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- Author or Editor: Gerald R. North x

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## 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.

## 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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## 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.

## 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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## Abstract

Optimal space–time signal processing is used to infer the amplitude of the large-scale, near-surface temperature response to the, “11 year” solar cycle. The estimation procedure involves the following stops. 1) By correlating 14 years of monthly total solar irradiance measurements made by the *Nimbus-7* satellite and monthly Wolf sunspot numbers, a monthly solar irradiance forcing function is constructed for the years 1894–1993. 2) Using this forcing function, a space-time waveform of the climate response for the same 100 years is generated from an energy balance climate model. 3) The space-time covariance statistics in the frequency band (16.67 yr)^{−1}–(7.14 yr)^{−1} are calculated using control runs from two different coupled ocean-atmosphere global climate models. 4) Using the results from the last two stops, an optimal filter is constructed and applied to observed surface temperature data for the years 1894–1993. 5) An estimate of the ratio of the real climate response, contained in the observed data, and the model generated climate response from step 2 is given, as well as an estimate of its uncertainty. A number of consistency checks are presented, such as using data from different regions of the earth to calculate this ratio and using data lagged up to ±5 yr. Our findings allow us to reject the null hypothesis. that no response to the solar cycle is present in the data, at a confidence level of 97.4%.

## Abstract

Optimal space–time signal processing is used to infer the amplitude of the large-scale, near-surface temperature response to the, “11 year” solar cycle. The estimation procedure involves the following stops. 1) By correlating 14 years of monthly total solar irradiance measurements made by the *Nimbus-7* satellite and monthly Wolf sunspot numbers, a monthly solar irradiance forcing function is constructed for the years 1894–1993. 2) Using this forcing function, a space-time waveform of the climate response for the same 100 years is generated from an energy balance climate model. 3) The space-time covariance statistics in the frequency band (16.67 yr)^{−1}–(7.14 yr)^{−1} are calculated using control runs from two different coupled ocean-atmosphere global climate models. 4) Using the results from the last two stops, an optimal filter is constructed and applied to observed surface temperature data for the years 1894–1993. 5) An estimate of the ratio of the real climate response, contained in the observed data, and the model generated climate response from step 2 is given, as well as an estimate of its uncertainty. A number of consistency checks are presented, such as using data from different regions of the earth to calculate this ratio and using data lagged up to ±5 yr. Our findings allow us to reject the null hypothesis. that no response to the solar cycle is present in the data, at a confidence level of 97.4%.

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The perception of the hypothesized greenhouse effect will differ dramatically depending upon the location on the earth at which the effect is analyzed. This is due mainly to two causes: 1) the warming signal depends upon the position on the earth, and 2) the natural variability of the warming has a strong position dependence. To demonstrate these phenomena, simulations were conducted of the surface temperature field with a simple stochastic climate model that has enough geographical resolution to see the geographic dependence. The model was tuned to reproduce the geographical distribution of the present climate, including its natural variability in both the variance and the space–time correlation structure. While such effects have been discussed elsewhere with even more realistic climate models, it is instructive to actually see simulations of time series laid side by side in order to easily compare their differences and similarities. Because of the model's simplicity, the causes of the variations are easy to analyze. Not surprisingly, some realizations of the temperature for some local areas show countertrends for a period of several decades in the presence of the greenhouse warming.

The perception of the hypothesized greenhouse effect will differ dramatically depending upon the location on the earth at which the effect is analyzed. This is due mainly to two causes: 1) the warming signal depends upon the position on the earth, and 2) the natural variability of the warming has a strong position dependence. To demonstrate these phenomena, simulations were conducted of the surface temperature field with a simple stochastic climate model that has enough geographical resolution to see the geographic dependence. The model was tuned to reproduce the geographical distribution of the present climate, including its natural variability in both the variance and the space–time correlation structure. While such effects have been discussed elsewhere with even more realistic climate models, it is instructive to actually see simulations of time series laid side by side in order to easily compare their differences and similarities. Because of the model's simplicity, the causes of the variations are easy to analyze. Not surprisingly, some realizations of the temperature for some local areas show countertrends for a period of several decades in the presence of the greenhouse warming.

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## 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.

## 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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## 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.

## 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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## 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.

## 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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## 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 slightly better performance of the present method than that of the predictors based on the maximum pattern correlations (CCA, MCA, and PCR) or the POP predictor. One-dimensional applications demonstrate the usefulness of the predictor. The NINO3 and the NINO3.4 sea surface temperature time series (3-month moving average) were forecasted reasonably up to the lead time of about 6 months. The prediction skill seems to be comparable to other more elaborate statistical methods. Two-dimensional prediction examples also demonstrate the utility of the new algorithm. The spatial patterns of SST anomaly field (3-month moving average) were forecasted reasonably up to about 6 months ahead. All these examples illustrate that the prediction algorithm is useful and computationally efficient for routine prediction practices.

## 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 slightly better performance of the present method than that of the predictors based on the maximum pattern correlations (CCA, MCA, and PCR) or the POP predictor. One-dimensional applications demonstrate the usefulness of the predictor. The NINO3 and the NINO3.4 sea surface temperature time series (3-month moving average) were forecasted reasonably up to the lead time of about 6 months. The prediction skill seems to be comparable to other more elaborate statistical methods. Two-dimensional prediction examples also demonstrate the utility of the new algorithm. The spatial patterns of SST anomaly field (3-month moving average) were forecasted reasonably up to about 6 months ahead. All these examples illustrate that the prediction algorithm is useful and computationally efficient for routine prediction practices.

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## Abstract

A simple Budyko-Sellers mean annual energy balance climate model with diffusive transport (North, 1975b) is extended to include a seasonal cycle. In the model the latitudinal distribution of the zonal average surface temperature is represented by a series of Legendre polynomials, while its time-dependence is represented by a Fourier sine-cosine series. The model has three parameters which are adjusted so that the observed amplitudes of the Northern Hemisphere's zonal mean surface temperature are recovered. In order to obtain the correct amplitude and phase of the surface temperature's seasonal oscillation, allowance must be made for the disparity between the thermal inertia of the atmosphere over continents and that of the ocean's mixed layer. Although the model parameters are adjusted to recover the surface temperature fields of the Northern Hemisphere, a test of the model's ability to produce the fields of the Southern Hemisphere indicates that the model responds properly to changes in boundary conditions.

The seasonal model is used to reveal how the annual mean climate and its sensitivity to changes in incident radiation differ from the predictions obtained with the corresponding mean annual model. Although the zonal temperatures obtained with the seasonal model are 1–3°C higher than those obtained with the mean annual model, the changes in the global average annual mean surface temperatures calculated with the two models are practically identical for a 1% decrease in solar constant. Furthermore, because the albedo changes in them are linked mainly to changes in surface temperature, both models respond in the same manner to changes in the incident solar radiation caused by changes in the earth's orbit. The distribution of the incident solar radiation in the models is shown to be insensitive to changes in the eccentricity and the longitude of perihelion and sensitive only to changes in the obliquity of the earth. For past orbital changes, both the seasonal and the mean annual model fail to produce glacial advances of the magnitude that are thought to have occurred.

## Abstract

A simple Budyko-Sellers mean annual energy balance climate model with diffusive transport (North, 1975b) is extended to include a seasonal cycle. In the model the latitudinal distribution of the zonal average surface temperature is represented by a series of Legendre polynomials, while its time-dependence is represented by a Fourier sine-cosine series. The model has three parameters which are adjusted so that the observed amplitudes of the Northern Hemisphere's zonal mean surface temperature are recovered. In order to obtain the correct amplitude and phase of the surface temperature's seasonal oscillation, allowance must be made for the disparity between the thermal inertia of the atmosphere over continents and that of the ocean's mixed layer. Although the model parameters are adjusted to recover the surface temperature fields of the Northern Hemisphere, a test of the model's ability to produce the fields of the Southern Hemisphere indicates that the model responds properly to changes in boundary conditions.

The seasonal model is used to reveal how the annual mean climate and its sensitivity to changes in incident radiation differ from the predictions obtained with the corresponding mean annual model. Although the zonal temperatures obtained with the seasonal model are 1–3°C higher than those obtained with the mean annual model, the changes in the global average annual mean surface temperatures calculated with the two models are practically identical for a 1% decrease in solar constant. Furthermore, because the albedo changes in them are linked mainly to changes in surface temperature, both models respond in the same manner to changes in the incident solar radiation caused by changes in the earth's orbit. The distribution of the incident solar radiation in the models is shown to be insensitive to changes in the eccentricity and the longitude of perihelion and sensitive only to changes in the obliquity of the earth. For past orbital changes, both the seasonal and the mean annual model fail to produce glacial advances of the magnitude that are thought to have occurred.

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## Abstract

Tropical wave phenomena have been examined in the last 520 days of two 15-year runs of a low-resolution general circulation model (CCMO). The model boundary conditions were simplified to all-land, perpetual equinox, and no topography. The two runs were for fixed soil moisture at 75% and 0% , the so-called “wet” and “dry” models. Both models develop well-defined ITCZs with low-level convergence erratically concentrated along the equator. Highly organized eastward-propagating waves are detectable in both models with different wave speeds depending on the presence of moisture. The wave amplitudes (in, e.g., vertical velocity) are many orders of magnitude stronger in the wet model. The waves have a definite transverse nature as precipitation (low-level convergence) patches tend to move systematically north and south across the equator. In the wet model the waves are distinctly nondispersive and the transit time for passage around the earth is about 50 days, consistent with the Madden–Julian frequency. The authors are also able to see most of the expected linear wave modes in spectral density plots in the frequency–wavenumber plant and compare them for the wet and dry cases.

## Abstract

Tropical wave phenomena have been examined in the last 520 days of two 15-year runs of a low-resolution general circulation model (CCMO). The model boundary conditions were simplified to all-land, perpetual equinox, and no topography. The two runs were for fixed soil moisture at 75% and 0% , the so-called “wet” and “dry” models. Both models develop well-defined ITCZs with low-level convergence erratically concentrated along the equator. Highly organized eastward-propagating waves are detectable in both models with different wave speeds depending on the presence of moisture. The wave amplitudes (in, e.g., vertical velocity) are many orders of magnitude stronger in the wet model. The waves have a definite transverse nature as precipitation (low-level convergence) patches tend to move systematically north and south across the equator. In the wet model the waves are distinctly nondispersive and the transit time for passage around the earth is about 50 days, consistent with the Madden–Julian frequency. The authors are also able to see most of the expected linear wave modes in spectral density plots in the frequency–wavenumber plant and compare them for the wet and dry cases.