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Cécile Penland

It seems that stochastic climate models are beginning to be fashionable. In this article, current theories of where noise comes from, its relation to chaos, and how temperamental a numerical treatment of noise in a climate model can be are all discussed. There are ways of avoiding common pitfalls.

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Cecile Penland

Abstract

The effects of random forcing and deterministic feedback are combined in a measured multivariate time series. It is shown here how the characteristics of the driving noise can be found after the deterministic effects have been identified by the principal oscillation pattern (POP) analysis. In addition, the POP analysis is extended to enable the prediction of the most probable meteorological pattern at some future time when the present pattern is known, and the conditional probability of finding the process at any location within a range of values given the value of the process at another location at an earlier time. Estimates of how well these predictions can be trusted are also given. The basic assumption of POP analysis is that the system can be optimally modeled by a linear Markov process.

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Cécile Penland

Abstract

Linear inverse modeling (LIM) is a statistical technique based on covariance statistics that estimates the best-fit linear Markov process to a multivariate time series. An integral, often-ignored part of the technique is a test of whether or not the linear assumptions are valid. One test for linearity is the so-called tau test. While this test can be trusted when it passes, it sometimes fails when it ought to pass. In this article, we discuss one of the reasons for spurious failure, the “Nyquist issue,” which occurs when the lagged covariance matrix used in the analysis is numerically performed at a lag greater than or nearly equal to half the period of a natural mode of variability represented in the time series. As an illustration relevant to a system with many degrees of freedom, but simple enough to solve analytically, we consider a four-dimensional system consisting of two modal pairs. Within this framework, we suggest one solution that can be applied if the time series are long enough. It is hoped that awareness of this issue can prevent misinterpretation of LIM results.

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Cécile Penland
and
Ludmila Matrosova

Abstract

The authors discuss forecast uncertainty, which should be employed for the purpose of judging prediction model performance, and actual forecast errors, which can be employed by users to judge the reliability of the forecasts. Realistic a priori estimates of forecast uncertainty for linear inverse modeling sea surface temperature predictions are presented and compared with the actual forecast errors occurring during the course of real-time predictions. The a priori estimates are of size similar to the actual errors except during the warmest phase of observed El Niño events.

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Cécile Penland
and
Michael Ghil

Abstract

Multivariate linear prediction based on single-lag inverse modeling is developed further and critically examined. The method is applied to the National Meteorological Center analyses of Northern Hemisphere 700-mb geopotential height anomalies, which have been filtered to eliminate periods shorter than 10 days. Empirically derived normal modes of the randomly forced linear system are usually correlated, even at zero lag, suggesting that combinations of modes should be used in predictions. Due to nonlinearities in the dynamics and the neglect of interactions with other pressure levels, the lag at which the analysis is performed is crucial; best predictions obtain when the autocovariances involved in the analysis are calculated at a lag comparable to the exponential decay times of the modes. Errors in prediction have a significant seasonal dependence, indicating that the annual cycle affects the higher-order statistics of the field. Optimized linear predictions using this method are useful for about half a day longer than predictions made by persistence.

Conditional probabilities are much more efficiently calculated using normal-mode parameters than from histograms, and yield similar results. Maps of the model's Fourier spectra—integrated over specified frequency intervals and consistent with the assumptions made in a linear analysis—agree with maps obtained from fast Fourier transforms of the data.

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Cécile Penland
and
Ludmila Matrosova

Abstract

The predictability of tropical Atlantic sea surface temperature on seasonal to interannual timescales by linear inverse modeling is quantified. The authors find that predictability of Caribbean Sea and north tropical Atlantic sea surface temperature anomalies (SSTAs) is enhanced when one uses global tropical SSTAs as predictors compared with using only tropical Atlantic predictors. This predictability advantage does not carry over into the equatorial and south tropical Atlantic; indeed, persistence is a competitive predictor in those regions.

To help resolve the issue of whether or not the dipole structure found by applying empirical orthogonal function analysis to tropical Atlantic SSTs is an artifact of the technique or a physically real structure, the authors combine empirically derived normal modes and their adjoints to form “influence functions,” maps highlighting the geographical areas to which the north tropical Atlantic and the south tropical Atlantic SSTs are most sensitive at specified lead times. When the analysis is confined to the Atlantic basin, the 6-month influence functions in the north and south tropical Atlantic tend to be of the opposite sign and evolve into clear dipoles within 6 months. When the analysis is performed on global tropical SSTs, the 6-month influence functions are connected to the El Niño phenomenon in the Pacific, with the strongest signal in the north tropical Atlantic. That is, while the south tropical Atlantic region is weakly sensitive to the optimal initial structure for growth of El Niño, SST anomaly in the Niño3 region is a strong 6-month predictor of SST anomaly in the north tropical Atlantic. The results suggest that the tropical Atlantic dipole is a real phenomenon rather than an artifact of EOF analysis but that the influence of the Indo–Pacific often disrupts the northern branch so that the dipole does not dominate tropical Atlantic dynamics on seasonal timescales.

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Cécile Penland
and
Ludmila Matrosova

Abstract

A dynamically based filter is used to separate tropical sea surface temperatures (SSTs) into three components: the evolving El Niño signal, the global tropical trend, and the background. The components thus isolated are not independent. On the contrary, this procedure allows us to see the importance of the interdecadal signal to the predictability of El Niño.

The data filtered in this way reveal El Niño signals in the equatorial Indian Ocean and in the north tropical Atlantic Ocean that are remarkably similar. A signature of El Niño in the south tropical Atlantic leads Niño-3.4 SST anomalies by about 9 months. The time series of a global tropical trend is found to have a very smooth parabolic structure. In unfiltered data, this trend conspires with El Niño to obscure a meridional tropical Atlantic dipole, which is significant in the filtered background SST data.

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Cécile Penland
and
Ludmila Matrosova

Abstract

Stochastic forcing due to unresolved processes adds energy to a measurable system. Although this energy is added randomly in time, conservation laws still apply. A balance condition for stochastically driven systems is discussed. This “fluctuation-dissipation relation” may be used either to deduce the geographical properties of the stochastic forcing from data given a model for the evolution of the macroscopic variables or to diagnose energy conservation in a stochastic numerical model.

The balance condition in its first role was applied to sea surface temperatures (SSTs) in the Indo-Pacific basin. A low-dimensional empirical dynamical model of SSTs was generated in such a way that observed statistical properties of the field are preserved. Experiments varying the stochastic forcing in this model indicated how the geographical characteristics of the forcing affect the distribution of variance among the various normal modes thereby determining the dominant timescales of the SST field. These results suggest that the south Indian Ocean and the equatorial Pacific close to the date line are important to the amplitude and timing of the warm phase of El Niño-Southern Oscillation.

Fourier spectra obtained from output of the stochastically forced linear model were found to agree with those obtained from COADS data when time series of equal length were compared. A discussion of how spectra from a multivariate linear system can be confused with those of a nonlinear system is presented.

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Cécile Penland
and
Theresa Magorian

Abstract

Linear inverse modeling is used to predict sea surface temperatures (SSTS) in the Niño 3 region. Predictors in three geographical locations are used: the tropical Pacific Ocean, the tropical Pacific and Indian oceans, and the global tropical oceans. Predictions did not depend crucially on any of these three domains, and evidence was found to support the assumption that linear dynamics dominates most of the record. The prediction model performs better when SST anomalies are rapidly evolving than during warm events when large anomalies persist. The rms prediction error at a lead time of 9 months is about half a degree Celsius.

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James A. Hansen
and
Cecile Penland

Abstract

The delicate (and computationally expensive) nature of stochastic numerical modeling naturally leads one to look for efficient and/or convenient methods for integrating stochastic differential equations. Concomitantly, one may wish to sensibly add stochastic terms to an existing deterministic model without having to rewrite that model. In this note, two possibilities in the context of the fourth-order Runge–Kutta (RK4) integration scheme are examined. The first approach entails a hybrid of deterministic and stochastic integration schemes. In these examples, the hybrid RK4 generates time series with the correct climatological probability distributions. However, it is doubtful that the resulting time series are approximate solutions to the stochastic equations at every time step. The second approach uses the standard RK4 integration method modified by appropriately scaling stochastic terms. This is shown to be a special case of the general stochastic Runge–Kutta schemes considered by Ruemelin and has global convergence of order one. Thus, it gives excellent results for cases in which real noise with small but finite correlation time is approximated as white. This restriction on the type of problems to which the stochastic RK4 can be applied is strongly compensated by its computational efficiency.

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