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Optimum selection method of initial guess values for maximum likelihood estimation of Sea Surface Temperature and Precipitable Water

Shigeyuki Takasaki, Masao matsumoto, Kiyoshi Tsuchiya
Remote Sensing and Image Research Center, Chiba University
1-33,Yayoi-cho,Chiba city, Chiba, 260 Japan.


Abstract
In order to estimate the perceptible water and the sea surface temperature simultaneously as the maximum likelihood solutions of linear zed radioactive transfer equation, it si important to select the optimum initial guess values for sea surface temperature, air temperature and perceptible water.

In this study, an attempt is made to select the optimum initial guess values by using the statistical method from several atmospheric conditions and sea surface temperature which are summarized from meteorological observation data.

based on that values, the perceptible water and sea surface temperature is estimated from MOS-1/VTIR data and compared with sea surface temperature and perceptible water data.

Introduction
A method was developed by Aoki [1] to estimate the sea surface temperature and perceptible water simultaneously using the observed data from satellite in the atmospheric window region, taking a priori knowledge of statistical characteristics of meteorological data into account. As the estimation errors in this method considerably depend on the initial guess values, the optimum initial guess values must be selected. Aoki determined them from the detailed meteorological data over 1° x 1° latitudinal/longitudinal area. However, it is difficult in general to obtain such detailed meteorological data.

In this paper, atmospheric condition around Japan is divided into ten groups .Initial guess values of air temperature and perceptible water are selected from radio sounding data, and at the same time, this statistical character for each group are calculated. from that values and sea surface temperature which is selected character for each group are calculated . From that values and sea surface temperature which is selected from the mean sea surface temperature in August(summer) and February (winter),simulation data of radiance upwelling to the satellite is calculated using computer code LOWTRAN7. After comparing simulation data with the satelite observed data MOS-1/VTIR, simulation data which has minimum distance from the observed data is selected as initial guess values of that observed date based on their values, sea surface temperature and precipitable water is estimated from MOS-1/VTIR data as the maximum likelihood solutions of linearized radioactive transfer equation and discussed about estimation error of sea surface temperature and perceptible water.

Maximum Likelihood Solutions
With the assumption of local thermodynamically equilibrium and negligble scattering effect by molecules, the spectral radiance Il and the mean effective radiance I upwelling from the cloudless area is written as


Where Bl, T, Ts, tl and p are spectral Planck function, air temperature, sea surface temperature, spectral transmittance and pressure respectively . F(l) is the spectral filter response of the radiometer channel. l1 and l2 are the lower and upper limits of the wavelength of the filter response function TB is the brightness temperature which is observed at the satelite. The subscript s refers to the surface. B and tare average planck function and average transmittance defined by the following equations.


Eq. (1) can be linerarized at the neighborhood of initial guess values of sea surface temperature, air temperature and perceptible water u.Aoki[1] represented the difference between the true and initial guess value with the following equation.




The superscript 0 refers to the initial guess values.

An attempt is made to apply Eq. (4) to VTIR channel 3 and 4 data, Denoting r, DTB and DTs with x1, x2, x3 and the coefficients of each term with kij (i=1,2,3 j=3,4) Eq. (4) can be expressed with the following equation.

y = Kx-----------------------(8)

y=(DTB3 DTB4)1, x = (x1 x2 x3) (t: transpose)------------------(9)


The values of r, T and DT, corresponding to x1 x2 and x3 are estimated as maximum likelihood solutions x3





Where Sx is the variance-covariance matrix of x, Sy is the variance matrix of the noise of the sensor or crrors due to sensor function in channel 3 and 4 of MOS-1VTIR[2]
  1. Optimum selection method of initial guess values

    Figure 1 shows the flow chart to select the optimum initial guess values. At first, ten reasonable atmospheric groups selected around Japan. The mean values of air temperature and perceptible water. Obtained from the radio sounding near or in the each area are adopted as initial guess values. Figure 2 shows the examples of vertical distribution of air temperature and perceptible water. Corresponding to atmospheric group, reasonable sea surface temperature values are selected from the mean sea surface temperature in August (summer) and February (winter) . From these data, initial guess value of brightness temperature T°B is calculated by using computer code LOWTRAN7 [3]

    On the other hands, in order to retrieve several cloudless sea category. MOS-1/VTRchannel 3 and arc classfied by means of cluster analysis (Figure 3) , and TB which is spatial mean value of brightness temperature TB is calculated for each category

    After that, the distance between T°B and is calculated, and T°B which has the minimum distance from is adopted as initial guess value for that category. Thus, initial guess values of air temperature, perceptible water and sea surface temperature are also decided. Based on these initial guess values, sea surface temperature and perceptible water for each category are simultaneously estimated as the maximum likelihood data and the variance of spatial distribution of sea surface temperature is approximated based on that of channel 3 brightness temperature for each category.








    Results of estimation
    Figure 4 and show the results of estimation of sea surface temperature and perceptible water respectively. The average RMS error between the observed values [4] and those estimated is also shown in Table. 1. The average RMS error for sea surface temperature is 1.13 [K] and correlation coefficient is 0.99. According to this result, sea surface temperature can satisfactorily be estimated by the method developed in this research. It should be pointed out that the RMS error of perceptible water in the Table 1 is computed under the assumption that perceptible water over the respective study area is nearly equal to that of the nearly radio sounding station. A little large error of 0.62 (cm) may be due to the above reason since the true perceptible water over the area may be a little different from that of nearly radio sounding station.





    Table Results of estimation of sea surface temperature (SST)and precipitable water (PW)
      Corr.
    Coef.    		 Bias
    error    		 RMS
    error
    SST
    PW    		 0.99
    0.95    		 -0.40(K)
    0.27(cm)    		 1.13(k)
    0.62(cm)

    Conclusion
    In this study , it is concluded that a method has been developed to select the optimum initial guess values for maximum likelihood estimation, and this method is effective to estimate sea surface temperature and precipitable water simultaneously from MOS-1/VTIR data, in spite of based on only the simple meteorological data.

    Acknowledgements
    The authors are grateful to Mr. K.Kajiwara and Mr. H.Okumura of Remote Sensing & Image Research Center, Chiba Univ. for data processing .

    References
    1. Aoki T., 1979: on the information content of the satellite measured infrared Radiation in the Atmospheric Window region, J. meteor Soc. Japan Vol57,No. ,73-78

    2. matsumoto,M,S. Takasaki and K. Tsuchiya 1989 : Simultaneous esimation of Sea Surface Temperature and Precipitable water Using MOS-1VTIR Data,proc. 9th Japanese Conference Remote Sens., 117 -120

    3. Xneizys,F.X.et al., 1988: Users Guide to LOWTRAN 7 , AFGL-TR-88-0177,pp 137,Air Force Geophysics lab.

    4. Matsumoto M. and K. Tsuchiya, 1990: Sea Surface Temperature and Precipitable water Estimation Using MOS-1/VTIR Data, J. Remote Sens . Soc Japan, Japan, Vol. 10. No. 1, 19-28 (* : Orginal text in Japanese )