Gate and Trmm

GATE and TRMM By Gerald R. North Texas A&M University Introduction 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: GATE and 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. The GARP (Global Atmospheric Research Program) had as one of its components the GARP Atlantic Tropical Experiment (GATE). The major aims were to spell out the details of convective energetics in tropical plumes. Houze and Betts (1981) provided an excellent review article on convection findings based upon GATE. GATE was a fully international program with many nations participating. Among the features of the 1 observing system was an array of stationary ships with precipitation radars aboard. These were positioned in a hexagonal configuration at roughly 8 deg N in the Tropical Atlantic. The ships and accompanying aircraft collected many kinds of data over several intensive observing periods lasting a few weeks each in the summer of 1974. The radars collected space-time snapshots of the precipitation fields at the surface and at higher elevations. The radar data were converted to rain rates and continuously calibrated with rain gauges aboard the ships. The eventual product that resulted was a multivariate time series taken at 15 minute intervals and averaged over 4 by 4 km boxes (Arkell and Hudlow, 1977; Hudlow and Patterson, 1979; Hudlow, 1979). The spatial array was 70 by 70 tiles providing an areal coverage of 280 by 280 km. (The GATE dataset actually covered a disk of diameter 400 km within which the 280 km square was fitted.) Some years after the completion of GATE there was a broad based workshop on Measurements of Precipitation from Satellites held at Goddard Space Flight Center and hosted by David Atlas and Otto Thiele (1981). One study in the volume of papers presented at this conference stands out in my mind as I later began to prepare presentations for TRMM. This was a little-noticed paper by Charles Laughlin (1981), a communications engineer who retired from NASA shortly after the workshop. In his watershed paper Laughlin worked up some important statistical summaries of the GATE precipitation data. Among his most important computations he presented a graph of the autocorrelation function of area averaged precipitation as a function of the area being considered. Larger areas had longer autocorrelation times and the largest area he considered (280 by 280 km) had an autocorrelation time of about 6 hours. Laughlin went 2 on to conclude that the diurnal cycle was small and that a satellite returning to the same averaging box every 12 hours could have tolerable sampling error characteristics (of the order of 10%). In formulating the first presentations for TRMM it was recognized that the most challenging issues before us were: 1) Statistical sampling errors due to the sparse visiting pattern of a single satellite. 2) Inevitable ‘beam filling’ errors due to the finite size of an individual microwave radiometer field-of-view (FOV) compared to the smaller important scales of tropical precipitation. 3) Ground truth for such a satellite observing system because of the difficulties in deploying the necessary ground observing system. The purpose of this chapter is to show how studies with the same GATE precipitation data set have helped us to sort out some of the difficulties with the TRMM observing system and to provide some insights into how these might be overcome. TRMM Observing System The original TRMM observing system as planned in the 1980’s was not so different from the final configuration except for the addition of a lightning instrument and an earth radiation budget instrument (Wilheit, 1986; Simpson et al., 1988). In most of the studies to be summarized in this chapter, the attention was be mostly focussed on the 19.35 GHz channel of the microwave radiometer. This instrument had an FOV of the order of 20 km across and the scanning was conical across track with a scan width of about 700 km (72 FOVs). There are other channels available on the radiometer and there are other 3 instruments on board, but in these studies we assumed that the primary measurement over oceans at least would come from the 19.35 GHz channel. This is our prototype measurement. Sampling Error Studies Laughlin constructed a univariate time series model for GATE area averages. He then estimated the sampling errors for a satellite visiting at different time intervals from a few hours to a few days with one week, two week and one month averages. For one month averages and a satellite visiting the whole averaging box, the errors were expected to be about 10%. Following Laughlin, the first GATE-related study to appear on TRMM sampling errors was that of McConnell and North (1987). In this study an imaginary satellite was flown over the GATE region at 12 hour intervals and the data recovered were used to estimate the three week long average of rain rate (GATE I was slightly less than three weeks and GATE II was a few weeks later in 1974 and of duration about 18 days). An ensemble of satellite overpasses was configured so that the first satellite overflew at 8 AM/ 8 PM, etc. The next member began at 9 AM/PM, then 10 AM/PM, etc. In this way an ensemble of 9 members was constructed. The month long averages of light, medium, heavy and extremely heavy categories were estimated from the ensemble. The estimates from the different ensemble members were of course correlated, but visual inspection of the estimates in bar graph form was rather revealing. The bar graphs showed that the sampling errors as indicated by the spread across the ensemble members was less than 10% for the lighter rain categories and only somewhat larger for the other 4 categories. Of course, there was no proof that the same kind of statistics would hold for another month, but the objection that the system might miss significant rain events by such a sampling scheme was considerably quieted. Later we found that individual rain storms in GATE lasted 12 to 18 hours and most of this lifetime might occur in a single averaging box. The graphics from the McConnell-North study were very useful in the early stages of support building for TRMM. Soon after the McConnell-North study came the study based on modeling the probability density functions for rain rates. Kedem, Chiu, and North (1990; later referred to as KCN; see also Kedem and Chiu, 1987). The idea is based upon the mixed probability density distribution – in this case, a log-normal distribution when raining, and a delta function at zero rain rate – the latter containing usually more than 90% of the probability. GATE data were used to verify the mixed log-normal distribution – it was already known that many parameters (areas, storm column heights, etc.) in tropical convection follow the log-normal (Lopez, 1977). This rainrate distribution was found to hold for a range of different averaging areas. A good approach to estimating the area average rain rate was to estimate the parameters of the mixed log-normal distribution. The main advantage of this system is that if the microwave retrieval algorithm `saturates’, i.e., it cannot really resolve high rain rates, then one can estimate the parameters of the whole distribution from a knowledge of only the lower rain rate portion of the distribution. In this KCN study it was also found that sampling errors for a TRMM-like configuration of the order of 10% of the mean rainrate for large area averages were not an unreasonable expectation. 5 An important side product of the KCN study was that Long S. Chiu discovered in the GATE data a linear relationship between instantaneous area average rain and the area covered by rain at a rate above a certain threshold (Chiu, 1988). Somewhat similar relationships had been known from previous studies by Doneaud, al. (1984); as pointed out by Atlas et al., (1990). Finally, the KCN study was the basis for one of the important retrieval strategies based on probability distribution estimation (Wilheit et al., 1991). The technique in KCN also leads to the multiple threshold technique proposed by Meneghini et al. (1989) for the TRMM standard product (3A26). One problem with the Laughlin, McConnell-North and KCN studies was that they were based on the GATE area which was only 280 by 280 km. The first data requirements for TRMM were for a 500 by 500 km averaging box for month long averages as opposed to only three weeks (Thiele, 1987). Hence some extrapolating to the larger box and longer averaging time had to be conducted. Moreover, the visits in these primitive studies were always ‘flush’, i.e., the satellite covered the entire averaging box on each visit. In reality, the swaths will only partially cover the box on an individual visit. There was a need to construct sampling models which could take these effects into account. One approach along these lines was to construct a stochastic rain field model with two spatial dimensions and temporally evolving. The model could be tuned to fit GATE data. This program was carried out by Thomas L. Bell. The idea of the model was inspired by turbulence theory in that a gaussian random field of vertical winds was produced as though from stationary, spatially homogeneous turbulence but with the space-time 6 spectrum to be determined. For areas where the vertical wind exceeded a certain threshold value, it was deemed ‘raining’. The exceedence variate was distributed like the tail of a gaussian distribution. Bell then converted this random ‘exceedence’ field to lognormal. Finally, the spectrum of the original underlying field was adjusted in both space and time to bring the rain field statistics (space-time autocorrelations, etc.) into agreement with those of the GATE data. This field was very similar in appearance to GATE rainfall data. It was patchy, and had the correct spatial correlation structure by design. It still did not have some features such as bands or waves. Nevertheless, it was a very good model for the purpose intended. The model is described in Bell (1987) and sampling error results are reported in Bell et al. (1990). One important conclusion was that based upon this model the error distributions tended to be gaussian even though the underlying fields were far from it. The beauty of this model was in its flexibility. One could produce a long control run of the model and save the output. Hence, more than a few weeks could be easily generated. This permitted many studies for different orbit altitudes, inclinations, etc. One could even make partial coverage visits based on real orbit calculations. A major problem was that the simulations were very computer intensive (for the mid-eighties). A much simpler model was that of Shin and North (1988). This model used realistic overflights generated from orbital calculations, but a univariate model of the area-averaged rain rate. The fraction of coverage for each visit was calculated and weights were assigned to each visit accordingly to estimate the month long average over the box. The results were in 7 reasonable agreement with those from the Bell model. Again the sampling errors for orbits in the parameter range envisioned by TRMM were on the order of 10%. North and Nakamoto (1989; referred to as NN) introduced a spectral approach to evaluating sampling errors for a general class of sampling designs. A formula for the mean square sampling errors was constructed based upon an integral over all space wavenumbers and frequencies. The integrand consisted of two factors, one a function only of the sampling design and the second factor depending only on the space-time spectral density of the field being sampled. The formula was interesting because it showed that the only field property needed to evaluate the mean square error was the space-time spectrum. This is especially interesting since it is only a second order statistic and even though the field may be highly non-gaussian (such as rain rates) the mean square error only depends on the second moment statistics of the field. North and Nakamoto presented error estimates for satellite overpasses and for some other configurations such as regular arrays of point gauges. Part of the paper of further interest is a rain rate model based on a damped diffusion model forced by space-time white noise. The two parameters of this model were tuned to GATE data. While the model solutions were not patchy like real rain, it could be made to have roughly the correct second moment statistics. The North-Nakamoto approach led to some space-time spectral analyses of GATE data. Examples include those of Nakamoto et al. (1990) and Polyak and North (1995). There were also a few studies of hydrological interest not directly related to GATE but stimulated by the GATE-based models (Graves et. al., 1993; Valdes et al., 1994). Valdes 8 et al. (1990) used several spectral estimates based upon different rain field models to obtain sampling errors. How can TRMM data be combined with other data? North et al. (1991) showed using the NN formalism that one could easily evaluate a design composed of a combination of satellite overpasses and ground-based gauges. A very good approximation consists of simply averaging the two together using the inverse variances as weights. There is a cross term because the two measurements are correlated, but this term was shown to be very small. The culmination of this series of papers was one by North et al. (1993) in which the errors were evaluated for configurations of several satellites combining TRMM with sunsynchronous orbiters. The NN rain rate model was used along with a stochastic model of the fractional visits. Once again, the sampling errors were in the tolerable range (5 to 10%). Many of the early results were summarized by North (1988). An issue always lurking in the TRMM estimation problem has been the diurnal cycle. Using GATE statistics Bell and Reid (1993) showed that to estimate the amplitude of the diurnal cycle might take many months of TRMM data. More recently, Shin and North (2000) investigated the contribution to sampling errors if the field being sampled were diurnally cyclostationary. This allows for the possibility that the variance might have a diurnal cycle along with the mean. This effect does increase sampling errors but only by a few percentage points. 9 A major problem for all the analyses based upon GATE data is that GATE may not be representative. Shin and North (1991) used ESMR-5 data to show that the autocorrelation lengths of rain rates from one FOV to another along the track were quite homogeneous across the tropical oceans. This property might not be too surprising since the sizes of tropical convective cells are not highly variable from one place to another over the tropical oceans. On the other hand, GATE was located in the ITCZ and this is not typical of most of the tropics. The rain rates in the ITCZ are much greater and all of the studies listed above were based upon those kinds of statistics. Tropical rain tends to be characterized over a month by a small number of heavy events. In the ITCZ there are many of these events in a month. Away from the equator the number of events in a month will be smaller. This will tend to make sampling errors larger. Bell and Kundu (1996, 2000) have developed a sampling error model much more appropriate for these kinds of situations. The percentage sampling errors (compared to the local mean) in their model go as R –1/2 where R is the monthly averaged rain rate at the particular location, season, etc. Bell and Kundu (2000) compare in detail a number of the studies referred to above as well as many based on other field experiments. The inverse square root scaling seems to be a satisfactory way of accounting for the evaluation of sampling errors outside the heavy rain areas associated with the ITCZ where GATE data were taken. 10 Beam Filling If only a single microwave frequency is used to estimate rain rate, there is an inevitable problem because of the nonlinearity of the formula relating microwave temperature and rain rate. The satellite-borne radiometer sees an area average of microwave brightness temperature over the FOV (typically 20 km across). But the FOV may contain patchy rain patterns much smaller than the FOV. The result of this is that the distribution of rain rates inside the FOV lead to a measurement of brightness temperature which when inverted does not give the average of the rain rate in the FOV. A bit of analysis shows that there will be a bias and a large random error in this kind of single channel retrieval. The use of additional channels can sometimes reduce the beam filling error but as long as the retrieval is nonlinear, no amount of channels can completely eliminate it. GATE data make an excellent test bed for the size of this kind of error. The first recognition (to my knowledge) of the beam filling error in these kinds of retrievals is in an unpublished report by Eric Smith and Stan Kidder (1978) based upon ESMR-5 data compared to GATE. The most revealing study was that of Chiu et al. (1990). In this paper the retrieval errors were estimated from GATE data with different size FOVs. A simple saturating exponential model of the T-R relationship was adopted: T(R) =A +B e-CR 11 Using the rain rate fields from GATE one could insert ensembles of fields inside the FOV to see how the inversion to rain rate worked. As expected the bias was about 40% and the random error accompanying the retrieval was similarly of the same order or larger. In the estimate of monthly averages the latter was not considered a problem, but in smaller aggregates it could be significant. Another extremely interesting study of the beam filling effect was by David Short in his PhD dissertation (Short, 1988) and in the paper, Short and North (1990). In this study, GATE data were compared to ESMR-5 brightness temperatures overpass by overpass. Because of small positioning errors, the two pictures had to be shifted until the best fit occurred. Once this was done, the expected brightness temperatures could be computed from the GATE data and compared to the satellite data. The satellite data could also be inverted to give the GATE data plus error. It was found that the errors were entirely consistent with the earlier findings by Chiu et al. (1990). In other words, ESMR-5 (19.35 GHz) errors in retrieving rain rates could be completely explained in terms of simple beam filling error. These studies showed that the prototype measurement based upon the 19.35 GHz channel with FOV about 20 km was a perfectly good means of retrieving rain rates from space if the bias could be removed perhaps by climatology. The addition of other channels could be used to ascertain the freezing level, etc., but the fundamental principle long before advanced by Wilheit (1986) was intact. 12 Another source of error related to beam filling is the random error from FOV to FOV. This kind of random error is usually assumed to be statistically independent (e.g., Bell and Kundu, 2000; see also Bell et al., 1990, and Wilheit, 1988). North and Polyak (1996) examined GATE data for neighboring 20 by 20 km boxes. The beam filling error is mostly accounted for by the variance of rain rate within the FOV (Chiu, et al., 1990; Short, 1988). If these variances are correlated from one FOV to another, the random part of the beam filling error will be correlated. Polyak and North (1996) found a correlation coefficient of the variances (when raining) between neighboring FOVs of between 0.35 and 0.50. This reduces the number of independent samples by about a factor of three. When a month of data are aggregated into 5 by 5 deg boxes, this can add a random error of the order of one to two percentage points to the sampling error. Ground Truth The third major concern raised in the early presentations promoting TRMM was ground truth (some prefer ground validation ). In fact, probably more resources have been devoted to this than to any other aspect of TRMM other than the actual satellite’s construction. Many problems beset the ground truth problem. First is that the Tropical Oceans are remote and especially difficult to work in for field campaigns. Even though several such campaigns have been undertaken since the launch of TRMM, they are necessarily of short duration and any comparisons of ground and space measurements will be troubled with sampling errors. 13 All ground based systems have their own set of biases and random errors. In fact, some have conjectured that the satellite may be a more accurate measure of rain rates than any ground based observing system. The main tool for ground truth is the surface-based precipitation radar. However, the radar retrieval of rain is usually based upon a Z-R relationship which is strongly dependent on rain type and additionally complicated by empirical parameters. Nevertheless, the radar is a very useful tool because of its spatial resolution, which is better than or equal to that of the satellite’s. One can examine spatial patterns of each observed field and from those draw conclusions, even if the calibration of the radar is not certain. Perhaps the closest to ‘truth’ is the point rain gauge. Unfortunately, over the oceans these are usually located on small islands and there is a clear bias associated with small landmasses located in large oceanic regions. Such a gauge will not be representative of the open ocean. Ships offer another alternative, but these are also beset with biases owing to location amidst the proximate geometrical structure of the vessel. One gauge configuration seems promising: the point gauge located on oceanic buoys. There is such a set of gauges located on the TOGA TAO array along the Equatorial East Pacific (McPhadden, 1995). Several of these gauges have collected more than 12 months of rain rate data and this may be enough to make a definitive ground truth test of TRMM. GATE data have afforded us some ideas about how the test might be conducted. First, the satellite and the gauge measure very different quantities: the gauge measures precipitation rate continuously in time at a point in space; the satellite measures an area 14 average over an individual FOV. There will be a random sampling error even if both systems are perfect. Furthermore, the location of the gauge will be positioned at a random location within the visiting FOV. Several theoretical studies have been undertaken to investigate the types of problems that might be encountered in such an approach. The main question to be addressed is how many visit pairs are required to assess a bias of 10% of the natural variability (Ha and North, 1994; North et al., 1994; Ha and North, 1998; Ha and North, 2000). The basic formalism for the comparison was worked out by North et al. (1994), and stochastic model examples were presented based on GATE data. It appeared that if both instruments were perfect but offset from one another by a bias, the number of pairs might require as much as a year of data. Ha and North (1994) investigated the gain to be made by having several gauges close together and even short microwave attenuation links cutting across the FOV. There was found to be some significant gain from having two gauges, but little more from the attenuation or more than two gauges. Ha and North (1999) conducted further studies along these lines with more realistic models which included the ‘patchiness’ feature of real rain. Finally, a study has been conducted with overflights of actual GATE data (Ha and North, 2000). The conclusion is that several months of data pairs would be required. Actually we know that the retrieval of an individual FOV rain rate will have a random beam filling error of approximately 50% (e.g., Chiu et al. 1990); therefore the 8 to 10 month requirement will probably be substantially increased. Conclusions 15 GATE data gave us a good idea about how tropical convection over the oceans works. But it was in a small limited region perhaps not representative of the tropical oceans as a whole. The emergence of ENSO as a dominant factor in seasonal forecasting led to an urgent need to extend this kind of measurement to the Pacific and throughout the tropics. Furthermore, the need to provide data bases to compare with climate model simulations has become an important priority. So the Tropical Rainfall Measuring Mission has been approved and successfully launched with now nearly four years of excellent data being archived. Some of the main problems associated with the TRMM observing system remain to this day, but the early studies with GATE data (although now somewhat dated by more recent, but less comprehensive field campaigns) were an essential ingredient in the successful understanding of the TRMM observing system. Acknowledgements The author wishes to thank the NASA TRMM Office for its continuous grant support since 1984. I further acknowledge my indebtedness to all of the co-authors listed in the references. Finally, I wish to give my special thanks to Joanne Simpson for her courage, doggedness, scientific leadership, friendship and encouragement over these years. 16 References: Arkell, R. and M. D. Hudlow, 1977: GATE International Meteorological Radar Atlas, U. S. Department of Commerce, NOAA, U. S. Government Printing Office, Washington, D. C. Atlas, D., D. Rosenfeld, and D. A. Short, 1990: The estimation of convective rainfall by area integrals. PartI: The theoretical and empirical basis. J. Geophys. Res., 95, 2153-2160. Atlas, D. and O. W. Thiele (Eds.), 1981: Precipitation Measurements from Space: Workshop Report. NASA-Goddard Space Flight Center, October, 1981. Bell, T. L., 1987: A space-time stochastic model of rainfall sor satellite remote-sensing studies, J. Geophys. Res., 92, 9631-9643. Bell, T. L., A. Abdullah, R. L. Martin and G. R. North, 1990: Sampling errors for satellite-derived tropical rainfall: Monte Carlo study using a space-time stochastic model. J. Geophys. Res., 95, 2195-2205. 17 Bell, T. L., and N. Reid, 1993: Detection of the diurnal cycle of tropical rainfall from satellite observations. J. Appl. Meteor., 32, 311-322. Bell, T. L., and P. K. Kundu, 1996: A study of the sampling error in satellite rainfall estimates using optimal averaging of data and a stochastic model. J. Climate, 9, 1251-1268. Bell, T. L., and P. K. Kundu, 1996: Dependence of satellite sampling error on monthly averaged rain rates: Comparison of simple models and recent studies. J. Climate, 13, 449-462. Chiu, L. S., G. R. North, and D. A. Short, 1989: Errors in satellite rainfall estimation due to non-uniform field of view of space-borne microwave sensors. In Microwave Remote Sensing of the Earth System. Alain Chedin (Ed.). Deepak Publishing, Hampton, VA, 95-109. Chiu, L. S., G. R. North, D. A. Short, and A. McConnell, 1990: Rain estimation from satellites: Effect of finite field of view. J. Geophys. Res., 95, 2177-2185. Chiu, L. S., 1988: Estimating rain rates from rain area, in Tropical Precipitation Measurements, J. Theon and N. Fugono (Eds.), Deepak Press, Hampton, Va. 361367. 18 Doneaud, A. a., S. I. Niscov, D. L. Priegnitz, and P. L. Smith, 1984: The area-time integral as an indicator for convective rain volumes. J. Climate Appl. Meteor., 23, 555-561. Graves, C. E., J. B. Valdes, S. S. P. Shen, and G. R. North, 1993: Evaluation of sampling errors of precipitation from spaceborne and ground sensors. J. Appl. Meteorol., 32, 374-385. Ha, E. and G. R. North, 1995: Model studies of the beam-filling error for rain rate retrieval with microwave radiometers, . J. Atmos. Oceanic Technol., 12, 268-281. Ha, E. and G. R. North, 1994: Use of multiple gauges and microwave attenuation of precipitation for satellite verification. J. Atmos. Oceanic Technol., 11, 629-636. Ha, E. and G. R. North, 1999: Error analysis for some ground validation designs for satellite observations of precipitation. J. Atmos. Oceanic Technol., 16, 1949-1957. Ha, E., G. R. North, Chulsang Yoo, and Kyung-Ja Ha, 2000: Evaluation of some ground truth designs for satellite estimates of rain rate. Submitted to . J. Atmos. Oceanic Technol. (August 2000). Houze, R. A., and A. K. Betts, 1981: Convection in GATE. Rev. Geophys., 19, 541-576. 19 Hudlow, M. D., 1979: Mean rainfall rates for the three phases of GATE. J. Appl. Meteorol., 18, 958-962. Hudlow, M. D., and V.L. Patterson, 1979: GATE Radar Rainfall Atlas, NOAA special report, 158pp, U. S. Government Printing Office, Washington, D. C. Kedem, B., L. S. Chiu, and G. R. North, 1990: Estimation of rain rate: Application to satellite observations. J. Geophys. Res., 95, 1965-1972. Kedem, B. and L. S. Chiu, 1987: On the lognormality of rain rates. Proc. Nat. Acad. Sci. USA, 84, 901-905. Laughlin, Charles R., 1981: On the effect of temporal sampling on the observations of mean rainfall. In: Precipitation Measurements From Space, Workshop Report, Edited by David Atlas and Otto Thiele, NASA Goddard Space Flight Center, October, 1981. Lopez, R. E., 1977: The lognormal distribution and cumulus cloud populations. Mon. Wea. Rev., 105, 865-872. McConnell, A., and G. R. North, 1987: Sampling errors in satellite estimates of tropical rain. J. Geophys. Res., 92 D8, 9567-9570. 20 McPhadden, M. J., 1995: The Tropical Atmosphere Ocean (TAO) Array is completed. Bull. Amer. Meteor. Soc., 76, 739-741. Meneghini, R., 1998: Application of a Threshold Method to Airborne–Spaceborne Attenuating-Wavelength Radars for the Estimation of Space–Time Rain-Rate Statistics. J. Appl. Meteor, 37, 924-938; Meneghini, R., and J. A. Jones, 1993: An Approach to Estimate the Areal Rain-Rate Distribution from Spaceborne Radar by the Use of Multiple Thresholds. J. Appl. Meteor., 32, 386-398. Nakamoto, S., J. B. Valdes, and G. R. North, 1990: Frequency-wavenumber spectrum for GATE Phase I rainfields. J. Appl. Meteorol., 29, 842-850. Nakamoto, S., J.-T. Wang, D. A. Short, and G. R. North, 1988: Estimation of lagged space-time correlations in rain data. In Tropical Rainfall Measurements. J. S. Theon and N. Fugono (Eds), A. Deepak Publishing, Hampton, VA. North, G. R., 1988: Survey of sampling problems for TRMM. In Tropical Rainfall Measurements. J. S. Theon and N. Fugono (Eds.). Deepak Publishing, Hampton, VA. 337-348. North, G. R., and S. Nakamoto, 1989: Formalism for comparing rain estimation designs. . J. Atmos. Oceanic Technol., 6, 985-992. 21 North, G. R., J. B. Valdes, E. Ha, and S. S. P. Shen, 1994: The ground-truth problem for satellite estimates of rain rate. J. Atmos. Oceanic Technol., 11, 629-636. 10351041. North, G. R., S. S. P. Shen, and R. Upson, 1991: Combining gages with satellite measurements for optimal estimates of area-time averaged rain rates. Water Resources Res., 10, 2785-2790. North, G. R., 1992: Characteristics of tropical precipitation important for its estimation by satellites. In Global Role of Tropical Rainfall, J. Theon, T. Matsuno, T. Sakata, and N. Fugono (Eds). Deepak Publishing. Hampton, VA. North, G. R., S. S. P. Shen, and R. Upson, 1993: Sampling errors in rainfall estimates by multiple satellites. J. Appl. Meteorol., 32, 399-410. Polyak, I., and G. R. North, 1995: The second moment climatology of the GATE rain rate data. Bull. Amer. Meteorol. Soc., 76, 535-550. Polyak, I., and G. R. North, 1996: Spatial correlation of beam-filling error in microwave rain-rate retrievals. J. Atmos. Oceanic Technol. , 13, 1101-1106. 22 Shin, D. B., and G. R. North, 2000: Errors incurred in sampling a cyclostationary field. J. Atmos. Oceanic Technol. , 17, 656-664. Shin, K.-S., and G. R. North, 1988: Sampling error study for rainfall estimate by satellite using a stochastic model. J. Appl. Meteorol., 27, 1218-1231. Shin, K.-S., and G. R. North, 1991: On the homogeneity of spatial correlation statistics of tropical rainfall. J. Geophys. Res., 96, 9273-9283. Short, D. A. 1988: Remote sensing of oceanic rain rates by passive microwave sensors: A statistical-physical approach. PhD dissertation, Texas A&M University, August 1988, College Station, Texas. Short, D. A., and G. R. North, 1990: The beam filling error in the Nimbus 5 Electronically Scanning Microwave Radiometer observations of global atlantic tropical Experiment rainfall. J. Geophys. Res., 95, (D3) 2187-2193. Smith, E. A., S. Q. Kidder, 1978: A multispectral satellite approach to rainfall estimates. Unpublished manuscript, Colorado State University, Fort Collins, 26pp. Plus tables and figures. Simpson, J., R. F. Adler, and G. R. North, 1988: A proposed Tropical Rainfall Measuring Mission (TRMM) satellite, Bull. Amer. Meteorol. Soc., 69, 278-295. 23 Thiele, O. W. (Ed.), 1987: On requirements for a satellite mission to measure tropical rainfall. NASA RP-1183, Washington, D. C. Valdes, J. B., S. Nakamoto, S. S. P. Shen, and G. R. North, 1990: Estimation of multidimensional precipitation parameters by areal estimates of oceanic rainfall. J. Geophys. Res., 95, 2101-2111. Valdes, J. B., E. Ha, C. Yoo, and G. R. North, 1994. Stochastic characterization of spacetime precipiation: Implications for remote sensing. Adv. in Wat. Resourses Res., 17, 47-59. Wilheit, T. T., 1986: Some comments on passive microwave measurement of rain. Bull Amer. Meteorol. Soc., 67, 1226-1232. Wilheit, T. T., 1988: Error analysis for the Tropical Rainfall Measuring Mission (TRMM). Tropical Rainfall Measurements, J. S. Theon and N. Fugono, Eds., A. Deepak, 377-385. Wilheit, T. T.,A. Chang, and L. S. Chiu, 1991: Retrieval of monthly rainfall indices from microwave radiometric measurements using probability distribution functions. J. Atmos. Oceanic Technol. , 8, 118-136. 24 25 View publication stats