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EXAMINATION OF 'GLOBAL ATMOSPHERIC TEMPERATURE MONITORING WITH SATELLITE MICROWAVE MEASUREMENTS': 1) THEORETICAL CONSIDERATIONS C. P R A B H A K A R A Code 913, NASA/Goddard Space Flight Center, Greenbelt, MD 20771, U.S.A. J. J. NUCCIARONE Hughes-STX Corporation (HSTX), Greenbelt, MD 20770, U.S.A. and JUNG-MOON Y O 0 Ewha Women's University, SeouI, South Korea

Abstract. In recent studies (Spencer and Christy, 1990; and Spencer et aI., 1990) it is suggested that observations at 53.74 GHz made by the Microwave Sounding Unit (MSU), flown on NOAA operational weather satellites, can yield a precise estimate of global mean temperäture and its change as a function of time. Hansen and Wilson (1993) question their interpretation of temporal changes on the grounds that the microwave observations could be influenced by the opacity of the variable constituents in the atmosphere. This issue has broad interest because of the importance of detection of global climatic ¢hange. In order to help resolve this issue, in this study we utilize a radiative transfer model to simulate: (a) the observations of MSU Channel 1 (Ch. 1) at 50.3 GHz, in the weakly absorbing region of the 60 GHz molecular oxygen absorption band; and (b) the observations of MSU Channel 2 (Ch. 2) at 53.74 GHz, in the moderately strong absorption region of the same band. This radiative transfer model includes extinction doe to clouds and rain in addition to absorption due to molecular oxygen and water vapor. The model simulations show that, over the oceans, extinction due to rain and clouds in Ch. 1 causes an increase in brightness temperature, while in Ch. 2 it causes a decrease. Over the land, however, both Ch. 1 and Ch. 2 show a decrease in brightness temperature due to rain and cloud extinction. These theoretical results are consistent with simultaneous observations in Ch. 1 and Ch. 2 made by MSU. Based on theory and observations we infer that a substantial number of the MSU observations at 53.74 GHz used by Spencer et al. contain rain and cloud contamination. As a result, their MSU derived global mean temperatures and long term trend is questionable.

Introduction Passive radiometer measurements made from satellites in the window region (10 to 13 #m) of the infrared spectrum are significantly contaminated by clouds and are not suitable for global mean temperature estimation. On the other hand, microwave radiometer measurements are not strongly affected by non-raining clouds. However the surface emissivity in the microwave region changes significantly between land and water bodies, and also depends on the relative amount of soff moisture (Schmugge, 1978; Wang and Schmugge, 1980). Hence the window regions of the microwave are also not well suited for global mean temperature sensing. For these Climatic Change 30: 349-366, 1995. @ 1995 KluwerAcademic Publishers. Printed in the Netherlands.

350

C. PRABHAKARA ET AL.

reasons, Spencer and Christy (1990) and Spencer et aI. (1990, hereafter referred to as SCG) considered Microwave Sounding Unit Channel 2 (Ch. 2) observations at 53.74 GHz for global mean temperature sensing. The emission upwelling to the top of the atmosphere from the surface, over land or ocean, is less than ~ 8% in this channel. Emission from atmospheric levels between approximately 800 and 400 mb primarily determines the Ch. 2 brightness temperature observed at the top of the atmosphere. From examination of Ch. 2 measurements SCG concluded that clouds and rain have no significant effect on these measurements, except in the case of scattering of microwave radiation by ice in heavy convective rain events. Such scattering is associated with a noticeable depression in the brightness temperature. According to SCG these obviously bad data constitute a small percentage of the total number of observations. Such data were screened out by them with a simple technique. Then the remaining data were used to estimate the global mean temperature. An analysis of Ch. 2 data from 1978 onward using this approach showed there to be no significant global trend in temperature during this observation period. Hansen and Wilson (1993) questioned the above conclusion drawn from MSU Ch. 2 observations. They suggest that the variable constituents of the atmosphere, such as water vapor, clouds, and especially rain, alter the vertical distribution of atmospheric opacity in the microwave region, and hence the Ch. 2 measurements do not always represent the temperature of a single fixed layer, ~ 800 to 400 mb, in the atmosphere. If this is correct, the Ch. 2 data may be seriously compromised for use as a measure of global mean temperature change. Since satellite sensing of global mean temperature is a valuable goal, we would like to resolve the differences in the views expressed above. An important issue in this regard is the adequacy of the method used by SCG to eliminate rain contamination. For this purpose we will examine the effects of clouds and rain in MSU Ch. 1 and Ch. 2 data with the help of radiative transfer theory. The reason for analyzing the measurements in these two channels is as follows. The Ch. 1 measurements at 50.3 GHz, which is in the window region of the 60 GHz molecular oxygen absorption band, respond quite strongly to the effects of clouds and rain in the lower troposphere. The magnitude of this rain and cloud effect in Ch. 2 is about an order of magnitude weaker and could be missed in the presence of meteorological variability or noise. With the help of theoretical analysis, simultaneous observations made by MSU in Ch. 1 and Ch. 2 can be scrutinized to isolate the hydrometeor effect in the presence of meteorological noise.

Theoretical Simulation of MSU Ch. 1 and Ch. 2 0 b s e r v a t i o n s

Tropical atmospheric conditions are used for the simulation of Ch. 1 and Ch. 2 brightness temperatures utilizing a radiative transfer theoretical model. Details of this theoretical model are presented in the appendix. The tropical atmosphere is

GLOBAL ATMOSPHERIC TEMPERATURE MONITORING

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assumed to have a surface temperature of 299 K, with a lapse rate of 5.4 K per km up to the tropopause at 18 km. The temperature increases at the rate of 4 K per km in the stratosphere. Columnar water vapor amount in the atmosphere is 5 g/cm 2 and is vertically distributed with a scale height of 3 km. The rain rate is assumed to be uniform with height up to the top of the rain column at 5 km. The cloud layer extends from 3 to 5 km with a liquid watet concentration CL (g/m 3) assumed to vary as a function of rain rate R as follows. CL =

{0.5R where R < 1 m m / h 0.5(1 + l o g R ) where R > 1 m m / h

(1)

This formulation of cloud liquid water as a function of rain rate resembles that adopted by Hinton et al. (1992). The surface emissivity in the microwave region over the land is assumed to be 1 (this excludes wet lands, rivers, lakes, and snow, where the emissivity can be significantly smaller than 1) at all incidence angles, while on the ocean surface (assumed to be smooth) it is allowed to vary as a function of the incidence angle according to the model of Chang and Wilheit (1989). In particular, the emissivity at 50.3 GHz over the ocean, at 0 ° incidence, is 0.5. In the model atmosphere, the temperature at altitudes above 5 km is less than 272 K. Utilizing this physical condition to simulate the extinction effect of ice in Ch. 1 and Ch. 2, we have assumed that when rain drops are present above 5 km they are in the frozen state. A schematic diagram of the tropical atmosphere showing the thermal stratification, rain below the clouds, and ice particles above the clouds is shown in Figure 1. With the above information the MSU measurements at nadir (i.e. the incidence angle at the surface is 0) in Ch. 1 (50.3 GHz) and Ch. 2 (53.74 GHz) are simulated as a function of rain rate with the theoretical model. Input information pertaining to the rain extinction, single scattering albedo, and phase function, is based on the study of Savage (1976). Savage has presented tables of these quantities for rain and frozen rain, for several discrete frequencies in the range 19 through 231 GHz. A Marshall-Palmer size distribution and a rain drop temperature of 0 °C are assumed in these input data. The absorption coefficient of water vapor, cloud liquid water, and transmission due to molecular oxygen, are derived from the study of Chang and Wilheit (1979). This input information is presented in Table I and Figure 2. From Table Ia we notice the single scatter albedo a; of frozen rain particles is close to 1 which leads to strong scattering as compared to rain drops. The distinct difference in the transmission of radiation at 50.3- and 53.74-GHz due to atmospheric 02 is evident from Figure 2. At 50.3 GHz the atmospheric 02 is nearly transparent while the channel at 53.74 GHz is almost opaque. The results of the computations of the brightness temperatures Tl (Ch. 1) and T2 (Ch. 2), obtained for the nadir direction over ocean and land are shown with the label No Iee in Figures 3a-d as a function of rain rate. Also in these figures computed values of T1 and 772 are shown with the label With Iee when there is a

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maximum value for clear sky and then steadily decrease with an increase in rain rate. The approximation of a plane parallel stratified atmosphere or horizontal homogeneity may be applicable to the scale ofcloud cells (~ 1 km). The MSU radiometer field of view (fov) is large, about 110 km, compared to this fine scale. As the rain rate varies on the scale of the cell size the distribution of rain in the for tends to be highly inhomogeneous in the horizontal. In order to obtain the brightness temperature in the fov containing an inhomogeneous rain rate we utilize a procedure analogous to the one developed by Hinton et al. (1992). It is assumed that on a fine scale ( ~ 1 km) rain is horizontally homogeneous. Further the spatial distribution of rain in thefor is described with the fine scale probability distribution of rain rate P(R). Based on the fact that f ö P(R)dR is equal to 1, we can define P(R)dR as the fractional area of the radiometerfovthat has rain rate R. Following Hinton et al. (1992), the probability distribution P(R) is assumed to be a gamma distribution that was proposed by Short and North (1990). For a rain rate R > 0, the gamma distribution is

P(R)

= [F(ct)fl~]-lR c~-I e x p - ( R / i l ) .

(2)

The advantage of this distribution is that the mean # and the standard deviation « are given by # = oz/3 and o.2 = c~/32. By choosing/3 = 10 and varying oz we have obtained a range, 0 to 8 mm/h, for thefov weighted rain rate. We note that the maximum rain rate of 8 mm/h, on a scale of 110 km, is about an order of magnitude larger than the climatological estimate over the rainy inter-tropical convergence regions over ocean (see e.g. Dorman and Bourke, 1979). With this probability distribution we have approximated the weighted mean brightness temperature as follows. (Th(R)) ~ To [1

/ ~ P(R)dR1 +

P~

Tb(R)P(R)dR

(3)

where R0 -- 0.0001 mm/h is assumed to be the cutoffpoint between the raining and non-raining area within the radiometerfov, and To is the brightness temperature measured by the radiometer when R = 0 mm h -~. An upper limit of R = 50 mm h -1 is used in the above integration. In Equation 3, assuming/3 is equal to 10, we find the fractional area f~ P(R)dR is 0.72 when # equals 1, while it is 0.47 when # equals 0.5. This shows the importance of the parameters of the gamma distribution in determining {Tb(R)). Again taking -R0 as the cutoff point between the raining and non-raining portions of the f o r we can approximate thefov-weightedmean rain rate {R) as:

=

P(R)RdR

o~/3 ~

(4)

0

Following the method given in Equations 3 and 4 we have computed the brightness temperatures (Tl/ and (T2> as a function of rain rate (R). These weighted mean

356


values are assumed to represent the MSU data when the rain rate distribution in thefov is inhomogeneous. In Figure 3 (Tl) and (T2) plotted as a function of (R) (the averaging notation '()' is not shown) are identified with the label 'weighted'. We note that the weighted T1 and T2 vary rauch more gradually with respect to R, while the unweighted T1 and T2 vary substantially. This is mainly due to the smoothing introduced by the weighting procedure. An important objective of this study is to compare T1 vs. T2 and thereby infer the effects of hydrometeors. For this purpose in Figures 4a and b plots of Tl vs. T2, representing ocean and land rain scenes, are shown. In these figures theoretical simulations with a plane parallel assumption and with an assumption of a gamma rain rate distribution are shown. From these figures, as stated earlier, we get a negative correlation between T1 and T2 over oceans and a positive correlation over land. The negative correlation over ocean is primarily a consequence of the sea surface emissivity. Based on the results shown in these figures it may be remarked that the negative slope of the regression line between T1 and T2 deduced over the tropical ocean is significantly increased by the horizontal inhomogeneity of the hydrometeors. The positive slope of this regression line over the land is not critically dependent on the nature of the horizontal distribution. The foregoing theoretically derived results and discussion demonstrate the effects of clouds and rain on the MSU Ch. 1 and Ch. 2 measurements. As a next step we utilize the radiative transfer formalism to analyze the sensitivity of these MSU channels, over oceans and land, to perturbations in surface temperature (~Ts), surface emissivity (&s), and atmospheric temperature (~T*). These sensitivities are presented in Table II. In these calculations the tropical model atmosphere, described earlier, is used. To ger the MSU Ch. 1 and Ch. 2 sensitivity to air temperature T*, the physical temperature of the air between the surface and 200 mb is uniformly changed by 1 K. The surface emissivity over land is taken as 1, and 0.5 over the ocean. The sensitivity to rain rate R is deduced from the calculations presented in Figure 3a-d. Specifically this sensitivity is based on the weighted mean values (Tl) and (T2) presented in these figures. In Table II the ratio r = (~T2/~T1) is shown as a sensitivity factor of Ch. 2 with respect to Ch. 1. Clearly, MSU Ch. 2 is most sensitive to atmospheric temperature. The ratio r is at least an order of magnitude weaker for rain, but rain yields the second largest ratio. From Table II we note that r due to rain has a negative sign over ocean, while all other ratios, over land and ocean, are positive. The implication of this result is a negative correlation between MSU Ch. 2 and Ch. 1 measurements over oceans can be used to infer the occurrence of extinction due to clouds, rain, and ice in the atmosphere. This signature is very helpful to pick up the rain signal over ocean in the presence of other meteorological variability or noise of other parameters (~Ts, Ges, 6T*).

357


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Land 255 . . . . . . . . . . . . . . . . . . . ' .............. ' .... 255 260 265 270 275 280 285 290 295 T 1 (K) Fig. 4. Plots of weighted and unweighted Tt vs. T2. Simulations weighted according to gamma rain rate probability are shown with a line of circles and dashes. The solid line indicates the plot of unweighted T1 vs. T2. The point corresponding to clear sky is indicated. The dashed line indicates no cloud and rain contamination. (a) Over ocean. (b) Over land.

Comparison with MSU Observations and Discussion A key issue, as stated in the introduction, is the adequacy of the method used by SCG to eliminate M S U Ch. 2 data contaminated by rain. To test the adequacy of this method, we have implemented it with the help of the detailed account given by Spencer and Christy (1992). This method takes advantage of the cross track scanning by MSU. To describe it in brief, first a correction for limb darkening is applied to the Ch. 2 data along a scan line. Then any measurement in that

358


TABLE II Theoretically derived sensitivity of MSU Ch. 1 and Ch. 2 brightness temperatures, in the nadir directions, to various perturbations Perturbation

BTs = 1 K &s = 0.1 BT* = 1 K 6Æ = 1 mrrdh

Land

Ocean

((~T2/~T~)

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~T2 (K)

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~T1 (K)

~T2 (K)

r =

0.75 17.4 0.26 -3.5

0.08 0.4 0.92 ~).5

1/10 1/44 3/1 1/7

0.37 17.4 0.34 4

0.04 0.4 0.96 -0.5

1/10 1/44 3/1 -1/8

scan, including the one at nadiß is screened out if it is more than 0.5 K cooler than the average of the two adjacent limb corrected measurements. This method, called hefe the screening test, is applied to the MSU nadir data in this study as a preliminary screening. After this preliminary screening, we evaluate the adequacy of this method. This study focuses on the data obtained at nadir (the incidence angle at the surface is 0), where the rain contamination can be detected more easily. We may point out this contamination is appreciable even when the nadir angle is close to 45 °. In Figure 5a all the measurements in Ch. 1 and Ch. 2 made by MSU in the nadir direction in a grid box of 3 ° latitude by 5 ° longimde centered at 158.5 ° E and 0.5 ° S over a rainy equatorial Pacific Ocean region during the month of March 1982 are plotted to show the negative correlation between them. When we apply the screening test described above, out of a total of 45 data points, 17 are eliminated. The remaining 28 data points, which passed the screening test are shown in Figure 5b. The negative correlation between Ch. 1 and Ch. 2 observations is still present in this screened set. When we compare the plots of Ch. 1 rs. Ch. 2 M S U observations shown in Figure 5b with the simulations shown in Figure 4a, we see a close similarity. Based on this comparison and the sensitivities presented in Table II we conclude that a large number of screened data shown in Figure 5b are not free from cloud and rain contamination. A similar analysis is done in Figures 5c and d with MSU nadir observations during March 1982 over a tropical land area of 3 ° latitude by 5 ° longitude over Zaire, centered at 25.5 ° E and 0.5 ° S. The positive correlation between Ch. 1 and Ch. 2 observed in Figure 5c persists after the data are screened as shown in Figure 5d. Theoretical simulations for land area shown in Figure 4b and the sensitivities given in Table II are consistent with the positive correlation. This leads us to a plausible conclusion that cloud and rain contamination is not completely eliminated over land areas by the screening method. However, this conclusion is not as definitive over land as it is over ocean, because changes in ~Ts, &s, ~ST* can contribute to the positive correlation.

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Based on the detailed analysis ofMSU Ch. 1 and Ch. 2 data in the tropics carried out in this study, we conclude that in rainy regions there are a substantial number of Ch. 2 data at 53.74 GHz that are contaminated by clouds, rain, and ice. These data were not successfully screened out by the test used by SCG. This point is elaborated further by the analysis of Figure 6, which shows Ch. 1 data taken along nadir during March 1982 over a large section of the Inter-tropical Convergence Zone in the Eastern Pacific. The number of observations as a function of the Ch. I brightness temperature is shown along with the percentage of the data that are screened out with the screening test (i.e., data that show more than a 0.5 degree cooler temperature at nadir in Ch. 2 with respect to the average of the adjacent two limb corrected observations). There a r e a total of 660 observations, and 553 passed the screening test. One notices distinctly from the figure that the percentage of the data screened out is large at large Ch. 1 brightness values and small at low values of Ch. 1 brighmess. Based on the theoretical simulations presented in Figure 4a, we infer that there is a small contamination in the Ch. 2 data over tropical oceans eren in the presence of clouds and light rain. This small contamination can be readily inferred with the help of the details given in the figure. By definition the line labeled no contamination goes through the point 'clear sky' and is parallel to the z axis. The curve labeled 'weighted' represents a gamma distribution of rain rate and is appropriate for this smdy. This curve deviates significantly from the no contamination line. This deviation becomes larger as Ch. 1 temperature rises. Since the Ch. 1 weighted mean brightness temperature is directly dependent upon rain rate (see Figure 3a) we conclude that this deviation is increasing with the rain rate. Based on this information we conclude there is in Ch. 2 a cloud and rain contamination on the


3 61

order of 0.5 K when Ch. 1 temperature is in the fange of 230-240 K. In addition, from Figure 6 where the frequency distribution of MSU channel 1 data is shown, we note a maximum number of observations are contained in this temperature interval. Only a small percentage of these observations, as shown in Figure 6, reveal a contamination in excess of 0.5 K. These few data are excluded by the screening test. The remaining large number of observations in this temperature interval, which have contamination smaller than 0.5 K, are not screened out. The cumulative effect of these large number of contaminated data is not negligible. In order to illustrate the nature of the data shown in Figure 6 we show plots of Ch. 1 vs. Ch. 2 of all (660) data in Figure 7a. After screening, the remaining (553) data are plotted in Figure 7b. The negative slope or the correlation between Ch. 1 and Ch. 2 persists eren after screening. Presence of the negative slope can be noticed even when Ch. 1 brightness is less than 240 K. Based on the data presented in Figures 6 and 7, one can conclude that the screening test is not completely eliminating the rain contaminated data. The screening test assumes that the rain contamination is localized to a single footprint. This constitutes a weakness in this method. For example, when all adjacent footprints of the MSU data are contaminated by clouds and rain to a similar degree, the screening test fails to detect contamination. But Ch. 1, with a large sensitivity to clouds and rain, can reveal the contamination. This is the advantage offered by a multispectral approach. In the tropics the atmosphere tends to be barotropic where propagating meteorological disturbances that produce rain are not accompanied by large changes in temperature. For this reason we are able to isolate the rain contamination in Ch. 2 data in the tropics. In the mid-latitudes there are baroclinic effects and significant thermal perturbations in the atmosphere occur along with precipitation. These perturbations can mask the rain signal in Ch. 2 (see Table II). Hence in MSU data taken from the mid-latitudes we could not demonstrate the rain contamination in a simple manner.

Conclusions The principal finding of this study is that MSU Ch. 2 data that were screened to eliminate rain contamination, and then used by SCG to obtain the global mean temperature, are not strictly a measure of the thermal state of the surface and the troposphere up to ~ 400 mb. These screened Ch. 2 data contain a contamination due to various constituents of the atmosphere, in particular Clouds, rain, and ice. This study is limited to an oceanic region in the tropics to establish in principle the nature of the cloud and rain contamination in the MSU Ch. 2 data. In a forthcoming study we plan to extend these results to demonstrate the hydrometeor effects on a global scale with maps and figures. Other possible contaminations, such as that due

362

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363

to surface emissivity, may have to be examined to get a complete understanding of the global mean temperature derived from MSU Ch. 2 data. Study of global mean temperature with satellite data is a valuable goal. Spencer et al. have made a significant first attempt to demonstrate the potential of satellite remote sensing. Global mean temperature changes with time are small, on the order of 0.1 K per decade. This requires a careful examination and understanding of all the factors influencing the satellite data at this level of accuracy. As our understanding of the physics of radiative transfer applicable to the various constituents of the atmosphere and the surface improves, we should be able to make progress in that direction.

Appendix Radiative Transfer Consideration The radiative transfer equation, including absorption and scattering processes in the atmosphere, and assuming local thermodynamic equilibrium and plane parallel stratification, can be written as follows at a given frequency of radiation in the microwave region (Chandrasekhar, 1960; Weinmann and Guetter, 1977):

# dTb(T' dr #) -- --Tb(T' #) -~- - ~

1 P(#, #')Tb(7-, # ' ) d # ' +

+ { 1 - aa(~)}r(~)

(1)

where: Tb (#, 7-) is the brightness temperature of radiation along a direction given by the direction cosine # at an optical depth r in the atmosphere. aa is the single scatter albedo of both the cloud particles and the atmosphere. co = k s / ( k s + ka + kg), ks and ha are the volume scattering and volume absorption coefficients of the particles, while kg represents the total volume absorption coefficient due to gases in the atmosphere. P is the azimuthally averaged scattering phase function. T is the local temperature of the atmosphere. The two boundary conditions required to solve Equation 1 are the brightness temperature downwelling on the top of the atmosphere, and upwelling at the bottom of the atmosphere. These are given below. Tb(#-,

~- = 0) = 3 K

~b(,+,

~ = ~~) = T~~~(~) + {1 -- ~ ~ ( ~ ) } T b ( , - ,

r=r~)

(2)

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Rain Rate (mm/h) Fig. A1. Comparison of T37, the brightness temperature at 37 GHz, over tropical ocean computed as a function of rain rate. The sea surface temperature is 298.8 K, the rain column height is 4.8 km, and the incidence angle is 50 ° . The solid line is the result deduced from the model used in this study and the dashed line is the result of Olson (1987). (a) Horizontal polarization. (b) Vertical polarization.

where T b ( # - , ~- = ~-s) is the downwelling brightness temperature incident on the surface. Ts and Cs are the surface temperature and emissivity. We solve Equation 1 numerically by the method of successive orders of scattering. This method involves computation of the source function and intensity using the recursion relationship between them (see e.g. Liou, 1980; Olson, 1987; Prabhakara et al., 1988) for each successive order of scattering. It is found that after ten orders of scattering the error in neglecting higher orders amounts to less than

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0.1 K in brightness temperature emerging at the top of the atmosphere. To test the accuracy of our method we have compared our results with those of Olson. A case with only rain and no cloud liquid water absorption is considered to demonstrate the agreement between the two methods. In our computation model the input information - i.e. the temperature and water vapor profiles in the atmosphere, the sea surface emissivity, absorption due to water vapor, extinction and single scattering albedo due to raindrops, and the altitude of the rain column - is adopted from Olson. The phase function is not supplied by Olson; we use the Henyey-Greenstein function. The asymmetry parameter, g, of the phase function used in these calculations is adopted from Savage (1976). The value of g at 37 GHz is small; about 0.02 for a rain rate of 1 mm/hr, and about 0.07 for a rain rate of 20 mm/hr. With these inputs the 37 GHz brightness temperatures in the vertical and horizontal polarization, upwelling at the top of the atmosphere, over ocean at an incidence angle of 50 ° are computed as a function of rain rate and are compared with the results of Olson in Figures A l a and b. In both of these model calculations it is assumed that the clouds and rain do not polarize the radiation. The agreement between these two sets of brightness temperatures, over the range of rain rates 0 to 16 mm h -1, is within ~ 2 K. This accuracy is satisfactory for the purpose of the present study because this small difference is systematic in nature. Specifically our interest is to estimate the increase in brightness temperature, AT, from 0 rain rate to any given rain rate. The small systematic difference (~ 2 K) is on the average about 5% of A T over the range of rain rates 0 to 16 mm h -1 (see Figures A l a and b).

Acknowledgements We are thankful to Dr. Robert S. Fraser for his valuable comments and for his help to improve the paper.

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