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## Abstract

This paper presents the methodologies and results of the multivariate modeling and two-dimensional spectral and correlation analysis of PRE-STORM rainfall gauge data. Estimated parameters of the models for the specific spatial averages clearly indicate the eastward and southeastward wave propagation of rainfall fluctuations. A relationship between the coefficients of the diffusion equation and the parameters of the stochastic model of rainfall fluctuations is derived that leads directly to the exclusive use of rainfall data to estimate advection speed (about 12 m s^{−1}) as well as other coefficients of the diffusion equation of the corresponding fields.

The statistical methodology developed here can be used for confirmation of physical models by comparison of the corresponding second-moment statistics of the observed and simulated data, for generating multiple samples of any size, for solving the inverse problem of the hydrodynamic equations, and for application in some other areas of meteorological and climatological data analysis and modeling.

## Abstract

This paper presents the methodologies and results of the multivariate modeling and two-dimensional spectral and correlation analysis of PRE-STORM rainfall gauge data. Estimated parameters of the models for the specific spatial averages clearly indicate the eastward and southeastward wave propagation of rainfall fluctuations. A relationship between the coefficients of the diffusion equation and the parameters of the stochastic model of rainfall fluctuations is derived that leads directly to the exclusive use of rainfall data to estimate advection speed (about 12 m s^{−1}) as well as other coefficients of the diffusion equation of the corresponding fields.

The statistical methodology developed here can be used for confirmation of physical models by comparison of the corresponding second-moment statistics of the observed and simulated data, for generating multiple samples of any size, for solving the inverse problem of the hydrodynamic equations, and for application in some other areas of meteorological and climatological data analysis and modeling.

## Abstract

A class of simple climate models including those of the Budyko-Sellers type are formulated from a variational principle. A functional is constructed for the zonally averaged mean annual temperature field such that extrema of the functional occur when the climate satisfies the usual energy-balance equation. Local minima of the functional correspond to stable solutions while saddle points correspond to unstable solutions. The technique can be used to construct approximate solutions from trial functions and to carry out finite-amplitude stability analyses. A spectral example is given in explicit detail.

## Abstract

A class of simple climate models including those of the Budyko-Sellers type are formulated from a variational principle. A functional is constructed for the zonally averaged mean annual temperature field such that extrema of the functional occur when the climate satisfies the usual energy-balance equation. Local minima of the functional correspond to stable solutions while saddle points correspond to unstable solutions. The technique can be used to construct approximate solutions from trial functions and to carry out finite-amplitude stability analyses. A spectral example is given in explicit detail.

## Abstract

In this paper point gauge measurements are analyzed as part of a ground truth design to validate satellite retrieval algorithms at the field-of-view spatial level (typically about 20 km). Even in the ideal case the ground and satellite measurements are fundamentally different, since the gauge can sample continuously in time but at a discrete point, while a satellite samples an area average but a snapshot in time. The design consists of comparing a sequence of pairs of measurements taken from the ground and from space. Since real rain is patchy, that is, its probability distribution has large nonzero contributions at zero rain rate, the following ground truth designs are proposed. Design 1 uses all pairs. Design 2 uses the pairs only when the field-of-view satellite average has rain. Design 3 uses the pairs only when the gauge has rain. For the nonwhite noise random field having a mixed distribution, the authors evaluate each design theoretically by deriving the ensemble mean and the mean-square error of differences between the two systems. It is found that design 3 has serious disadvantage as a ground truth design due to its large design bias. It is also shown that there is a relationship between the mean-square error of design 1 and design 2. These results generalize those presented recently by Ha and North for the Bernoulli white noise random field. The strategy developed in this study is applied to a real rain rate field. For the Global Atmospheric Program (GARP) Atlantic Tropical Experiment (GATE) data, it is found that by combining 50 data pairs (containing rain) of the satellite to the ground site, the expected error can be reduced to about 10% of the standard deviation of the fluctuations of the system alone. For the less realistic case of a white noise random field, the number of data pairs is about 100. Hence, the use of more realistic fields means that only about half as many pairs are needed to detect a 10% bias.

## Abstract

In this paper point gauge measurements are analyzed as part of a ground truth design to validate satellite retrieval algorithms at the field-of-view spatial level (typically about 20 km). Even in the ideal case the ground and satellite measurements are fundamentally different, since the gauge can sample continuously in time but at a discrete point, while a satellite samples an area average but a snapshot in time. The design consists of comparing a sequence of pairs of measurements taken from the ground and from space. Since real rain is patchy, that is, its probability distribution has large nonzero contributions at zero rain rate, the following ground truth designs are proposed. Design 1 uses all pairs. Design 2 uses the pairs only when the field-of-view satellite average has rain. Design 3 uses the pairs only when the gauge has rain. For the nonwhite noise random field having a mixed distribution, the authors evaluate each design theoretically by deriving the ensemble mean and the mean-square error of differences between the two systems. It is found that design 3 has serious disadvantage as a ground truth design due to its large design bias. It is also shown that there is a relationship between the mean-square error of design 1 and design 2. These results generalize those presented recently by Ha and North for the Bernoulli white noise random field. The strategy developed in this study is applied to a real rain rate field. For the Global Atmospheric Program (GARP) Atlantic Tropical Experiment (GATE) data, it is found that by combining 50 data pairs (containing rain) of the satellite to the ground site, the expected error can be reduced to about 10% of the standard deviation of the fluctuations of the system alone. For the less realistic case of a white noise random field, the number of data pairs is about 100. Hence, the use of more realistic fields means that only about half as many pairs are needed to detect a 10% bias.

## Abstract

This paper presents an analysis of rainfall data based on the radar echoes collected in the vicinity of Darwin, Australia, during the special observation periods in 1988. The Darwin rainfall data are available in the form of hourly averaged grids of size 141 × 141 with an areal resolution of 2 km × 2 km. The data are available for approximately 19 days in the first subset and for 22 days in the second. Since the rainfall data were taken over both the land and the ocean, separate analyses were performed for land and ocean surfaces; thus, three univariate time series (for land, ocean, and combination) are presented for each set. Time series analysis was performed in both time and frequency domains, and both the correlogram and periodogram showed the presence of a strong diurnal cycle in all the time series. Considerable variations can be seen in the diurnal cycles of these time series. To analyze the effect of the diurnal cycle on the sampling errors, flush visits of idealized satellites were simulated. The root-mean-square (rms) errors were especially large for satellites with sampling intervals of 6 and 12 h (about 20% of the mean for the box size of 280 km × 280 km, for 20 days). The rms errors were very large (∼65%) for a sampling interval of 24 h, which is a possibility for the Defense Military Satellite Program satellites. The sampling errors were only 5%–10% for non-sun-synchronous orbiters. This result should be considered for satellite mission planning purposes.

## Abstract

This paper presents an analysis of rainfall data based on the radar echoes collected in the vicinity of Darwin, Australia, during the special observation periods in 1988. The Darwin rainfall data are available in the form of hourly averaged grids of size 141 × 141 with an areal resolution of 2 km × 2 km. The data are available for approximately 19 days in the first subset and for 22 days in the second. Since the rainfall data were taken over both the land and the ocean, separate analyses were performed for land and ocean surfaces; thus, three univariate time series (for land, ocean, and combination) are presented for each set. Time series analysis was performed in both time and frequency domains, and both the correlogram and periodogram showed the presence of a strong diurnal cycle in all the time series. Considerable variations can be seen in the diurnal cycles of these time series. To analyze the effect of the diurnal cycle on the sampling errors, flush visits of idealized satellites were simulated. The root-mean-square (rms) errors were especially large for satellites with sampling intervals of 6 and 12 h (about 20% of the mean for the box size of 280 km × 280 km, for 20 days). The rms errors were very large (∼65%) for a sampling interval of 24 h, which is a possibility for the Defense Military Satellite Program satellites. The sampling errors were only 5%–10% for non-sun-synchronous orbiters. This result should be considered for satellite mission planning purposes.

## 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 Bloch's 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.

## 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 Bloch's 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.

The Tropical Rainfall Measuring Mission (TRMM) satellite is planned for an operational duration of at least three years, beginning in the mid-1990's. The main scientific goals for it are to determine the distribution and variability of precipitation and latent-heat release on a monthly average over areas of about 10^{5} km^{2}, for use in improving short-term climate models, global circulation models and in understanding the hydrological cycle, particularly as it is affected by tropical oceanic rainfall and its variability.

The TRMM satellite's instrumentation will consist of the first quantitative spaceborne weather radar, a multichannel passive microwave radiometer and an AVHRR (Advanced Very High Resolution Radiometer). The satellite's orbit will be low altitude (about 320 km) for high resolution and low inclination (30° to 35°) in order to visit each sampling area in the tropics about twice daily at a different hour of the day. A strong validation effort is planned with several key ground sites to be instrumented with calibrated multiparameter rain radars.

Mission goals and science issues are summarized. Research progress on rain retrieval algorithms is described. Radar and passive microwave algorithms are discussed and the use of radiative models in conjunction with cloud dynamical-microphysical models is emphasized especially. Algorithms are being and will continue to be tested and improved using microwave instruments on high-altitude aircraft overflying precipitating convective systems, located in the range of well-calibrated radars.

The Tropical Rainfall Measuring Mission (TRMM) satellite is planned for an operational duration of at least three years, beginning in the mid-1990's. The main scientific goals for it are to determine the distribution and variability of precipitation and latent-heat release on a monthly average over areas of about 10^{5} km^{2}, for use in improving short-term climate models, global circulation models and in understanding the hydrological cycle, particularly as it is affected by tropical oceanic rainfall and its variability.

The TRMM satellite's instrumentation will consist of the first quantitative spaceborne weather radar, a multichannel passive microwave radiometer and an AVHRR (Advanced Very High Resolution Radiometer). The satellite's orbit will be low altitude (about 320 km) for high resolution and low inclination (30° to 35°) in order to visit each sampling area in the tropics about twice daily at a different hour of the day. A strong validation effort is planned with several key ground sites to be instrumented with calibrated multiparameter rain radars.

Mission goals and science issues are summarized. Research progress on rain retrieval algorithms is described. Radar and passive microwave algorithms are discussed and the use of radiative models in conjunction with cloud dynamical-microphysical models is emphasized especially. Algorithms are being and will continue to be tested and improved using microwave instruments on high-altitude aircraft overflying precipitating convective systems, located in the range of well-calibrated radars.

## 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.

## 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.

## 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 configuration. Comparisons with the 100-yr U.K. dataset show that correlations for the time series of the global temperature anomaly average between the full dataset and this study's sparse configurations are rather high. For example, the 63-station Angell-Korshover network with uniform weighting explains 92.7% of the total variance, whereas the same network with optimal weighting can lead to 97.8% explained total variance of the U.K. dataset.

## 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 configuration. Comparisons with the 100-yr U.K. dataset show that correlations for the time series of the global temperature anomaly average between the full dataset and this study's sparse configurations are rather high. For example, the 63-station Angell-Korshover network with uniform weighting explains 92.7% of the total variance, whereas the same network with optimal weighting can lead to 97.8% explained total variance of the U.K. dataset.

## Abstract

In this study the magnitude and the temporal and spatial correlation scales of background fluctuations generated by three climate models, two different coupled ocean-atmosphere general circulation models and one energy balance model, were examined. These second-moment statistics of the models were compared with each other and with those of the observation data in several frequency bands. This exercise shows some discordance between the models and the observations and also significant discrepancy among different numerical models. The authors also calculated the empirical orthogonal functions and eigenvalues because these am important ingredients for formulating estimation and detection algorithms. There are significant model to model variations both in the shape of eigenfunctions and in the spectrum of eigenvalues. Also, consistency between the modeled eigenfunctions and eigenvalues and those of the observations are rather poor, especially in the low-frequency bands.

## Abstract

In this study the magnitude and the temporal and spatial correlation scales of background fluctuations generated by three climate models, two different coupled ocean-atmosphere general circulation models and one energy balance model, were examined. These second-moment statistics of the models were compared with each other and with those of the observation data in several frequency bands. This exercise shows some discordance between the models and the observations and also significant discrepancy among different numerical models. The authors also calculated the empirical orthogonal functions and eigenvalues because these am important ingredients for formulating estimation and detection algorithms. There are significant model to model variations both in the shape of eigenfunctions and in the spectrum of eigenvalues. Also, consistency between the modeled eigenfunctions and eigenvalues and those of the observations are rather poor, especially in the low-frequency bands.

## Abstract

Empirical studies of total outgoing infrared radiation IR and surface temperature *T* have shown them to be well correlated for large time and space scales. An analysis of one year of Nimbus-6 data shows that the simple form IR = *A* + *BT* (with *A* = 204 W m^{−2}, *B* = 1.93 W m^{−2}K^{−1}) explains 90% of the area-weighted variance in the annual mean and annual cycle of the zonally averaged IR field. The geographical distribution of the annual cycle in IR shows a large amplitude over the continental interiors, as is found in the observed temperature field, and the ratio of the large amplitudes (*B _{local}
*) is approximately 2 W m

^{−2}K

^{−1}. This helps to explain our recent success in modeling the geographical distribution of the annual cycle in

*T*with a two-dimensional, time-dependent energy balance climate model (EBCM) which makes use of the

*A*+

*BT*rule. The parameterization works well in regions where the thermal inertia is small and the annual cycles of

*T*and IR are large and in phase. Those regions where

*B*differs markedly from 2 W m

_{local}^{−2}K

^{−1}are where the IR is strongly affected by the cloudiness of seasonal precipitation regimes. This effect is especially evident over the tropical oceans where the parameterization fails; but that is where the thermal inertia is large, the seasonal cycle in

*T*is small, and even large errors in the radiative cooling approximation will have little impact on seasonal cycle simulations by simple climate models.

## Abstract

Empirical studies of total outgoing infrared radiation IR and surface temperature *T* have shown them to be well correlated for large time and space scales. An analysis of one year of Nimbus-6 data shows that the simple form IR = *A* + *BT* (with *A* = 204 W m^{−2}, *B* = 1.93 W m^{−2}K^{−1}) explains 90% of the area-weighted variance in the annual mean and annual cycle of the zonally averaged IR field. The geographical distribution of the annual cycle in IR shows a large amplitude over the continental interiors, as is found in the observed temperature field, and the ratio of the large amplitudes (*B _{local}
*) is approximately 2 W m

^{−2}K

^{−1}. This helps to explain our recent success in modeling the geographical distribution of the annual cycle in

*T*with a two-dimensional, time-dependent energy balance climate model (EBCM) which makes use of the

*A*+

*BT*rule. The parameterization works well in regions where the thermal inertia is small and the annual cycles of

*T*and IR are large and in phase. Those regions where

*B*differs markedly from 2 W m

_{local}^{−2}K

^{−1}are where the IR is strongly affected by the cloudiness of seasonal precipitation regimes. This effect is especially evident over the tropical oceans where the parameterization fails; but that is where the thermal inertia is large, the seasonal cycle in

*T*is small, and even large errors in the radiative cooling approximation will have little impact on seasonal cycle simulations by simple climate models.