data in section 6. A summary and conclusion are given in the last section.

2. Background on spectral analyses of stationary time series

This section provides a brief summary on some standard concepts used in spectral analysis of time series. These concepts set the basis for, and help with the understanding of, the methodology of the interpolated patterns presented in the next section. In addition, most of this section can also serve as a brief review and can be used for educational purposes in weather/climate data analysis. Let xt, t = 1, 2, . . . , be a stationary time series with power spectrum fx( ); that is,
i1520-0442-21-24-6724-e1
for −πωπ. The variance of (xt) can be obtained from (1) as
i1520-0442-21-24-6724-e2
where γx( ) in (1) and (2) is the autocovariance function of the time series. Therefore, fx(ω) can be interpreted as the contribution to the time series variance from the frequency band centered around ω. A basic concept in stationary time series analysis is that of filtering. A linear filter with transfer function h( ) is a simple convolution operator, which in the discrete case takes the form
i1520-0442-21-24-6724-e3
where the weights hk, k = 0, ±1, . . . , are known as the coefficients of the impulse response function, ψ(z) = Σkhkzk, and B is the backward shift operator; that is, Bxt = xt−1. The covariance function of the filtered time series can easily be obtained using (3) and the covariance function γx( ) of (xt) as
i1520-0442-21-24-6724-e4
Note that (4) can also be written using integrals instead, that is, ∫∫h(u)h(υ)γx(τ + uυ)dudυ, as in the continuous case, after writing the transfer function as a sum of Dirac pulses:
i1520-0442-21-24-6724-e5
where δk( ) is the Dirac pulse at k. This form is sometimes convenient and easy to use (see the appendix). The Fourier transform of (4) yields the following relationship between the respective power spectra:
i1520-0442-21-24-6724-e6
where the frequency response function Γ(ω) is the Fourier transform of the impulse response function and takes the form
i1520-0442-21-24-6724-e7
As a simple illustration consider, for example, the difference operator yt = 1/2∇xt, where ∇xt = xtxt−1. The impulse response function, obtained using (3), is ψ(z) = 1/2 − z/2, and, using (7), we get |Γ(ω)| = |sin(ω/2)|. Thus, the signal with a period of twice the sampling interval, that is, the Nyquist period, does not get attenuated. Signals with periods of 5 times the sampling interval, for example, get attenuated by a factor of 0.6. This explains why Bjerknes (1964) used the differenced yearly values of the North Atlantic Oscillation to extract the 2–5-yr trend.
These concepts can easily extend to the multivariate case. In fact, if xt, t = 1, 2, . . . , is a multivariate stationary time series with autocovariance matrix
i1520-0442-21-24-6724-eq1
where E( ) stands for the expectation operator, then the cross-spectrum matrix 𝗙x(ω) of (xt) is similar to that shown in Eq. (1) and is given by
i1520-0442-21-24-6724-e8
and has the properties of being positive semidefinite and Hermitian. The remaining concepts of the unidimensional case also extend naturally to the multivariate case. For example, Eqs. (2) and (6) become, respectively,
i1520-0442-21-24-6724-e9
i1520-0442-21-24-6724-e10
where Γ( ) is the frequency response function of the linear filter, similar to (7), except that the weights of the filter operator are now matrices. The superscripts “*” and “T” stand for the complex conjugate and transpose operators, respectively.

3. Optimally interpolated patterns

a. Patterns derivation

Let again xt, t = 1, 2, . . . , be a stationary, multivariate, p-dimensional time series with covariance matrix 𝗖 and spectral density matrix 𝗙. Interpolation is related, but not identical, to prediction. Whereas prediction attempts to estimate xt+τ given the past values xt−k, k ≥ 0, of the series, interpolation aims at interpolating the time series for various reasons, for example, because of missing values. Details on interpolation can be found, for example, in Grenander and Rosenblatt (1957), Bonnet (1965), and Hannan (1970). The interpolation of xt, given xtj, for j ≠ 0, is then estimated by a conditional expectation and is given by the expression
i1520-0442-21-24-6724-e11
where h(z) = Σj≠0𝗔jzj, minimizing the mean square error
i1520-0442-21-24-6724-e12
Now, Eq. (11) is also similar to a linear filter with frequency response function given, in a similar manner to (7), by 𝗛(ω) = h(e). The interpolation error is then
i1520-0442-21-24-6724-e13
where 𝗜p is the p × p identity matrix, and the error can also be seen as an output from a linear filter. It has 𝗜p − 𝗛(ω) as frequency response function and hence, using (10), also has [𝗜p − 𝗛(ω)]𝗙(ω)[𝗜p − 𝗛(ω)]*T as spectral density matrix. The interpolation error covariance matrix, obtained using (2) and also (9), is therefore
i1520-0442-21-24-6724-e14
The minimization of (12) is therefore equivalent to the minimization of tr(Σ). The optimal error covariance matrix (14) minimizing (12) takes, in fact, a more elegant expression (Hannan 1970), which depends only on the spectral density matrix of the field, and is given by
i1520-0442-21-24-6724-e15
(see the appendix for an outline of the proof). In addition, the error covariance matrix Σ is nonsingular, and the optimal interpolation filter or frequency response function takes the following form:
i1520-0442-21-24-6724-e16
The OIPs, or interpolated EOFs, are then given by the eigenvectors of the error covariance matrix Σ in (15) arranged in decreasing (or increasing) order. So, for example, the eigenvector of Σ with the largest eigenvalue corresponds to the pattern whose time series has the largest interpolation error variance. In other words, the orthogonal OIP patterns are ranked by the magnitude of the expected error that would occur were a given pattern used to estimate, by interpolation, the data at a single time step. This ranks the patterns in terms of long-term variability, since such patterns will be less useful in time domain interpolation, and hence produce large interpolation errors.

b. A simple illustration

Here I give a simple illustrative example where (15) can be calculated analytically. Consider the simple p-dimensional stochastic model
i1520-0442-21-24-6724-e17
where u is a constant vector, (αt) is a univariate stationary time series with spectral density function g(ω), and (εt) is a p-dimensional random noise (uncorrelated in time) and independent of (αt) with covariance matrix Σε. Also, for simplicity, (αt) and u are supposed to have unit variance and unit length, respectively. Now, the spectral density matrix 𝗙(ω) of (17) is then given by
i1520-0442-21-24-6724-e18
To proceed, the covariance matrix Σε is supposed to be of full rank, which allows an easy computation of the inverse of (18) to yield
i1520-0442-21-24-6724-e19
which is also of the same type as (18), with α = 2πg(ω). After integrating and inverting (19), one gets the interpolation error covariance matrix
i1520-0442-21-24-6724-e20
Remember that when uTππ𝗙(ω) u and uTΣu are maximized they yield, respectively, EOFs and OIPs. It is clear from (20) that the OIPs take into account the shape of the power spectrum. It is also clear from (20) that the leading OIPs, corresponding to the maximum of the interpolation error variance, have, like EOFs, large-scale features. Since these patterns are frequency domain based they rather are associated with low-frequency variation. Therefore, high-frequency features minimize the interpolation error variance.

Now by comparing the interpolation error covariance matrix Σ with the data covariance matrix 𝗖 = ∫ππ𝗙(ω) , one can see that

  1. If u is an eigenvector of Σε, then the EOFs and OIPs are identical. This is easy to check since the remaining eigenvectors of Σε are orthogonal to u.

  2. If, however, u is not an eigenvector of Σε, then the EOFs and OIPs are different.

In the latter case, the contribution of u to the spectrum of the data covariance matrix 𝗖 and the interpolation error covariance matrix Σ, that is,
i1520-0442-21-24-6724-e21
i1520-0442-21-24-6724-e22
have been compared using two types of spectra g(ω), namely a first-order autoregressive [AR(1)] (short memory) process for which
i1520-0442-21-24-6724-e23
for |ϕ| < 1, and a Gaussian stationary long-memory, also known as fractionally differenced, process (Granger and Joyeux 1980; Hosking 1981) for which
i1520-0442-21-24-6724-e24
for 0 < δ < 1/2. Figure 1 shows the term β/(1 − βuTΣε−1u) in (22) as a function of uTΣε−1u for different values of ϕ; Eq. (23), ranging from 0 to 0.9 (dashed–dotted) and different values of δ; Eq. (24), ranging from 0 to 0.5 (continuous). The horizontal line, with a value of 1, corresponds to the white noise case δ = ϕ = 0. For the red noise model the values are always greater than 0.5, but for the long-memory process they are greater than about 0.82, and they are greater than 1 for small values of uTΣε−1u. So, for an arbitrary u (not necessarily an eigenvector of Σε) we find that, although it may not be an eigenvector of Σ, its contribution to the spectrum of the latter matrix Σ is either close to or larger than its contribution to the spectrum of the covariance matrix 𝗖 for a long-memory process. In other words, the leading eigenvalue of Σ is more constrained by the low-frequency signal in the data than is the leading eigenvalue of 𝗖.

4. Numerical aspects

In reality the spectral density matrix is not as easily obtainable in general as in the simple case presented in the previous section, and numerical approximations become the only tool to estimate it. From a finite sample, xt, t = 1, 2, . . . , n, of our climate field an estimate (ωp) of the spectral density matrix is first obtained at the discrete frequency values ωk = (2πk/n), with k = −[(n − 1/2)], . . . , [(n/2)], where [x] stands for the integer part of x. A trapezoidal rule is then used to approximate the integral in (15) and obtain an estimator of Σ−1 as
i1520-0442-21-24-6724-e25
The estimator (ωk) is obtained using the (multivariate) periodogram 𝗜 (ωk):
i1520-0442-21-24-6724-e26
where (ωk) = 1/nΣnt=1xtekt is the Fourier transform of the field xt, t = 1, 2, . . . , n, at frequency ωk.
It is well known, however, that the raw periodogram is not a good estimator of the power spectrum (Jenkins and Watts 1968; Chatfield 1996). The standard way to obtain a consistent estimator of the power spectrum is through smoothing of the periodogram. In this case the estimated spectral density matrix takes the form
i1520-0442-21-24-6724-e27
where Λ( ) is a smoothing spectral window. In addition, the estimated spectral density matrix (27) will be in general full rank, and leads to the estimation of the error covariance matrix Σ (15) using (27). Note that the estimators (26) are obtained at the discrete frequencies ωk = 2πk/n for k = −[n − 1/2], . . . , [n/2]. The spectral window Λ( ) is a symmetric kernel function that integrates to unity and decays away from the origin.

5. Illustration with a simple low-dimension stochastic model

Perhaps the simplest example of a low-frequency model that can be used is a long-term linear trend contaminated with stochastic noise. A simple low-dimension time series model containing a linear trend is considered here to illustrate the OIP method and contrast it with the EOF method. The model is a three-variable time series given by
i1520-0442-21-24-6724-e28
where α = 0.004, ε1t, ε2t, and ε3t are three first-order AR(1), models with respective lag-1 autocorrelations 0.5, 0.6, and 0.3. A similar example was used by Hannachi (2007) to identify trend patterns in climate data. A realization of this system is shown in Figs. 2a–c. Both EOF and OIP analyses are applied to this realization. The obtained PCs, arranged in decreasing order of their explained variance, are shown in Figs. 2d–f. The long-term (linear) trend is spread over all PCs (Figs. 2d–f). The EOF method is unable, in fact, to capture this trend because it seeks to maximize variance (Hannachi 2007). The OIP time series (Figs. 2g–i), however, reveal different features where the trend is now captured entirely by the leading time series (Fig. 2g).

Because the results presented above stem from one single realization, a significance test, using Monte Carlo approach, is performed next to check that these results are not due to sampling. A set of 100 realizations is first computed using (28), then, for each one, PCs and OIPs are computed. Finally, the correlation coefficients between the obtained PCs and the linear trend, and similarly for the OIP time series, are calculated. Figure 3 shows the histograms of the absolute value of the obtained correlation coefficients using the PCs (Figs. 3a–c) and OIPs (Figs. 3d–f). Figure 3 shows evidence that the long-term trend, that is, the lowest frequency in the model, is identified by all PCs (Figs. 3a–c), but it is consistently captured solely by the leading OIP time series (Figs. 3d–f).

A similar comparison between OIPs and correlation-based EOFs has also been conducted using a slightly different example (not shown). The conclusion is similar to that of the above example where, again, the trend was found in all PCs, and was correctly captured by the leading OIP time series.

6. Application to reanalyses

a. Data description

Two datasets are used in this investigation; the monthly SLP and monthly sea surface temperature (SST). The SLP data come from the National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalyses (Kalnay et al. 1996; Kistler et al. 2001). They are available on a 2.5° × 2.5° regular grid, and span the period January 1948 to December 2006. The second dataset consists of SSTs derived from Reynolds and Smith (1994). The data are available on a 2° × 2° regular grid and span the period January 1945 to December 2004. For both datasets, the mean annual cycle is first calculated by averaging the monthly data over the years then subtracted from the data to yield SLP and SST anomalies.

For SLP the method is applied to two distinct regions. In the first analysis, the focus is on analyzing the winter season defined by December–February (DJF) over the Northern Hemisphere (NH). The data are therefore obtained by concatenating the winter monthly means for all years, that is, December 1948 to February 2006, north of 20°N. In the second analysis, the focus is on the Southern Oscillation (SO) region. The monthly SLP data over the tropics, for the region 15°S–15°N around the globe, are considered. For SST the focus is on the Indian Ocean where SST anomalies over the region 24°N–30°S and 30°–20°E are considered.

b. Northern Hemisphere winter SLP

The computation of the spectral density matrix (27) and its inverse at all frequencies [see Eq. (25)] is prohibitively expensive when the whole physical space is used. To reduce the space dimension an EOF analysis is first applied then the leading m EOFs/PCs retained. The EOFs and PCs of the detrended SLP data have also been computed, but they are found to be virtually identical to those obtained using the nondetrended anomalies (see, e.g., Hannachi 2007 for a discussion on EOFs and trends). Sensitivity to changes in m has also been performed. The number of EOFs used is varied from m = 5, explaining around 60% of the total winter variance, to m = 20, accounting for about 93%. Various spectral windows have been used to compute (ω), Eq. (27), including moving average, Bartlett and Parzen windows (e.g., von Storch and Zwiers 1999). The results are found to be nonsensitive to the choice of the smoothing window, and the results shown are based on the Bartlett window.

Figure 4a shows an example of the leading OIP obtained using m = 5 EOFs. The corresponding error variance in percentage of total error variance is around 95%, with only 2% for the next one (Fig. 4b). OIP1 (Fig. 4a) shows unambiguously the familiar North Atlantic Oscillation (NAO), whereas OIP2 (Fig. 4b) shows the Pacific pattern. The third OIP (not shown) represents the Scandinavian pattern with error variance around 1%. For comparison, the leading two EOFs of the same SLP data are also shown (Figs. 4c,d). For example, the leading EOF (Fig. 4c) shows the Arctic Oscillation (Thompson and Wallace 1998; Wallace and Thompson 2002.) The correlation coefficient between OIP 1 time series and two NAO indices are also computed. The first NAO index comes from the Climate and Global Dynamic Division of NCAR (additional information is available online at http://www.cgd.ucar.edu/cas/jhurrell/indices.html) and is a monthly station-based index obtained from the difference of normalized SLP between Ponta Delgata, Azores, and Stykkisholmur/Reykjavik, Iceland. The obtained correlation is about 0.72. The second NAO index is obtained via Varimax EOF rotation (Hannachi et al. 2007) of the SLP EOFs, and the correlation is found to be 0.93. The correlation between OIP 2 time series and the Pacific pattern time series, obtained also via EOF rotation, is also computed and is about 0.95.

From the spectrum of the error covariance matrix, it seems that the first OIP, and perhaps the second one also, have the lowest frequency and are expected to be robust. In fact, when m increases it is found that the spectrum of Σ̂ starts to isolate the leading two patterns as the ones most associated with the lowest frequencies. Figure 5 shows the spectrum of Σ̂ for m = 20, where the error bars are based on the rule of thumb of North et al. (1982). So the leading two OIPs come out as the main low-frequency modes. The leading two OIP patterns and associated interpolation principal components (IPCs) are quite robust to changes in the number of EOFs used to compute the interpolation error covariance matrix. Figure 6 shows the correlations between OIP1 and associated IPC1 time series, obtained with m = 5 (Fig. 4a) and the same OIP, and associated IPC, but for different values of m = 6, 7, . . . , 20. The same correlations are also shown for the next two OIPs and associated IPCs. The spatial cross correlations between the different OIP1 patterns (and associated IPC1), obtained using different values of m = 5, 10, . . . , 30, are shown in Table 1. The same correlations obtained with OIP2 (and associated IPC2) are shown in Table 2. Figure 7 shows the power spectra of IPCs 1–5, scaled to unit variance, using two estimation methods: the Welch (Fig. 7a) and a smoother one, the Burg method, based on autoregressive modeling (Fig. 7b). One can see the high power, at low frequency, decreasing as one goes from IPC1 to IPC5. A similar picture was also obtained for all other values of m considered here (not shown).

The NAO is known to have power at all frequencies, with a particularly high power at low frequency. For example, decadal trends are apparent in the historical records of NAO (Deser and Blackmon 1993; Kushnir 1994.) Stephenson et al. (2000) suggest that NAO exhibits long-range dependence, which provides a better fit than red noise or random walk. On the Pacific side, Percival et al. (2001) show that the Aleutian low variability can be equally modeled as a short-memory (red noise) and a long-memory (or fractionally differenced) process. As pointed out by Percival et al. (2001), a long-memory paradigm for the North Pacific can be used as a characterization of regime-like behavior (Haines and Hannachi 1995). Although OIP method identifies these patterns with the lowest frequency in the data, the method is not suitable to distinguish between short- and long-memory processes. It is known that very low-frequency signals can arise from (a sum of) short-memory processes (Maraun et al. 2004).

c. Tropical SLP

SLP dynamics in the tropics differ from that in the midlatitudes because of the strong air–sea interaction, which leads to El Niño–Southern Oscillation (ENSO) mode, which is the mode with the lowest frequency in the tropics. Although ENSO is well captured by the leading tropical sea surface temperature (SST) EOF, the atmospheric part, that is, the SO, is not. In fact, the leading EOF of the tropical SLP (not shown) has one sign over the whole tropical region (see e.g., Dommenget 2007).

In a similar way to the previous section, OIPs are computed here using NCEP–NCAR tropical monthly SLP but restricted to the tropical channel 15°S–15°N. The tropical SLP pressure field has been detrended prior to computing EOFs, given the significant trend in tropical SLP. But the results are found to be similar to those obtained from the nondetrended SLP anomalies. Unlike the NH SLP, the leading OIP of the tropical SLP is well above the rest of the error covariance spectrum, for all values of m = 5, 10, . . . , 30, considered here. Figure 8 shows an example of the error covariance matrix spectrum (m = 20), in percentage of total error variance, where the leading eigenvalue is around 75% of the total error variance. The next one, around 12%, is also separated from the remaining eigenvalues, but is much smaller than the leading eigenvalue. Figure 9a shows an example of the leading SLP IOP pattern, obtained with m = 20. The figure shows the SO mode with a seesaw between east Pacific and west Pacific/Africa–North Atlantic reflecting the Walker circulation. This pattern is similar to the leading mode of SLP EOF rotation toward the largest difference relative to a fitted isotropic diffusion process reported in Dommenget (2007). For comparison, the second OIP is also shown (Fig. 9b), which does not look much like a Walker-type oscillation. Note, for example, that OIP2 does not have a seesaw feature, given the same sign of the pattern in west Pacific and Indian Ocean, with an opposite sign in tropical Atlantic and Indonesia. The leading two EOFs of the tropical SLP are also shown (Figs. 9c,d) for comparison. The leading EOF (Fig. 9c), for example, shows a monopole structure.

The leading OIP pattern is also quite robust to changes in the number m of EOFs used. Table 3, like Table 1, shows pairwise spatial (and also temporal) correlations between OIP1 patterns (and associated IPCs) for different values of m. So clearly, OIP1 represents the major low-frequency mode in tropical SLP, which represents the SO. In fact, Fig. 10 shows a plot of Niño-3 SST anomaly index (available online at http://www.cpc.ncep.noaa.gov/data/indices/) from 1950 to 2004 and the leading IPC time series. Table 4 shows the correlation coefficients of Niño-3 index with IPC1 and IPC2 for different values of m. Note that the index is more correlated with PC2 (0.55) than with PC1 (barely 0.5). Correlations using the SO index, instead, are also calculated, but the results are very similar to those shown in Table 4, and therefore are not reported. Figure 11 shows the power spectrum of IPC1 and IPC2, with focus on periods greater than 10 months. It can be seen that IPC1 has higher power than IPC2, particularly for periods greater than about 2 yr.

d. The Indian Ocean SST

Climate variability in the Indian Ocean is the subject of many discussions regarding its dynamics and external effects, such as ENSO (Webster et al. 1999; Saji et al. 1999; Baquero et al. 2002; Behera et al. 2003; Dommenget and Latif 2003). Although most of these discussions concentrate on interannual variability, here the intraseasonal variability is also included by looking at monthly SST anomalies over the Indian basin for the period 1945–2004. SST variability over the Indian Ocean is dominated by a basinwide warming, as shown by the leading EOF (Fig. 12a) and associated PC (Fig. 12b) of the SST anomalies, which tends to hinder any attempt to detect a possible internal mode of variability or teleconnection. The data are detrended, by removing a linear trend, and OIP patterns are computed. Note that EOF1 (Fig. 12a) remains the dominant mode of variability even after the data have been detrended. When the number m of retained EOFs is small (m ≤ 5) the leading OIP is still dominated by EOF1 structure, which explains a substantial amount of variance. When m increases the leading eigenvalue of the interpolation error covariance matrix becomes separated from the rest of the spectrum and the associated OIP starts to show a robust large-scale structure. Figure 13 shows an example of the error covariance matrix spectrum for m = 70. The leading eigenvalue is around 87% of the total error variance, and the next one is only about 6%. The associated OIP (Fig. 14a) shows a kind of tripole with the same sign on the eastern, south- and northwestern sides of the basin, and a boomerang-shaped opposite sign in the middle. This pattern has correlation coefficients of 0.29 and 0.35 with EOF2 and EOF4, respectively. Figure 14b shows the associated time series (continuous) along with the Niño-3 index (dotted). There is a clear coherency between IPC1 and the Niño-3 index, particularly around the warm phase of ENSO. The IPC1 time series lags Niño-3 index, and the maximum lagged correlation, 0.42, is obtained for a lag τ = 6 months. The second IPC is correlated with Niño-3 index at zero lag, with correlation coefficient 0.47, and the associated pattern is more like a north–south dipole (not shown). When the effect of ENSO was removed by regression the leading OIP remained unchanged, because of the lagged relationship with ENSO, but the second OIP disappeared. A similar analysis was also performed with the tropical Pacific SST anomalies, where El Niño pattern shows up as the leading OIP pattern (not shown).

7. Summary and conclusions

Spectral analysis of gridded weather–climate data constitutes a natural surrogate to the widespread time domain approaches. Very few spectral domain approaches have been developed and applied to gridded atmospheric data. This manuscript presents a new methodology to obtain orthogonal patterns from gridded climate data using a spectral-based approach. The method is based on interpolation rather than extrapolation or prediction. The method finds orthogonal patterns that optimize the interpolation error variance, and are hence labeled optimally interpolated patterns and associated interpolation principal components. The patterns are given by the eigenvectors of the interpolation error covariance matrix, which is obtained from an integration, over the spectral domain, of the inverse of the cross-spectrum matrix. The patterns are ranked according to the error variance obtained when a given pattern is used to estimate, by interpolation, the data at a single time step. The procedure effectively ranks the patterns in terms of long-term variability, since such patterns will be less useful in time domain interpolation, and so produce large interpolation error variance. So, the spectral-based OIP patterns, associated with the largest error interpolation variance, correspond to modes of lowest frequencies.

The method has been applied first to a stochastic low-dimension toy model. The model is a three-variable time series containing a linear trend plus an autocorrelated noise. With the EOF method the trend is found in all PCs, showing that the method failed to capture this low-frequency mode. The OIP method, however, successfully identified the trend mode as its leading pattern. The approach has been applied next to various reanalyses datasets. Two datasets are considered here: the NCEP–NCAR sea level pressure and the Reynolds sea surface temperature. Monthly SLP and SST anomalies, with respect to the respective mean annual cycles, are computed. Two main regions; the Northern Hemisphere, north of 20°N, and the tropical channel, 15S°–15°N are focused on using SLP. For SST the focus is on the Indian Ocean (30°S–24°N and 30E°–120°E).

The application to NH SLP shows two leading OIP patterns, well separated from the remaining OIPs. The first OIP shows the familiar NAO pattern, and the second one comes out as the North Pacific pattern, or the Aleutian mode. Both patterns have power at low frequency, with more power for NAO. These findings are in good agreement with previous investigations where, for example, the NAO was found to be well approximated by a long-memory process (Wunch 1999; Stephenson et al. 2000), and the Pacific pattern can be modeled equally as a short- and long-memory, and also as a regime-like process (Percival et al. 2001).

The application to the tropical SLP yields one robust leading pattern. The leading OIP shows the Southern Oscillation pattern. The associated leading OIP time series is highly correlated with the Niño-3 index. The leading OIP pattern is also similar to the leading SLP rotated pattern maximally departing from an isotropic diffusion process (Dommenget 2007).

The method is finally applied to the Indian Ocean SST anomalies. The data have been detrended to remove the effect of the basinwide warming. For a small number of retained EOFs the leading OIP is still dominated by the EOF1 structure. As the number of retained EOFs increases the leading OIP pattern starts to show a robust large-scale feature. The leading and well separated OIP has one sign in the eastern and south- and northwestern sides, and a boomerang-shaped opposite sign in the remaining part of the basin. The leading OIP has a 6-month lag with ENSO. The simultaneous effect of ENSO is absorbed more by the second OIP, which has a more north–south dipolar structure.

Because the OIP method ranks the patterns in terms of long-term variability, it could provide a better alternative to other standard methods. For instance, if the aim is to emphasize the more slowly varying patterns of climate variation, the OIP method offers itself naturally. One could, of course, apply standards methods, for example, EOFs, to a low-pass filtered data. Such approaches, however, require arbitrary choices about the filter to be applied hence the more organic approach of OIPs.

Acknowledgments

Some of this work was performed while the author was visiting the Walker Institute in the Department of Meteorology, Reading University. The author would like to thank two anonymous reviewers for their constructive comments.

REFERENCES

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APPENDIX

The Optimal Interpolation Error Covariance Matrix

We give below an outline of the proof for Eq. (15). We let xk = (xk1, . . . , xkp)T, k = 0, ±1, ±2, . . . be a sequence of zero mean second-order random vectors, that is, with components having finite variances. Let also Ht be the space spanned by the sequence {xkj, j = 1, . . . , p, k = 0, ±1, ±2, . . . , kt}, known also as random function (or random field). Basically, Ht is composed of finite linear combinations of elements of this random function. Then Ht has the structure of a Hilbert space with respect to a generalized inner product:
i1520-0442-21-24-6724-ea1
where u and v are elements of Ht, and the superscripts “*” and “T” stand, respectively, for the complex conjugate and the transpose operators (see, e.g., Parzen 1959; Koopmans 1974; Priestly 1981). Now, the estimator t in (11) can be seen as the (orthogonal) projection of xt onto Ht, with respect to the inner product (A1). Therefore xtt is orthogonal to xs for st, and also to t. The first of these two properties yields
i1520-0442-21-24-6724-ea2
Now, the cross-covariance matrix, expressed in the lhs of (A2), can also be expressed in the spectral domain, as detailed next, using the spectral density matrix 𝗙(ω) of the field (xt) and the multivariate frequency response function 𝗜p − 𝗛(ω) of xtt, see Eq. (13).
Now, let us go back for a moment to the unidimensional case for simplicity, and consider Eq. (3) again. As for Eq. (4), we can also obtain the cross-covariance function γxy( ) between (xt) and (yt), which reads
i1520-0442-21-24-6724-ea3
where the transfer function h( ) is given by (5). The Fourier transform of (A3) yields the cross-spectrum fxy( ) of (xt) and (yt):
i1520-0442-21-24-6724-ea4
where Γ( ) is the frequency response function of the filter. The inverse Fourier transform of the cross spectrum yields, as in (2),
i1520-0442-21-24-6724-ea5
Now, we can simply extend (A5) to the multivariate case and apply it to (A2) to obtain
i1520-0442-21-24-6724-ea6
Equation (A6) necessarily implies that the matrix inside the integral is independent of ω, that is,
i1520-0442-21-24-6724-ea7
where 𝗔 is a constant matrix. The second orthogonality condition, that is, E[(xtt)t*T] = 𝗢, also implies a similar relationship, namely,
i1520-0442-21-24-6724-ea8
Note that the frequency response function of t is 𝗛(ω)𝗙(ω). Now, by expanding expression (14), that is,
i1520-0442-21-24-6724-ea9
and using the last two orthogonality properties (A7) and (A8), one gets
i1520-0442-21-24-6724-ea10
Finally, by substituting the expression of 𝗜p − 𝗛(ω), obtained from (A7), into (A9), and remembering that 𝗔 is real [Eq. (A10)] and that 𝗙(ω) is Hermitian yields
i1520-0442-21-24-6724-ea11
where the invertibility of Σ is guaranteed by the integrability of 𝗙−1.

Fig. 1.
Fig. 1.

The term β/(1 − βuTΣε−1u), Eqs. (20) and (22), vs uTΣε−1u for two types of power spectra: a short-memory red noise process (dotted–dashed), and a long-memory or fractionally differenced process (continuous). For the red noise, the values of ϕ [Eq. (23)] are from 0 to 0.9 with an increment of 0.15, and for the long memory δ = 0, 0.1, . . . , 0.5.

Citation: Journal of Climate 21, 24; 10.1175/2008JCLI2328.1

Fig. 2.
Fig. 2.

A sample of time series of (a) xt, (b) yt, and (c) zt generated using (28), (d)–(f) the PCs of these data, and (g)–(i) the OIP time series of the same data.

Citation: Journal of Climate 21, 24; 10.1175/2008JCLI2328.1

Fig. 3.
Fig. 3.

Histograms of the (absolute value of the) correlation coefficients (a)–(c) between the linear trend [see Eq. (28)] and the PCs and (d)–(f) between the same linear trend and the OIP time series of 100 realizations of model (28).

Citation: Journal of Climate 21, 24; 10.1175/2008JCLI2328.1

Fig. 4.
Fig. 4.

The leading two OIPs obtained using (a), (b) the leading m = 5 PCs and (c), (d) the leading two EOFs of NH winter SLP.

Citation: Journal of Climate 21, 24; 10.1175/2008JCLI2328.1

Fig. 5.
Fig. 5.

The spectrum of the interpolation error covariance matrix, shown in percentage of the total interpolation error variance, when the leading m = 20 NH winter SLP EOFs are used.

Citation: Journal of Climate 21, 24; 10.1175/2008JCLI2328.1

Fig. 6.
Fig. 6.

Correlation of the leading three OIPs spatial patterns (thin) and associated IPCs time series (bold) obtained using m = 5 EOFs with the same OIP patterns and associated IPCs for m = 5, 6, . . . , 25 EOFs.

Citation: Journal of Climate 21, 24; 10.1175/2008JCLI2328.1

Fig. 7.
Fig. 7.

Power spectra of the leading five IPCs, obtained with m = 5 NH winter SLP EOFs, using the (a) Welch method and (b) Burg method.

Citation: Journal of Climate 21, 24; 10.1175/2008JCLI2328.1

Fig. 8.
Fig. 8.

As in Fig. 5, but using the tropical sea level pressure data.

Citation: Journal of Climate 21, 24; 10.1175/2008JCLI2328.1

Fig. 9.
Fig. 9.

As in Fig. 4, but for tropical seal level pressure data.

Citation: Journal of Climate 21, 24; 10.1175/2008JCLI2328.1

Fig. 10.
Fig. 10.

Niño-3 SST anomaly index (1950–2004) and the tropical sea level pressure IPC1, obtained with m = 20. Both time series are scaled.

Citation: Journal of Climate 21, 24; 10.1175/2008JCLI2328.1

Fig. 11.
Fig. 11.

Power spectra (2 month−1), using Welch estimates, of tropical SLP IPC1 and IPC2 for m = 20. Only periods greater than 10 months are shown.

Citation: Journal of Climate 21, 24; 10.1175/2008JCLI2328.1

Fig. 12.
Fig. 12.

(a) The leading EOF and (b) associated PC of the Indian Ocean SST anomalies.

Citation: Journal of Climate 21, 24; 10.1175/2008JCLI2328.1

Fig. 13.
Fig. 13.

As in Fig. 8, but for the detrended Indian Ocean SST anomalies with m = 70. Only the first 40 eigenvalues are shown.

Citation: Journal of Climate 21, 24; 10.1175/2008JCLI2328.1

Fig. 14.
Fig. 14.

(a) The leading OIP of the detrended Indian Ocean SST anomalies and (b) the associated IPC time series (continuous) along with the Niño-3 index (dotted) for the same period.

Citation: Journal of Climate 21, 24; 10.1175/2008JCLI2328.1

Table 1.

Cross correlations between the leading spatial OIP1 patterns for m = 5, 10, . . . , 30 (below the diagonal) and the associated leading IPC (above the diagonal) of Northern Hemispheric winter sea level pressure.

Table 1.
Table 2.

Same as Table 1 but for the second OIP pattern and associated IPC.

Table 2.
Table 3.

Same as Table 1 but for the tropical sea level pressure.

Table 3.
Table 4.

Correlation coefficients of Niño-3 SST index (1950–2004) with the leading two tropical SLP IPCs for various values of the number m of EOFs used. The correlation of the index with SLP PC1 is 0.5 and it is 0.55 with PC2.

Table 4.

Supplementary Materials

Save
  • Baquero, A., M. Latif, and S. Legutke, 2002: On dipolelike variability of sea surface temperature in the tropical Indian Ocean. J. Climate, 15 , 13581368.

    • Search Google Scholar
    • Export Citation
  • Barnett, T. P., 1983: Interaction of the monsoon and Pacific trade wind system at interannual time scales. Part I: The equatorial case. Mon. Wea. Rev., 111 , 756773.

    • Search Google Scholar
    • Export Citation
  • Behera, S. K., S. A. Rao, H. N. Saji, and T. Yamagata, 2003: Comments on “A cautionary note on the interpretation of EOFs.”. J. Climate, 16 , 10871093.

    • Search Google Scholar
    • Export Citation
  • Bjerknes, J., 1964: Atlantic air-sea interaction. Adv. Geophys., 10 , 182.

  • Bonnet, G., 1965: Theorie de l’information–sur l’interpolation optimale d’une fonction aléatoire èchantillonnée. C. R. Acad. Sci. Paris, 260 , 297343.

    • Search Google Scholar
    • Export Citation
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