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Jian-Ping Huang
and
Gerald R. North

Abstract

Due to the variety of periodic or quasi-periodic deterministic forcings (e.g., diurnal cycle, seasonal cycle, Milankovitch cycles, etc.), most climate fluctuations may be modeled as cyclostationary processes since their properties are modulated by these cycles. Difficulties in using conventional spectral analysis to explore the seasonal variation of climate fluctuations have indicated the need for some new statistical techniques. It is suggested here that the cyclic spectral analysis he used for interpreting such fluctuations. The technique is adapted from cyclostationarity theory in signal processing. To demonstrate the usefulness of this technique, a very simple cyclostationarity stochastic climate model is constructed. The results show that the seasonal cycle strongly modulates the amplitude of the covariance and spectrum. The seasonal variation of intraseasonal oscillations in the Tropics has also been studied on a zonally symmetric all-land planet in the absence of external forcing. The idealized planet has no ocean no topography. A 15-year length seasonal run of the atmosphere is analyzed with the NCAR Community Climate Model (CCM2, R15). Analysis of the simulation data indicates the presence of intraseaonal oscillations in the Tropics, which are also localized in the time of year.

Both examples suggest that these techniques might be useful for analysis of fluctuations that exhibit locality in both frequency and season.

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Kwang-Y. Kim
and
Gerald R. North

Abstract

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.

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Robert F. Cahalan
and
Gerald R. North

Abstract

This paper treats the stability of steady-state solutions of some simple, latitude-dependent, energy-balance climate models. For north-south symmetric solutions of models with an ice-cap-type albedo feed-back, and for the sum of horizontal transport and infrared radiation given by a linear operator, it is possible to prove a “slope-stability” theorem; i.e., if the local slope of the steady-state icelinc latitude versus solar constant curve is positive (negative) the steady-state solution is stable (unstable). Certain rather weak restrictions on the albedo function and on the heat transport are required for the proof, and their physical basis is discussed in the text.

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Kwang-Y. Kim
and
Gerald R. North

Abstract

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.

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Gerald R. North
and
Kwang-Y. Kim

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.

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Lai-Yung Leung
and
Gerald R. North

Abstract

Atmospheric variability an a zonally symmetric planet in the absence of external forcing anomalies is studied. With idealized boundary conditions such as the absence of ocean and topography, and by using perpetual equinox solar forcing, a 15-year long stationary time series of the atmosphere is simulated with the NCAR Community Climate Model (CCM0). This provides sufficient time samples for realistic study of the properties of the atmosphere. Zonally averaged and space-time statistics for the surface air temperature field on this planet are presented. Such statistics can serve as noise climatologies for climate sensitivity experiments, allowing the effects of changes of external forcing on the atmosphere to be asssessed.

In search of a simple statistical model for atmospheric variability, the space-time spectra obtained from the CCM simulation are fitted statistically with a stochastic energy balance model. The space-time spectra for three zonal wavenumbers are found to be fitted satisfactorily by the stochastic model with only five parameters (a heat diffusion coefficient, a constant zonal advection speed, a radiative damping constant and two parameters for blue spatial noise amplitudes). The estimated parameters agree with previously obtained values. This suggests that useful statistics for large-scale atmospheric variability may be obtained from simple statistical models. With the method of analysis provided in this study, the ability of the stochastic model for describing atmospheric variability on a more realistic planet (including geography and seasonal cycle) can be tested. This may involve comparing space-time statistics from the stochastic model with observed quantities and by using empirical orthogonal functions as a basis set for expansion.

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Kwang Y. Kim
and
Gerald R. North

Abstract

This study makes use of a simple stochastic energy balance climate model that resolves the land–sea distribution and that includes a crude upwelling-diffusion deep ocean to study the natural variability of the surface temperature in different frequency bands. This is done by computing the eigenfunctions of the space-time lagged covariance function. The resulting frequency-dependent theoretical orthogonal functions (fdTOFs) are compared with the corresponding frequency-dependent empirical orthogonal functions (fdEOFs) derived from 40 years of data. The computed and modeled eigenvalues are consistent with the difference mainly explained by sampling error due to the short observational record. The magnitude of expected sampling errors is demonstrated by a series of Monte Carlo simulations with the model. The sampling error for the eigenvalues features a strong bias that appears in the simulations and apparently in the data. Component-by-component pattern correlations between the fdEOFs and the fdTOFs vary from 0.81 to 0.28 for the first ten components. Monte Carlo simulations show that the sampling error could be an important source of error especially in the low (interannual) frequency band. However, sampling error alone cannot satisfactorily explain the difference between the model and observations. Rather, model inaccuracy and/or spatial bias of observations seem to be important sources of error. The fdTOFs are expected to be useful in estimation/prediction/detection studies.

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Lai-Yung Leung
and
Gerald R. North

Abstract

This paper introduces the use of information theory in characterizing climate predictability. Specifically, the concepts of entropy and transinformation are employed. Entropy measures the amount of uncertainty in our knowledge of the state of the climate system. Transinformation represents the information gained about an anomaly at any time t with knowledge of the size of the initial anomaly. It has many desirable properties that can be used as a measure of the predictability of the climate system. These concepts when applied to climate predictability are illustrated through a simple stochastic climate model (an energy balance model forced by noise). The transinformation is found to depict the degradation of information about an anomaly despite the fact that we have perfect knowledge of the initial state. Its usefulness, especially when generalized to other climate models, is discussed.

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Dong-Bin Shin
and
Gerald R. North

Abstract

Low earth-orbiting satellites such as the Tropical Rainfall Measuring Mission (TRMM) estimate month-long averages of precipitation (or other fields). A difficulty is that such a satellite sensor returns to the same spot on the planet at discrete intervals, about 11 or 12 h apart. This discrete sampling leads to a sampling error that is the one of the largest components of the error budget. Previous studies have examined this type of error for stationary random fields, but this paper examines the possibility that the field has a diurnally varying standard deviation, a property likely to occur in precipitation fields. This is a special case of the more general cyclostationary field.

In this paper the authors investigate the mean square error (mse) for the monthly averaging case derived from the satellites whose revisiting intervals are 12 h (sun synchronous) and off 12 h (11.75 h). In addition, the authors take the diurnal cycle of the standard deviation to be a constant plus a single sinusoid, either diurnal or semidiurnal.

The authors have derived an mse formula consisting of three parts: the errors from the stationary background, the cyclostationary part, and a cross-term between them. The separate parts of the mse allow the authors to assess the contribution of the cyclostationary error to the total mse.

The results indicate that the cyclostationary errors due to the diurnal variation appear small for both a 12-h and an off-12-h (11.75 h) revisiting satellite. In addition, the cyclostationary error amounts are similar to each other. For the semidiurnally varying field, the cyclostationary errors increase rapidly as the magnitude of the variance cycle increases for both the 12-h and off-12-h revisting satellites. However, the off-12-h sampling shows the cyclostationary error to be less than that of the exact 12-h sampling.

Furthermore, the authors have evaluated the cyclostationary error as a function of the phase of the satellite visit as it is shifted from the phase of the diurnal cycles (the sun-synchronous case or the start of the month for the off-12-h case). It is found that the cyclostationary error observed from the off-12-h satellite is much less sensitive to the phase shift than the cyclostationary error from the exact 12-h satellite.

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Gerald R. North
and
Ilya Polyak

Abstract

In this paper the authors consider the possibility of correlations between the random part of the so-called beam-filling error between neighboring fields of view in the microwave retrieval of rain rate over oceans. The study is based upon the GARP (Global Atmospheric Research Program) Atlantic Tropical Experiment (GATE) rain-rate dataset, and it is found that there is a correlation of between 0.35 and 0.50 between the errors in adjacent rainy fields of view. The net effect of this correlation is reducing the number of statistically independent terms accumulated in forming area and time averages of rain-rate estimates. In GATE-like rain areas, this reduction can be of the order of a factor of 3, making accumulated standard error percentages increase by a factor of the order of √3. For the Tropical Rainfall Measuring Mission using the microwave radiometer alone. this could increase the accumulated random part of the beam-filling error for month-long 5°×5° boxes from about 1.2% to 2%. The effect is larger for less rainy areas away from the equatorial zone.

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