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- Author or Editor: Gerald R. North x

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## 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
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## 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
}

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## 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.

## 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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## Abstract

We Present a simple Budyko-Sellers type climate model which is forced by a heating term whose time dependence is white noise and whose space-separated autocorrelation is independent of position and orientation on the sphere (statistical homogeneity). Such models with diffusive transport are analytically soluble by expansion into spherical harmonies. The modes are dynamically and statistically independent. Each satisfies a simple Langevin equation having a scale-dependent characteristic time. Climate anomalies in these models have an interval of predictability which can be explicitly computed. The predictability interval is independent of the wavenumber spectrum of the forcing in this class of models. We present the predictability results for all scales and discuss the implications for more realistic models.

## Abstract

We Present a simple Budyko-Sellers type climate model which is forced by a heating term whose time dependence is white noise and whose space-separated autocorrelation is independent of position and orientation on the sphere (statistical homogeneity). Such models with diffusive transport are analytically soluble by expansion into spherical harmonies. The modes are dynamically and statistically independent. Each satisfies a simple Langevin equation having a scale-dependent characteristic time. Climate anomalies in these models have an interval of predictability which can be explicitly computed. The predictability interval is independent of the wavenumber spectrum of the forcing in this class of models. We present the predictability results for all scales and discuss the implications for more realistic models.

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## 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.

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## 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.

## 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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## 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.

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## Abstract

In this paper both a microwave attenuation measurement along a horizontal line and multiple point gauge measurements are analyzed as possible ground-truth designs to validate satellite precipitation retrieval algorithms at the held of view spatial level (typically about 20 km). The design consists of comparing a sequence of pairs of contemporaneous measurements taken from the ground and from space. The authors examine theoretically the variance of expected differences between the two systems. The line average measurement leads to a smaller mean-square error compared to the case of a single point gauge, since some of the small-scale variability of the rain field is smoothed away by the line integration. The multiple paint gauge measurements also give smaller mean-square error than that of a single point gauge. The centroid of the line and point gauge configurations are considered to be located randomly inside the field of view for different overpasses. A space-time spectral formalism is used with a noise-forced diffusive rain field to find the mean-square error. By considering instantaneous ground and satellite measurement pairs over about 50 visits when raining, we can reduce the expected error to approximately 10% of the standard deviation of climatological variability. This is considered to be a useful level of tolerance for identifying biases in the retrieval algorithms. It is found that the multiple point gauges (especially two gauges) are the economical ground-truth design compared to the microwave attenuation based on the mean-square error comparison. The major finding of this study is that a significant improvement over the point gauge is obtained by adding a single additional piece of information; adding more gauges or extending the line of attenuation is not an important improvement.

## Abstract

In this paper both a microwave attenuation measurement along a horizontal line and multiple point gauge measurements are analyzed as possible ground-truth designs to validate satellite precipitation retrieval algorithms at the held of view spatial level (typically about 20 km). The design consists of comparing a sequence of pairs of contemporaneous measurements taken from the ground and from space. The authors examine theoretically the variance of expected differences between the two systems. The line average measurement leads to a smaller mean-square error compared to the case of a single point gauge, since some of the small-scale variability of the rain field is smoothed away by the line integration. The multiple paint gauge measurements also give smaller mean-square error than that of a single point gauge. The centroid of the line and point gauge configurations are considered to be located randomly inside the field of view for different overpasses. A space-time spectral formalism is used with a noise-forced diffusive rain field to find the mean-square error. By considering instantaneous ground and satellite measurement pairs over about 50 visits when raining, we can reduce the expected error to approximately 10% of the standard deviation of climatological variability. This is considered to be a useful level of tolerance for identifying biases in the retrieval algorithms. It is found that the multiple point gauges (especially two gauges) are the economical ground-truth design compared to the microwave attenuation based on the mean-square error comparison. The major finding of this study is that a significant improvement over the point gauge is obtained by adding a single additional piece of information; adding more gauges or extending the line of attenuation is not an important improvement.

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## 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.

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## 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.

## 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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## 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.