INTERNATIONAL JOURNAL OF CLIMATOLOGY, VOL. 16, 6 6 3 4 7 5 ( 1 996)

COMPARISON OF TWO MODELS RELATING PRECIPITABLE WATER TO SURFACE HUMIDITY USING GLOBALLY DISTRIBUTED

RADIOSONDE DATA OVER LAND SURFACES

B. J. CHOUDHURY Laboratory for Hydrospheric Processes, Hydrological Sciences Branch, NASAIGoddard Space Flight Center; Greenbelt, MD 20771, USA

Received 22 February 1995 Accepted 18 September 1995

ABSTRACT

Relationships between precipitable water (W; nun) and surface vapour pressure (eo; hPa), and surface dew point temperature ( t d ; "C) are evaluated using mean monthly radiosonde data for 24 months at 45 globally distributed locations both for individual locations and for the pooled data from all locations. The data selected cover a wide range of climatic conditions and complete seasonality for most of the locations. The physical basis of the models relating W to eo or td is reviewed briefly. Two semi- empirical models, namely, W= Cleo + C, and In W= c $ d + C4, gave highly significant correlations both for individual locations and for the pooled data (3 = 0.94, C1 = 1.70, C2 = - 0.1, C, = 0.058, and C4 = 2.42 for the pooled data), but examination of the residuals showed some inadequacy of these models for the pooled data.

KEY WORDS: precipitable water; vapour pressure; dew point temperature; regression.

INTRODUCTION

A quantitative knowledge of precipitable water is required in a range of studies, including estimation of incident longwave and solar radiation (Monteith, 196 1 ; Idso, 1969), initialization of numerical weather prediction models and analysis of weather systems (Lowry and Glahn, 1969), and atmospheric corrections to satellite observations in different wavelengths (Soufflet et al., 1991; Choudhury, 1993). The sparsity and infrequent nature of radiosonde data needed for calculating precipitable water has led to the development of a number of models for estimating precipitable water from surface humidity observations (vapour pressure, dew point temperature, absolute humidity, and mixing ratio). Although relations between precipitable water and surface humidity variables can be observed due to vertical mixing of the air mass (Reitan, 1963; Smith, 1966), decoupling of the atmospheric layers under inversion conditions can significantly affect such relations (Schwarz, 1968; Glahn, 1973; Karalis, 1974; Tomasi, 1977). The success of predicting precipitable water from surface humidity will depend upon the extent that surface humidity variables can characterize the variabilities of the atmospheric moisture profile.

Models relating precipitable water and surface humidity variables are reviewed briefly in the next section and their interrelationships are noted. The objective of this study is to compare two models for estimating precipitable water from surface humidity variables (vapour pressure and dew point temperature) using 24-monthly mean radiosonde data at 45 globally distributed locations. Relationships between precipitable water and surface humidity variables are evaluated both for individual locations and for the pooled data from all locations by calculating various statistics. The performance of these models has been assessed by calculating skewness and root-mean-square error. The present study extends previous model comparisons by Tuller (1968) for a global data set, and by Ojo (1970) over Nigeria, Tuller (1 977) over New Zealand, Karalis (1974) over Athens, Mohamed and Frangi (1983) over the Sahel and Adedokun (1989) over Central and West Africa.

After reviewing the physical basis of the models, the data and methodology are presented, followed by results and discussion. The data selected for the present analysis cover a wide range of climatic conditions and also captured the seasonality.

CCC 0899-84 18/96/060663-13 0 1996 by the Royal Meteorological Society

664 B. J. CHOUDHURY

Table I. Selected summary of regression slope (C,) and intercept (Cd) for equation (2) for the annual period

Author Slope Intercept Comment

Reitan (1 963) 0.061 2.413 15 locations, USA Lowry and Glahn (1 969) 0.058 2.3 18 56 locations, USA Ojo (1970) 0.050 2.573 5 locations, West Africa McGee (1 974) 0.066 2.265 7 locations, South Africa Tuller (1 977) 0.07 2.09 Christchurch (see reference for other data) Mohamed and Frangi (1 983) 0.058 2.533 Niamey (Niger)

REVIEW OF THE MODELS

The model proposed by Hann (1 906) related precipitable water W (units of mm) and surface vapour pressure eo (units of ma) as:

W = Cleo ( 1 4 where C, = 1.73.

A more generalized relation between Wand eo was found by Hay (1 970) for 33 locations over Canada (monthly mean data for the period 1957-1 964) and Tuller (1 977) for two locations over New Zealand (daily data for 1974):

W = Cleo + C,

where the mean slopes (C, ) were 1.61 and 1.53, respectively, for Canada and New Zealand, and average intercepts (C,) were 1.4 mm and - 1.9 mm. The explained variances (2) were about 98 per cent and 74 per cent, respectively. Such a linear model has been used by Garrison (1992) and others to compute K

(lb)

Reitan (1963) found the following relation between Wand dew point temperature td (“C):

In W = c 3 f d + C4 (2) where C, = 0-061 and C4 = 2.41 for 36 monthly mean data at 15 locations over the USA, with the explained variance of 98 per cent.

The above form of the relationship found by Reitan has been studied more extensively than has equation (1) (cf. Schwarz, 1968; Lowry and Glahn, 1969; Reber and Swope, 1972; Karalis, 1974; McGee, 1974; Tuller, 1977; Daoo et al., 1982; Mohamed and Frangi, 1983; Revuelta et al., 1985). A selected summary of C3 and C4 based on the analysis of data for the annual period is given in Table I. By considering a mixing ratio profile as

w = wo(P/Po)~ (3) where wo and PO are, respectively, the surface values of the mixing ratio and air pressure, w and P are the corresponding values at some height in the atmosphere, and A is a parameter characterizing the profile, Smith (1966) derived the following relationship between Wand wo:

(4)

wo = 0.622(e0/P0) ( 5 )

Cl = 6*36/(A + 1) (6)

w = Powo/Ig(A + 1)pJ where g is acceleration due to gravity and pw is the density of water. Because wo is related approximately to eo as

it was possible to derive equation (la) by inserting equation ( 5 ) into (4) to express C1, appearing in equation (la) as:

With the following equation relating eo to tb due to Tetens (1930),

eo = 6 . 11 eXp( 17.27td/(237.3 -I- td)} (7) Smith (1966) derived the following generalized relation between In Wand td by taking the logarithm of equation (4) to explain the correlation observed by Reitan:

In W = 0.071td + (3.674 - In(l+ 1)) (8)

RELATIONSHIP OF PRECIPITABLE WATER TO SURFACE HUMIDITY 665

(The factor (237.3 + td) appearing in equation (7) was taken as 244°C.) Seasonal and latitudinal variations of I, and hence of C, in equation (2), have been noted in Smith (1966) and Viswanadham (1981), although such variations have not always been observed (Lowry and Glahn, 1969). A linear relation between Wand eo or In Wand td should hold, except for the variabilities introduced by A. Representation of vapour pressure other than by equation (7) (Tabata, 1973) would provide a somewhat different relationship between Wand t d .

It is possible to use height variations of atmospheric moisture parameters other than the mixing ratio to obtain relationships between W and surface humidity (Reitan, 1963). With the following profile for vapour pressure variation with height z (km) by Yamamoto (1 949):

e = eoexp{-(5.8 x 103y/T$ + 0.055)} (9)

where y is the temperature lapse rate (K km-I) and To is the surface temperature (K), and air temperature variation with height as (Brutsaert, 1975)

T = TO expt -(ylTo)zJ (10)

one obtains the following absolute humidity (p) profile:

p = IMveo/(RTo)J exp(-z/W

where Mv is the molar weight of water vapour, R is the universal gas constant, and H is the scale height (km) for absolute humidity, given by

H = 1/[5.8 x 103y/Ti - y/To + 0.0051 (12)

W = I MvH/ ( R To PW) 1 eo (13)

CI = IMvff/(RTopw)} (14)

Now, one can obtain the following relation between Wand eo by integrating equation (1 1):

where pw is the density of water. Comparison of equations (1 a) and (1 3) gives the coefficient Cl of equation (1 a) as:

which can have seasonal and spatial variations through y and To, similar to that introduced through the parameter 2 in equation (6). For values of the parameters in equations (12) and (14) for the US Standard Atmosphere, namely y = 6.5 K km-' and To = 288 K, one obtains C, = 1.55, which is close to the values obtained from regression analysis of Wand eo quoted above. A slightly modified form for equation (14) would be obtained if instead of considering a single exponential representation (equation 1 1) one considers a double exponential representation of the absolute humidity profile (Tomasi and Paccagnella, 1988; Parameswaran and Krishna Murthy, 1990).

It is worth noting that a finite intercept C, appearing in equation (lb) does not appear directly in the model derivations. Also, the semi-empirical derivation of equation (2) as given by equation (8) follows from equation (1 a) or (4), i.e. when CZ = 0. Thus, equations (lb) and (2) could be considered to represent independent models, to a degree. A finite positive intercept (C,) may be realized when the effective scale height (H) is considered to increase for very low values of eo due to inversion (a decrease in the effective value of y in equation 12).

Although the above relations between Wand surface humidity variables can be obtained theoretically, Monteith (1 96 l), Idso ( 1 969) and Mohamed and Frangi (1983) have found significant correlations between In Wand the square root of vapour pressure (e2 .7 as:

In w = Cseo0.5 + CS (1 5) where the coefficients C5 and c6 were 0.67 and 0.46 in Monteith, 0.57 and 0.97 in Idso, and 0.47 and 1.41 in Mohamed and Frangi. It is interesting to note that a higher value of c6 is associated with a lower value of C5 (i.e. a negative correlation between C5 and C6). Although a relation of the form of equation (15) may not be derived directly, one could approximate he0 or td by a linear function of e2.5 for certain ranges of vapour pressure, and thus equation (1 5) may be considered as an approximation obtainable from equation (la) or (2). For example, in the vapour pressure range of 5 to 18 hPa (pertinent in Monteith) one could approximate he0 (by a least square analysis) as 0.23 + 0.64e00'5, and for the range of 8 to 28 hPa (pertinent in Idso) the approximation could be 0-74 + 0.50e00'5. These least-squares approximations illustrate the appearance of a higher intercept (c6) with a lower slope (C5) (i.e. a

666 B. J. CHOUDHURY

negative correlation between C5 and C6) in equation (15) as stated above. Note that the studies conducted by Monteith (1961) and Idso (1969) pertained to estimating the optical depth of water vapour from the surface observations rather than to estimating precipitable water per se.

Adedokun (1 986) tested several log-linear and log-log relations between precipitable water and specific humidity (q, in units of g kg-') for observations over West African stations, which showed that a log-log relation had the highest correlation and lower coefficient of variation. The relation obtained was:

In W = C7 lnq + C8 (16)

where C7 = 1.388 and c8 = 0.026 for average aerological data representing the mean characteristics over eight stations, with the explained variance of 96 per cent. This study (Adedokun, 1986) and a later study (Adedokun, 1989) with an expanded data set (monthly mean upper air data from 13 West and Central African stations for the period 1971-1984) found that equation (16) performs better overall than a In W-td model such as equation (2), except during the Harmattan season (low humidity).

Equation (1 6) would be consistent with equation (1 a) only if C7 does not deviate from 1 .O within a prescribed confidence level; otherwise the W-eo relationship implicit in equation (16) would be non-linear. Adedokun (1986) found C7 = 1.039 and c8 = 0.997 for stations within the northern West African zone, but C7 = 3.1 54 and c8 = 5.04 for stations within the southern West African zone, and noted that no single relationship was found suitable for both zones. It is interesting to note that equation (16) could be expressed as equation (1 a) for the northern zone because C7 was found to be close to 1 , and, with C7 = 1, one can calculate C1 = 622 exp (cg)/Po. Now, for a surface pressure (Po) of 1000 hPa one obtains C1 = 1 -69, which is in good agreement with the previous estimates of C1 quoted above. Because In W is unlikely to be negative brecipitable water less than 1 mm would have to be an exceptional case), this non-linear model (equation 16) can give highly variable values of C7 and c8 for locations having a seasonal variation of q from less than 1 (i.e. eo less than about 1.5 hPa) to greater than 1, because of the logarithmic dependence on q. The results of Hay (1970) over Canada leading to equation (1 b), which are also found in this study (see below), show that while W remains greater than 1 mm, eo decreases below 1 hPa during the winter at many locations in the high northern latitudes. Thus, applicability of equation (16) would appear to be limited.

The above review of the models relating W to surface humidity variables was attempted in order to provide a background for semi-empirical basis of and possible interrelationships among the models. Radiosonde data at globally distributed locations have been used to evaluate the models give by equations (1 b) and (2). The objective was to assess the nature of variability of the model parameters. These models have been evaluated for individual locations and also with the pooled data from all locations. The locations selected for this study cover a broad range of climatic conditions found within 68"N to 40"s.

DATA AND METHODS

Radiosonde data used in this study are the mean monthly values of air temperature and dew point depression at the standard pressure levels published by NOAA ( 1 985-1 99 1). The station selection attempted to satisfy the following: (i) to be away from extensive water bodies so as to minimize coastal influence (Glahn, 1973; Karalis, 1974; Adedokun, 1989), (ii) availability of air and dew point depression data up to the pressure level of 500 hPa (relative humidity has been set to zero at 300 hPa in calculating the precipitable water), (iii) station height be not more than 1 km a.s.1. in order to minimize possible altitude influence on the relationships between Wand surface humidity (Tuller, 1968; Hay, 1970), and (iv) providing a broad range of climatic conditions. The 45 stations selected, their latitude and longitude, time of observations of the radiosonde data, climatic condition as designated by the Koppen classification (Miiller, 1982) and mean values of the precipitable water and vapour pressure are given in Table 11. The locations are also shown in Figure 1. The accuracy of the surface and upper air data has been assessed subjectively by comparing with other data (nearby stations, same month of other years), and reasonable effort has been made to capture a complete seasonal cycle and fairly uniform distribution over the range in the data selection. All reported data have not been included in the analysis.

Different climatic regimes are not represented equally by the stations selected because of the availability of the data and the spatial extent of these regions. The number of stations from cold snowy and moist all year (Koppen classification: Df), temperate rainy (Cf), and desert (BW) climates are more than the stations from tropical savanna

RELATIONSHIP OF PRECIPITABLE WATER TO SURFACE HUMIDITY 667

Table 11. Name of locations, latitude and longitude (in degrees, positive for north and east), time of observations code (1 for 0000 GMT, 2 for 1200 GMT and 3 for 0000 and 1200 GMT) where available, climatic condition of the location as given by the Koppen classification (Muller, I982), number of months of data used N, average precipitable water W (mm) and vapour pressure

eo ( h W

Name (Latitude, Time Koppen N W e0

Alice Springs (-23.8, 133.9) - BS 24 16 9

Bamako (12.5, -8.0) 3 AW 24 28 17 Berlin (52.5, 13.4) 2 Cf 24 12 9 Bismarck (46.8, - 100.8) 2 Df 24 12 7 Brasilia (-15.9, -47.9) 2 Aw 24 27 17 Camp0 Grande (-20.5, -54.6) 2 Aw 24 23 14

Cita (52.0, 113.3) 1 Dw 24 7 4 Fort Smith (60.1, -111.9) 2 Df 24 9 5

G u an g zh 0 u (23.1, 113.3) 1 c w 24 41 20 Hail (27.4, 41.7) 3 BW 24 9 I

Hyderabad (17.4, 78.5) 3 BS 24 32 20 In Salah (27.2, 2.5) 1 /3 BW 24 15 9 Jakutsk (62.1, 129.7) 1 Df 24 7 4 Kalgoorlie (-30.8, 121.4) - BW 24 17 11 Khartoum (15.6, 32.5) 2 BW 24 26 14 Kolpasev (58.3, 82.9) 1 Df 24 9 5

Lucknow (26.7, 80.9) 3 c w 24 33 21 Madrid (40.5, -3.6) 3 c s 24 12 9

Minsk (53.9, 27.5) 1 Df 24 14 9

Mt Isa (-20.6, 139.5) - BS 24 22 13 Muenchen (48.2, 11.5) 3 Cf 24 14 9 Nancy (48.7, 6.2) 2 Cf 24 14 9 Nashville (36.3, -86.6) 2 Cf 24 23 13

Niamey (13.5, 2.2) 3 BS 24 29 17 Noman Wells (65.3, -126.8) 2 Df 24 9 4 Olenek (68.5, 112.5) 1 Df 24 6 3

Riyadh (24.9, 46.7) 2 BW 24 13 7 Turuhansk (65.8, 88.0) 1 Df 24 8 4

Upington (-28.6, 21.3) I BW 24 15 9 Vilhena (-12.8, -60.1) 2 Aw 24 32 20 Wagga (-35.1, 147.5) - Cf 24 18 12 Washington (39.0, -77.5) 2 Cf 24 20 11

longitude)

Ashabad (38.0, 58.3) 1 BS 24 16 8

Chiang Mai (1 8.8, 99.0) 1 c w 24 43 23

Giles (-25.0, 128.3) - BW 24 14 8

Harkov (49.9, 36.3) 1 Df 24 13 8

Kustanaj (53.2, 63.6) 1 Df 24 11 6 Longview (32.3, -94.6) 2 Cf 24 28 16

Manaus (-3.1, -60.0) 2 Af 24 47 26 Maniwaki (46.4, -75.9) 2 Df 24 13 8

Moree (-29.5, 149.8) 1 Cf 24 21 14

Neuquen (-39.0, -68.1) 2 BW 24 11 8

Orenburg (51.7, 55.1) 1 Df 24 11 6 Resistencia (-27.4, -59.0) 2 Cf 24 27 18

Ubon Ratchathani (15.3, 104.9) 1 Aw 24 46 25

(Aw), steppe (BS), warm temperate with dry winter (Cw) or with dry summer (Cs), cold snowy with dry winter (Dw), and tropical rainforest (Af) climates. High values of Wand eo are found in Af, Aw, and BS climates, whereas low values of Wand eo are found in Dw, Df, and BW climates. All stations satisfy the selection criteria except for Brasilia (Aw), which is at 1061 m a.s.1. This station was, nevertheless, selected to increase the number of stations with higher values of IY The data for all months of a year are represented for most stations; exceptions are that

668 B. J. CHOUDHURY

Figure 1. Location of stations used in the present analysis

February and March are missing for Camp0 Grande, May for Maniwaki and Norman Wells, July for Madrid, September for Longview, November for Vilhena, and December for Alice Springs, Resistencia, and Wagga.

Ordinary least-squares linear regression was used to determine the parameters for the models given by equations (Ib) and (2) by calculating the explained variance (?), slope and intercept together with their standard deviations and root-mean-square error (Draper and Smith, 198 1). These statistics were calculated for individual stations and all stations taken together (pooled data). Residuals have been plotted and skewness has been calculated to assess any model deficiency.

Histogram frequency distributions of the data values (Wand eo) used in the present analysis are shown in Figure 2. The distributions of these data values are seen to be not uniform although such a distribution was desirable for the linear regression analysis of Wand eo. Although extreme values appearing in isolation can bias such analysis, we see that the extreme values in the data set are tied to a tail rather than appearing in isolation. Also, in the regression analysis of In Wversus td it would be desirable to have a uniform distribution for In Wrather than for which would be difficult to create. Nevertheless, as noted above, the present data set is less represented by stations from tropical climates having high values of W

RESULTS AND DISCUSSION

The linear regression results for both models (equations 1 b and 2) for all locations and for the pooled data are given in Table 111. These two models differ little with respect to the magnitude of the correlation coefficient (r) at any location and also for the pooled data, and these coefficients are significant at the 1 per cent level in all cases. These coefficients are rather low for locations with small seasonal variability of W (cf. Riyadh and Manaus), but the explained variance (2) for most locations is higher than 85 per cent.

The root-mean-square error (RMSE) for model 1 (equation lb) is 2 mm or less for 28 locations, but it exceeds 3 mm for six locations. The RMSE for the pooled data is 3.2 mm. The intercept (C2) for model 1 does not differ from 0.0 at the 0.05 significance level (t-test) for 18 locations, whereas it is found to be positive for 19 locations and negative for eight locations. Except for two locations (Wagga and Moree), these negative intercepts are seen for locations having high values of W from tropical climates (cf. Hyderabad, Lucknow, and Manaus). Positive intercepts are generally seen for locations in cold snowy and temperate moist climates with low values of W (Cf, Df, and Dw), with some exceptions (cf. Guangzhou and Niamey). The sign and the magnitude of the intercepts together with the RMSE for three stations (Fort Smith, Maniwaki, and Norman Wells) given in Table 111 agree well with those obtained by Hay (1 970), although the slopes obtained in this study are about 10 per cent lower than those obtained by Hay (1 970). Of the 27 locations for which the intercept differs from zero at the 0-05 significance level, we find a high negative correlation (r = -0.92) between the intercept and the slope for 25 locations for which ? is greater than 85 per cent (namely, C2=28.8 - 17.6C1). For these 25 locations, there is also a significant (1 per cent level)

RELATIONSHIP OF PRECIPITABLE WATER TO SURFACE HUMIDITY 669

0 150

P E z' 100

50

0 5 10 15 20 25 30 Vapor Pressure (hPa)

0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0

Precipitable Water (rnm)

Figure 2. Histogram frequency distribution of precipitable water (mm) and surface vapour pressure (hF'a) data used in the analysis

positive correlation (r = 0.6 1) between the slope and mean vapour pressure (namely, C1 = 1.35 + 04XZ7'eo). Thus, locations with higher slope generally have lower intercept, and also locations having higher slope generally have higher mean vapour pressure. This latter correlation between slope and vapour pressure, although not high, would suggest that the relationship between W and eo has some non-linearity, which will be elaborated below.

The results for model 2 (equation 2) given in Table 111 can be compared with those quoted in Table I. The slopes for locations within the USA agree well with the previous estimates, although the intercepts in Table 111 are a little higher. Some differences with the previous estimates are also seen for the West African (Bamako, Niamey, and In Salah) and South African (Upington) locations. In comparing the results for Niamey in Tables I and 111 it is pertinent to note that Mohamed and Frangi's (1983) results are based upon daily data, whereas the present results are based upon mean monthly data. The nature of data can affect the relationship because of the logarithmic dependence on W Also, it is pertinent to note that because of the non-linearity in the relationship between vapour pressure and dew point temperature (equation 7),mean monthly dew point temperature will not exactly provide the mean monthly vapour pressure. The slope and the intercept of the relationship for the pooled data are very close to those obtained previously for the USA (Table I). For example, the relationship obtained by Lowry and Glahn (1969) using daily radiosonde data would predict a Wvalue of 14 mm at a dew point temperature of 5.8"C, with 95 per cent probability that the true value of W would be between 8 and 26 mm. At the same dew point temperature, the relationship obtained in this study would predict W to be 16 mm, with 95 per cent probability for it to be between 1 1 and 23 mm.

The performance of the two models for individual locations, as might be judged objectively by the explained variance (?), show that model 1 (equation 1 b) is better than model 2 (equation 2) for 23 locations, although model 2 is better for 1 1 locations. Both models performed equally for 1 1 locations and for the pooled data. By examining

670 B. J. CHOUDHURY

Table 111. Regression statistic for model 1 (relating precipitable water to vapour pressure; equation 1 b) and model 2 (relating logarithm of precipitable water to dew point temperature; equation 2) for various locations. The data given are the explained variance (?), slope with its standard deviation in parenthesis, intercept with its standard deviation in parenthesis and root-mean-

square error (RMSE). Intercepts with asterisk are not different from zero at 0.05 significance level

Name Model ? Slope Intercept RMSE

Alice Springs

Askabad

Bamako

Berlin

Bismarck

Brasilia

Camp0 Grande

Chiang Mai

Cita

Fort Smith

Giles

G u an g zh o u

Hail

Harkov

Hyderabad

In Salah

Jakutsk

Kalgoorlie

KhartOUm

Kolpasev

Kustanaj

Longview

Lucknow

Madrid

Manaus

Maniwaki

Minsk

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 I 2 1 2 1 2

0.65 0.64 0.93 0.93 0.92 0.90 0.95 0.94 0.98 0.99 0.88 0.92 0.90 0.90 0.97 0.97 0.99 0.97 0.99 0.97 0.75 0.80 0.96 0.95 0.61 0.69 0.99 0.97 0.93 0.92 0.77 0.79 0.99 0.97 0.77 0.8 1 0.93 0.90 0.99 0.98 0.99 0.97 0.95 0.93 0.93 0.93 0.89 0.9 1 0.55 0.53 0.99 0.99 0.98 0.98

1.64(0.26) 0.065(0.010) 1.88(0.11) 0.067(0.004) 1.43(0.09) 0.052(0.004) 1.5 l(0.07) 0.074(0.004) 1.57(0.04) 0.059(0.002) 2.23(0.18) 0.097(0.006) 1.86(0.13) 0.074(0.005) 2.16(0.08) 0.072(0.003) 1.51(0.03) 0.060(0.002) 1.44(0.04) 0.049(0.002) 1.52(0.18) 0.065(0.007) 1.69(0.08) 0.055(0.003) 1.44(0.25) 0.088(0.0l3) 1.57(0.04) 0.062(0.003) 1.93(0.12) 0.074(0.005) 1.43(0.17) 0.066(0.007) 1.49(0.03) 0.047(0.002) 1.82(0.21) 0.082(0.009) 1.83(0.11) 0.064(0.004) 1.41(0.02) 0.053(0.002) 1.52(0.03) 0.055(0.002) 1.52(0.08) 0.057(0.003) 1.97(0.12) 0.077(0.005) 1.39(0.10) 0.076(0.005) 4.15(0.8) 0.142(0.028) 1.24(0.03) 0~052(0~001) 1.59(0.05) 0.067(0.002)

1 .I *(2.4) 2.38(0.07) 0.4*( 1 .O) 2.48(0.02) 3.4( 1.6) 2.53(0.06) - 1.3*(0.7) 2.09(0.03) 1.9(0.3) 2.45(0.02) -1 1.2(3.0) 1.8 l(0.09) -2.8*( 1.9) 2.25(0.06) -6.4(2.0) 2.34(0.05) 0.9(0.1) 2.31(0.04)

2.40(0.03) 1.4*(1.7) 2.35(0.04) 7.0( 1.6) 2.74(0.05) - 1.4*( 1.8) 1.97(0.04) l.O(O.4) 2.36(0.02) -5.4(2.4) 2.20(0.08) 1.5*( 1.6) 2.29(0.05) 1.4(0.2) 2.36(0.05) -2.8*(2.3) 2.15(0.07) 0.1 *( 1.6) 2.48(0.06) 1.8(0.2) 2.36(0.02) 2.0(0.3) 2.44(0.02) 3.2( 1.4) 2.5 l(0.05) -7.8(2.6) 2.07(0.08) -0.9*( 1 .O) 2.02(0.03) -61.2(21.1) 0.75(0.62) 2.8(0.3) 2.37(0.01) 0.2*(0.5) 2.30(0.02)

2.1(0.2)

3.5 0.22 1.4 0.08 3.4 0.15 1.1 0.09 0.9 0.07 2.6 0.09 2.0 0.09 2.0 0.05 0.5 0.15 0.6 0.1 1 2.6 0.16 3.1 0.09 1.6 0.17 0.9 0.10 2.8 0.09 2.4 0.16 0.7 0.17 2.0 0.1 1 2.7 0.13 0.5 0.09 0.8 0.1 1 2.8 0.13 4.4 0.13 1.1 0.09 3.3 0.07 0.7 0.07 1 .o 0.07

(continued)

RELATIONSHIP OF PRECIPITABLE WATER TO SURFACE HUMIDITY 67 1

Table 111. (continued)

Name Model 3 Slope Intercept RMSE Moree

Mt Isa

Muenchen

Nancy

Nashville

Neuquen

Niamey

Norman Wells

Olenek

Orenburg

Resistencia

Riyadh

Turuhansk

Ubon Ratchathani

Upington

Vilhena

Washington

All stations

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 I 2 1 2 1 2 1 2 1 2 1 2 1 2

0.96 0.95 0.92 0.91 0.95 0.95 0.93 0.94 0.98 0.97 0.70 0.70 0.91 0.94 0.99 0.97 0.99 0.97 0.98 0.97 0.90 0.89 0.36 0.36 1 .O 0.98 0.93 0.93 0.87 0.88 0.91 0.90 0.89 0.89 0.96 0.97 0.94 0.94

1.79(0.08) 0.084(0.004) 1.67(0.10) 0.063(0.004) 1.53(0.07) 0.076(0.004) 1.39(0.08) 0.063(0.003) 1.50(0.05) 0.054(0*002) 1 .l4(0.16) 0.058(0.008) 1.3 l(0.09) 0.046(0402) 1.56(0.04) 0.047(0.002) 1.58(0.03) 0~050(0~002) 1.63(0.05) 0.058(0.002) 1.33(0.10) 0.058(0.004) 1.20(0.34) 0.048(0.0 14) 1.57(0.02) 0.053(0.002)

0.07(0.004) 1.65(0.14) 0.073(0.006) 2.49(0.17) 0.098(0.007) 1.77(0.13) O.OSl(0.006) 1.54(0.06) 0.055(0.002) 1.70(0.01) 0~058(0~0Ol)

2.01(0.12)

-4.3( 1.2)

1.3*(1.4) 2.01 (0.05)

244(0.05) -0.7*(0.7) 2.14(0.03) 1.4*(0.8) 2.29(0.02) 4.0(0.7) 2.57(0.02)

2.19(0.04) 6.3( 1.7) 2.68(0.04) 2.4(0.2) 2.49(0.04) 1.3(0.l) 2.35(0.04) I *5(0.4) 2.44(0.02) 2.6*( 1.8) 2.36(0.07) 4.6(2.2) 2.47(0.03) 1.8(0.1) 2.43(0.03) -3.6*(3.0) 2.36(0.09) -0.1 *( 1.4) 2.27(0.04) - 17.6(3.5) 1.73(0.12) -3q1.6) 2.09(0.06) 3.4(0.8) 2.56(0.02)

2.42(0.01)

2.2*(1.3)

-0.1*(0.2)

1.4 0.08 2.6 0.13 1.2 0.10 1.3 0.08 1.5#15; 0.08 1.7 0.15 3.7 0.12 0.7 0.13 0.4 0.15 1 .O 0.1 1 2.5 0.10 1.9 0.16 0.4 0.09 2.7 0.007 2.6 0.16 2.6 0.09 1.4 0.08 1.9 0.09 3.2 0.19

the locations where one model performed better than the other it appears that climatic condition may not be a significant determinant of the performance of these two models. It is rather outstanding to note that the slope (C,) of model 1 obtained in this study for the pooled data differs little from that obtained by Hann (1906).

Scatter plots and residuals of the pooled data for models 1 and 2 are shown in Figures 3 (a and b) and 4 (a and b), respectively. Although the explained variances (3) are quite high (94 per cent), these figures indicate some deficiencies. Relationships between Wand eo or In Wand td (Figures 3(a) and 4(a)) appear to have some non- linearity, and the residuals (Figures 3(b) and 4(b)) are not distributed uniformly. The skewness calculated for models 1 and 2 are, respectively, 0.20 and -0.13. The t test will reject skewness to be zero for model 1 , but not for model 2, at the 0.05 significance level (Zar, 1974). Thus, the relationship between Wand eo for the pooled data deviates from linearity at the 0.05 significance level. Also, although the calculated value of C, for the pooled data does not differ from zero at the 0.05 significance level, the precipitable water is seen to remain finite at very low value of eo (Figure 3(a)).

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Vapor Pressure (hPa)

Figure 3. Scatter plots of (a) precipitable water and vapour pressure for all locations, (b) residuals and predicted precipitable water

For globally distributed observations over the oceans, Liu (1986) found the following non-linear relationship for estimating the surface mixing ratio wo (g kg-') from precipitable water:

wo = 3.818724W' +O*1897219(W')2 +0.1891893(W')3 - 0.07549036(W')4 +0*006088244(W')' (17)

where W' = (WAO). Some of the characteristics of this equation were considered to be due to boundary layer thermodynamics of different air masses over the oceans.

Although the present analysis also suggests non-linearity in the relation between Wand eo (or wO), one important characteristic of equation (1 7) that would separate it from the present results over the land surfaces is that wo = 0 may not imply W= 0, although these limits can only be approached but not reached in realistic situations. A model such as equation (lb), with a finite positive value of C,, appears to be a more realistic description of the W - eo

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SUMMARY AND CONCLUSIONS

Mean monthly radiosonde data for 24 months at 45 globally distributed locations were analysed in order to evaluate models relating precipitable water ( W; mm) to two surface humidity variables, namely vapour pressure (eo; hPa) and dew point temperature ( td ; “C). The locations selected covered a broad range of climatic conditions, and the models evaluated were of the form of linear regression relationships: W = Cleo + C2 and In W = C3td + C,. The latter model could be derived from the former model when C2 = 0. The parameters of these models (C, to C,) were determined through least-squares analysis for each location and for the aggregated data from all locations.

Both models explained similar amounts of variance (3) for individual locations and for the aggregated data, and these I’ values ranged between 35.699.5 per cent for individual locations (the 3 value exceeded 85 per cent for 37 locations) and 94.0 per cent for the aggregated data. The C2 values for 27 locations were found to differ from zero at the 0.05 significance level, and thus the two models maintain some degree of independence at individual locations. An objective appraisal of the models based upon the explained variance (2) showed that model 1 performed better at 23 locations, whereas model 2 performed better at 11 locations. Both models performed equally well at the other 1 1 locations. Thus, for the selected locations taken individually, model 1 performed a little better than model 2. An examination of the locations where one model performed better than the other suggests that climatic condition may not be considered as a significant determinant in the performance of these models. There was also no clear geographical bias in determining the performance. The residuals for the aggregated data were not distributed uniformly, and the skewness for the W versus eo relationship was higher than that for In W versus td . Thus, a non- linear model appeared to be more appropriate for the W versus eo relationship.

The present study showed that surface humidity at a monthly time-scale could be a rather significant indicator of the total precipitable water for globally distributed locations having a wide range of climatic conditions. Robustness of model 1 with essentially zero intercept for the pooled data suggests that an exponential model of water vapour profile (equation 9) with a prescribed scale height could be considered as a good overall approximation for the state of water vapour, but systematic deviations would have to be recognized; most significant being in the northern latitudes (winter) and in the tropics, as indicated by the appearance of a finite intercept (C2) and non-linearity in the W - eo relationship. Water vapour remains confined mostly in the upper atmosphere during winter in the northern latitudes (giving the finite intercept), and damping of water vapour is weaker in the tropics (giving the non-linearity) (cf. Parameswaran and Krishna Murthy, 1990). Thus, calculations of the radiative characteristics of the atmosphere, such as atmospheric longwave radiation, based upon surface observations or an exponential water vapour profile with a prescribed scale height appropriate for temperature conditions can give systematically biased results in the tropics and in the winter conditions (see p. 58, Budyko, 1974). Results of the present analysis may aide better parameterization of the water vapour profile and thus the radiative characteristics for climatological studies.

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