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Kyung-Sup Shin
,
Phil E. Riba
, and
Gerald R. North

Abstract

This paper presents a new simple retrieval algorithm for estimating area-time averaged rain rates over tropical oceans by using single channel microwave measurements from satellites. The algorithm was tested by using the Nimbus-5 Electrically Scanning Microwave Radiometer (ESMR-5) and a simple microwave radiative transfer model to retrieve seasonal 5° × 5° area averaged rainrate over the tropical Atlantic and Pacific from December 1973 to November 1974.

The brightness temperatures were collected and analyzed into histograms for each season and in each grid box from December 1973 to November 1974. The histograms suggest a normal distribution of background noise plus a skewed rain distribution at the higher brightness temperatures. By using a statistical estimation procedure based upon normally distributed background noise, the rain distribution was separated from the raw histogram. The radiative transfer model was applied to the rain-only distribution to retrieve area-time averaged rainrates throughout the tropics. An adjustment for the beam filling error was incorporated in the procedure.

Despite limitations of single channel information, the retrieved seasonal rain rates agree well in the open ocean with expectations based upon previous estimates of tropical rainfall over the oceans. We suggest that the beam filling correction factor is the most important, but least understood parameter in the retrieval process.

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Shoichiro Nakamoto
,
Juan B. Valdés
, and
Gerald R. North

Abstract

The oceanic rainfall frequency-wavenumber spectrum and its associated space-time correlation have been evaluated from subsets of GATE Phase 1 data. The records, of a duration of 4 days, were sampled at 15 minute intervals in 4 × 4 km grid boxes ova a 400 km diameter hexagon.

In the low frequencies-low wavenumber region the results coincide with those obtained by using the stochastic model proposed by North and Nakamoto. From the derived spectrum the inherent time and space scales of the stochastic model were determined to be approximately 13 hours and 36 km. The space-time correlation function evaluated from the function-wavenumber spectrum and that obtained directly from GATE Phase I records agreed.

The formalism proposed by North and Nakamoto was taken together with the derived spectrum to compute the mean square sampling error due to intermittent visits of a spaceborne sensor. The sampling error was estimated to be on the order of 10%, for monthly mean rainfall averaged over 500 × 500 km boxes which meets the scientific requirements of the TRMM mission. This result is consistent with those previously reported in the literature.

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Ilya Polyak
,
Gerald R. North
, and
Juan B. Valdes

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.

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Vishwas V. Soman
,
Juan B. Valdés
, and
Gerald R. North

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.

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Gerald R. North
,
Samuel S. P. Shen
, and
Robert Upson

Abstract

This paper examines the sampling characteristics of combining data collected by several low-orbiting satellites attempting to estimate the space–time average of rain rates. The several satellites can have different orbital and swath-width parameters. The satellite overpasses are allowed to make partial coverage snapshots of the grid box with each overpass. Such partial visits are considered in an approximate way, letting each intersection area fraction of the grid box by a particular satellite swath be a random variable with mean and variance parameters computed from exact orbit calculations. The derivation procedure is based upon the spectral minimum mean-square error formalism introduced by North and Nakamoto. By using a simple parametric form for the space–time spectral density, simple formulas are derived for a large number of examples, including the combination of the Tropical Rainfall Measuring Mission with an operational sun-synchronous orbiter. The approximations and results are discussed and directions for future research are summarized.

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Hye-Kyung Cho
,
Kenneth P. Bowman
, and
Gerald R. North

Abstract

This study investigates the spatial characteristics of nonzero rain rates to develop a probability density function (PDF) model of precipitation using rainfall data from the Tropical Rainfall Measuring Mission (TRMM) satellite. The minimum χ 2 method is used to find a good estimator for the rain-rate distribution between the gamma and lognormal distributions, which are popularly used in the simulation of the rain-rate PDF. Results are sensitive to the choice of dynamic range, but both the gamma and lognormal distributions match well with the PDF of rainfall data. Comparison with sample means shows that the parametric mean from the lognormal distribution overestimates the sample mean, whereas the gamma distribution underestimates it. These differences are caused by the inflated tail in the lognormal distribution and the small shape parameter in the gamma distribution. If shape constraint is given, the difference between the sample mean and the parametric mean from the fitted gamma distribution decreases significantly, although the resulting χ 2 values slightly increase. Of interest is that a consistent regional preference between two test functions is found. The gamma fits outperform the lognormal fits in wet regions, whereas the lognormal fits are better than the gamma fits for dry regions. Results can be improved with a specific model assumption depending on mean rain rates, but the results presented in this study can be easily applied to develop the rainfall retrieval algorithm and to find the proper statistics in the rainfall data.

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Kwang-Y. Kim
,
Gerald R. North
, and
Jianping Huang

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.

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Yue Li
,
Ping Yang
,
Gerald R. North
, and
Andrew Dessler

Abstract

The fixed anvil temperature (FAT) hypothesis is examined based on the Aqua Moderate Resolution Imaging Spectroradiometer (MODIS)-based cloud-top temperature (CTT) in conjunction with the tropical atmospheric profiles and sea surface temperature (SST) from the European Centre for Medium-Range Weather Forecasts (ECMWF) Interim Reanalysis. Consistent with the physical governing mechanism of the FAT hypothesis, the peak clear-sky diabatic subsidence and convergence profiles are located at roughly the same level (200 hPa) as the peak in the cloud profile, which is fundamentally determined by the rapid decrease of water vapor concentration above this level. The geographical maxima of cloud fraction agree well with those of water vapor, clear-sky cooling rates, and diabatic convergence at 200 hPa. The use of direct CTT measurements suggests the CTT in specific Pacific basins exhibit different characteristics as the frequency distribution of the tropical SST varies from boreal winter to summer. When averaging over the tropics as a whole, the CTT distributions are approximately unchanged primarily because of cancellation by the variations associated with individual regions. An analysis of the response of the tropical mean CTT anomaly time series to the SST indicates that a possible negative relationship is present, whereas the relationship tends to be positive over the tropical western Pacific and Indian Oceans. In addition, it is suggested to interpret the FAT hypothesis, and the more recent proportionately higher anvil temperature (PHAT) hypothesis, by using the temperature at the maximum cloud detrainment level instead of the CTT.

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Gerald R. North
,
Louis Howard
,
David Pollard
, and
Bruce Wielicki

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.

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Kwang-Y. Kim
,
Gerald R. North
, and
Samuel S. Shen

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.

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