The Effect of Spatial Resolution on the Correlation Structure of Gate III Rainfall Fields

The presen叫t study uses a fine scale data set from GATE ph祖 e III to i 扭 nves品 t i ga叫 .t s pa訕也 i al eπ 'ects on rainfall cha缸 ra缸 ct扭 er吋 is刮 tic臼 s and its lagged aut岫 oc叩 or叮 Tela訕 tion. Due to the 直 ner s truct机 ure of the present data set, the physical processes invloved for a shorter distance scale can be identified, which has crucial information in the estimation for the sampling eπors of a rain field. It is found that for a rainrate less than 4 mm/ hr, a 1 km by 10 and 4 km by 40 data set will not make any noticeable difference to accumulated rainfall statistics. This is directly implied by stratiform rainfall associated with the mesoscale circulation system. The rainfall within that physical region is continuous with a ramra 個 less th祖 6 mm/hr. A rainfall rate greater than 40 mm/hr contributes 20 % to the total rainfall in 1 km by 10 d叫 a, in contrast to an 8 % contribution from the 4 km by 40 data. The area-averaged processes suppress extreme rainfall considerably (the extreme would have come from the convective scale). The variance and the autocorrelation calculated in this study reveal that the slope of the variance and autocorrelation for a smaller scale are different from those estimated through a larg叮叮 ea data. This is an indication that the convective scale rainfall field caused this change, and the existence of a horizontal integral length scale for the shorter scale end is pο曲 s ula扭 d. The implications of a horizontal integral length scale and suggestions for further research are discussed.


INTRODUCTION 51
It was understood that outside the tropics, the primary energy source of synoptic disturbances is available potential energy with a strong latitudinal temperature gradient.In the tropics, on the other hand, the storage of available potential energy is very small due to the very weak temperature gradients.The latent heat release associated with convective rainfall appears to be the primary 1 Institute of Atmospheric Physics, National Central University, Chung-Li, Taiwan 32054, R. 0. C.
2 Division of Oceanic Development, Japan Marine Science and Technology Center, Yoko• suka, 298 Japan.
TAO Vo l.2, No.l energy source for disturbance there.Most latent heat release in the tropics oc curs in convective cloud systems.This large sum of latent heat released through precipitation is vital to a large-scale general circulation pattern, and the water associated with it is essential for human activity.However, the estimation of precipitation over the tropics is a difficult problem.Unlike many meteorological parameters, rainfall is discontinuous in space and time and exhibits a large nat ural variability.Current observation systems, such as rain gauges and radar, are generally limited to the measurement of precipitation over land.The oceanic rainfall is more frequently extrapolated from other data rather than measured directly� Satellites have been regarded as a means to circumvent some of the defi ciencies of gauge and radar measurement of rainfall.Satellites have the ability to access remote areas of the world on a regular basis and this is a great advantage.However, satellite systems present their own problems.For example, rainfall rates must be indirectly inferred using remote sensing methods.A number of techniques have been developed to infer rainfallfrom visible or infrared data or to relate it more physically to radiation emitted at microwave frequencies.
Sampling errors due to temporal gaps dominate the error budget of a global rainfall dataset exclusively constructed by a specified satellite.The sam pling errors may be defined as the difference between a "true" estimate one would obtain from a temporally continuous data set and an estimate from the discontinuous data.Radar or gauge observed data must be used to estimate the sampling errors and the implication for errors in satellite-measured datasets can then be identified.One of the most frequently used observational data sets for this type of research is from the GATE (Global Atmospheric Research Program,   At_ !�ntic Tropical Experiment). .
During the GATE in 1974 , the approach of combining satellite, radar, and aircraft, with an exceptionally dense array of ships launching upper-air balloons was adopted, to probe the convective cloud field.These data were supplemented by standard synoptic observations, surface-based cloud photog raphy, an� a variety of boundary layer measurements.The GATE dataset offered an unprecedented opportunity to investigate the evolution of convective clouds and the rainfall characteristics associated with these events.Many stud ies had been conducted using both real data taken from the GATE experiment and using stochastic models of rain rates based on the observed data to inves-, tigate the lagged space-time correlation of rain fields (McConnell and North,   1987; Laughlin, 1981; Hudlow and Patterson, 1979)  The lagged space-time correlation of the rain rate is needed for the esti mation of sampling errors in finite rain-gauge networks, and in the estimation of bias due to beam-filling.The spatial autocorrelation of the rain rate was derived from the GATE phase I and Phase II data.The correlation field is well where Cr (s) is the spatial autocorrelation and s is the spatial distance (km).
That is the spatial autocorrelation function is described by a power law s-2/3• The variance of the area-averaged rain rate decreases much more slowly and is nearly A-1/3 where A stands for an area (Laughlin, 1981).This finding about the variance is consistent with the s-2/3 spatial correlation power law .This implies that the concept of a conventional horizontal integral length scale is questionable.If there were a finite correlation length, the variance of the area averaged rain rate would have to decrease as A -1• If there were no integral length scale for the smallest scale, the variance of the small area-averaged rain field would be large, which then indicates large sampling errors in the estimation of a true correlation.However, the s-2/3 power law of spatial autocorrelation is based on data with a minimum resolution of 4 km x 4 km.The physical events which have a scale smaller than 4 km could not be resolved in the study when deriving Eq. ( 1) .Hence, how the s-2/3 power law formulation in Eq. ( 1) is affected by the smaller-scale events is uncertain.After the GATE experiment, one of the major findings is the identification of mesoscale disturbances embedded in a convective cloud cluster system (Houze and Betts, 1981).In the GATE region, cloud cluster systems have a lifecycle of 12 to 24 hours evolving from individual convective cells to a mature organized mesoscale system and finally di_ ssipating.Rainfall deriving from the mesoscale anvil region is mostly continuous and can last for several hours at a rate of 2,..., , 6 mm/hr, in contrast to the convective core region where rainfall rates are in the range of 10,... .., 100 mm/hr (Leary and Houze, 1979).The characteristic length scale of the convective core region is only on the order of several km, the mesoscale anvil region has a spatial domain of about 100 km.So the convective core regions may not be totally resolved by a 4 km by 4 km data set.• This convective disturbance may thus play a role in the spatial lagged correlation statistics.Therefore this study attempts to reexamine the lagged space-time autocorrelation based on a fine scale data set which can resolve the small convective events, in order to investigate the correlation structure for the smaller-scale in order to estimate of the rainfall statistics, and compare them with the conclusions drawn from studies with a coarser grid data.An attempt will be made to physically interpret the statistical results of this study through the observed cloud cluster events.Section 2 will describe the data set and the techniques used in the anal ysis.The results and discussion are presented in Sec. 3. Section 4 compares TAO Vol.2, No.I the present study with other previous studies and a generalized hypothesis is postulated.Conclusions and suggestions for further research follow (Sec.5).

DATA AND ANALYSIS TECHNIQUE
The data set u • sed in this study is based on radar observations from the Canadian ship named "Quadra" during Phase 3 of the GATE experiment.The GATE radar data has been studied extensively.However, most of the radar rainfall analyses were based on the smoothed data set constructed by Hudlow and Patterson (1979).can be estimated by the formula as follows. (2) where the overbar represents an average through all the available data used in the study, namely, (3) with m as the spatial lag.X can be any meteorological variable of interest.
Here the rain rate is used.The subscript of n (i) stands for the data point in the spatial (temporal) domain.The lagged autocorrelation function (ACRF) c; (0) is the ACVF for zero lag.We name this scheme the "method B" scheme.
There exist different estimators to estimate the ACVF, for example, it can be estimated as where and We call this scheme the "method A" scheme.
(5 ) (6) The main difference between these two estimates of the ACRF is that method B uses all the possible information contained in the dataset to form an ensemble mean, as indicated in Eq. (3), while method A only used the data in a given time segment to form the mean, as represented by Eq. (6).Further discussion can be found in the Appendix.
For a consistent estimator of the ACVF or ACRF, we want the limit of large N not to lead to a value different from the true ensemble average.
In rainfall data, one often has a finite strip of data N, but many different independent realizations of the strip.In such a case, we need to know which is the better estimate of the ACVF.Trenberth (1984) has considered these two estimators with the application to large scale meteorological fields in mind.He found that method B was the better estimator for such an application.Since his analysis assumed Gaussian statistics which do not hold for rain fall statistics, we have to reexamine this estimator problem.We construct a more generalized model which can treat either the Gaussian or non-Gaussian noise processes.An analytic expression for the bias generated through different methods under a hypothesized ARl red noise process is derived.By looking at the estimators for an ARl process within the limit of large N, the asymp totic estimation of the bias generated by different schemes may be written (see where b(m) is the bias for lag m and a is the ARl parameter.For method A, N + is the number of data points in a given segment.For method B, N + is TAO Vol.2, No. l the number of data points over the total segments [or N x P as in Eq. (3)].
Bias generated by method B is much smaller than that generated by method A. This demonstrates that method Bis a better estimator for the estimation of the rainfall statistics.Trenberth (1984) also pointed out method B is a better estimator for his formulation.Our generalized study finds that the Gaussian or non-Gaussian noise only affected the covariance function.It doesn't affect the correlation function.Therefore Eq. ( 4) is used in this study to estimate the ACRF.

RESULTS AND DISCUSSION
Before showing the results of the spatial autocorrelation function com puted from the present dataset, some of the mean statistics are presented first to identify the effects of area-averaging on the fundamental rainfield structure.

a. Mean
Figure 1 shows the probability density function of the rainfield, based on the present dataset (which has 1 km by 1° resolution) and the data which are reconstructed from the present set to have 4 km by 4° resolution.The length of al° sweep angle depends on it's radius.It varies from about 0.85 km (around a radius of 50 km) to 1.7 km (around a radius of 100 km).The largest probability for the rain rate mode is on the order of 4 mm/hr.I km by 1° and 4 km by 4° have a similar probability structure, and the latter has a weaker probability distribution at the 4 mm/ hr.rate.The corresponding contribution to the total rainfall by these 2 resolutions is shown in Fig. 2. In the 1 km by 1° dataset, the contribution from a rainrate greater than 40 mm/hr contributes about 20 % of the total rainfall, in contrast to the 8 % contribution from the 4 km by 4° resolution.This means that extreme rainfall occurs in a smaller area and the area averaging process suppresses these extreme values.The contribution to the rainfall from the smaller rain rate (say, less than 6 mm/ hr) is quite comparable in these two resolutions.Thi� leads to the confirmation that the rainfall from the anvil regions is mostly homogeneous and at rates smaller than 6 mm/hr.This point is clearly identified in Fig. 3. Figure 3 shows the a�cumulated rainfall contribution from different rain rates for the two resolutions used in Fig. I.For a rain rate::; IO mm/hr, they contribute about 50 % of total rainfall for a 1 km by 1° dataset.Due to smoothing at the 4 km by 4° resolution, it takes a higher rain rate for 1 km by 1° data to achieve the same level of contribution to total rainfall.For example, the accumulated 10 mm/hr rainrate in the 4 km by 4° resolution contributes 55 % of the total rainfall, in contrast to 50 % from the 1 km by 1° resolution.
Below a 6 mm/hr rain rate, the contributions from these two resolutions  rain rate can be explained by the mechanisms involved in the convective cloud system.It is known that in the GATE region, mature cloud clusters not only has a convective core region, they also have an anvil region associated with them.
The rainfall from an anvil region is estimated at a rate of 2 rv 6 mm/ hr and it usually lasts for several hours.Because of the large horizontal area.occupiedby the anvil region, these two resolutions actually resolved the physically similar behavior at a smaller rain rate branch.
Only in the convective core region, which is associated with higher rain.:.fall, does the contribution to the total accumulated rainfall through these two resolutions begin to differ.The two different characteristic rainfall mechanisms in the GATE cloud clusters do have an impact on the accumulated rainfall contribution with respect to different spatial resolutions.This suggests that it would be interesting to investigate rainfall characteristics in other climatologi cally different regimes to see if this phenomenon persists.Such information will be useful for generalizing the conclusions drawn based on the GATE data.

b. Variances
The variances of these two resolutions with different radii of radar mea surement are shown in Table 1.The general pattern is that the 1 km by 1° spatial resolution has large variances as exp ected.However, the point which really concerns us is the slope of this increase from a larger spatial domain to                 : l l :   The contribution of various rainrates to the total rainfall for 1 km by 1° and 4 km by 4° resolutions.
a smaller region.Table 2 shows the estimated slope between these two res olutions.Some of the data in Table 2 is plotted in Fig. 4a.The slopes as shown from • Table 2 are all concentrated in the neighborhood of 0. 166, with 0.024 as the standard deviation.In general, the variance is in logarithmics where C is an empirical constant, A is the area under investigation and V is the logarithm of the variance asociated with the area A.
The slope is apparently larger than those deduced from GATE phases I and II which used the 4 km by 4 km grid as the smallest spatial resolution.
Figure 4b from North (1987) shows that the decrease of variance for an area averaged rain rate decreases at about power law �1/3.The finding is consistent with the -2/3 spatial correlation power law found by Bell (1987).The e�pMted difference in the variance from the -.3392 power law is 0.41.

TAO
Vol.2, No.l As illustrated in Fig. 4a, the solid line should be the estimated variance increase as the power law of -0.33, and tJteCijashed line is the actual increase given by the present study.Here the variance which approached from the even larger area is assumed to be at a rate of -0.33 power law.Our present study indicates that the change of vaoo�<JRiwt the area-averaged rain rate between 1 km by 1° and 4 km by 4° decreasE?sat a .Power la:w of -0.166, or we may write it as Vi s proportional to A-�!1�"S9 �1'-�P J �hi all�b W er suggests that the variance associated with the smaller spatfa� e le�olution f��l� off more slowly.That could be an indication of the existence 'iJ'r iR integraf1�ri_l gth icale.This is in contrast to the spatial region which does not?J e � m to ha�� 8 a?prefJ h able horizontal length scale.We are hypothesizing that ther � exists a H?J"r�zont�f)ength scale somewhat smaller than 4 km.Physically, this may be rel &t�.a to t il�EiP ainfall characteristic associated with the convective scale.The next �€dt ion c!B'Rfu ins more discussion on this.The-lagged autocorrelation of these two resoluti i� s c are computed from Eq. (3) .Figure 5 shows the lagged autocorrelation of t h� 0 rlt infield with respect to the distance for two resolutions • at a radar sweep radius 'cfr 102 km.•The 4 km by 4° data has a higher correlation distance for a given correlati6n value than the 1 km by 1° data set.This fact can be illustrated by the e-folding distance.The e-folding distance of 1 km by 1° data is about 5 km, whiMqt4J.e 4 km by The plot of some of the data points from Table 2.The solid line indicates that variance increased at -0.33 power law, while the dashed line is the actual computation to reach the 1 km by 1° area.
4° data is about 25 km.The decay of the autocorrelation in the logarithm for the 4 km by 4 ° data on the large correlation end is close to a straight line with a slope of around -0.6.Over the small correlation end, the shape of the autocorrelation is not a straight line.For a straight line in a log-log plot, the ACRF(s) is proportional to s-k, and k is the estimated slope in the log...: log plot.
For. a large correlation (or smaller distance) , the 4 km by 4° autocorrelation behaves as a -0.6 power law.

Fig. 6a
The logarithmic plot of lagged ACRF for a 1 km by 1° resolution, for various radii from the radar measurement center.
shorter distance end, the lagged ACRF are all clos� to a straight line, and are quite different from those of the 1 km by.1° resolution.This slower decay of the ACRF at the shorter distance end has significant implications.This aspect is consistent with the variance results discussed ear lier.This is also an indication that an integral length scale exists somewhere for a shorter distance to make the characteristics of the ACRF differ when compared to the outer distance end, due to the physical characteristic of the convective scale associated with the rainfield.The typical convective rain scale is only on the order of several km.On the other hand, the rainfall from the mesoscale stratiform cloud region will extend for a much larger outer distance.
Ci �.a . 78km radius  For the small correlation end, the slope of the logarithmic ACRF is also far from a straight line, both in 4 km by 4° and 1 km by 1° data.We attributed this departure from a straight line is due to the smallness of the sample size used in the present study, which only included 8 partial days of rainfall data, in contrast to the 18 days of data used by Laughlin (1981) andBell (1987).
Due to the small size of the data, the results for large spatial distances have not been able to respond to the ample information contained in the horizontal stratiform rainfall region.This will be improved a great deal when more data (in a temporal sense) is included.

COMPARISON AND IMPL ICATIONS OF THE RESULT
The present study used the GATE phase III data from "Quadra" to es timate the lagged autocorrelation.The original data is in a 1 km by 1° grid.The 4 km by 4° data set used here is constructed from the original data by simple area weighted averaging.Some previous studies of this problem used data from Phases I and II of the GATE (Laughlin, 1981;McConnel and North, 1987;Bell, 1987) .A 4 km by 4 km grid was used as the basic data set.Physi cal characteristics having a scale smaller than 4 km can't be resolved entirely.Figure 7 from Bell (1987) shows the spatial correlation of the GATE I rain fall.The corresponding lagged spatial ACRF in logarithmics is shown in Fig. 8 (North, 1987).Their results show a s-0 • 58 power law decay of the ACRF, and the long-tail of a spatial correlation with a value of about 0.2 extends to a farther distance (at least up to 70 km) .Also the power law decay of ACRF can extend to around 70 km before the fitting begins to deteriorate.Laughlin (1981) studied the effect of spatial averaging on the temporal autocorrelation in rainfall fields.He found the e-folding time scale changed from around 1 hr (in 4 by 4 km2 ) to more than 10 hr (in 280 by 280 km2) .The temporal correlation has an implicit implication on the spatial correlation.This indicates that the larger area-averaged data should contain a slower correlation decay in the outer distance.
Results from the present study illustrate the correlation characteristics of the rainfall data for the shorter distance end.The spatial ACRF estimated by 4 km by 4 ° data in the present study shows a power law decay of s-k, k is betwe e n (-0.667 and -0.58), which occurs in a comparable structure with the results shown by North (1987).However, the results from a 1 km by 1° dataset reveal e d a different characteristic of the A C RF for the shorter end.There exist ranges where the ACRF decay is slower than the power law'established for the larger area (namely, 4 km by 4° and beyond).The actual slope of this decay in the shorter end varies from radius to radius, but in general, the trend to have a slower decay rate for ACRF exists, and this slower d ecay of the lagged ACRF is consistent with the A -0•166 power law estimated through variance analysis.
As discussed in previous sections, this point is itdopted as the basis of our postulation.We thus postulated that the slower decay of the spatial ACRF for the shorter end is due to the existence of a horizontal integral length scale.This length scale is around 2 � 3 km.Physically, we interpreted this le�gth scale t o be due to the rainfall characteristic on the .convectivescale.This observation couldn't have been determined with a 4 km by 4° or 4 km by 4 km data.
The characteristic length scale of the convective scale is on the order of several km.The larger data set has a great impact in suppressing the convective scale rainfall characteristics .The spatial correlation of GATE I rainfall from Bell {1987).
Vol.2, No.1 So as determined from the GATE study (Houze and Betts , 1981), the con vective scale, as well as the mesoscale structure, is an integral part of tropical cloud clusters.The rainfall charact e ristic associated with these two distinctively different branches have a profound impact on the spatial ACRF structure.The long-tailed structure found by Bell (1987} is associated with continous strati form rainfall produced from the mesoscale circulation. The existence of a horizontal integral length scale has a great significance when relating the problems that the TRMM (Tropical Rainfall Measurement Mission, see Simpson et al., 1988 for more details) project encounters.In short, the TRMM is a proposed project to measure tropical rainfall from space through microwave channels and radar.Sampling errors for this project, as well as rain gauge measurement, depends on the characteristics of tropical cloud clusters.Present results indicates the existence of an integral length scale.T h is also implies more n u mbers of independent samples, which will reduce the.rainfall sampling errors estimated t h rough the proposed TR M M satellite.The logarithmic plot of the lagged ACRF based on GATE phase I data, which had a.
4 km by 4 fem resolution as a function of the.distance, from North (1�87).

RESEARCH
The present study used a finer data set from the GATE phase III to investigate the spatial effects on rainfall characteristics and its lagged autocor,... relation.Because of the finer structure of this present data set, we are able to identify the physical processes invloved in a shorter distance scale, which has crucial information for estimating sampling errors in the TRMM project.
It is found that for a rainrate Jess than 4 mm/hr, the 1 km by 1° and 4 km by 4° data set will not make any noticeable difference to accumulated rainfall statistics.This is a direct implication f rom the stratiform rainfall as sociated with the mesoscale circulation ( or rainfall from the mesoscale anvil) .
The rainfall within that physical region is continuous with a rainrate less than 6 mm/hr.A rainfall rate larger than 40 mm/hr contributes 20 % of the total rainfall in a 1 km by 1° dataset, in contrast to' the 8' % contribution from the 4 km by 4° data set.The area-averaged process.suppresses the extreme rainfall considerably (the extreme would have come from the c .on�ective scale) .This is another way to indicate that convective scale J,"ainfall always has a smaller spatial scale and an extremely large rainrate.
The variance and the autocorrelation calculated in this study revealed that the slope of the variance and.autocorrelation for the _ smaller scale are different from those estimated through the large area data.This is an l ridication that the convective scale rainfall field caused this change.The existence o f an integral Vol.2, No.1 length scale is a reasonable consequence of this scale of disturbance.
The slow decay of the autocorrelation found by Bell (1987) for large dis tances (say, up to 70 km) is due to the characteristics of homogeneous stratiform rainfall.This fact leads to an interesting problem, since all the studies so far used data which contains these two different scales of motion within tropical cloud clusters (Laughlin, 1981;Bell, 1987;McConnel and North, 1987;and the present study) .It will be helpful then to investigate how spatial effects on the autocorrelation field within an area which contains only stratiform rain fall.With this observational study and some rainfall modeling studies that simulate rainfall from the convective scale, we may learn more about the in teraction of rainfall statistics (convective scale and mesoscale) as well as how these two physically different processes evolve , reaching the final rainfall field as envisioned from the tropical region.These areas will be the topics for further research.

Fig. 6b Same
Fig. 6b Fig. 7 Fig. 8 Houze and Betts (1981)by Hudlow and Patterson was 4 km by 4 km and the time interval was 15 minutes.This resolution is not capable of deducing behavior smaller than 4 km.The present data set is based on radar observations at 1 km by I 0 of sweep angle and at 5 minute intervals.Only 8 Julian days of data are used in this analysis (day245, 246, 252, 253, 254,  260, 261, 262).The detailed structure of the convective events on these days have been studied by many investigators, e.g,Houze and Betts (1981), Leary and Houze (1979), Esbensen et al. (1988 ) .
If we have P data segments in a fixed time interval, and for each segment, there are N numbers of data points, the lagged autocovariance function(ACVF)

Table 2 .
The estimated slope for the variances at 1 km by 1° (Dl) resolution to reach the area associated with 4 km by 4° (D4) resolution.