Editorial Type:
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Article Type:
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Intraseasonal Oscillation Indices from Complex EOFs

Gerd Bürger Institute of Environmental Science and Geography, University of Potsdam, Potsdam, Germany

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Online Publication:
24 Nov 2020
Print Publication:
01 Jan 2021
DOI:
https://doi.org/10.1175/JCLI-D-20-0427.1
Page(s):
107–122
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Abstract

Indices of oscillatory behavior are conveniently obtained by projecting the fields in question into a phase space of a few (mostly just two) dimensions; empirical orthogonal functions (EOFs) or other, more dynamical, modes are typically used for the projection. If sufficiently coherent and in quadrature, the projected variables simply describe a rotating vector in the phase space, which then serves as the basis for predictions. Using the boreal summer intraseasonal oscillation (BSISO) as a test case, an alternative procedure is introduced: it augments the original fields with their Hilbert transform (HT) to form a complex series and projects it onto its (single) dominant EOF. The real and imaginary parts of the corresponding complex pattern and index are compared with those of the original (real) EOF. The new index explains slightly less variance of the physical fields than the original, but it is much more coherent, partly from its use of future information by the HT. Because the latter is in the way of real-time monitoring, the index can only be used in cases with predicted physical fields, for which it promises to be superior. By developing a causal approximation of the HT, a real-time variant of the index is obtained whose coherency is comparable to the noncausal version, but with smaller explained variance of the physical fields. In test cases the new index compares well to other indices of BSISO. The potential for using both indices as an alternative is discussed.

© 2020 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Gerd Bürger, gbuerger@uni-potsdam.de

Abstract

Indices of oscillatory behavior are conveniently obtained by projecting the fields in question into a phase space of a few (mostly just two) dimensions; empirical orthogonal functions (EOFs) or other, more dynamical, modes are typically used for the projection. If sufficiently coherent and in quadrature, the projected variables simply describe a rotating vector in the phase space, which then serves as the basis for predictions. Using the boreal summer intraseasonal oscillation (BSISO) as a test case, an alternative procedure is introduced: it augments the original fields with their Hilbert transform (HT) to form a complex series and projects it onto its (single) dominant EOF. The real and imaginary parts of the corresponding complex pattern and index are compared with those of the original (real) EOF. The new index explains slightly less variance of the physical fields than the original, but it is much more coherent, partly from its use of future information by the HT. Because the latter is in the way of real-time monitoring, the index can only be used in cases with predicted physical fields, for which it promises to be superior. By developing a causal approximation of the HT, a real-time variant of the index is obtained whose coherency is comparable to the noncausal version, but with smaller explained variance of the physical fields. In test cases the new index compares well to other indices of BSISO. The potential for using both indices as an alternative is discussed.

© 2020 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Gerd Bürger, gbuerger@uni-potsdam.de
Keywords: Madden-Julian oscillation; Oscillations; Empirical orthogonal functions; Filtering techniques; Statistical techniques; Forecasting techniques

1. Introduction

The monitoring and prediction of oscillatory climatic features is often done with the help of a projection down into a phase space of a few, mostly just two, dimensions, in which the oscillation is represented in a rather simple way by a, more or less, rotating vector. Numerous examples of such an “index” exist, such as El Niño–Southern Oscillation (ENSO; cf. Xu and von Storch 1990), the quasi-biennial oscillation (QBO; cf. Wallace et al. 1993), and the Madden–Julian oscillation [MJO; cf. Wheeler and Hendon 2004; see also Box 14.1, Table 1 of Christensen et al. (2013)]. By doing so it is, explicitly or implicitly, hoped that two things hold:

  • V: the index explains enough original (physical) variability, and

  • P: the index is itself sufficiently regular (and predictable).

Needless to say, there exists a trade-off between the two, and one cannot be had without losing the other: If an index follows each and every climatic or weather fluctuation, random or not, it gradually loses its character of being a predictable oscillation; conversely, if an index is defined that is too regular and too predictable, it will most certainly have little amplitude and cannot explain much of the original variation. The index prediction has therefore two aspects: the first is the index prediction itself and the second is the prediction of the physical variables from the index. If measured in terms of variance explained, both terms can be coupled in a multiplicative way, and it is this coupled predictability P · V that is relevant for applications.

It is clear that reducing an often complex dynamical mechanism to a simple oscillation, which is essentially determined by only one spatial and one temporal degree of freedom, leaves out many of the nonlinear aspects of a phenomenon, as well as features that are imposed by varying boundary conditions (cf. Roundy 2015). One must be aware that by focusing on a single dynamical phenomenon and reducing it to a simple oscillation, the fraction of variance explained by the oscillation can be quite small.

The reduction to phase space requires both physical insight and a statistical machinery: the most appropriate variables must be selected, as parsimoniously as possible, from the most appropriate space and time frame, and then fed into a suitable transfer function. Here again a multitude of techniques exist, among which are empirical orthogonal functions, real or complex (EOFs or CEOFs), and real or complex principal oscillation patterns [POPs or CPOPs, which have recently resurfaced under the name dynamic mode decomposition (DMD)], and several relatives, all of which are reviewed in von Storch and Zwiers (2001).

Among the most studied indices is the real-time multivariate index of the MJO (RMM), which is conveniently shown as a phase plot of EOF-projected fields of outgoing longwave radiation (OLR) and winds at levels 850 and 200 hPa (U850, U200) (Wheeler and Hendon 2004). Empirical and dynamical predictability of the MJO, and from it various linked weather phenomena along the tropical belt and globally, have frequently been studied and are reviewed in Lee et al. (2016) or Kim et al. (2018). Closely related to the MJO is the boreal summer intraseasonal oscillation (BSISO). A corresponding index was defined by Lee et al. (2013, hereafter L13), from OLR and U850 fields. The challenge faced by L13 for the index definition was to disentangle the mostly zonal characteristic of the MJO, dominant in winter, and the Asian summer monsoon, which has a strong meridional component. The BSISO has two modes of variability, one between 30 and 60 days and one between 10 and 20 days.

Classical MJO and BSISO indices are monitored in real time and predicted operationally by numerous agencies. Conditioned on their predicted phase, medium-term weather predictions are regularly produced along the tropical belt and, through teleconnection, outside of it (Lee et al. 2017; Vitart 2017). There are some alternatives to measuring the BSISO, such as the extended EOF-based index of Kikuchi et al. (2012), the OLR-based MJO index (OMI) by Kiladis et al. (2014), or the precipitation-based index (PII) by Wang (2020); all indices are obtained as principal components of the temporally smoothed physical fields. The smoothing broadly follows Wheeler and Kiladis (1999), who employ a 30–96-day bandpass filter (“MJO band”) that leaves about 10% of the total tropical OLR variation. This renders the indices more coherent at the price of losing explanatory power for the total variations.

L13 proceed by keeping the higher frequencies and their explanatory power, and it marks the point of departure for this study. I will introduce a BSISO index that is far more coherent than in L13 but that is free of any temporal smoothing. By augmenting the OLR and U850 fields with their Hilbert transform (HT) and forming a complex field array, the index is obtained as a projection on its dominant complex EOF. Being derived from the HT, the index cannot monitor the BSISO in real time, though. By defining a causal version of the HT, a real-time BSISO index is obtained. These new HT-based indices are compared to each other as well as to the EOF-based BSISO index of L13 and the alternatives OMI, PII, and RMM.

2. Methods and data

a. EOFs

Despite being fairly common, I recall here the definition of EOFs in some detail since it is needed for a better understanding of the following. Suppose we are given a p-dimensional vector time series x(t), t = 1, …, n; this may include variables of mixed type, as in L13 or here in section 2e. By forming the cross-covariance matrix C, with angle brackets (“⟨ ⟩”) denoting expectation,
Cjk=〈xj⁡(t)xk⁡(t)〉
and its decomposition into orthonormal eigenvectors E and eigenvalues contained in the diagonal matrix λ,
C=EλE′,
the series x can be represented as
x⁡(t)=∑j≤pξj⁡(t)Ej,
with corresponding principal components (PCs) ξj(t); conversely, the PCs are obtained as
ξ⁡(t)=∑j≤pxj⁡(t)Ej′,
from which it follows that
λjk=〈ξj⁡(t)ξk⁡(t)〉,
with λ being diagonal. Equation (5) shows that the ξj(t), j ≤ p, are mutually uncorrelated (orthogonal), and variances are given by the eigenvalues λjj.

The main purpose of EOFs here is to project the normally high-dimensional physical variables x(t) onto a few dominant EOF patterns, using Eq. (4).

b. The Hilbert transform and CEOFs

EOFs are derived solely from covariance information and therefore do not encode temporal information in a genuine way (they are, for example, invariant under shuffling of the time coordinate). Capturing time-bound features such as propagating waves is therefore not possible directly with EOFs (Cressie and Wikle 2015, p. 294), and can only be verified after an EOF analysis between two or more patterns, by way of autocorrelation or cross-spectral analysis of the PCs ξj(t). In classical mechanics, the most natural way to represent a propagating feature is by using the canonical coordinates known from the Hamiltonian formulation: each state variable is augmented by a momentum (or tendency) variable, and in the two-dimensional reference frame an oscillation is simply a point rotating in a circle. From a data point of view, this augmentation can be achieved by way of the HT. The HT is best explained in the frequency domain. Denoting the Fourier transform by the superscript F, the HT y of x is given by
yF⁡(ω)=−isgn⁡(ω)xF⁡(ω),
using the sign function sgn. The imaginary factor in Eq. (6) simply means that the frequency components of x are shifted by an angle of π/2, putting y into quadrature to x. In the time domain, the HT is given by
y⁡(t)=1π∫−∞∞x⁡(τ)t−τdτ.
From Eq. (6) it is evident that the two signals x and y are perfectly coherent across all frequencies. Using the new complex variable z(t) = x(t) + iy(t), the above procedures of Eqs. (1)–(5) are repeated, and corresponding complex EOFs Ej, complex PCs ξj(t), 1 ≤ j ≤ p, and corresponding real-valued eigenvalues are obtained representing variances.

c. A two-dimensional BSISO index from one complex EOF

L13 define the BSISO index to be the first and second PC of the OLR and U850 fields (see section 2e). By definition, the PCs are uncorrelated, and if the corresponding modes dominate the spectrum they are in quadrature as well, one lagging the other by a quarter of the oscillation period. Using this mutually lagged information, the second is predicted by the first, which is predicted by the second, and so on, until the memory fades. In phase space, the corresponding point then circles around the origin, performing a “life cycle” of the BSISO, as illustrated below. But this is only a simple graphical description of the real physical processes involved, as predictability is ultimately rooted in the dynamical interplay of tropical sea surface temperatures, winds, and convection.

By using L13 as a point of divergence, I propose now to conduct the analogous method in complex space, using the complex time series z(t) in place of the real-valued x(t). In this case, patterns and coefficients are related by complex operations (projection and expansion). Specifically, the EOF expansion of Eq. (3) reads, for the real and imaginary parts, x and y, indicated by superscript r and i, respectively:
x⁡(t)=∑j≤pξjr⁡(t)Ejr+ξji⁡(t)Eji,y⁡(t)=∑j≤pξji⁡(t)Ejr−ξjr⁡(t)Eji,
and the projection of Eq. (4):
ξr⁡(t)=∑j≤pxj⁡(t)Ej′r−yj⁡(t)Ej′i,ξi⁡(t)=∑j≤pyj⁡(t)Ej′r+xj⁡(t)Ej′i,
with the prime (′) denoting conjugate transpose throughout. This means that real and imaginary parts of the patterns and components interact in a more complicated manner than in the real-valued case of EOFs. To distinguish the complex from the real case the term CEOF is used here and elsewhere. A major difference, in terms of the real patterns, lies in the fact that the EOFs are constrained to be orthogonal (angle of 90°) while the CEOFs are not (see section 3 below).

I note in passing that Kim et al. (2009) use the term “CEOF” to denote the combined EOFs from multiple variables; this is unfortunate because the term was introduced decades ago and is since then well known to stand for complex EOFs (Barnett 1983; von Storch et al. 1995).

d. A real-time (causal) Hilbert transform

The merits of using state and momentum for the BSISO, however, render real-time monitoring prohibitive. The reason is that knowing, or estimating, the exact momentum (phase) of a particular oscillation requires future knowledge [cf. Eq. (7)]. Interestingly, despite its importance there does not seem to be much reported effort to obtain or approximate a real-time HT of a measured signal. Bürger (1993) contains such an attempt by using, for any time τ, only past observations x(t), t ≤ τ. Here I use a slight modification of that approach. It starts with approximating z(t) as a linear dynamical system (or autoregressive process),
z⁡(t+1)=z(t)S+ε⁡(t),
assuming that errors ε(t) are uncorrelated and identically distributed, to find the least squares estimate of the system matrix S to be
S=〈z⁡(t)′z⁡(t)〉−1〈z⁡(t)′z⁡(t+1)〉.
It is worth noting that the eigenmodes of the system (i.e., eigenvalues and eigenvectors of S) are the so-called complex POPs (CPOPs; Bürger 1993; von Storch et al. 1995), and are an early equivalent of the so-called dynamic mode decomposition (cf. Tu et al. 2014). Suppose now that a calibration period has been separated in which S was estimated. In terms of real and imaginary parts, Eq. (10) reads
x⁡(t+1)=x(t)Sr−y(t)Si+εr⁡(t)y⁡(t+1)=y(t)Sr+x(t)Si+εi⁡(t).
Instead of solving Eq. (12) for S with x and y being known [leading to the solution of Eq. (11)], Bürger (1993) solves the equation for the entire series y, with x and S being known. Although it was then applied in an EOF-reduced space, the fact that an entire time series needed to be optimized rendered the procedure numerically challenging at the time. Diverting from that approach, here I proceed as follows: Given x and S, the imaginary part of Eq. (12) is solved recursively, that is
y⁡(t+1)=y(t)Sr+x(t)Si.
The real part of Eq. (12) is ignored, hence; potential consequences thereof are discussed in section 4. With no error term, it is assumed that the matrix S perfectly represents the dynamics of the system, which means in reverse that the outcome will not represent the HT perfectly. Formally, the index depends on the initial state y(0), but as shown in section 3, memory of y(0) is lost after just a few days, so that the definition in Eq. (13) is more or less unique. As a safeguard against fitting random effects, the estimation of S in Eq. (11) is not done in the full p-dimensional space of the physical variables (cf. section 2a) but in the reduced space of a limited number of PCs. Cross-validation is used to determine the optimal degree of reduction (cf. section 2e).

The PCs obtained from z(t) = x(t) + iy(t) using Eq. (9) are real time; the real and imaginary parts of the first PC, (ηr(t), ηi(t)), constitute a corresponding real-time version of the complex BSISO index, denoted here as rCEOF.

e. Data

Following L13, the BSISO is defined from daily fields of OLR and U850 over the area 40°–160°E and 10°S–40°N in 2.5° resolution, which gives for each field a dimension of 49 × 21 = 1029. The study period is from 1979 to 2019, which thus gives a 41-yr-long daily record of dimension 2 × 1029 = 2058. The annual cycle (mean and first three harmonics) is removed as well as the running mean of the past 120 days (to avoid interannual variability). From this step onward, data are restricted to the period from from May to October. Weighted by their area-averaged standard deviation, the fields are concatenated and subjected to an EOF and CEOF analysis. The most dominant eigenvalues in both analyses are all well separated (cf. North et al. 1982), and the first two of the EOF PCs, in this study as in L13 called PC1(t) and PC2(t), constitute what in L13 is called the BSISO1 index and represents the northward component relevant for the monsoon; the BSISO2 of L13 is ignored here. Of all CEOF patterns, the real and imaginary parts of the first show the greatest similarity to EOF1 and EOF2 so that, dropping the subscript 1 from now on, the new CEOF-based BSISO index becomes the (real-valued) pair (ξr(t), ξi(t)) with corresponding (real-valued) patterns Er and Ei.

Estimation of the causal HT [from Eqs. (11) and (13)] requires the separation of a calibration period from the full dataset, for which the years between 1981 and 1995 are chosen. The degree of reduction (number of retained EOFs) is determined by cross-validating the resulting complex correlation between original and reconstructed physical fields. Figure 1 shows that the explained variance quite smoothly depends on the degree of reduction, so that overfitting is not to be expected from this procedure; Fig. 1, moreover, does not depend strongly on the choice of the calibration and validation period (as long as both are sufficiently long). Of the original 2058 potential EOFs, the leading 200 were retained; the corresponding noncausal PCs reproduce 90%, and the causal 63% (calibration) and 40% (validation) of the total (complex) field variance.

Fig. 1.
Fig. 1.

Cumulative explained (complex) variance of the complexified OLR and U850 fields, approximated by the real-time HT in dependence of retained EOFs. For comparison, the cumulative EV of the corresponding full HT approximation is shown.

Citation: Journal of Climate 34, 1; 10.1175/JCLI-D-20-0427.1

In the following, the properties of the three BSISO indices, the classical of L13 (PC1, PC2), the full HT-based (ξr(t), ξi(t)), and the causal HT version (ηr(t), ηi(t)) will be analyzed and compared, especially with respect to their ability to monitor and predict the state of the BSISO. To ease the language, I shall simply refer to EOF, CEOF, and rCEOF to mean the corresponding BSISO index. To warrant a just comparison, all EOF patterns, real and complex, were estimated from the same calibration period, and corresponding PCs obtained from the physical fields by projection. All subsequent comparisons are based on the validation period 1996 to 2010.

The RMM was downloaded from http://www.bom.gov.au/climate/mjo, OMI from https://www.esrl.noaa.gov/psd/mjo/mjoindex, and PII from https://github.com/wangsg2526/PII_data.

3. Comparison of the BSISO index variants

The study L13 being the point of departure, I will present results along the lines of that study and mostly focus on a comparison of the EOF- and CEOF-based indices; for completeness, the other indices (OMI, PII, RMM) will also be considered briefly. Figure 2 shows EOF1 and EOF2 along with CEOFr and CEOFi. While EOF1 and CEOFr are basically identical, CEOFi is shifted somewhat southwest to EOF2, with its western center located directly over India, and on the other hand more northeastwardly aligned in the Pacific portion. Moreover, its northeastern center is more concentrated toward southeast China. Overall, the CEOFr/i patterns appear to be closer (more similar) to each other than the EOF1/2 patterns. This is only natural because the orthogonality constraint applies only to EOF1/2 but not to CEOFr/i, whose vectorized spatial structures have an angle of about 60° as compared to the 90° for EOF1/2. The corresponding PCs, and thus the BSISO indices as defined by L13 and here, are shown in Fig. 3 for the year 2010. It is noticeable that both first PCs, PC1 and ξr, are very similar, corresponding to the similar spatial patterns in Fig. 2, but the second PCs (PC2, ξi, and ηi) differ. Quite obviously, especially from the phase plot (see also “life cycles” below), the (r)CEOF PCs display a much more organized behavior, in particular for the larger amplitudes, with rCEOF being not as organized as CEOF but significantly more than EOF. The cycling about the origin has a correspondence in the time series, which are much more in quadrature for the CEOFs. This is verified by calculating the lagged correlations between the EOF and (r)CEOF PCs, based on the independent validation period; the latter fact brings about that PCs are not per se in quadrature (i.e., have zero correlation at lag zero), as it is the case for the calibration period. And as Fig. 4 shows, only CEOF happens to be strictly in quadrature. All three PC pairs represent a propagation in phase space with an approximate half-period (distance between extremes) of 19 (EOF), 14 (CEOF), and again 19 (rCEOF) days; this does not necessarily entail a propagation in physical space, though (Wang et al. 2018; this will be discussed later around Fig. 11 herein) The corresponding maximum (absolute) correlations are higher for the complex variants, with ρ = 0.59 and ρ = 0.69 for the (r)CEOF case, respectively, and ρ = 0.34 for the EOF (as in L13). So much memory in the system entails considerable predictability, enabling empirical prediction schemes similar to those of von Storch and Xu (1990) or of Lo and Hendon (2000). More will be said on that in section 3b.

Fig. 2.
Fig. 2.

(top) The two dominant EOF patterns of OLR; (bottom) real and imaginary part of the dominant CEOF pattern. Both patterns are normalized (as Euclidean vectors).

Citation: Journal of Climate 34, 1; 10.1175/JCLI-D-20-0427.1

Fig. 3.
Fig. 3.

Principal components for the year 2010 of the (top) EOF and CEOF patterns from Fig. 1, with (middle) full and (bottom) real-time version, using (left) a time series and (right) a phase plot.

Citation: Journal of Climate 34, 1; 10.1175/JCLI-D-20-0427.1

Fig. 4.
Fig. 4.

Lagged correlation between PC pairs for EOF (red), CEOF (blue), and rCEOF (green).

Citation: Journal of Climate 34, 1; 10.1175/JCLI-D-20-0427.1

Before further studying the rCEOF and the other variants, the initial state dependence of the rCEOF PC ηi needs to be assessed. Starting with an ensemble of 10 members randomly distributed about zero with one standard deviation of the calibration period ξi, Fig. 5 demonstrates that they all converge quickly. The evolution is practically unique after about 20 days, for which ηi is thus well defined by Eq. (13). The bivariate (real and imaginary part) correlation between ξ and η is 0.82, which is smaller than for the respective real-time versions of Kiladis et al. (2014) and Wang (2020), but still satisfactory (later I will use complex correlations; cf. section 3a); and as Fig. 6 shows, ηi is strongly coherent with its noncausal counterpart ξi, albeit with a small negative phase shift in the BSISO frequency range whose nature is unknown. It can be concluded that the rCEOF is a close real-time approximation of the full CEOF.

Fig. 5.
Fig. 5.

Dependence of rCEOF on the initial state, starting with a perturbation of 1σ of ξi(t).

Citation: Journal of Climate 34, 1; 10.1175/JCLI-D-20-0427.1

Fig. 6.
Fig. 6.

(top) Squared coherency and (bottom) phase spectrum between causal and full (noncausal) Hilbert transform, ξi and ηi.

Citation: Journal of Climate 34, 1; 10.1175/JCLI-D-20-0427.1

The greater coherency of the complex PCs is further confirmed using spectral analysis, similar to Figs. 5 and 6 of L13. In Fig. 7, the peak spectral densities of the EOFs are near a period of 50 days, roughly corresponding to the 2 × 19 day estimate of Fig. 4; the complex variants show both a longer periodicity of about 60 days. The rCEOF PC ηi, however, displays a marked variability increase especially for the longer periods, which will be discussed further below. In both variants, the complex PCs show a much higher squared coherency, with CEOF values approaching unity and those of rCEOF above 0.8 near the peak frequency. Near that frequency, the phase is at π/2 for CEOF (in quadrature), and for rCEOF and EOF slightly below and slightly above, respectively, which roughly corresponds to the autocorrelation values of Fig. 4.

Fig. 7.
Fig. 7.

(a),(b) Spectrum of EOF and (r)CEOF PCs and (c),(d) squared coherency and phase spectrum between the respective PCs.

Citation: Journal of Climate 34, 1; 10.1175/JCLI-D-20-0427.1

The life cycles are best described by a two-dimensional phase plot, with coordinates being for each of the three variants (PC1, PC2), (ξr, ξi), and (ηr, ηi), respectively. Figure 8 displays the mean over all BSISO evolutions that start from a strong event in any of 8 evenly distributed sectors or phases, P1, …, P8, the strength being given by a normalized amplitude of at least 1.5. From that event, the BSISO tends to spiral inward during a certain episode, the length of which is controlled by the damping. Along that spiraling, the BSISO passes the four main directions in phase space (i.e., geographic patterns) as marked by the coordinate axes, and thus spirals through a sequence A → B → −A → −B → A …, with patterns A and B that are either (EOF1, EOF2) or (CEOFr, CEOFi). It is obvious from Fig. 8 that EOF and CEOF are markedly distinguished by the length of the episodes. While EOF is strongly damped with almost no time to reach the next sector, the CEOF travels at least two full sectors with sufficient strength. The episode length of the rCEOF scheme is somewhat between EOF and CEOF. The geographic representation of this is shown in Fig. 9. After adopting the convention to start the cycle with P1 having no significant contribution in the western North Pacific (WNP), the EOF evolution is generally comparable to L13, Fig. 9. As the CEOF and rCEOF patterns are practically identical, I shall only deal with CEOF here. Although the main intraseasonal transition of the monsoon occurs in all variants from the Indian subcontinent (P1) over the South China Sea (SCS, P5) to the WNP (P8), the EOF and CEOF evolutions deviate in certain significant details from each other. Two main features are most notable: First, the Pacific portion shows a southeastward direction for EOF while for CEOF it is northeastward, as known already from Fig. 2 (see P1, P4, and P5). And second, for the CEOF the Indian monsoon remains stronger for a much longer time (P3, P4), the corrollary being extended drier periods through P7 and P8.

Fig. 8.
Fig. 8.

Phase plots showing the BSISO life cycle from EOFs, CEOFs, and rCEOFs. The black dots indicate phase P1 (see text). Weak amplitudes (≤1) are shown as a gray disc.

Citation: Journal of Climate 34, 1; 10.1175/JCLI-D-20-0427.1

Fig. 9.
Fig. 9.

OLR life cycle of BSISO as reconstructed from the main phase states (gray dots) of Fig. 8 (with correspondingly normalized amplitudes), based on (left) EOFs and (right) CEOFs. The rCEOFs are practically identical to the CEOFs and therefore not shown.

Citation: Journal of Climate 34, 1; 10.1175/JCLI-D-20-0427.1

One must be aware that Figs. 8 and 9 carry no direct information on phase velocities, as, for example, the CEOFs may simply pass P3 and P4 quicker than the EOFs. This is verified in Fig. 10, which shows for the same sectors P1, …, P8 the climatological phase velocity of the BSISO index. For all variants it does not change much between sectors, with 0.05, 0.1, and 0.2 rad day−1 for EOF, rCEOF, and CEOF, respectively; this roughly corresponds to periods of 124, 62, and 31 days per full cycle. Especially for EOF this is in considerable disagreement to the estimates from Fig. 4 (lagged correlations) and Fig. 7 (spectrum) of about 2 × 19 = 38 days. But since the power spectrum is broader for EOF, more neighboring frequencies and different phase velocities contribute to the mean periodicity.

Fig. 10.
Fig. 10.

Polar plot of phase velocity. Colors identical to Fig. 8.

Citation: Journal of Climate 34, 1; 10.1175/JCLI-D-20-0427.1

How the propagation displayed in the phase plots plays out in physical terms is shown in Fig. 11. It shows the northward propagation of the OLR signals, as observed and as reconstructed from the indices using the respective physical (EOF) patterns [cf. Eqs. (3) and (8)] and bandpass filtered with a 20–70-day window. The propagation is identified as a lag correlation between zonal (80°–90°E) averages of those fields and a fixed reference field, which is chosen here to be the section bounded by 5°–10°N, 80°–90°E over the Indian Ocean, identical to the one used by Wang et al. (2018). The observations show a clear propagating signal roughly from the equator to 15°N, corresponding to 1° northward per day. This propagation is not found in the EOF reconstruction, which shows a clear standing oscillation characterized by the vertical isolines of equal correlations. Compared to that, northward traveling is obvious for both the full and the real-time CEOF reconstructions. With the full CEOF version being slightly faster and the real-time version being slightly slower than the observed traveling rate of 1° day−1, the correspondence is convincingly strong; only the reversal seen north of 20°N has no counterpart in the observations. Overall, the propagation characteristics of CEOF are comparable to that of OMI and PII (cf. Wang et al. 2018, Wang 2020).

Fig. 11.
Fig. 11.

OLR propagation characteristics of OBS and as reconstructed from EOFs, CEOFs, and rCEOFs, measured by lagged correlations to a reference region over the Indian Ocean (5°–10°N, 80°–90°E). The latitudinal coordinate describes zonal averages between 80° and 90°E.

Citation: Journal of Climate 34, 1; 10.1175/JCLI-D-20-0427.1

Figure 12 displays the pattern of daily OLR variance explained by the three BSISO indices. All patterns have maxima southwest of India and southeast of China, with details depending on the index. The amount of total daily variance explained is 12.2% by EOF (maximum by definition) followed by CEOF with 9.3% and rCEOF with 4.6%; the variance explained over India is nevertheless higher for CEOF, which is likely related to the different second CEOF pattern (Fig. 2) and is also visible in the different CEOF life cycle (Fig. 9).

Fig. 12.
Fig. 12.

Explained variance (%) of daily OLR from EOFs, CEOFs, and rCEOFs.

Citation: Journal of Climate 34, 1; 10.1175/JCLI-D-20-0427.1

a. Relation to other indices

Now I compare the indices with the other prominent BSISO indices described earlier, that is, the OMI, PII, and RMM. In previous work (Kiladis et al. 2014; Wang 2020), bivariate correlations as defined in Lin et al. (2008) had been used as a similarity measure; these are normal correlations but applied to the concatenated pair of components. Because oscillations from time-lagged processes, as they are conveniently displayed in two-dimensional phase plots, are best described in the complex plane,1 I have chosen to use instead complex correlations.2 Because they “absorb” oscillatory behavior, complex auto- and cross-correlations are more adapt for assessing predictability, as only they show the decay that one expects with increasing time lag (= lead time). As in the HT (cf. 2b), complexification is achieved by transforming any two-dimensional real-valued index (x(t), y(t)) into a one-dimensional complex process z(t) = x(t) + iy(t), and correlations are calculated just as in the real-valued case.

Table 1 summarizes all pairwise cross-correlations of the indices, based on the common period 2002–17. The table reveals that all pairs are moderately similar, but OMI and PII stand out with very high correlation of 0.89. I may add that because the optimal lags are low throughout, the corresponding (real-valued) bivariate correlations turn out to be similar but generally a little lower.3

Table 1.

Absolute values of complex correlations and optimal lags between the five BSISO indices, based on the summer season (May–October) of the years 2002–17. Values depend on phase space orientation (P2/3 = Indian Ocean, etc.). Here they are adjusted from Wang (2020, Fig. 6 therein) and Chen et al. (2019, Fig. 11 therein).

Table 1.

b. Toward the coupled predictability P⋅V

Assessing the full predictability of an index requires the use of numerical weather prediction models and how they predict the physical fields and the index derived from them (Jie et al. 2017; Wang et al. 2019; see also http://s2sprediction.net/xwiki/bin/view/Phase2/MJOTel); this may include (for realistic models) dynamical evolutions of the phenomenon in question (here BSISO) that are not captured by the index. For the heavily smoothed OMI and PII indices this involves forecast data well beyond the subseasonal range, which are often not available. Following Kikuchi et al. (2012), Kiladis et al. (2014) and Wang (2020) therefore derive real-time variants of OMI and PII by employing a special windowing technique that is tapered to each respective target date. This real-time index is then derived from the predicted fields and verified against the observed real-time index, and Wang (2020) also validates against the full index (which uses the noncausal smoothing). If seasonal predictions were available one could estimate the full indices from them, with a somewhat reduced quality at issue time (due to forecast imperfections), and at valid time verification is postponed to a somewhat later date when all fields used for the filtering have been observed. One should at least take into account the possibility that using the full predicted index produces higher skill as the real-time variant.

Without atmospheric predictor fields, one is left with predicting the index by itself, using persistence. In this case, of course, real-time indices are mandatory, so that a real-world prediction is best approximated by calculating the lagged correlations between the real-time and the full index. Like in the previous section, complex correlations are most appropriate for that. Figure 13 shows the results in terms of absolute correlations and corresponding phase evolution. As could be expected, the time-filtered indices OMI and PII are, in that order, best predictable, falling under |ρ| = 0.6 only after about 20 days (note that OLR is likely more autocorrelated than precipitation). RMM has shorter memory and is generally comparable to the (full) CEOF, both falling under |ρ| = 0.6 after about 8 days; EOF values are below |ρ| = 0.2 already after about 10 days. Note that because EOF and RMM are fully real-time they start at |ρ| = 1. The phase grows quite uniformly at a speed of about 0.13 rad day−1, which corresponds to 47 days for one cycle. The results of Fig. 13 can be used as a first, quite heuristic assessment of the coupled predictability P · V mentioned in section 1. This is done by interpreting P, like V, as a fraction of explained variance, which can then be multiplied with V to obtain the coupled result. Translating the correlations of Fig. 13 (upper panel) into explained variances is done as follows: by stationarity, variances are generally equal under time shifts, from which it follows that P is related to |ρ| as P = 2|ρ| − 1, as derived in the appendix. As an estimate for the V term, the total daily (spatiotemporal) variance explained by an index is used here. That estimate is only available for EOF and CEOF, with V = 12.2% and V = 9.3%, respectively; RMM, as a meridional average index, cannot be directly compared here, and I am not aware of any estimate for OMI and PII for the unfiltered daily OLR fields.4 Figure 14 shows for EOF and CEOF the resulting coupled total predictability P · V for the first 20 days. One sees that the smaller V value of CEOF is outweighed by its longer memory after about 2 days.

Fig. 13.
Fig. 13.

Skill of real-time indices predicting their own full variant, depending on time lag, measured by (top) absolute complex correlation and (bottom) phase.

Citation: Journal of Climate 34, 1; 10.1175/JCLI-D-20-0427.1

Fig. 14.
Fig. 14.

Heuristic for the coupled predictability P · V, depending on time lag, of the total physical fields by the EOF and CEOF indices, measured in terms of explained variance (P derived from Fig. 13, V from Fig. 12).

Citation: Journal of Climate 34, 1; 10.1175/JCLI-D-20-0427.1

c. Case 1: The extreme tropical cyclone activity of 2018

Chen et al. (2019) report extremely high tropical cyclone activity over East Asia during the summer of 2018, with centers in the SCS and WNP. Ranked as the second highest since 1951, I will use it as a test case to analyze how the different indices have monitored such activity. To obtain a first visual impression, Fig. 15 displays the index evolutions during the year of 2018; for clarity we only show one component of each index. While the overall variability is similar among all indices, especially at the dominant periodicity near 30 days, there are also differences among them, mostly in the early part of the year and during late summer. A fairly coherent evolution occurs in the month of July where all indices show a positive anomaly. Following Figs. 8 and 11 of Chen et al. (2019), which display for the months of July and August 2018 the RMM and EOF indices, Fig. 16 shows for all five indices of Table 1 the normalized phase plot for July, the month of strongest activity. While RMM shows a strong maximum in the Maritime Continent (MC)—this is RMM phase 5, here between India and SCS—the EOF is strongest in the SCS, with amplitudes approaching 2, and remains strong for the rest of the month where activity propagates to the WNP, so that according to Chen et al. (2019) the anomalous cyclonic activity in that period is better monitored by the EOF. Compared to that, the OMI and PII indices are weaker, albeit above the standard unit; due to their time filtering operation, both indices, along with their real-time variants, evolve much more smoothly than the EOF index; while the OMI is more concentrated toward India, PII activity occurs mainly in the SCS; full and real-time variants are similar for both. The CEOF index grows in the MC and monitors persistent high activity from the SCS to the WNP. The rCEOF starts only in the SCS, but then sharply grows to very strong amplitudes into the WNP; the later start may be related to the phase shift that was seen in Fig. 6.

Fig. 15.
Fig. 15.

Time series of the indices (only one component) for the year 2018.

Citation: Journal of Climate 34, 1; 10.1175/JCLI-D-20-0427.1

Fig. 16.
Fig. 16.

Index evolution for July of the year 2018, thin lines displaying the respective real-time variant. The bullet marks 1 July. Below-unit amplitudes are indicated by the gray disk, and amplitudes of 2 are shown by the dashed circle. OMI is colored pink; PII is cyan.

Citation: Journal of Climate 34, 1; 10.1175/JCLI-D-20-0427.1

Since the CEOF signal is similarly concentrated in the SCS/WNP region but evolves more coherent than EOF, it can be argued that the strong cyclonic activity in July 2018 in that area, as reported by Chen et al. (2019), is most clearly represented by CEOF.

d. Case 2: The YOTC period

As a final example I show a case from the Year of the Tropical Convection (YOTC; cf. Waliser et al. 2011) study period (2008–10). In Fig. 17, the five indices are plotted in phase space as in the previous figure, for November 2009, which is, as one may note, outside of the calibration period of both EOF and CEOF. The figure should be compared, for example, to Fig. 8 of Kiladis et al. (2014), and the overall understanding is that an event is initiated at the end of October that is soon amplified over the Indian Ocean and maintained through the MC and WNP. On that background, the EOF displays a somewhat erratic behavior, with an early Indian Ocean evolution that is backward (clockwise) in phase space, to return to the normal phase evolution, with little amplitude, into the SCS. The RMM shows a fairly regular, above normal evolution that starts in the western Indian Ocean and ends in the SCS. This is similar for the OMI and PII indices, except that their evolution precedes the RMM by almost one sector and accordingly reaches the WNP at the end of November. Both variants of the CEOF are quite similar, and with respect to phase angle lie between the RMM and OMI/PII.

Fig. 17.
Fig. 17.

As in Fig. 16, but for November 2009.

Citation: Journal of Climate 34, 1; 10.1175/JCLI-D-20-0427.1

For CEOF an interesting episode occurs in the middle of the month, with amplitudes dropping to unit levels, which is also reflected in the RMM but not in the OMI/PII. It is unclear whether this reflects a synoptic event that happened to project strongly onto the CEOF/RMM patterns, or whether it is indicative of an intermediate “break phase” of the ongoing oscillation. Further investigations are needed.

4. Conclusions

The study introduced the new two-dimensional BSISO index CEOF based on complexified physical fields using the full Hilbert transform (HT), along with its real-time companion rCEOF. They are derived from and thoroughly compared to the real-valued “classical” version EOF. The EOF explains a larger portion of OLR variance, but its components PC1 and PC2 are only weakly coherent. Conversely, the (r)CEOF variants explain less of the variability but are much more coherent. Both results are not really surprising since EOFs are defined to explain maximum variance, and HT, along with its derived principal components, are defined to be perfectly coherent and in quadrature. Therefore, by recalling the main two criteria mentioned right at the start in 1, the comparison comes down to weighing between P, the index itself being predictable (through memory, cf. Fig. 13), and V, its ability to explain variance in the physical fields (cf. Fig. 12). Toward this goal, I have introduced the heuristic “P · V” scheme that measures the coupled predictability of an index, and according to that scheme the CEOF is better predictable than EOF after day 2. But especially if future information of the physical fields is available [e.g., when they are model-predicted as in Jie et al. (2017) or Wang et al. (2018)], the full potential of the noncausal HT can be used, for which case I consider the corresponding CEOF index to be superior; in that case, the dynamical predictions should also help to better estimate the current BSISO state. Following the same P · V argument, the OMI and PII indices, with their remarkable predictability P, would in turn have to be preferred over CEOF, provided enough daily variance V is left to explain the physical OLR fields; this needs to be shown. For OMI and PII, the balance between P and V is determined by the degree of filtering, which increases P at the price of V. I am a bit concerned, though, that their P · V values would be somewhat inflated due to the in-sample calibration of the filtering procedure. A more comprehensive account following the P · V paradigm, be it based on the real-time indices or on the full indices obtained from dynamical prediction models, is beyond the current scope and enough material for an extra study, perhaps along the lines of, for example, Marshall et al. (2016) or Lee et al. (2017). The results would fit nicely into the Subseasonal-to-Seasonal (S2S) Prediction Project, http://s2sprediction.net/.

Northward propagation of (r)CEOF was comparable to OMI and PII. The case studies for November 2009 and July 2018 revealed a tendency of rCEOF to be somewhat phase-shifted relative to CEOF, a bias that was also seen in the spectrum; otherwise the amplitude of all major events was monitored by the CEOF, with some uncertainty remaining in all indices with respect to the phase.

Another important caveat of rCEOF is the fairly high spectral power for the causal component ηi toward longer periods, which I have so far not been able to understand. A possible explanation is the truncation needed to remove sampling noise and obtain a stable system matrix, which may amplify only the longer periods. This should not affect the evolution of the phase, however, which crucially determines the conditional probabilities for, and thus the prediction of, the physical variables.

From a physical point of view, one must note that EOF and CEOF do indeed describe different processes, as demonstrated by the patterns visited by the BSISO life cycle. We have seen (Fig. 2) that the imposed orthogonality of the EOF patterns leads to a second pattern that is quite different from the CEOFi, which has a stronger center over central India. The corresponding life cycles, passing in a linear way through patterns P1 → P2 → −P1 → −P2 → P1 …, will be significantly different especially with respect to India, as seen in Fig. 9. This definitely deserves a deeper analysis from both an observational and a modeling point of view.

There is an interesting side aspect with respect to the causal HT variant. The average behavior of the PCs shown in Fig. 8 unveils the dynamics behind the BSISO, which can be understood as the mode of a damped linear system excited by random fluctuations. Such systems are conveniently simulated as a vector autoregressive process, and have been studied extensively in the 1990s under the name of principal oscillation patterns (POPs), interestingly with the MJO being their first use case (von Storch et al. 1995). According to that scheme, the BSISO appears as an eigenmode (=POP) of the system, with a complex eigenvalue ρe2πif that defines a characteristic damping rate ρ and frequency f of an oscillation. It means that, once having reached a strong amplitude, the BSISO will spiral inward with e-folding time of −1/ln(ρ) and period 1/f. On this background, empirical index predictions using regression on the lagged PCs, such as those for the MJO by Marshall et al. (2016), are equivalent to the POP prediction of the MJO reported in von Storch et al. (1995) [equivalent to the real form of Eq. (10)]. As has been more thoroughly discussed by Bürger (1993), a single complex POP (CPOP), just like a CEOF, can describe oscillations of any kind, whether it is standing or traveling, or even any combination thereof; this is what we probably see in Fig. 9, and it is a simple consequence of their (real) patterns being free of the orthogonality constraint. But to emphasize again, all these approaches describe only a first-order (linear) approximation of the average BSISO; the full extent of the BSISO remains a dynamical phenomenon governed by multiple time scales and a complicated coupling of tropical sea surface temperatures, winds, and convection.

The causal HT was defined here using the recursive approach of Eq. (13), which (in my view) has served its purpose and is simpler and more elegant than the original full optimization approach of Bürger (1993). But as now, almost 30 years later, numerical limitations should be out of the way one may want to give that approach another try and check whether the results are similar, or if perhaps that older variant turns out to be superior in terms of BSISO monitoring and predicting. Especially the extra constraint from the real part of Eq. (12) may prove beneficial with respect to the described deficits (i.e., the phase bias and the excessive spectral power at longer periods).

To demonstrate the use of complex EOFs I have chosen the BSISO because it fits well as a test case and was easy to adapt technically. But since the approach is fairly general, there is not much in the way of applying it to other indices, most notably the MJO (cf. Wheeler and Hendon 2004) or other EOF-based, two-dimensional indices such as the MISO (Suhas et al. 2013) or the QBO and ENSO (Wilhelmsen et al. 2018).

Acknowledgments

The study improved considerably through the help of three anonymous reviewers; it was supported through the German Research Foundation (DFG SHIVA, project 265653116) and the SaWaM project (https://bmbf-grow.de/de/verbundprojekte/sawam). The main code is available at https://gitlab.com/gbuerger/iso.

Data availability statement

The OLR data are available at https://www.esrl.noaa.gov/psd/data/gridded/data.olrdcdr.interp.html and U850 at http://www.cpc.ncep.noaa.gov/products/wesley/reanalysis2/.

APPENDIX

EV and ρ under Equal Variance

The expected squared error (with angle brackets “⟨ ⟩” denoting expectation) between two random variables X and Y is given as
〈⁡(X−Y)2〉=〈X2〉−2〈XY〉+〈Y2〉.
Letting ⟨X2⟩ = ⟨Y2⟩, it follows that
〈⁡(X−Y)2〉〈X2〉=1−2〈XY〉〈X2〉+1,
and therefore
EV=1−〈⁡(X−Y)2〉〈X2〉=2ρ−1.

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1

Oscillations are solutions of linear dynamical systems associated with complex eigenvalues.

2

For two complex random variables, x and y, their complex correlation is ρ = xy*/(|x||y|), with the asterisk (*) denoting complex conjugation.

3

One should not be confused about the apparent inconsistency (intransitivity) of the optimal lags because, for example, from the relative lags of OMI and PII to RMM (0 and 3) one cannot deduce their mutual optimal lag of 3 (it is 0); apart from the usual estimation error this is due to the fact that only perfect correlations are transitive; consequently, there is no unique time shift per index that brings them all into optimal concordance.

4

The OMI and PII references report excessive trials to obtain optimal filter weights. Strict predictability studies require independent data, so this must be factored in as it otherwise creates artificial skill.

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Journal of Climate

  • Barnett, T. P., 1983: Interaction of the monsoon and Pacific trade wind system at interannual time scales. Part I: The equatorial zone. Mon. Wea. Rev., 111, 756–773, https://doi.org/10.1175/1520-0493(1983)111<0756:IOTMAP>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Bürger, G., 1993: Complex principal oscillation pattern analysis. J. Climate, 6, 1972–1986, https://doi.org/10.1175/1520-0442(1993)006<1972:CPOPA>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Chen, L., Z. Gong, J. Wu, and W. Li, 2019: Extremely active tropical cyclone activities over the western North Pacific and South China Sea in summer 2018: Joint effects of decaying La Niña and intraseasonal oscillation. J. Meteor. Res., 33, 609–626, https://doi.org/10.1007/s13351-019-9009-x.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Christensen, J. H., and Coauthors, 2013: Climate phenomena and their relevance for future regional climate change. Climate Change 2013: The Physical Science Basis, T. F. Stocker et al., Eds., Cambridge University Press, 1217–1308.

  • Cressie, N., and C. K. Wikle, 2015: Statistics for Spatio-Temporal Data. John Wiley & Sons, 624 pp.

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  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    Cumulative explained (complex) variance of the complexified OLR and U850 fields, approximated by the real-time HT in dependence of retained EOFs. For comparison, the cumulative EV of the corresponding full HT approximation is shown.

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    (top) The two dominant EOF patterns of OLR; (bottom) real and imaginary part of the dominant CEOF pattern. Both patterns are normalized (as Euclidean vectors).

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    Principal components for the year 2010 of the (top) EOF and CEOF patterns from Fig. 1, with (middle) full and (bottom) real-time version, using (left) a time series and (right) a phase plot.

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    Lagged correlation between PC pairs for EOF (red), CEOF (blue), and rCEOF (green).

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    Dependence of rCEOF on the initial state, starting with a perturbation of 1σ of ξi(t).

  • <sc>Fig</sc>. 6.
    View in gallery
    Fig. 6.

    (top) Squared coherency and (bottom) phase spectrum between causal and full (noncausal) Hilbert transform, ξi and ηi.

  • <sc>Fig</sc>. 7.
    View in gallery
    Fig. 7.

    (a),(b) Spectrum of EOF and (r)CEOF PCs and (c),(d) squared coherency and phase spectrum between the respective PCs.

  • <sc>Fig</sc>. 8.
    View in gallery
    Fig. 8.

    Phase plots showing the BSISO life cycle from EOFs, CEOFs, and rCEOFs. The black dots indicate phase P1 (see text). Weak amplitudes (≤1) are shown as a gray disc.

  • <sc>Fig</sc>. 9.
    View in gallery
    Fig. 9.

    OLR life cycle of BSISO as reconstructed from the main phase states (gray dots) of Fig. 8 (with correspondingly normalized amplitudes), based on (left) EOFs and (right) CEOFs. The rCEOFs are practically identical to the CEOFs and therefore not shown.

  • <sc>Fig</sc>. 10.
    View in gallery
    Fig. 10.

    Polar plot of phase velocity. Colors identical to Fig. 8.

  • <sc>Fig</sc>. 11.
    View in gallery
    Fig. 11.

    OLR propagation characteristics of OBS and as reconstructed from EOFs, CEOFs, and rCEOFs, measured by lagged correlations to a reference region over the Indian Ocean (5°–10°N, 80°–90°E). The latitudinal coordinate describes zonal averages between 80° and 90°E.

  • <sc>Fig</sc>. 12.
    View in gallery
    Fig. 12.

    Explained variance (%) of daily OLR from EOFs, CEOFs, and rCEOFs.

  • <sc>Fig</sc>. 13.
    View in gallery
    Fig. 13.

    Skill of real-time indices predicting their own full variant, depending on time lag, measured by (top) absolute complex correlation and (bottom) phase.

  • <sc>Fig</sc>. 14.
    View in gallery
    Fig. 14.

    Heuristic for the coupled predictability P · V, depending on time lag, of the total physical fields by the EOF and CEOF indices, measured in terms of explained variance (P derived from Fig. 13, V from Fig. 12).

  • <sc>Fig</sc>. 15.
    View in gallery
    Fig. 15.

    Time series of the indices (only one component) for the year 2018.

  • <sc>Fig</sc>. 16.
    View in gallery
    Fig. 16.

    Index evolution for July of the year 2018, thin lines displaying the respective real-time variant. The bullet marks 1 July. Below-unit amplitudes are indicated by the gray disk, and amplitudes of 2 are shown by the dashed circle. OMI is colored pink; PII is cyan.

  • <sc>Fig</sc>. 17.
    View in gallery
    Fig. 17.

    As in Fig. 16, but for November 2009.

Fig. 1.

Cumulative explained (complex) variance of the complexified OLR and U850 fields, approximated by the real-time HT in dependence of retained EOFs. For comparison, the cumulative EV of the corresponding full HT approximation is shown.
Fig. 2.

(top) The two dominant EOF patterns of OLR; (bottom) real and imaginary part of the dominant CEOF pattern. Both patterns are normalized (as Euclidean vectors).
Fig. 3.

Principal components for the year 2010 of the (top) EOF and CEOF patterns from Fig. 1, with (middle) full and (bottom) real-time version, using (left) a time series and (right) a phase plot.
Fig. 4.

Lagged correlation between PC pairs for EOF (red), CEOF (blue), and rCEOF (green).
Fig. 5.

Dependence of rCEOF on the initial state, starting with a perturbation of 1σ of ξi(t).
Fig. 6.

(top) Squared coherency and (bottom) phase spectrum between causal and full (noncausal) Hilbert transform, ξi and ηi.
Fig. 7.

(a),(b) Spectrum of EOF and (r)CEOF PCs and (c),(d) squared coherency and phase spectrum between the respective PCs.
Fig. 8.

Phase plots showing the BSISO life cycle from EOFs, CEOFs, and rCEOFs. The black dots indicate phase P1 (see text). Weak amplitudes (≤1) are shown as a gray disc.
Fig. 9.

OLR life cycle of BSISO as reconstructed from the main phase states (gray dots) of Fig. 8 (with correspondingly normalized amplitudes), based on (left) EOFs and (right) CEOFs. The rCEOFs are practically identical to the CEOFs and therefore not shown.
Fig. 10.

Polar plot of phase velocity. Colors identical to Fig. 8.
Fig. 11.

OLR propagation characteristics of OBS and as reconstructed from EOFs, CEOFs, and rCEOFs, measured by lagged correlations to a reference region over the Indian Ocean (5°–10°N, 80°–90°E). The latitudinal coordinate describes zonal averages between 80° and 90°E.
Fig. 12.

Explained variance (%) of daily OLR from EOFs, CEOFs, and rCEOFs.
Fig. 13.

Skill of real-time indices predicting their own full variant, depending on time lag, measured by (top) absolute complex correlation and (bottom) phase.
Fig. 14.

Heuristic for the coupled predictability P · V, depending on time lag, of the total physical fields by the EOF and CEOF indices, measured in terms of explained variance (P derived from Fig. 13, V from Fig. 12).
Fig. 15.

Time series of the indices (only one component) for the year 2018.
Fig. 16.

Index evolution for July of the year 2018, thin lines displaying the respective real-time variant. The bullet marks 1 July. Below-unit amplitudes are indicated by the gray disk, and amplitudes of 2 are shown by the dashed circle. OMI is colored pink; PII is cyan.
Fig. 17.

As in Fig. 16, but for November 2009.
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