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A universal regression retrieval method of the ground-based microwave remote sensing of precipitable water vapor and path-integrated cloud liquid water content
.\TX IOSPHEt~I(' RESEAR('H ELSEVIER
Atmospheric Research 34 (1994) 309-322
A universal regression retrieval method of the ground-based microwave remote sensing of precipitable water vapor and path-integrated cloud liquid water content Chong Wei, Daren Lu Institute of Atmospheric Physics, Academia Sinica, Beijing, China 100029 (Received January 4, 1993; revised and accepted May 12, 1993)
Abstract
An attempt has been made to obtain two respective universal equations, applying to different regions, elevations and seasons in the world, for remote sensing the atmospheric precipitable water vapor (Q) and the path-integrated cloud liquid water content (L) by using a ground-based dual-frequency (20.6 and 31.4 GHz) microwave radiometer (GBDFMR). To do so, a set of a priori radiosonde data with a total of 2742 cases was selected in typical seasons (winter and summer) at eight radiosonde stations (Beijing, Guangzhou, Guam, Yap, Lhasa, Zhangye, Nagqu and Lijiang) typical of the climates of mid-latitude mainland, tropical marine, plain, plateau and mountain, respectively. A cloud model was constructed in a way much the same as that by Decker et al. ( 1978, pp. 17891790) and an ensemble of cloudy- and clear-day mixed samples were elaborately constructed. Based on this ensemble, numerical simulation was done for each case by using a microwave radiation transfer model to compute the radiometric brightness temperatures Tbt and Tb2as well as the dependent variables L and Q. The simulated Tbl and Tb2 together with surface air temperature, surface humidity, surface pressure, the index of clear or cloudy day, cloud base height and their combinations were used as the candidates for the predictors of multivariate regression. The stepwise regression and ridge regression techniques ensure the two respective resultant regression equations to be steady, optimal and feasible for retrieval of L and Q. The tests on these equations by using temporal and spatial extrapolation samples with total of 1020 cases show that they have quite good accuracies for predicting the Q and L and can be used operationally. This work suggests a broad prospect in the application of GBDFMR in cloud liquid water and precipitable water vapor measurements.
0169-8095/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSD1 0169-8095 (93 ) E0113-D
310
C. Wei, D. Lu /Atmospheric Research 34 (1994) 309-322
1. Introduction
Water vapor and cloud liquid water in the atmosphere as important links of the budgets of atmospheric energy and water substance are essential factors in meteorologcal and climatologcal research. Their accurate and real-time measurement is one of the focuses of atmospheric probing's attentions. Ground-based microwave remote sensing, being a branch of remote sensing and having merits of itself in comparison with other measurement methods such as radiosonde, has been known as a feasible means for the measurements ofprecipitable water vapor (Q) and path-integrated cloud liquid water content (L) (Hogg et al., 1983; Heggli et al., 1987; Wei et al., 1992 ). Nowadays, physical and statistical methods are two basic types of ground-based dual-wavelength microwave radiometric methods in deducing path-integrated cloud liquid water L and precipitable water vapor Q, the statistical one is more feasible. As regards the statistical method, there are some difficulties to be dealt with, for example, it is very difficult to simultaneously obtain both the ground-based radiometer measurements and the water vapor and cloud liquid water data, which are necessary for the establishment of the statistical relationships to retrieve the L and Q. Besides, the radiosonde does not measure liquid water and we have no other conventional means to do it. One solution is to use a set of a priori statistical radiosonde data and insert some assumed values of cloud liquid water content in the levels of the radiosonde profile, at which the relative humidities exceed certain threshold, to construct a cloud model for simulating the variables by using radiation transfer equation calculation (Decker et al., 1978; Huang et al., 1987). Thus, the statistical relationships between these simulated variables can be established. Up to now all statistical relationships are based on a priori local (in situ) radiosonde data in order to keep relative higher retrieval accuracy. However, this makes some limitations of radiometric application in a vast area, especially in the area without radiosonde stations. Recently, we have been attempting to obtain the universal regression equations (URE's) for retrieval of L and Q by using GBDFMR, applying to various regions, seasons and elevations, in order to make a progress in radiometric application. As the first step to achieve this objective, in another paper (Lu, D.R. et al., to be published) we have discussed the universal regression retrieval of precipitable water in the situation of clear sky, obtained an accurate and feasible universal regression equation for retrieving the precipitable water vapor Q and made some comparisons between two frequency-pairs (22.2 and 35.0 GHz and 20.6 and 31.65 GHz). As a second step, in this paper we try to study the retrieval method of liquid water and precipitable water for both cloudy and clear days. The universal retrieval for both L and Q obeys the similar way as that for mere clear days but has its own specific key points such as how to set up the cloud model, how to choose the regression samples and how to construct the ensemble of clear and cloudy mixed samples. In this paper we focus our discussion on these key points and only choose the relatively better frequency-pair (20.6 and 31.65
C. Wei, D. Lu / Atmospheric Research 34 (1994) 309-322
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Ghz) that has been shown in the above-mentioned paper as a results of the comparison. In Section 2 the cloud model and the cloudy microwave radiation transfer calculation are discussed. The option of the regression samples and the construction of the cloudy and clear mixed ensemble are introduced in Section 3. The stepwise regression analysis is given in Section 4 and the ridge regression in Section 5. Finally, the last two sections give the evaluation on the resultant universal regression equations and the discussion. 2. Cloud model and microwave radiation transfer in cloudy day
The GBDFMR-received microwave brightness temperature Tb (f) in the zenith direction, the precipitable water of atmosphere Q and the path-integrated cloud liquid water L can be written: c~
Q= ~t2H20(z) dz,
(1)
0 oo
L= ~g2cld(Z)dz,
(2)
0
and Tb(f) = T~ X e x p ( - f
(aH2o(f,,z)+ao2(fz)+acld(f,z))*dz
0 oo
+ fT(z)dz* (a.~o(f,,z) +ao~(f,z) +acld(f,z) ) 0
*exp(- i
(aH2o(f,z')+ao2(f,z' )+acld(f,z')dz'
),
(3)
0
wherefis the central frequency of GBDFMR. T~ the cosmic background brightness temperature (2.7 K). T(z) is the air temperature at level z. 12H20 and g2cld are the densities of water vapor and cloud liquid water, respectively, aH2o, ao2 and ac,d the volume absorption coefficients of water vapor, oxygen and cloud liquid water at level z and frequency f, respectively. The former two are functions of atmospheric temperature, humidity and pressure and therefore can be derived by using radiosonde data. For the a~,d, since no routine observational liquid water data are available, the simulated data have to be used. Actually, a method similar to Decker's (1978 ) is used in this paper. That is, the ensembles are obtained by inserting clouds into a priori set of radiosonde data if at any point the relative humidity exceeded a given criterion. Base heights are taken to be the (interpo-
C. Wei, D. Lu / Atmospheric Research 34 (1994) 309-322
312
lated) levels at which the humidity first exceeded the criterion and thicknesses are determined using the point at which the humidity dropped below the criterion. Some profiles will contain two or more cloud layers. The liquid water content within a given cloud is assumed as a function of height and the total thickness of this cloud. Obviously, for a specific ensemble of radiosonde data the number of "cloudy" cases and the configuration of the "cloud layer" for certain cases vary with the particular value of the humidity criterion. Please keep this in mind and we will use it in next section. Thus, the simulated GBDFMR-measurements Tb ~ ) and Tb (f2) as well as the L and Q can be obtained by using Eqs. ( 1 ) - (3) based on the cloudy model.
3. The sample option for universal regression and the construction of the statistical ensemble 3.1. The option of the regression predictors
From Eqs. 1 to 3, it can be seen that L and Q are related with the radiometric measurements Tb~ and 7"82in quite a complicated way, determined by many factors such as the atmospheric profiles of temperature, humidity and pressure, the macro- and micro-features of clouds, etc. The objective of this study is the regression equations to retrieve L and Q globally in all seasons, thus, as many as possiTable 1 The construction of the ensemble of universal regression samples Station
Guam
Yap
Guangzhou
Beij ing
Longitude Latitude Elevation ( m )
144° 5 0 ' E 13 ° 3 3 ' N 111
138°05'E 09 ° 3 9 ' N 77
113 ° 19'E 23 ° 0 8 ' N 7
116 ° 17'E 39 ° 5 6 ' N 55
Seasons
sum
win
sum
win
sum
win
sum
win
Total no. of samp. Cloudy n u m b e r Cloudy/Total Overcast ratio R.H.C. a
114 76 0.87 0.87 0.85
338 192 0.57 0.57 0.85
98 72 0.73 0.73 0.85
207 122 0.59 0.59 0.85
144 103 0.72 0.72 0.85
213 158 0.74 0.74 0.80
48 36 0.75 0.75 0.90
226 40 0.18 0.18 0.85
45 103 5 4 1 0
21 11 1 2 1 0
FD b
0 < L ~ < 100 100 3000
20 47 4 1 4 0
94 88 4 3 3 0
18 39 8 6 1 0
36 71 6 8 1 0
34 53 5 9 1 1
28 6 3 3 0 0
C. Wei, D. Lu /Atmospheric Research 34 (1994) 309-322
313
ble predictors having influence on the L and/or Q should be chosen. Considering that the profiles are not easy to be measured, some easy-measured surface meteorological variables and their combinations are chosen as the predictors. According to a previous work (Huang et al., 1987 ), the cloud integrated liquid water L has a non-linear relationship with the brightness temperature at 31.65 GHz in cloudy sky. Consequently, not only Tb~ and Tb2 but also (Tb~)2 and (Tbz)e are used as predictors. Summarily, the following twelve variables are chosen as the predictors: The-the brightness temperature at 20.6 GHz, Tb2-that for 31.65 GHz, Ts-the surface air temperature, Es- the surface absolute humidity, Ps-the surface pressure, C B H - the cloud base height, Exist-the index of clear or cloudy day ( = 0 clear, = 1 c l o u d y ) , Es/P~, Tbl/Ts, TB2/Ts, ( T b l ) 2 and (Tb2) 2. All these variables can be obtained by using the radiosonde data and the cloud model calculations.
3.2. The representativeness of regression samples In order to make the regression samples involve various conditions of the global atmosphere as many as possible, eight radiosonde stations typical of the climates of mid-latitude continent, tropical marine, plain, plateau, and mountain are selected. The historical radiosonde data with a total of 2742 cases in winter and summer are selected from these stations with at least 300 cases for each station. The details for these samples are listed in Table 1.
Lhasa
Nagqu
Zhangye
Lijlang
91o08'E 29o40'N 3650
92°04'E 31°29'N 4508
100o26'E 38°56'N 1483
100°23'E 26o52'N 2394
sum
win
sum
win
sum
win
sum
win
sum
win
Total
176 141 0.80 0.80 0.70
200 10 0.05 0.05 0.70
147 110 0.75 0.75 0.70
198 18 0.09 0.09 0.70
113 24 0.21 0.21 0.90
200 15 0.08 0.08 0.80
125 96 0.77 0.77 0.80
195 31 0.16 0.16 0.70
965 658 0.68 0.68
1777 586 0.33 0.33
2742 1244 0.45 0.45
12 61 4 36 13 15
1 8 0 1 0 0
19 52 6 31 2 0
7 11 0 0 0 0
11 8 3 2 0 0
5 5 2 3 0 0
4 54 15 14 5 4
aRelative humidity criterion bFrequency distribution of L (no. of cases) and L in g / m 2.
Amount
16 13 2 0 0 0
139 325 46 101 27 20
232 305 22 22 5 0
371 630 68 123 32 20
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3.3. The overcast ratio and the construction of the mixed ensemble of clear-andcloudy samples The G B D F M R usually operates in an unmanned way, and no other additional cloud identification observations can be used. Accordingly, the regression relationship should produce the resultant L and Q in a natural way, with which the cloudy or clear day information can be obtained automatically. This can be done by choosing the regression samples in due proportion of clear and cloudy cases. Firstly, the climatical statistics of the overcast ratio for each station in winter and summer is made, respectively. The statistical data are taken from the "Chinese Surface Meteorological Monthly Bulletin" and from the "Japanese Daily Synoptic Map". The daily mean total cloud amount and low-level cloud amount in two monthes of each season are used from 1979 to 1986. If a daily mean low-level cloud amount is more than 4 or a total cloud amount more than 7, the day is counted in "overcast", otherwise it is considered as "clear". The statistical ratio of the number of overcast days to the number of total days during the statistical period is defined as the overcast ratio, which varies from station to station and season to season. The values for each sub-ensemble from certain station and in certain season are listed in Table 1. Secondly, the previously-mentioned (in Section 2 ) humidity criterion for each sub-ensemble is modulated to obtain a set of mixed regression samples of cloudy and clear sky in the proportion of the overcast ratio of each sub-ensemble. Finally, all the sub-ensembles are merged into one ensemble for the universal regression. The relative humidity criterions are also listed in Table 1, as well as the statistical features of the path-integrated cloud liquid water L. From the Table, it can be seen that the statistical distribution of L in the constructed ensemble is much the same as that from the previous G B D F M R measurements (Wei et al., 1989; 1992).
4. Stepwise regression analysis (the least square fit) for L and Q 4.1. The multivariate regression model and stepwise regression Let Ybe a dependent variable and x~, x2,..., x m be the m independent variables. The n sets of measurement data for the variables are:
(Xlt,X2t,'",Xm,; Yt), t = 1,2,...,n.
(4)
Y is related with all the independent variables by a regression matrix equation:
Y=Xfl+~, where
(5)
315
C Wei, D, L u / A t m o s p h e r i c R e s e a r c h 34 (1994) 3 0 9 - 3 2 2
|
m
m
Yl Y2
m
m
m
...
Xlm
ro
El
1
x22
.-.
x2m
rl
~2
x21
•
.
,
x=
8--
•
. °
Yn_
m
Xl2
•
Y=
m
1 xll
- 1
Xn I
Xn2
""
Xnrn -
_rm _
•
,
(6) •
_~-n .
The least square estimate is usually used to provide the optimal fit to the coefficiencies r = (rio, fli ,..., rm)'. The estimated vector b ~ 8 ^ fits the following regular equation
(X'X)r ^ =X'Y
(7)
therefore
b' = ( bo,b , ,...,bin) and
b~r ^ =(x'x)-'x'Y.
(8)
So the (empirical) regression equation is Y^ =bo
+blxl +bzx2 +...-4-bmxm.
When the error e is unbiased and its components ei are independent with each other, 8 ^ is the unbiased estimate o f t , and its variance-covariance matrix is
D(r^ )=(Tz(x'x) -~
(9)
w h e r e (72 is the common variance of e,.
For a set of regression samples, the standard residuum Se can be used as a judgement on the fitting effectiveness. The less the Se is, the better the fitting effectiveness is, Se=L~= "
(Y,-Y,)2/(n-m-l)]
1/2.
Another judgement factor, the multicorrelation coefficient Re, can also be used. The closer to unity the Re is, the better the fitting effectiveness is. Generally, a resultant regression equation should only include those independent variables (predictors) strongly influencing the dependent variable. To do so, the stepwise regression analysis is made, that is, introducing the independent variables having significant influnce on the dependent variables step by step. As soon as a new variable is introduced into the equation, all previously introduced variables should be checked again and those variables which become less significant will be rejected from the equation immediately. By this way the resultant regression equation can be ensured to have the significant variables only.
-0.324496 -0.386305 -0.228998 -0.224890
-134.547 -36.8941 -22.5702 -76.2782
2 3 4 5
2 3 4 5
-12.9199 - 19.0112 -21.0554 - 179.314
0.135671 0.131368 0.103939 0.103337
Ol (Tbl)
27.5298 24.9557 72.4290 171.338
-0.0642343 -0.0538219 -0.0390047 -0.0174627
B2 ( Tb2)
0.107465 0.0770664 0.0929689
0.000319972 0.000303044
n3(Tb21 )
-
-0.0000599453 -0.000189374 -0.000186911
B4(T22 )
47452.6
-
-
- 12690.7 -42502.0
-5.85447
L)
Bs(Tb,/T~)B'5(Tb2/
0.979800 0.987076 0.988712 0.993312
0.997231 0.997650 0.999787 0.999809
R~
98.028 78.587 73.471 56.604
0.1437 0.1324 0.03988 0.03779
S¢d
(12)
(11)
No. o f Eqs.
aDependent Variable bThe n u m b e r of independent variables CBo is constant term of regression equation and B 1-B 5 the relevant regression coefficients corresponding to the independent variables shown in brackets, dThe units of Se is in cm for Q and in g / m 2 for L.
Q
O0
DV ~ No. b Regression coefficient ~
Table 2 The stepwise regression analysis (Least Square Estimate) for Q and L
t,,a
ta.a
C. Wei, D. Lu / Atmospheric Research 34 (1994) 309-322
317
4.2. The results of stepwise regression At the significance level of 2.0, the above-mentioned twelve variables are all significant for the dependent variables Q and L, respectively. For the 12-variableregression equation of Q, the Se is 0.03704 cm and Re is 0.99982, while for that of L, Se=45.64 g / m 2 (0.00456 cm) and Re=0.99567. However, the twelve variables have disparate contributions to the dependents. In order to make sure, a series of regression equations with different number of variables out of the twelve are compared, and some results are listed in Table 2. It is found that the Eqs. ( 11 ) and (12), each of which include respective five variables, have good enough effectiveness. The residua of Eqs. ( 11 ) and ( 12 ) are much close to those of the above-mentioned respective 12-variable-equations. In Eqs. (11) and (12) only three variables (Tbl , Tb2 and Ts) among the five are directly-measured and the other two are their combinations. Moreover, Tbl and Tb2 are the GBDFMR measurements and Ts is the most easily measured meteorological variable. The less the number of direct measurement variables is, the less the introduced-measurement errors are. Considering all above aspects, we only use the variables in Eqs. ( 11 ) and ( 12 ) as the predictors for the URE of Q and L, respectively. They are Tbl , Tb2 , (Tbl)2, (Tb2)2 and T b 2 / T s for Q, and Tbl, Tb2, (Tbl) z, Tbl/T~and Tb2/TsforL.
5. The ridge regression analysis for L and Q
5.1. The ridge estimate of the regression coefficients In Eqs. ( 11 ) and (12), there are high correlativity between the predictors. Both for Q and L, out of the total five, there are four correlation coefficients being more than 90% between the independent variables in each equation. Therefore, the determinant of the coefficient matrix XX' of the regular Eq. (7) may be close to zero, so that the X is "illness" and the variance, which is related to ( X ' X ) - 1 based on Eq. (9), of the least-square estimated regression coefficients may become too large to keep the regression equation steady. Ridge regression analysis (Hoel et al., 1970a, b) is one way to overcome this difficulty. The method is putting a relative small positive k in each diagonal element of the coefficient matrix X'X of the regular Eq. (7) when the regression coefficient fl is estimated:
r ^ (k) = ( X ' X + k I ) - ' X ' Y
(10)
where k is known as ridge parameter. In comparison with the least square estimate r ^ (0), the ridge estimate has a little bit less fitting effectiveness for the regression samples used for setting up the regression equations, that is, with a bit larger Se. However, it has been proved theoretically that it is possible to make the total root mean square error offl ^ (k) less than that offl ^ (0) when the value of
318
C. Wei, D. Lu / Atmospheric Research 34 (1994) 309-322 [ l l l l [ l l l r ] l l l l l r [ I
C~ 1.0
,F
,,,
, r,
, i , f ,r
14
~
, I b
10
TbL
,d
5
"U
,,
a
Tb2
0.5
f
0
Tb2
e~
-5 T/Ts z -0.5 0
5
10
Ridge Parameter
15
(X25XI0
')
0
k
5
10
Ridge Parameter
15
(X25XI0
~)
k
Fig. 1. The Ridge Track Map. (a) For the ridge regression equations of Q and (b) for those of L.
k is properly chosen, so that the ridge regression equation has better effectiveness for the new test samples.
5.2. The main results of ridge regression estimate One of the key points for ridge regression analysis is the option of k. Shown in Table 3 is the coefficients and the residua for different ridge regression equations of Q and L corresponding to the different ridge parameter k. From the Table it can be seen that when k = 0 the values of each coefficient are the same as those in Eqs. ( 11 ) and ( 12 ) shown in Table 2 (the least square estimate situation ). The regression coefficients vary and Se increases when the value of k increases. The variations of the normalized coefficients with k are shown in Fig. 1a and b, respectively, which are known as ridge track map. It can be seen that when k varies from zero to 0.005 in Fig. la and from 0 to 0.00005 in Fig. lb each coefficient varies rapidly, and when k goes beyond the scopes the relevant regression coefficients vary slowly. Considering that the Se in Table 3 should be limited in a relatively small value, 0.0025 and 0.000025 are chosen as the respective ridge parameters, and their corresponding Eqs. (13) and (14) (see Table 3) as the final universal equations for Q and L, respectively. 6. The test of the universal regression equations
6.1. Testing samples Radiosonde data with total of 1020 cases are used to construct the cloudy- and
0 0.0025 0.0050 0.0075 0.0100
0 0.000025 0.000050 0.000075 0.000100
a
L
-76.2782 -59.4002 -51.2152 -46.4912 -43.4843
-0.222490 -0.200341 -0.182765 -0.187807 -0.154600
Oo
-179.314 -118.862 -88.8330 -70.9923 -59.2374
0.103337 0.0980471 0.0942337 0.0913035 0.0889568
Ol (Tbl)
Regression coefficienff
171.338 123.350 98.9081 83.9664 73.8140
-0.0174627 -0.0139841 -0.0124467 -0.0113130 -0.0104269
O2(Tb2)
0.0929689 0.0934947 0.0941587 0.0948311 0.0954756
0.000303044 0.000346449 0.000377063 0.000399228 0.000415813
B3(T~l )
-
-0.000186911 -0.000213026 -0.000230834 -0.000243489 -0.000252753
B4(T22)
-47452.6 29471.4 20547.6 15251.7 11766.7
-
-42502.0 -28400.7 -21235.4 - 16866.6 - 13906.4
-5.85447 -5.62776 -5.20482 -4.87445 -4.61039
Ts)
O 5 ( T b , / T s ) BPs(Tb2/
aDependent variable bRidge parameter CFor the definition of Bo-Bs, reference to Table 2.dFor the unit of Se, reference to Table 2.
L
kb
DV a
Table 3 The ridge regression analysis for Q and L
56.6043 59.6402 63.2527 65.9907 68.0309
0.0377896 0.0439400 0.0545500 0.0630900 0.0948600
&d
(14)
(13)
No. of Eqs.
I t,,a t-o
C. Wei, D. Lu / Atmospheric Research 34 (1994) 309-322
320
clear-mixed testing samples by the same way as stated in Section 2 and 3 but the relative humidity criterion is equal to constant 85% rather than modulated. There are two kinds of samples in the testing ensemble, the first with total of 708 cases is known as temporal extrapolation samples which are from the same eight stations as those used to set up the URE but in different years or in different seasons. The second with total of 312 cases is spatial extrapolation samples from seven new stations also having the representativeness of climate, season and location. According to stations or seasons, the total test samples are divided into 21 sets each with 24 to 115 cases more or less (see in Table 4).
6.2. Standard of testing The root mean square deviation (RMS) is used as the test standard of the absolute accuracy and the average relative deviation (Rel) as that of relative accuracy for each of the 21 sets of the testing samples. For Q both absolute and relative tests are made, and for L, only the absolute accuracy test is given since Table 4 The test samples and test effect Station
Longitude
Latitude
Elevation Particular time of Total No. of Test effect (m) data no. of cloudy samples cases Q Year Season RMS Rel (cm) (%)
L
RMS ( g / m 2)
Naqiu
92°04'E 100°26'E Guam 144°50'E Guangzhou 113°19'E Lhasa 91°08'E Yap 138°05'E Beijing 116°17'E Beijing 116°17'E Lijiang 100°26'E Lijiang 100°26'E Xi'an 108°56'E Xi'an 108°56'E Urumqi 37°37'E Urumqi 87°37'E Shanghai 121 ° 2 6 ' E Shanghai 121 ° 2 6 ' E Kunming 102°41'E Kunming 102°41'E Chichijima 1 4 2 ° 1 1 ' E Denver 104°52'W Washington 7 7 ° 0 2 ' W
Zhangye
Total
31°29'N 38°56'N 13°33'N 23°08'N 29°40'N 09°39'N 39°56'N 39°56'N 28°52'N 26°52'N 34° 1 8 ' N 34° 1 8 ' N 43°47'N 43°47'N 31°I0'N 31 ° I 0 ' N 25°01'N 25°01'N 27°05'N 39°45'N 38°51 ' N
4508 1483 111 7 3650 77 55 55 2394 2394 398 398 918 918 5 5 1891 1891 4.1 1625 84
1981 1981 1981 1981 1981 1981/82 1981 1981 1981 1981 1980 1980 1980 1980 1980 1980 1980 1980 1980 1961 1961
sum. sum. sum. win. sum. sum. spr. aut. aut. spr. win. sum. win. sum. win. sum. win. sum. win. sum. sum.
89 103 101 115 101 89 28 29 24 29 30 28 28 27 26 30 28 28 30 31 26
29 24 89 36 28 60 2 6 4 13 7 20 12 1 15 21 12 27 23 8 5
1020
442
0.0530 3.9 0.0473 1.4 0.0388 0.55 0.0584 2.3 0.0388 1.6 0.0543 0.64 0.0304 2.13 0.0432 4.0 0.0796 2.1 0.0320 1.0 0.0205 2.8 0.0504 0.89 0.0277 7.3 0.0227 1.2 0.0369 3.4 0.0356 0.4 0.0137 1.3 0.0629 1.6 0.1021 3.7 0.0644 2.6 0.0403 1.1
34.71 47.34 57.33 31.06 37.53 59.58 44.16 44.35 19.62 37.28 42.73 79.79 29.39 4.02 44.16 66.33 4.97 102.25 77.70 28.47 20.01
C. Wei, D. Lu / Atmospheric Research 34 (1994) 309-322
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the values of L equal zero for the clear cases in the mixed test samples and the corresponding Rel are meaningless.
6.3. Testing results The testing results are shown in Table 4. For the 21 sets of samples the RMS of Q are in the range of 0.0137 to 0.102 cm and the Rel are all less than 4% except for the set of Wulumqi winter with the value of 7.29%. Considering the facts that the testing samples vary over a wide range of times, seasons, locations and elevations, and that the radiosonde measurement of humidity has an error about 3% to 10% itself, it can be said that the effect of the URE (13) is fairly good. For L, the RMS are within 50 g / m 2 (0.005 cm) for most sets and the largest value is only 102.25 g / m 2 (0.0120 cm), which is even better than some local statistical regression results (Wei et al., 1989 ). Summarily, the URE's for both L and Q are all of perfect accuracy.
7. Conclusions and discussion ( 1 ) Two respective universal regression equations for GBDFMR-retrieval of the precipitable water Q and path-integrated cloud liquid water content L have been obtained by using stepwise regression and ridge regression techniques based on an elaborately-constructed ensemble of clear- and cloudy-mixed samples. The numerical tests show that the retrieval accuracy of Q by the present URE is fairly high, even better than that of some local statistical regression equation (e.g. Wei et al., 1989 ), which just have two independent variables and use a set of local (in situ) historical radiosonde data as the regression samples. The retrieved L also has high accuracy and is feasible and realizable for most application purposes. (2) In comparison with the test result of the Q-URE in the authors' other paper (to be published) which only discusses the situation of clear sky, the present QURE has higher accuracy. The reason is that we introduce the more non-linear terms (Tbl)2 and (Tb2) 2 in this work rather than only the linear terms Tbl and Tb2 as in the previous paper. (3) In this paper, the measurement errors are not considered though it is easy to do. The samples both for regression and for testing are still limited as concerns "global", so the results are still preliminary.
Acknowledgment The reasearch discussed in this paper was supported by the National Natural Science Foundation of China. We are grateful to Professor Feng Shiyong of the Institute of Systematical Sciences, Academia Sinica for his advice and to Mr. Wang Xiaogang for his program assistance.
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References Decker M.T., Westwater, E.R. and Guiraud, F.O., 1978. Experimental evaluation of ground-based microwave radiometric sensing of atmospheric temperature and water vapor profiles. Appl. Meteorol., 17: 1788-1795. Heggli, M.F., Rauber, R.M. and Snider, J.B., 1987. Field evoluation of a dual-channel microwave radiometer designed for measurements of integrated water vapor and cloud liquid water in the atmosphere. J. Atmos. Ocean. Technol., 4 ( l ): 204-213. Hoel, A.E. and Kennard, R.W., 1970a. Ridge regression biased estimation for non-orthogonal problems. Technometrics, 12: 55-67. Hoel, A.E. and Kennard, R.W., 1970b. Ridge regression applications to non-orthogonal problems. Technometrics, 12: 69-82. Hogg, D.C., Guiraud, F.O., Snider, J.B., Decker, M.T. and Westwater, E.R., 1983. A steerable dualchannel microwave radiometer for measurement of water vapor and liquid in the troposphere. J. Clim. Appl. Meteorol., 22 ( 5 ): 789-806. Huang, R.H. and Zhou, S.X., 1987. Remote sensing of precipitable water vapor and liquid water of cloudy atmosphere by dual-wavelength microwave radiometer. Chin. J. Atmos. Sci., 11: 397-403 ( in Chinese). Lu D.R., Wei, C., Xin, M.X., Feng, S.Y. and Wang, X.G., 1993. The universal regression retrieval of atmospheric precipitable water in clear sky by using ground-based dual-wavelength microwave radiometer. Chin. J. Atmos. Sci., 17 ( 6 ): 721-731 (both in Chinese and in English ). Wei C., Leighton, H.G. and Rogers, R.R., 1989. A comparison of several radiometric methods of deducing path-integrated cloud liquid water. J. Atmos. Ocean. Technol., 6 (6): 1001-1012. Wei, C., Xin, M.X., Wang, P.C., Zhou, S.X., Xuang, Y.G., Chen, Y., Lin, H. and Lu, D.R., 1992. Analysis of precipitable water and cloud liquid water content over the equatorial area of the westera pasific on the basis of remote sening. Chin. J. Atmos. Sci., 16( l ): 118-128 (Both in Chinese and English ).
Atmospheric Research 34 (1994) 309-322
A universal regression retrieval method of the ground-based microwave remote sensing of precipitable water vapor and path-integrated cloud liquid water content Chong Wei, Daren Lu Institute of Atmospheric Physics, Academia Sinica, Beijing, China 100029 (Received January 4, 1993; revised and accepted May 12, 1993)
Abstract
An attempt has been made to obtain two respective universal equations, applying to different regions, elevations and seasons in the world, for remote sensing the atmospheric precipitable water vapor (Q) and the path-integrated cloud liquid water content (L) by using a ground-based dual-frequency (20.6 and 31.4 GHz) microwave radiometer (GBDFMR). To do so, a set of a priori radiosonde data with a total of 2742 cases was selected in typical seasons (winter and summer) at eight radiosonde stations (Beijing, Guangzhou, Guam, Yap, Lhasa, Zhangye, Nagqu and Lijiang) typical of the climates of mid-latitude mainland, tropical marine, plain, plateau and mountain, respectively. A cloud model was constructed in a way much the same as that by Decker et al. ( 1978, pp. 17891790) and an ensemble of cloudy- and clear-day mixed samples were elaborately constructed. Based on this ensemble, numerical simulation was done for each case by using a microwave radiation transfer model to compute the radiometric brightness temperatures Tbt and Tb2as well as the dependent variables L and Q. The simulated Tbl and Tb2 together with surface air temperature, surface humidity, surface pressure, the index of clear or cloudy day, cloud base height and their combinations were used as the candidates for the predictors of multivariate regression. The stepwise regression and ridge regression techniques ensure the two respective resultant regression equations to be steady, optimal and feasible for retrieval of L and Q. The tests on these equations by using temporal and spatial extrapolation samples with total of 1020 cases show that they have quite good accuracies for predicting the Q and L and can be used operationally. This work suggests a broad prospect in the application of GBDFMR in cloud liquid water and precipitable water vapor measurements.
0169-8095/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSD1 0169-8095 (93 ) E0113-D
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C. Wei, D. Lu /Atmospheric Research 34 (1994) 309-322
1. Introduction
Water vapor and cloud liquid water in the atmosphere as important links of the budgets of atmospheric energy and water substance are essential factors in meteorologcal and climatologcal research. Their accurate and real-time measurement is one of the focuses of atmospheric probing's attentions. Ground-based microwave remote sensing, being a branch of remote sensing and having merits of itself in comparison with other measurement methods such as radiosonde, has been known as a feasible means for the measurements ofprecipitable water vapor (Q) and path-integrated cloud liquid water content (L) (Hogg et al., 1983; Heggli et al., 1987; Wei et al., 1992 ). Nowadays, physical and statistical methods are two basic types of ground-based dual-wavelength microwave radiometric methods in deducing path-integrated cloud liquid water L and precipitable water vapor Q, the statistical one is more feasible. As regards the statistical method, there are some difficulties to be dealt with, for example, it is very difficult to simultaneously obtain both the ground-based radiometer measurements and the water vapor and cloud liquid water data, which are necessary for the establishment of the statistical relationships to retrieve the L and Q. Besides, the radiosonde does not measure liquid water and we have no other conventional means to do it. One solution is to use a set of a priori statistical radiosonde data and insert some assumed values of cloud liquid water content in the levels of the radiosonde profile, at which the relative humidities exceed certain threshold, to construct a cloud model for simulating the variables by using radiation transfer equation calculation (Decker et al., 1978; Huang et al., 1987). Thus, the statistical relationships between these simulated variables can be established. Up to now all statistical relationships are based on a priori local (in situ) radiosonde data in order to keep relative higher retrieval accuracy. However, this makes some limitations of radiometric application in a vast area, especially in the area without radiosonde stations. Recently, we have been attempting to obtain the universal regression equations (URE's) for retrieval of L and Q by using GBDFMR, applying to various regions, seasons and elevations, in order to make a progress in radiometric application. As the first step to achieve this objective, in another paper (Lu, D.R. et al., to be published) we have discussed the universal regression retrieval of precipitable water in the situation of clear sky, obtained an accurate and feasible universal regression equation for retrieving the precipitable water vapor Q and made some comparisons between two frequency-pairs (22.2 and 35.0 GHz and 20.6 and 31.65 GHz). As a second step, in this paper we try to study the retrieval method of liquid water and precipitable water for both cloudy and clear days. The universal retrieval for both L and Q obeys the similar way as that for mere clear days but has its own specific key points such as how to set up the cloud model, how to choose the regression samples and how to construct the ensemble of clear and cloudy mixed samples. In this paper we focus our discussion on these key points and only choose the relatively better frequency-pair (20.6 and 31.65
C. Wei, D. Lu / Atmospheric Research 34 (1994) 309-322
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Ghz) that has been shown in the above-mentioned paper as a results of the comparison. In Section 2 the cloud model and the cloudy microwave radiation transfer calculation are discussed. The option of the regression samples and the construction of the cloudy and clear mixed ensemble are introduced in Section 3. The stepwise regression analysis is given in Section 4 and the ridge regression in Section 5. Finally, the last two sections give the evaluation on the resultant universal regression equations and the discussion. 2. Cloud model and microwave radiation transfer in cloudy day
The GBDFMR-received microwave brightness temperature Tb (f) in the zenith direction, the precipitable water of atmosphere Q and the path-integrated cloud liquid water L can be written: c~
Q= ~t2H20(z) dz,
(1)
0 oo
L= ~g2cld(Z)dz,
(2)
0
and Tb(f) = T~ X e x p ( - f
(aH2o(f,,z)+ao2(fz)+acld(f,z))*dz
0 oo
+ fT(z)dz* (a.~o(f,,z) +ao~(f,z) +acld(f,z) ) 0
*exp(- i
(aH2o(f,z')+ao2(f,z' )+acld(f,z')dz'
),
(3)
0
wherefis the central frequency of GBDFMR. T~ the cosmic background brightness temperature (2.7 K). T(z) is the air temperature at level z. 12H20 and g2cld are the densities of water vapor and cloud liquid water, respectively, aH2o, ao2 and ac,d the volume absorption coefficients of water vapor, oxygen and cloud liquid water at level z and frequency f, respectively. The former two are functions of atmospheric temperature, humidity and pressure and therefore can be derived by using radiosonde data. For the a~,d, since no routine observational liquid water data are available, the simulated data have to be used. Actually, a method similar to Decker's (1978 ) is used in this paper. That is, the ensembles are obtained by inserting clouds into a priori set of radiosonde data if at any point the relative humidity exceeded a given criterion. Base heights are taken to be the (interpo-
C. Wei, D. Lu / Atmospheric Research 34 (1994) 309-322
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lated) levels at which the humidity first exceeded the criterion and thicknesses are determined using the point at which the humidity dropped below the criterion. Some profiles will contain two or more cloud layers. The liquid water content within a given cloud is assumed as a function of height and the total thickness of this cloud. Obviously, for a specific ensemble of radiosonde data the number of "cloudy" cases and the configuration of the "cloud layer" for certain cases vary with the particular value of the humidity criterion. Please keep this in mind and we will use it in next section. Thus, the simulated GBDFMR-measurements Tb ~ ) and Tb (f2) as well as the L and Q can be obtained by using Eqs. ( 1 ) - (3) based on the cloudy model.
3. The sample option for universal regression and the construction of the statistical ensemble 3.1. The option of the regression predictors
From Eqs. 1 to 3, it can be seen that L and Q are related with the radiometric measurements Tb~ and 7"82in quite a complicated way, determined by many factors such as the atmospheric profiles of temperature, humidity and pressure, the macro- and micro-features of clouds, etc. The objective of this study is the regression equations to retrieve L and Q globally in all seasons, thus, as many as possiTable 1 The construction of the ensemble of universal regression samples Station
Guam
Yap
Guangzhou
Beij ing
Longitude Latitude Elevation ( m )
144° 5 0 ' E 13 ° 3 3 ' N 111
138°05'E 09 ° 3 9 ' N 77
113 ° 19'E 23 ° 0 8 ' N 7
116 ° 17'E 39 ° 5 6 ' N 55
Seasons
sum
win
sum
win
sum
win
sum
win
Total no. of samp. Cloudy n u m b e r Cloudy/Total Overcast ratio R.H.C. a
114 76 0.87 0.87 0.85
338 192 0.57 0.57 0.85
98 72 0.73 0.73 0.85
207 122 0.59 0.59 0.85
144 103 0.72 0.72 0.85
213 158 0.74 0.74 0.80
48 36 0.75 0.75 0.90
226 40 0.18 0.18 0.85
45 103 5 4 1 0
21 11 1 2 1 0
FD b
0 < L ~ < 100 100
20 47 4 1 4 0
94 88 4 3 3 0
18 39 8 6 1 0
36 71 6 8 1 0
34 53 5 9 1 1
28 6 3 3 0 0
C. Wei, D. Lu /Atmospheric Research 34 (1994) 309-322
313
ble predictors having influence on the L and/or Q should be chosen. Considering that the profiles are not easy to be measured, some easy-measured surface meteorological variables and their combinations are chosen as the predictors. According to a previous work (Huang et al., 1987 ), the cloud integrated liquid water L has a non-linear relationship with the brightness temperature at 31.65 GHz in cloudy sky. Consequently, not only Tb~ and Tb2 but also (Tb~)2 and (Tbz)e are used as predictors. Summarily, the following twelve variables are chosen as the predictors: The-the brightness temperature at 20.6 GHz, Tb2-that for 31.65 GHz, Ts-the surface air temperature, Es- the surface absolute humidity, Ps-the surface pressure, C B H - the cloud base height, Exist-the index of clear or cloudy day ( = 0 clear, = 1 c l o u d y ) , Es/P~, Tbl/Ts, TB2/Ts, ( T b l ) 2 and (Tb2) 2. All these variables can be obtained by using the radiosonde data and the cloud model calculations.
3.2. The representativeness of regression samples In order to make the regression samples involve various conditions of the global atmosphere as many as possible, eight radiosonde stations typical of the climates of mid-latitude continent, tropical marine, plain, plateau, and mountain are selected. The historical radiosonde data with a total of 2742 cases in winter and summer are selected from these stations with at least 300 cases for each station. The details for these samples are listed in Table 1.
Lhasa
Nagqu
Zhangye
Lijlang
91o08'E 29o40'N 3650
92°04'E 31°29'N 4508
100o26'E 38°56'N 1483
100°23'E 26o52'N 2394
sum
win
sum
win
sum
win
sum
win
sum
win
Total
176 141 0.80 0.80 0.70
200 10 0.05 0.05 0.70
147 110 0.75 0.75 0.70
198 18 0.09 0.09 0.70
113 24 0.21 0.21 0.90
200 15 0.08 0.08 0.80
125 96 0.77 0.77 0.80
195 31 0.16 0.16 0.70
965 658 0.68 0.68
1777 586 0.33 0.33
2742 1244 0.45 0.45
12 61 4 36 13 15
1 8 0 1 0 0
19 52 6 31 2 0
7 11 0 0 0 0
11 8 3 2 0 0
5 5 2 3 0 0
4 54 15 14 5 4
aRelative humidity criterion bFrequency distribution of L (no. of cases) and L in g / m 2.
Amount
16 13 2 0 0 0
139 325 46 101 27 20
232 305 22 22 5 0
371 630 68 123 32 20
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314
3.3. The overcast ratio and the construction of the mixed ensemble of clear-andcloudy samples The G B D F M R usually operates in an unmanned way, and no other additional cloud identification observations can be used. Accordingly, the regression relationship should produce the resultant L and Q in a natural way, with which the cloudy or clear day information can be obtained automatically. This can be done by choosing the regression samples in due proportion of clear and cloudy cases. Firstly, the climatical statistics of the overcast ratio for each station in winter and summer is made, respectively. The statistical data are taken from the "Chinese Surface Meteorological Monthly Bulletin" and from the "Japanese Daily Synoptic Map". The daily mean total cloud amount and low-level cloud amount in two monthes of each season are used from 1979 to 1986. If a daily mean low-level cloud amount is more than 4 or a total cloud amount more than 7, the day is counted in "overcast", otherwise it is considered as "clear". The statistical ratio of the number of overcast days to the number of total days during the statistical period is defined as the overcast ratio, which varies from station to station and season to season. The values for each sub-ensemble from certain station and in certain season are listed in Table 1. Secondly, the previously-mentioned (in Section 2 ) humidity criterion for each sub-ensemble is modulated to obtain a set of mixed regression samples of cloudy and clear sky in the proportion of the overcast ratio of each sub-ensemble. Finally, all the sub-ensembles are merged into one ensemble for the universal regression. The relative humidity criterions are also listed in Table 1, as well as the statistical features of the path-integrated cloud liquid water L. From the Table, it can be seen that the statistical distribution of L in the constructed ensemble is much the same as that from the previous G B D F M R measurements (Wei et al., 1989; 1992).
4. Stepwise regression analysis (the least square fit) for L and Q 4.1. The multivariate regression model and stepwise regression Let Ybe a dependent variable and x~, x2,..., x m be the m independent variables. The n sets of measurement data for the variables are:
(Xlt,X2t,'",Xm,; Yt), t = 1,2,...,n.
(4)
Y is related with all the independent variables by a regression matrix equation:
Y=Xfl+~, where
(5)
315
C Wei, D, L u / A t m o s p h e r i c R e s e a r c h 34 (1994) 3 0 9 - 3 2 2
|
m
m
Yl Y2
m
m
m
...
Xlm
ro
El
1
x22
.-.
x2m
rl
~2
x21
•
.
,
x=
8--
•
. °
Yn_
m
Xl2
•
Y=
m
1 xll
- 1
Xn I
Xn2
""
Xnrn -
_rm _
•
,
(6) •
_~-n .
The least square estimate is usually used to provide the optimal fit to the coefficiencies r = (rio, fli ,..., rm)'. The estimated vector b ~ 8 ^ fits the following regular equation
(X'X)r ^ =X'Y
(7)
therefore
b' = ( bo,b , ,...,bin) and
b~r ^ =(x'x)-'x'Y.
(8)
So the (empirical) regression equation is Y^ =bo
+blxl +bzx2 +...-4-bmxm.
When the error e is unbiased and its components ei are independent with each other, 8 ^ is the unbiased estimate o f t , and its variance-covariance matrix is
D(r^ )=(Tz(x'x) -~
(9)
w h e r e (72 is the common variance of e,.
For a set of regression samples, the standard residuum Se can be used as a judgement on the fitting effectiveness. The less the Se is, the better the fitting effectiveness is, Se=L~= "
(Y,-Y,)2/(n-m-l)]
1/2.
Another judgement factor, the multicorrelation coefficient Re, can also be used. The closer to unity the Re is, the better the fitting effectiveness is. Generally, a resultant regression equation should only include those independent variables (predictors) strongly influencing the dependent variable. To do so, the stepwise regression analysis is made, that is, introducing the independent variables having significant influnce on the dependent variables step by step. As soon as a new variable is introduced into the equation, all previously introduced variables should be checked again and those variables which become less significant will be rejected from the equation immediately. By this way the resultant regression equation can be ensured to have the significant variables only.
-0.324496 -0.386305 -0.228998 -0.224890
-134.547 -36.8941 -22.5702 -76.2782
2 3 4 5
2 3 4 5
-12.9199 - 19.0112 -21.0554 - 179.314
0.135671 0.131368 0.103939 0.103337
Ol (Tbl)
27.5298 24.9557 72.4290 171.338
-0.0642343 -0.0538219 -0.0390047 -0.0174627
B2 ( Tb2)
0.107465 0.0770664 0.0929689
0.000319972 0.000303044
n3(Tb21 )
-
-0.0000599453 -0.000189374 -0.000186911
B4(T22 )
47452.6
-
-
- 12690.7 -42502.0
-5.85447
L)
Bs(Tb,/T~)B'5(Tb2/
0.979800 0.987076 0.988712 0.993312
0.997231 0.997650 0.999787 0.999809
R~
98.028 78.587 73.471 56.604
0.1437 0.1324 0.03988 0.03779
S¢d
(12)
(11)
No. o f Eqs.
aDependent Variable bThe n u m b e r of independent variables CBo is constant term of regression equation and B 1-B 5 the relevant regression coefficients corresponding to the independent variables shown in brackets, dThe units of Se is in cm for Q and in g / m 2 for L.
Q
O0
DV ~ No. b Regression coefficient ~
Table 2 The stepwise regression analysis (Least Square Estimate) for Q and L
t,,a
ta.a
C. Wei, D. Lu / Atmospheric Research 34 (1994) 309-322
317
4.2. The results of stepwise regression At the significance level of 2.0, the above-mentioned twelve variables are all significant for the dependent variables Q and L, respectively. For the 12-variableregression equation of Q, the Se is 0.03704 cm and Re is 0.99982, while for that of L, Se=45.64 g / m 2 (0.00456 cm) and Re=0.99567. However, the twelve variables have disparate contributions to the dependents. In order to make sure, a series of regression equations with different number of variables out of the twelve are compared, and some results are listed in Table 2. It is found that the Eqs. ( 11 ) and (12), each of which include respective five variables, have good enough effectiveness. The residua of Eqs. ( 11 ) and ( 12 ) are much close to those of the above-mentioned respective 12-variable-equations. In Eqs. (11) and (12) only three variables (Tbl , Tb2 and Ts) among the five are directly-measured and the other two are their combinations. Moreover, Tbl and Tb2 are the GBDFMR measurements and Ts is the most easily measured meteorological variable. The less the number of direct measurement variables is, the less the introduced-measurement errors are. Considering all above aspects, we only use the variables in Eqs. ( 11 ) and ( 12 ) as the predictors for the URE of Q and L, respectively. They are Tbl , Tb2 , (Tbl)2, (Tb2)2 and T b 2 / T s for Q, and Tbl, Tb2, (Tbl) z, Tbl/T~and Tb2/TsforL.
5. The ridge regression analysis for L and Q
5.1. The ridge estimate of the regression coefficients In Eqs. ( 11 ) and (12), there are high correlativity between the predictors. Both for Q and L, out of the total five, there are four correlation coefficients being more than 90% between the independent variables in each equation. Therefore, the determinant of the coefficient matrix XX' of the regular Eq. (7) may be close to zero, so that the X is "illness" and the variance, which is related to ( X ' X ) - 1 based on Eq. (9), of the least-square estimated regression coefficients may become too large to keep the regression equation steady. Ridge regression analysis (Hoel et al., 1970a, b) is one way to overcome this difficulty. The method is putting a relative small positive k in each diagonal element of the coefficient matrix X'X of the regular Eq. (7) when the regression coefficient fl is estimated:
r ^ (k) = ( X ' X + k I ) - ' X ' Y
(10)
where k is known as ridge parameter. In comparison with the least square estimate r ^ (0), the ridge estimate has a little bit less fitting effectiveness for the regression samples used for setting up the regression equations, that is, with a bit larger Se. However, it has been proved theoretically that it is possible to make the total root mean square error offl ^ (k) less than that offl ^ (0) when the value of
318
C. Wei, D. Lu / Atmospheric Research 34 (1994) 309-322 [ l l l l [ l l l r ] l l l l l r [ I
C~ 1.0
,F
,,,
, r,
, i , f ,r
14
~
, I b
10
TbL
,d
5
"U
,,
a
Tb2
0.5
f
0
Tb2
e~
-5 T/Ts z -0.5 0
5
10
Ridge Parameter
15
(X25XI0
')
0
k
5
10
Ridge Parameter
15
(X25XI0
~)
k
Fig. 1. The Ridge Track Map. (a) For the ridge regression equations of Q and (b) for those of L.
k is properly chosen, so that the ridge regression equation has better effectiveness for the new test samples.
5.2. The main results of ridge regression estimate One of the key points for ridge regression analysis is the option of k. Shown in Table 3 is the coefficients and the residua for different ridge regression equations of Q and L corresponding to the different ridge parameter k. From the Table it can be seen that when k = 0 the values of each coefficient are the same as those in Eqs. ( 11 ) and ( 12 ) shown in Table 2 (the least square estimate situation ). The regression coefficients vary and Se increases when the value of k increases. The variations of the normalized coefficients with k are shown in Fig. 1a and b, respectively, which are known as ridge track map. It can be seen that when k varies from zero to 0.005 in Fig. la and from 0 to 0.00005 in Fig. lb each coefficient varies rapidly, and when k goes beyond the scopes the relevant regression coefficients vary slowly. Considering that the Se in Table 3 should be limited in a relatively small value, 0.0025 and 0.000025 are chosen as the respective ridge parameters, and their corresponding Eqs. (13) and (14) (see Table 3) as the final universal equations for Q and L, respectively. 6. The test of the universal regression equations
6.1. Testing samples Radiosonde data with total of 1020 cases are used to construct the cloudy- and
0 0.0025 0.0050 0.0075 0.0100
0 0.000025 0.000050 0.000075 0.000100
a
L
-76.2782 -59.4002 -51.2152 -46.4912 -43.4843
-0.222490 -0.200341 -0.182765 -0.187807 -0.154600
Oo
-179.314 -118.862 -88.8330 -70.9923 -59.2374
0.103337 0.0980471 0.0942337 0.0913035 0.0889568
Ol (Tbl)
Regression coefficienff
171.338 123.350 98.9081 83.9664 73.8140
-0.0174627 -0.0139841 -0.0124467 -0.0113130 -0.0104269
O2(Tb2)
0.0929689 0.0934947 0.0941587 0.0948311 0.0954756
0.000303044 0.000346449 0.000377063 0.000399228 0.000415813
B3(T~l )
-
-0.000186911 -0.000213026 -0.000230834 -0.000243489 -0.000252753
B4(T22)
-47452.6 29471.4 20547.6 15251.7 11766.7
-
-42502.0 -28400.7 -21235.4 - 16866.6 - 13906.4
-5.85447 -5.62776 -5.20482 -4.87445 -4.61039
Ts)
O 5 ( T b , / T s ) BPs(Tb2/
aDependent variable bRidge parameter CFor the definition of Bo-Bs, reference to Table 2.dFor the unit of Se, reference to Table 2.
L
kb
DV a
Table 3 The ridge regression analysis for Q and L
56.6043 59.6402 63.2527 65.9907 68.0309
0.0377896 0.0439400 0.0545500 0.0630900 0.0948600
&d
(14)
(13)
No. of Eqs.
I t,,a t-o
C. Wei, D. Lu / Atmospheric Research 34 (1994) 309-322
320
clear-mixed testing samples by the same way as stated in Section 2 and 3 but the relative humidity criterion is equal to constant 85% rather than modulated. There are two kinds of samples in the testing ensemble, the first with total of 708 cases is known as temporal extrapolation samples which are from the same eight stations as those used to set up the URE but in different years or in different seasons. The second with total of 312 cases is spatial extrapolation samples from seven new stations also having the representativeness of climate, season and location. According to stations or seasons, the total test samples are divided into 21 sets each with 24 to 115 cases more or less (see in Table 4).
6.2. Standard of testing The root mean square deviation (RMS) is used as the test standard of the absolute accuracy and the average relative deviation (Rel) as that of relative accuracy for each of the 21 sets of the testing samples. For Q both absolute and relative tests are made, and for L, only the absolute accuracy test is given since Table 4 The test samples and test effect Station
Longitude
Latitude
Elevation Particular time of Total No. of Test effect (m) data no. of cloudy samples cases Q Year Season RMS Rel (cm) (%)
L
RMS ( g / m 2)
Naqiu
92°04'E 100°26'E Guam 144°50'E Guangzhou 113°19'E Lhasa 91°08'E Yap 138°05'E Beijing 116°17'E Beijing 116°17'E Lijiang 100°26'E Lijiang 100°26'E Xi'an 108°56'E Xi'an 108°56'E Urumqi 37°37'E Urumqi 87°37'E Shanghai 121 ° 2 6 ' E Shanghai 121 ° 2 6 ' E Kunming 102°41'E Kunming 102°41'E Chichijima 1 4 2 ° 1 1 ' E Denver 104°52'W Washington 7 7 ° 0 2 ' W
Zhangye
Total
31°29'N 38°56'N 13°33'N 23°08'N 29°40'N 09°39'N 39°56'N 39°56'N 28°52'N 26°52'N 34° 1 8 ' N 34° 1 8 ' N 43°47'N 43°47'N 31°I0'N 31 ° I 0 ' N 25°01'N 25°01'N 27°05'N 39°45'N 38°51 ' N
4508 1483 111 7 3650 77 55 55 2394 2394 398 398 918 918 5 5 1891 1891 4.1 1625 84
1981 1981 1981 1981 1981 1981/82 1981 1981 1981 1981 1980 1980 1980 1980 1980 1980 1980 1980 1980 1961 1961
sum. sum. sum. win. sum. sum. spr. aut. aut. spr. win. sum. win. sum. win. sum. win. sum. win. sum. sum.
89 103 101 115 101 89 28 29 24 29 30 28 28 27 26 30 28 28 30 31 26
29 24 89 36 28 60 2 6 4 13 7 20 12 1 15 21 12 27 23 8 5
1020
442
0.0530 3.9 0.0473 1.4 0.0388 0.55 0.0584 2.3 0.0388 1.6 0.0543 0.64 0.0304 2.13 0.0432 4.0 0.0796 2.1 0.0320 1.0 0.0205 2.8 0.0504 0.89 0.0277 7.3 0.0227 1.2 0.0369 3.4 0.0356 0.4 0.0137 1.3 0.0629 1.6 0.1021 3.7 0.0644 2.6 0.0403 1.1
34.71 47.34 57.33 31.06 37.53 59.58 44.16 44.35 19.62 37.28 42.73 79.79 29.39 4.02 44.16 66.33 4.97 102.25 77.70 28.47 20.01
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the values of L equal zero for the clear cases in the mixed test samples and the corresponding Rel are meaningless.
6.3. Testing results The testing results are shown in Table 4. For the 21 sets of samples the RMS of Q are in the range of 0.0137 to 0.102 cm and the Rel are all less than 4% except for the set of Wulumqi winter with the value of 7.29%. Considering the facts that the testing samples vary over a wide range of times, seasons, locations and elevations, and that the radiosonde measurement of humidity has an error about 3% to 10% itself, it can be said that the effect of the URE (13) is fairly good. For L, the RMS are within 50 g / m 2 (0.005 cm) for most sets and the largest value is only 102.25 g / m 2 (0.0120 cm), which is even better than some local statistical regression results (Wei et al., 1989 ). Summarily, the URE's for both L and Q are all of perfect accuracy.
7. Conclusions and discussion ( 1 ) Two respective universal regression equations for GBDFMR-retrieval of the precipitable water Q and path-integrated cloud liquid water content L have been obtained by using stepwise regression and ridge regression techniques based on an elaborately-constructed ensemble of clear- and cloudy-mixed samples. The numerical tests show that the retrieval accuracy of Q by the present URE is fairly high, even better than that of some local statistical regression equation (e.g. Wei et al., 1989 ), which just have two independent variables and use a set of local (in situ) historical radiosonde data as the regression samples. The retrieved L also has high accuracy and is feasible and realizable for most application purposes. (2) In comparison with the test result of the Q-URE in the authors' other paper (to be published) which only discusses the situation of clear sky, the present QURE has higher accuracy. The reason is that we introduce the more non-linear terms (Tbl)2 and (Tb2) 2 in this work rather than only the linear terms Tbl and Tb2 as in the previous paper. (3) In this paper, the measurement errors are not considered though it is easy to do. The samples both for regression and for testing are still limited as concerns "global", so the results are still preliminary.
Acknowledgment The reasearch discussed in this paper was supported by the National Natural Science Foundation of China. We are grateful to Professor Feng Shiyong of the Institute of Systematical Sciences, Academia Sinica for his advice and to Mr. Wang Xiaogang for his program assistance.
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