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## Abstract

It is natural that a book chapter honoring Joanne Simpson draw the connection between the two most important tropical meteorological observing programs in the history of meteorology: the Global Atmospheric Research Program Atlantic Tropical Experiment (GATE) and the Tropical Rainfal Measuring Mission (TRMM). Both programs were dominated by the influences of Joanne Simpson. When TRMM data are all in, these two grand experiments will have given us more information about the behavior of tropical convection and precipitation over the tropical oceans than all other tropical field campaigns combined. But some may not know how GATE data played a key role in demonstrating the feasibility of a mission like TRMM. This chapter will present a review of a number of studies that connect GATE precipitation data with TRMM, especially in the early planning stages.

## Abstract

It is natural that a book chapter honoring Joanne Simpson draw the connection between the two most important tropical meteorological observing programs in the history of meteorology: the Global Atmospheric Research Program Atlantic Tropical Experiment (GATE) and the Tropical Rainfal Measuring Mission (TRMM). Both programs were dominated by the influences of Joanne Simpson. When TRMM data are all in, these two grand experiments will have given us more information about the behavior of tropical convection and precipitation over the tropical oceans than all other tropical field campaigns combined. But some may not know how GATE data played a key role in demonstrating the feasibility of a mission like TRMM. This chapter will present a review of a number of studies that connect GATE precipitation data with TRMM, especially in the early planning stages.

## Abstract

Simple climate models employing diffusive heat transport and ice cap albedo feedback have equilibrium solutions with no stable ice cap smaller than a certain finite size. For the usual parameters used in these models the minimum cap has a radius of about 20 degrees on a great circle. Although it is traditional to remove this peculiar feature by various *ad hoc* mechanisms, it is of interest because of its relevance to ice age theories. This paper explains why the phenomenon occurs in these models by solving them in a physically appealing way. If an ice-free solution has a thermal minimum and if the minimum temperature is just above the critical value for formation of ice, then the artificial addition of a patch of ice leads to a widespread depression of the temperature below the critical freezing temperature; therefore, a second stable solution will exist whose spatial extent is determined by the range of the influence function of a point sink of heat, due to the albedo shift in the patch. The range of influence is determined by the characteristic length in the problem which in turn is determined by the distance a heat anomaly can be displaced by random walk during the characteristic time scale for radiative relaxation; this length is typically 20–30 degrees on a great circle. Mathematical detail is provided as well as a discussion of why the various mechanisms previously introduced to eliminate the phenomenon work. Finally, a discussion of the relevance of these results to nature is presented.

## Abstract

Simple climate models employing diffusive heat transport and ice cap albedo feedback have equilibrium solutions with no stable ice cap smaller than a certain finite size. For the usual parameters used in these models the minimum cap has a radius of about 20 degrees on a great circle. Although it is traditional to remove this peculiar feature by various *ad hoc* mechanisms, it is of interest because of its relevance to ice age theories. This paper explains why the phenomenon occurs in these models by solving them in a physically appealing way. If an ice-free solution has a thermal minimum and if the minimum temperature is just above the critical value for formation of ice, then the artificial addition of a patch of ice leads to a widespread depression of the temperature below the critical freezing temperature; therefore, a second stable solution will exist whose spatial extent is determined by the range of the influence function of a point sink of heat, due to the albedo shift in the patch. The range of influence is determined by the characteristic length in the problem which in turn is determined by the distance a heat anomaly can be displaced by random walk during the characteristic time scale for radiative relaxation; this length is typically 20–30 degrees on a great circle. Mathematical detail is provided as well as a discussion of why the various mechanisms previously introduced to eliminate the phenomenon work. Finally, a discussion of the relevance of these results to nature is presented.

## Abstract

A class of mean annual, zonally averaged energy-balance climate models of the Budyko-Sellers type are studied by a spectral (expansion in Legendre polynomials) method. Models with constant thermal diffusion coefficient can be solved exactly, The solution is approached by a rapidly converging sequence with each succeeding approximant taking into account information from ever smaller space and time scales. The first two modes represent a good approximation to the exact solution as well as to the present climate. The two-mode approximation to a number of more general models are shown to be either formally or approximately equivalent to the same truncation in the constant diffusion case. In particular, the transport parameterization used by Budyko is precisely equivalent to the two-mode truncation of thermal diffusion. Details of the dynamics do not influence the first two modes which fortunately seem adequate for the study of global climate change. Estimated ice age temperatures and ice line latitude agree well with the model if the solar constant is reduced by 1.3%.

## Abstract

A class of mean annual, zonally averaged energy-balance climate models of the Budyko-Sellers type are studied by a spectral (expansion in Legendre polynomials) method. Models with constant thermal diffusion coefficient can be solved exactly, The solution is approached by a rapidly converging sequence with each succeeding approximant taking into account information from ever smaller space and time scales. The first two modes represent a good approximation to the exact solution as well as to the present climate. The two-mode approximation to a number of more general models are shown to be either formally or approximately equivalent to the same truncation in the constant diffusion case. In particular, the transport parameterization used by Budyko is precisely equivalent to the two-mode truncation of thermal diffusion. Details of the dynamics do not influence the first two modes which fortunately seem adequate for the study of global climate change. Estimated ice age temperatures and ice line latitude agree well with the model if the solar constant is reduced by 1.3%.

## Abstract

A simple radiative balance climate model is presented which includes the ice feedback mechanism, zonal averaging, constant homogeneous cloudiness, and ordinary diffusive thermal heat transfer. The simplest version of the model with only one free parameter is solved explicitly in terms of hypergeometric functions and is used to study ice sheet latitude as a function of solar constant. A multiple branch structure of this function is found and discussed along with comparison to earlier results. A stability analysis about the equilibrium solutions shows that the present climate as well as an ice-covered earth are stable while an intermediate solution is unstable for small perturbations away from equilibrium.

## Abstract

A simple radiative balance climate model is presented which includes the ice feedback mechanism, zonal averaging, constant homogeneous cloudiness, and ordinary diffusive thermal heat transfer. The simplest version of the model with only one free parameter is solved explicitly in terms of hypergeometric functions and is used to study ice sheet latitude as a function of solar constant. A multiple branch structure of this function is found and discussed along with comparison to earlier results. A stability analysis about the equilibrium solutions shows that the present climate as well as an ice-covered earth are stable while an intermediate solution is unstable for small perturbations away from equilibrium.

## Abstract

An attempt to provide physical insight into the empirical orthogonal function (EOF) representation of data fields by the study of fields generated by linear stochastic models is presented in this paper. In a large class of these models, the EOFs at individual Fourier frequencies coincide with the orthogonal mechanical modes of the system-provided they exist. The precise mathematical criteria for this coincidence are derived and a physical interpretation is provided. A scheme possibly useful in forecasting is formally constructed for representing any stochastic field by a linear Hermitian model forced by noise.

## Abstract

An attempt to provide physical insight into the empirical orthogonal function (EOF) representation of data fields by the study of fields generated by linear stochastic models is presented in this paper. In a large class of these models, the EOFs at individual Fourier frequencies coincide with the orthogonal mechanical modes of the system-provided they exist. The precise mathematical criteria for this coincidence are derived and a physical interpretation is provided. A scheme possibly useful in forecasting is formally constructed for representing any stochastic field by a linear Hermitian model forced by noise.

## Abstract

Space-time averages of rain rates are needed in several applications. Nevertheless, they are difficult to estimate because the methods invariably leave gaps in the measurements in space or time. A formalism is developed which makes use of the frequency-wavenumber spectrum of the rain field. The mean square error of the estimate is expressed as an integral over frequency and two-dimensional wavenumber of an integrand consisting of two factors, a design-dependent-filter multiplied by the space-time spectrum of the rain rate field. Such a formalism helps to separate the design issues from the peculiarities of rain rate random fields. Two cases are worked out in detail: a low orbiting satellite which takes cell-wide snapshots at discrete intervals and a network of raingages which are gappy in space but continuous in time.

## Abstract

Space-time averages of rain rates are needed in several applications. Nevertheless, they are difficult to estimate because the methods invariably leave gaps in the measurements in space or time. A formalism is developed which makes use of the frequency-wavenumber spectrum of the rain field. The mean square error of the estimate is expressed as an integral over frequency and two-dimensional wavenumber of an integrand consisting of two factors, a design-dependent-filter multiplied by the space-time spectrum of the rain rate field. Such a formalism helps to separate the design issues from the peculiarities of rain rate random fields. Two cases are worked out in detail: a low orbiting satellite which takes cell-wide snapshots at discrete intervals and a network of raingages which are gappy in space but continuous in time.

## Abstract

Estimates of the amplitudes of the forced responses of the surface temperature field over the last century are provided by a signal processing scheme utilizing space–time empirical orthogonal functions for several combinations of station sites and record intervals taken from the last century. These century-long signal fingerprints come mainly from energy balance model calculations, which are shown to be very close to smoothed ensemble average runs from a coupled ocean–atmosphere model (Hadley Centre Model). The space–time lagged covariance matrices of natural variability come from 100-yr control runs from several well-known coupled ocean–atmosphere models as well as a 10 000-yr run from the stochastic energy balance climate model (EBCM). Evidence is found for robust, but weaker than expected signals from the greenhouse [amplitude ∼65% of that expected for a rather insensitive model (EBCM: *T*_{2×CO2}

## Abstract

Estimates of the amplitudes of the forced responses of the surface temperature field over the last century are provided by a signal processing scheme utilizing space–time empirical orthogonal functions for several combinations of station sites and record intervals taken from the last century. These century-long signal fingerprints come mainly from energy balance model calculations, which are shown to be very close to smoothed ensemble average runs from a coupled ocean–atmosphere model (Hadley Centre Model). The space–time lagged covariance matrices of natural variability come from 100-yr control runs from several well-known coupled ocean–atmosphere models as well as a 10 000-yr run from the stochastic energy balance climate model (EBCM). Evidence is found for robust, but weaker than expected signals from the greenhouse [amplitude ∼65% of that expected for a rather insensitive model (EBCM: *T*_{2×CO2}

## Abstract

In this paper point gauges are used in an analysis of hypothetical ground validation experiments for satellite-based estimates of precipitation rates. The ground and satellite measurements are fundamentally different since the gauge can sample continuously in time but at a discrete point, while the satellite samples an area average (typically 20 km across) 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 has a large nonzero contribution 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, and design 3 uses the pairs only when the gauge has rain. The error distribution of each design is derived theoretically for a Bernoulli spatial random field with different horizontal resolutions. It is found that design 3 cannot be used as a ground-truth design due to its large design bias. The mean-square error is used as an index of accuracy in estimating the ground measurement by satellite measurement. It is shown that there is a relationship between the mean-square error of design 1 and design 2 for the Bernoulli random field. Using this technique, the authors derive the number of satellite overpasses necessary to detect a satellite retrieval bias, which is as large as 10% of the natural variability.

## Abstract

In this paper point gauges are used in an analysis of hypothetical ground validation experiments for satellite-based estimates of precipitation rates. The ground and satellite measurements are fundamentally different since the gauge can sample continuously in time but at a discrete point, while the satellite samples an area average (typically 20 km across) 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 has a large nonzero contribution 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, and design 3 uses the pairs only when the gauge has rain. The error distribution of each design is derived theoretically for a Bernoulli spatial random field with different horizontal resolutions. It is found that design 3 cannot be used as a ground-truth design due to its large design bias. The mean-square error is used as an index of accuracy in estimating the ground measurement by satellite measurement. It is shown that there is a relationship between the mean-square error of design 1 and design 2 for the Bernoulli random field. Using this technique, the authors derive the number of satellite overpasses necessary to detect a satellite retrieval bias, which is as large as 10% of the natural variability.

## 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.

## 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.

## Abstract

Low-frequency (<20 GHz) single-channel microwave retrievals of rain rate encounter the problem of beam-filling error. This error stems from the fact that the relationship between microwave brightness temperature and rain rate is nonlinear, coupled with the fact that the field of view is large or comparable to important sales of variability of the rain field. This means that one may not simply insert the area average of the brightness temperature into the formula for rain rate without incurring both bias and random error. The statistical heterogeneity of the rain-rate field in the footprint of the instrument is key to determining the nature of these errors. This paper makes use of a series of random rain-rate fields to study the size of the bias and random error associated with beam filling. A number of examples are analyzed in detail: the binomially distributed field, the gamma, the Gaussian, the mixed gamma, the lognormal. and the mixed lognormal (“mixed” here means there is a finite probability of no rain rate at a point of space-time). Of particular interest are the applicability of a simple error formula due to Chiu and collaborators and a formula that might hold in the large field of view limit. It is found that the simple formula holds for Gaussian rain-rate fields but begins to fail for highly skewed fields such as the mixed lognormal. While not conclusively demonstrated here, it is suggested that the notion of climatologically adjusting the retrievals to remove the beam-filling bias is a reasonable proposition.

## Abstract

Low-frequency (<20 GHz) single-channel microwave retrievals of rain rate encounter the problem of beam-filling error. This error stems from the fact that the relationship between microwave brightness temperature and rain rate is nonlinear, coupled with the fact that the field of view is large or comparable to important sales of variability of the rain field. This means that one may not simply insert the area average of the brightness temperature into the formula for rain rate without incurring both bias and random error. The statistical heterogeneity of the rain-rate field in the footprint of the instrument is key to determining the nature of these errors. This paper makes use of a series of random rain-rate fields to study the size of the bias and random error associated with beam filling. A number of examples are analyzed in detail: the binomially distributed field, the gamma, the Gaussian, the mixed gamma, the lognormal. and the mixed lognormal (“mixed” here means there is a finite probability of no rain rate at a point of space-time). Of particular interest are the applicability of a simple error formula due to Chiu and collaborators and a formula that might hold in the large field of view limit. It is found that the simple formula holds for Gaussian rain-rate fields but begins to fail for highly skewed fields such as the mixed lognormal. While not conclusively demonstrated here, it is suggested that the notion of climatologically adjusting the retrievals to remove the beam-filling bias is a reasonable proposition.