Spatial relationship between cloud-cover and rainfall fields: a statistical approach combining satellite and ground data

Transcription

1 Hydrologie Applications of Space Technology (Proceedings of the Cocoa Beach Workshop, Florida, August 985). IAHS Publ. no. 6, 986. Spatial relationship between cloud-cover and rainfall fields: a statistical approach combining satellite and ground data J, D, CREUTIN, P. U\CMBA & CH, OBLED Institut de Mécanique de Grenoble, LA n 6, BP 68, F-3842 Saint Martin D'Hères, Cedex, France Abstract This study is devoted to the spatial estimation of rainfall depths over a region of the Middle-East using both the measurements from a scarce raingage network and images from meteorological satellites. In a first step, a nephanalysis is performed on IR.and VIS images. A regression between the ground measurements and the cloud-cover due to the different cloud types allows to derive guess-fields at time steps from 5 to 5 days, corresponding to rainy events. A second step consists in an optimal combination of the measurements from the two devices through the so-called cokriging system which is a simple extension of kriging interpolation when several measurement sets are available. Introduction A rainfall estimation study has been carried out over a region of the Middle-East, in order to assess the replenishment rate of an aquifer. As the scarce raingage network did not allow an accurate evaluation of the rainfall amounts, an attempt was made to assess rainfall with the help of both ground data and images of meteorological satellites. Various studies on rainfall estimation by satellite (E.C. BARRETT, 977 ; A. VAN DIJK, 98 ; D. ATLAS et al., 982) use rainfall production coefficients related to different cloud-types. This work is an attempt to complement this classical approach by a geostatistical combination of satellite and ground measurements. Thus, after a brief description of the data set that has been used, the implementation of a mean square regression between cloud-cover and ground measurements is shown to produce a first estimation of the rainfall depths which can be considered as a guess-field ; then, cokriging systems is detailed and shown to be appropriate to combine the satellite guessfield and the gage measurements ; finally the improvement provided by the use of remote sensed data is illustrated comparing maps yielded on one hand by cokriging and on the other hand by simple kriging of the gages measurements alone. Data set used Image set. The year 979 was retained for the study because a geostationary satellite, GOES Indian Ocean, was on orbit at that time. Additional images were provided by a polar orbiting satellite, DMSP (Defense Meteorological Satellite Program). Both sets were available only as paper prints. The images that were purchased, cover a period of 6 months of 979 only, because during the rest of the year, nearly no precipitation occurred. As the series provided by the two satellites let appear

2 82 J.D.Creutin et al. numerous gaps, they had to be merged, in order to get a greater daily frequency of images. Even so, no more than 6 IR and/or VIS images could be used within a day,usually 3. Raingage records. As for the ground data, 45 raingages recorded continuously the daily rainfall amounts during the whole year 979 (Fig. 5). Their distribution over the region was irregular, leaving some large areas without any measurement. Satellite data conditioning (derivation of rainfall guess fields from nebulosity) Cloud cover digitalization. As images were available as paper prints, the information they contained has to be digitalized. However, prints allow a visual classification of clouds into different types (nephanalysis), which would be difficult to implement automatically because of the accounting of parameters such as pattern, texture... Then simple relationships between rainfall depths and digitalized cloud-cover can be derived, providing a first estimation of rainfall fields. The implementation of nephanalysis implies using a grid of digitalization. A set of square cells of width was considered as sufficiently accurate. The classification of clouds in this study led to the identification of 3 types related to rainfall occurence : isolated convective clouds, convective areas of cyclonic disturbances and heterogeneous patterns of clouds. Each cloud-type was quantified separately as percent coverage of each cell by this cloud type. Method used to identify rainfall parameters. The simplest relationship between rainfall depth and cloud-type that may be found is linear, such as : N z = p. C. + e i=l X X where z is the rainfall amount, C. the cloud cover for the i cloud-type among N and e the residual error, each of these values being measured over a cell of the grid, within a day of a period. Such parameters can be found by minimizing the error according to a critérium. The least square critérium has been chosen in this case-study. As raingage measurements are not spatially representative values, they had to be averaged over the cell by taking the arithmetic mean of the inside point values. However, the number of gages per cell had to be considered as a weight attributed to each cell, in order to privilege well instrumented cells. The number of images per day was given a similar weighting role in the attempt to favor well sampled days. Once the regression has been performed, its quality can be assessed through the correlation coefficient and through the pattern of the regression plot which has to conform to the linearity hypothesis. If a significant relationship can be found, it allows a rainfall estimation even in the cells that are unprovided with ground measurements. Implementation of linear and non-linear regression over various timesteps. First, daily data were processed for each month separately. This led to very low correlation coefficients (less than.6) and plots (fig. ) with many outlayers. Such a result reflects the poor representativeness of gages, whose number is 4 at most, in meshes about km wide. In addition, an uncertainty of several days in rainfall records was evidenced in several cases, through it could not be systematically detected. In order to counter-balance recording errors and to obtain rainfall

3 Spatial relationship between cloud-cover S rainfall fields 83 FIG Daily amounts; three images a day 93 events and 3 meshes gaged giving 96 dots). unit=miii FIG.2 Amounts of rainy events; three images a day (93 dots). FIG.3 Amounts of rainy events; one image a day (93 dots). FIGS,2 & 3 Regression plots between observed (X-axis) and estimated (Y-axis) rainfall for Jan Dashed lines are the best linear relationships computed by regression method; solid lines show perfect agreement. depths of better spatial structure, monthly regressions were performed with sets of data cumulated over periods of various length. For 4 months among 6, the correlation coefficients and plotted patterns improved, presenting an optimum for rainy periods, 5 to 5 days long and separated by dry days (Fig. 2). Further improvements were awaited from non linear relationships such as : p. C.. l l i (Z P. C. + E ). in order to account for the greater variability of rainfall compared to that of cloud-cover. However, as table shows, none of these methods

4 84 J.D.Creutin et al. produced markedly improved results, and so the linear model can be considered as sufficiently suitable to this case-study. As more images were available for the month of January 979, the sensitiveness to image frequency was tested for this period. As clouds appeared mostly as isolated cumulonimbi, the cloud-cover was assessed with one, two or three images for each day when they developed. Regression performed with such cloud-cover values provided correlation coefficientsgrowing from.75 to.82 as image frequency increased from to 3 a day (Fig. 3). This is due to a better accounting of growth and motion of isolated cumulonimbi. Table - Correlation coefficients measuring the quality of the described regressions; () linear regression, (2) square root of rainfall depth and (3) square of cloud-cover unlinear regressions.! i Oct. I Jan..46 Daily data Daily data Monthly cumulated data I! Data cumulated over 3 periods () (2) I (3) I.74 Spatial gage-satellite rainfall estimation combination After the above appropriate satellite data conditionning step two sets of measurements are now available to estimate the rainfall depth at the ground level : the gage network that punctually provides direct measurements and the satellite that provides a rainfall guess-field regularly distributed in space. These two devices appear to be complementary : the ponctuai precision of the gages allowing satellite guess-field calibration and satellite coverage compensating for ground network scarcity. A geostatiscal approach seems to be a good way to combine these two sources of information since the rainfall estimation derived from it would take into account the statistical properties of each measurement set and of their combination. After a brief description of the so-called cokriging system-simple extension of kriging '(now a well known technique in hydrosciences, see J.P. DELHMME; 978 or P. DELFINER, 975) when several measurement sets are available, the statistical structure of the gage and satellite measurements will be analysed ; finally a comparison will be performed from the various maps corresponding to the interpolation derived from gages only or to a combination of satellite and gages data. Cokriging : theoretical background. The purpose of this objective analysis technique is to provide an unbiased linear estimator minimizing the rnean # squared error of reconstitution. Let z Q(x ) be the estimated value, at a given point x of the studied area, given by the following linear combination :

5 Spatial relationship between cloud-cover S rainfall fields 85 n i G, G, N a, S, Z G (x o> Z X z (x.) + j; X z (x ) () = = where z and z represent the two types of measurements (ground and satellite) and (x<5, i=l,n) and ( x^ a=,n) the corresponding networks of points or cells. If this estimated value is required to be unbiased, its expectation must fit the expectation of the true value of the phenomenon : E Z G ( V E z (x ) o Since expectation is linear, replacement of () into (2) leads to: (2)?. ', X E z (x G ) G l + " x a E zfx S ) S a = E zfx ) G o = a=l When the variables z and z are supposed to be intrinsically stationary (i.e. second order stationarity of the first order increments of the variables) their expectations can be assumed locally constant in space (E z (x) = m and E z (x) = m ) and equation (3) yields the following system G G b o (3) n Z i=l N Z a= X : X e (4) If the estimated value () is also required to be optimal, in the sense of the minimization of the estimation variance, then the following partial derivatives must be set equal to zero : N E(z*(x )-z(x )) 2 =2 X J C r (x G,x G )+2 n (x G,x S )-2C (x G,x ) (5) GS K 8X i=l and N,a E(z*(x G o )-z(x )) 2 =2 o Z \ B C(x S,x S ) + 2 S a f î Z. G S i a X i C r^(x G,x S )-2C(x S,x)(6) G S a o 3X B=l = the various direct and cross covariance are denoted C, C and C (for instance C (x,x') = E z (x).z (x ). Annulation of the above derivatives combined with equations (4) gives the cokriging system where represents Lagrangian multipliers : C G (x G,x G ) <WV*!P i C GS ( V C S (X!' x ) a x ) B x. x a = = C GS ( V X o» ^G C G (x G,X o ) (6) If the conditions of system (6) are fulfilled, the estimation variance can be written as follows : E(z*(x o )-z(x o )) 2 = C G (x o,x o ) -? X^.xJ) -Z X«C GS (x a S,x o )- Pc = a=l providing a good indicator of the expected precision of the estimator. (7)

6 86 J.D.Creutin et al. If only intrinsic stationnarity is assumed the covariances have to be replaced by variograms, in the same way as in the simple kriging context; for instance the cross variogram gets the following formulation : Y GS (x,x') = X E (z Q (x) - z G (x')).(z s (x)-z s (x')) When the covariances exist and are symetrical, the classical relationship: Y GS U,x' ) = C QS (x,x) -C GS (x,x') also valid for direct variograms, allows to demonstrate simply that an equivalent system to system (6) can be obtained replacing C by-y'. This remark is also valid for expression (7). Structure analysis. The different rainfall fields studied have, as a strong common characteristic, a general shape presenting a peak value located on the South East part of the region and a large rainfree area covering the North West remainding part. This general trend of the phenomenon (also called drift) is consistently evaluated by both ground and satellite devices. The most appropriate way to reduce the effect of such a drift on the structure identification would be to normalize the data using the climatological mean values over a large period. Unfortunately the available sample of events is too small to exhibit ensemble results for such a multirealization approach. So, each field is considered separately and, in order to cancel drift effects, the less affected direction is chosen (generally N45E) to compute the experimental variograms (i.e. only the couples of measurement points approximately oriented in that direction are selected). As shown in table 2 and illustrated in Figure 4, the experimental Table 2 - Description of the fitted models for the direct and cross variograms (for each variogram the sill, the range and the nugget effect are to be read succcessively). Direct variograms Cross vario gram Event Ground Satelli te Ground vs s at. jan jan jjan Mars i Octobre

7 Spatial relationship between cloud-cover & rainfall fields 87 cross and direct variograms have been modelled using a spherical model (i.e. showing a stabilization of its variations after a given distancethe range, around a given value-the sill). October, km January 5-26, k m FIG.4 Experimental direct and cross variograms for two events; solid lines represent the theoretical model fitting experimental values. It can be noted that ranges for satellite variograms are systematically larger than for ground variograms ; this fact is connected with the integrated or block averaged nature of the satellite measurements ; this should also lead to systematically lower sills : this is less evident in this experiment context where drift effects are certainly not fully filtered. The cross structure is generally significant in the sense that sill values of cross variograms are of the same order magnitude that for the direct variograms ; the range values are consistent. This structure analysis indicating a fairly good cofluctuation of the two signals, enables to foresee a good efficiency of the following attempt to combine these two kinds of rainfall measurements (i.e. there is a good chance that satellite data would help to fill the gap between the ground values). Results. The various results obtained from the different steps of this study may be compared through contour maps of the rainfall fields estimated by (i) an appropriate processing of the satellite pictures (see

8 88 J.D.Creutin et al. part III), (ii) a classical interpolation of the ground measurements (actually performed by kriging) and (iii) a combination of these two estimations through cokriging. For practical display reasons, figure 5 only shows the maps corresponding to one of the most caracteristic events (979 January 5-26). On one hand, a good agreement can be noted between the general shapes of the ground and satellite estimations even if satellite significantly iquifer lihiit sao.o aju 3oo.o «au ras FIG.5 Contour maps of rainfall depths for a selected event. Map () is derived from satellite picture, (2) from gage only interpolation S (3) from a combination of data from both devices.

9 Spatial relationship between cloud-cover S rainfall fields 89 overestimates the rainfall in the Eastern part of the study area (comparison of maps and 2). On the other hand, the introduction of satellite information in ground interpolation leads to significant modifications of the isolines in the regions where the ground network density in the lowest. The addition of satellite information may also be appreciated by the plot of standard deviations of estimation provided either by kriging or cokriging (see formula 7). This indicator of the expected error of prediction is mapped for the selected event (see figure 6) showing, after the introduction of satellite information, a significant reduction of the area where the standard deviation exceeds 2 mm. FIG.6 Contour maps of the estimation standard deviations predicted by () cokriging and (2) kriging for a selected event. Concluding remarks This study confirms that rainfall depth assesment from satellite data produces coherent results even when the rainfall parameters are computed by simple linear regression provided that the considered time step is large enough. However, this classical statistical approach appears to be suitably

10 9 J.D.Creutin et al. complemented by using geostatistical concepts in order (i) to appreciate the coherence between satellite and gage measurements (by means of structure analysis) and (ii) to provide,, at unrecorded points, an "optimal" estimation of the rainfall depth (using the cokriging system). Acknowledgments We are grateful to Bureau de Recherches Géologiques et Minières, which provided us the opportunity of carrying out this study. It supplied us satellite images as well as raingage recordings. The DMSP images were provided by the NOAA-CIRES World Data Center for Glaciology in Boulder (United States). As for the GOES.. images, they are collected by CMS-Météorologie Nationale de Lannion (France) during the GATE experiment. References Atlas, D., Eckerman, J., Meneghini, R., Moore, R.K., 982, The outlook for precipitation measurements from space: Atmosphere Ocean, 2, p Barret, E.C., 97, The assessment of rainfall in north-eastern Oman through the integration of observations from conventional and satellite sources: Consultant's report to the Food and Agriculture Organization, 55 p. Delhomme, J.P., 978, Kriging in hydrosciences: Advan Water ResourcesimO, (5), p Delfiner, P., 975, Linear estimation of non stationary spatial phenomena: Advanced geostatistics in the Mining Industry, M. Guarascio et al. Ed., Hingham- Mass., p Van Dijk, A., 98, Precipitation assessment from environmental satellite data for north-west Libya including the Grefara Plain: Consultant's report to the Food and Agriculture Organization, 45 p.