Kwang-Y. Kim

EOFs of Harmonizable Cyclostationary Processes

While approximate cyclostationary processes are commonly found in climatic and geophysical studies, one great disincentive for using cyclostationary empirical orthogonal functions is their computational burden. This is especially so for the three-dimensional, space‐time case. This paper discusses a simple method of computing approximate cyclostationary empirical orthogonal functions based on the theory of harmonizable cyclostationary processes. The new method is computationally much more efficient than that of Kim et al. In the new method, cyclostationary empirical orthogonal functions are easier to understand. Namely, they are naturally defined as the products of Bloch functions (inner modes) and Fourier functions (outer modes), which otherwise are the result of the factorization theorem. Bloch functions are simply the principal components (PC) of the multivariate coefficient time series, which are generally correlated. They represent the normal modes of the nested fluctuations of harmonizable cyclostationary processes. Under the assumption of independent PC time series, Bloch functions are computed independently of the outer modes, which results in a tremendous speedup in computation.

EOFs of One-Dimensional Cyclostationary Time Series: Computations, Examples, and Stochastic Modeling

Abstract Many climatic time series seem to be a mixture of unpredictable fluctuations and changes that occur at a known frequency, as in the case of the annual cycle. Such a time series is called a cyclostationary process. The lagged covariance statistics of a cyclostationary process are periodic in time with the frequency of the nested undulations, and the eigenfunctions are no longer Fourier functions. In this study, examination is made of the properties of cyclostationary empirical orthogonal functions (CSEOFs) and a computational algorithm is developed based on Blochs theorem for the one-dimensional case. Simple examples are discussed to test the algorithm and clarify the nature and interpretation of CSEOFs. Finally, a stochastic model has been constructed, which reasonably reproduces the cyclostationary statistics of a 100-yr series of the globally averaged, observed surface air temperature field. The simulated CSEOFs and the associated eigenvalues compare fairly with those of the observational data.

A Comparison Study of EOF Techniques: Analysis of Nonstationary Data with Periodic Statistics

Abstract Identification of independent physical/dynamical modes and corresponding principal component time series is an important aspect of climate studies for they serve as a tool for detecting and predicting climate changes. While there are a number of different eigen techniques their performance for identifying independent modes varies. Considered here are comparison tests of eight eigen techniques in identifying independent patterns from a dataset. A particular emphasis is given to cyclostationary processes such as deforming and moving patterns with cyclic statistics. Such processes are fairly common in climatology and geophysics. Two eigen techniques that are based on the cyclostationarity assumption—cyclostationary empirical orthogonal functions (EOFs) and periodically extended EOFs—perform better in identifying moving and deforming patterns than techniques based on the stationarity assumption. Application to a tropical Pacific surface temperature field indicates that the first dominant pattern and ...

Detection of Forced Climate Signals. Part 1: Filter Theory

Abstract This paper considers the construction of a linear smoothing filter for estimation of the forced part of a change in a climatological field such as the surface temperature. The filter is optimal in the sense that it suppresses the natural variability or “noise” relative to the forced part or “signal” to the maximum extent possible. The technique is adapted from standard signal processing theory. The present treatment takes into account the spatial as well as the temporal variability of both the signal and the noise. In this paper we take the signals waveform in space-time to be a given deterministic field in space and lime. Formulation of the expression for the minimum mean-squared error for the problem together with a no-bias constraint leads to an integral equation whose solution is the filter. The problem can be solved analytically in terms of the space-time empirical orthogonal function basis set and its eigenvalue spectrum for the natural fluctuations and the projection amplitudes of the sig...

Spectral Approach to Optimal Estimation of the Global Average Temperature

Abstract Making use of EOF analysis and statistical optimal averaging techniques, the problem of random sampling error in estimating the global average temperature by a network of surface stations has been investigated. The EOF representation makes it unnecessary to use simplified empirical models of the correlation structure of temperature anomalies. If an adjustable weight is assigned to each station according to the criterion of minimum mean-square error, a formula for this error can be derived that consists of a sum of contributions from successive EOF modes. The EOFs were calculated from both observed data and a noise-forced EBM for the problem of one-year and five-year averages. The mean square statistical sampling error depends on the spatial distribution of the stations, length of the averaging interval, and the choice of the weight for each station data stream. Examples used here include four symmetric configurations of 4 × 4, 6 × 4, 9 × 7, and 20 × 10 stations and the Angell-Korshover configurat...

Detection of Forced Climate Signals. Part II: Simulation Results

Abstract This paper considers some tests of the procedures suggested in Part I on the detection of forced climate signals embedded in natural variability. The optimal filters are constructed from simulations of signals and natural variability in a noise-forced energy balance model that explicitly resolves land-sea geography and that has an upwelling-diffusion deep ocean. Filters are considered for the climate forcing of faint sunspot signals and for the greenhouse warming problem. In each case, the results are promising in that signal-to-noise ratios of unity or greater might be achievable. Rather than conclusive arguments, them exercises are meant to bring out key aspects of the detection problem that deserve the most attention and which parts of the procedure are most sensitive to assumptions.

EOF-Based Linear Prediction Algorithm: Examples

Abstract Considered here are examples of statistical prediction based on the algorithm developed by Kim and North. The predictor is constructed in terms of space–time EOFs of data and prediction domains. These EOFs are essentially a different representation of the covariance matrix, which is derived from past observational data. The two sets of EOFs contain information on how to extend the data domain into prediction domain (i.e., statistical prediction) with minimum error variance. The performance of the predictor is similar to that of an optimal autoregressive model since both methods are based on the minimization of prediction error variance. Four different prediction techniques—canonical correlation analysis (CCA), maximum covariance analysis (MCA), principal component regression (PCR), and principal oscillation pattern (POP)—have been compared with the present method. A comparison shows that oscillation patterns in a dataset can faithfully be extended in terms of temporal EOFs, resulting in a slightl...

Modeling the climate effect of unrestricted greenhouse emissions over the next 10,000 years

Although emission controls for greenhouse gases may limit their future rise, there is a finite chance that a substantial fraction of the available fossil fuel reservoir will be utilized. Geochemical models suggest that the atmospheric perturbation would then last for thousands of years. The authors use a well-calibrated energy balance model, coupled to an upwelling-diffusion deep-ocean model, to estimate the temperature effects of such unrestricted emissions. They calculate that greenhouse warming will peak between {approximately} 2200-2400, with a global temperature increase 4-13{degrees}C greater than present. The high-end estimate is as much as 100% larger than estimates of the temperature increase for the ice-free Cretaceous (100 Ma) warm period. Warming of as much as 2-5{degrees}C persists as late as 10,000 years AP. The {open_quotes}out-year{close_quotes} projections are larger than Milankovitch effects occurring over the same interval, indicating that unrestricted emissions could lead to conditions in which anthropogenic warming dominates climate variation for the next 10,000 years. 23 refs., 5 figs.

EOF-Based Linear Prediction Algorithm: Theory

This study considers the theory of a general three-dimensional (space and time) statistical prediction/extrapolation algorithm. The predictor is in the form of a linear data filter. The prediction kernel is based on the minimization of prediction error and its construction requires the covariance statistics of a predictand field. The algorithm is formulated in terms of the spatiotemporal EOFs of the predictand field. This EOF representation facilitates the selection of useful physical modes for prediction. Limited tests have been conducted concerning the sensitivity of the prediction algorithm with respect to its construction parameters and the record length of available data for constructing a covariance matrix. Tests reveal that the performance of the predictor is fairly insensitive to a wide range of the construction parameters. The accuracy of the filter, however, depends strongly on the accuracy of the covariance matrix, which critically depends on the length of available data. This inaccuracy implies suboptimal performance of the prediction filter. Simple examples demonstrate the utility of the new algorithm.

Optimal estimation of spherical harmonic components from a sample with spatially nonuniform covariance statistics

Abstract An optimal estimation technique is presented to estimate spherical harmonic coefficients. This technique is based on the minimization of the mean square error. This optimal estimation technique consists of computing optimal weights for a given network of sampling points. Empirical orthogonal functions (E0Fs) are an essential ingredient in formulating the estimation technique of the field of which the second-moment statistics are non-uniform over the sphere. The EOFs are computed using the United Kingdom dataset of global gridded temperatures based on station data. The utility of the technique is further demonstrated by computing a set of spherical harmonic coefficients from the 100-yr long surface temperature fluctuations of the United Kingdom dataset. Next, the validity of the mean-square error formulas is tested by actually calculating an ensemble average of mean-square estimation error. Finally, the technique is extended to estimate the amplitudes of the EOFS.