Comparative Study on Sea Surface Temperature Estimation with Thermal Infrared Radiometer Data among Conventional MCSST, Split Window and Conjugate Gradient Based Methods

Comparative study on Sea Surface Temperature: SST estimations among the conventional Multi-Channel Seat Surface Temperature: MCSST, split window method and the proposed Conjugate Gradient based method: CGM with Thermal Infrared Radiometer: TIR data through simulations is conducted. Utilizing the proposed linearized inversion of radiative transfer equation, SST can be estimated. SST estimation accuracy of the proposed method is compared to the conventional regression based method (Split Window and MCSST method). Through the simulation study, it is found that the proposed CGM based method is superior to the conventional regression based method.


INTRODUCTION
Sea Surface Temperature: SST estimation with thermal infrared radiometer onboard satellites is well known and widely used in a variety of research fields, in particular climate changes, global warming, etc. SST estimation methods are proposed [1]- [4].Most of these are based on regressive analysis and use several spectral bands in Thermal Infrared: TIR wavelength region.The most dominant atmospheric factor is precipitable water.Using the different wavelength TIR bands whose influences due to water vapor are different, it is possible to reduce the influence.The most popular method is Multi Channel Sea Surface Temperature: MCSST [5].Also previously proposed SST estimation methods are summarized by I. Barton [6].In the same time, comparative study among the previously proposed methods is well reported [7].
Based on radiative transfer equation, inversion based SST estimation method is proposed [8].Nonlinear radiative transfer equation is linearized then optimum combination of wavelength regions are selected [9].Other than that, Geographic Information System: GIS based neural network is proposed for SST estimation method [10].In this paper, linearized inversion based SST estimation method is utilized.Conjugate gradient method is applied to solve the linearized radiative transfer equation.
The following section describes the proposed SST estimation method with some theoretical background followed by some experiments with the conventional regression based methods.Then conclusion is described together with some discussions.

A. Theoretical Background on SST Estimation with Thermal
Infrared Radiometer Data Radiation from a blackbody with physical temperature of T is expressed in equation ( 1)

Where
The contribution from the atmosphere can be expressed as follows, (2) Where And Is called as optical depth of the atmosphere.

B. At Sensor Radiance of Thermal Infrared Radiometer
For sea surface observation with TIR radiometers onboard remote sensing satellites, radiance includes three components, www.ijarai.thesai.org the contribution from sea surface, the contribution from the reflected radiance at sea surface and the contribution from the atmosphere.
(3) Where Spectral response function means spectral sensitivity function of spectral bands of TIR onboard satellites.In general, emissivity of sea surface in TIR wavelength region is almost 1.Therefore, the second term of the equation ( 3) can be neglected. (4)

C. Approximation of At Sensor Radiance
Assuming spectral response function in the spectral wavelength region of spectral band is 1, then the equation ( 4) and be rewritten as follows, (5) The second term of equation ( 5) can be approximated as follows, (6) Where I ai denotes representative of spectral band i of radiance.Atmospheric transparency can be rewritten as follows, (7) Where u denotes precipitable water while m denotes slant length between sea surface and TIR instrument onboard satellites.In the TIR wavelength region, precipitable water is major absorbing continuants in the atmosphere.Through simulation studies with radiative transfer code of MODTRAN with six atmospheric models (Tropic, Mid.Latitude Summer, Mid.Latitude Winter, Sub Arctic Summer, Sub Arctic Winter and 1976 US Standard Atmosphere), the coefficients are obtained as shown in Table 1.

TABLE I. COEFFICIENTS OF EQUATION (7) OBTAINED WITH MODTRAN OF ATMOSPHERIC SOFTWARE CODE
Then I ai is calculated as follows, (8) The coefficients in the equation ( 8) are calculated with MODTRAN in the same manner which is mentioned above.Table 2 shows the results.Consequently, radiance of spectral band i can be expressed as follows, (9) In order to avoid divergence of the solution, the following conditional equation is introduced.Then the unknown variables are estimated through iterations.

F. Estimation of Hessian
The first derivatives of the cost function are expressed as follows,

G. Conventional Regression Based Method of Split Window
Split window method is based on regression with some training dataset which consists of truth data of SST and TIR spectral band data.
Namely, once regression analysis is made, then SST can be estimated with TIR spectral band data (TIR i ).Regression analysis uses the following regressive equation, Where c i denotes regressive coefficients for spectral band i.

H. Conventional MCSST
Multi Channel Sea Surface Temperature: MCSST is the method for SST estimation with NOAA/AVHRR (National Oceanic and Atmospheric Administration / Advanced Very High Resolution Radiometer) data.MCSST is based on split window method.AVHRR consist five channels, two channels in visible band, one channel in shortwave infrared band, and two channels in thermal infrared bands (Band 4 and 5).In general, MCSST is expressed in equation ( 28).where T S is given in °C and T 4 and T 5 are in Kelvin.The RMSE and the bias is 1.03°C and -0.51°C for daytime, and 0.73°C and 0.19°C for nighttime, respectively.

A. Experimental Conditions
The following parameters are set for the experiments with MODTRAN obtaining at sensor radiance of spectral TIR band data.www.ijarai.thesai.orgThen SST is estimated with the proposed method and the conventional split window method using the calculated at sensor radiance.

B. Example of Solution Behavior
RMSE is estimated with a variety of precipitable water and representative radiance of TIR band.One of the examples of solution behavior is illustrated in Figure 1.In general, RMSE (SST estimation error) is proportional to the precipitable water.On the other hands, the relation between representative radiance of TIR band and RMSE shows complicated characteristics, in particular for small representative radiance regions.This implies that it is not easy to find the global optimum in the solution space.In other words, it is easy to fall in one solution of local minima.

C. Evaluated Averaged SST Estimation Error
Averaged SST estimation error of MCSST method, Split Window method, and Conjugate Gradient method for the basic atmospheric models, Tropic (Figure 2), Mid.Latitude Summer (Figure 3) and Winter (Figure 4), Sub Arctic Summer (Figure 5) and Winter (Figure 6) as well as 1976 US standard atmosphere (Figure 7).For all cases, the proposed conjugate gradient method is superior to the other conventional methods of MCSST and Split Window, in particular for thick atmosphere (precipitable water rich atmosphere).Furthermore, MCSST has systematic errors of which the SST estimation error increases in accordance with increasing of SST.
Although only the difference between MCSST and Split Window methods is the regression coefficients, Split Window method is superior to the MCSST method because the regression coefficients are different among the atmospheric models for Split Window method; MCSST method uses the same coefficients for all atmospheric models.

IV. CONCLUSION
Comparative study on Sea Surface Temperature: SST estimations among the conventional Multi-Channel Seat Surface Temperature: MCSST, split window method and the proposed Conjugate Gradient based method: CGM with Thermal Infrared Radiometer: TIR data through simulations is conducted.Utilizing the proposed linearized inversion of radiative transfer equation, SST can be estimated.SST estimation accuracy of the proposed method is compared to the conventional regression based method (Split Window method and MCSST).
Through the experiments, it is found that the proposed conjugate gradient method is superior to the other conventional methods of MCSST and Split Window for all cases, in particular for thick atmosphere (precipitable water rich atmosphere).
Furthermore, MCSST has systematic errors of which the SST estimation error increases in accordance with increasing of SST.Although only the difference between MCSST and Split Window methods is the regression coefficients, Split Window method is superior to the MCSST method because the regression coefficients are different among the atmospheric models for Split Window method; MCSST method uses the same coefficients for all atmospheric models.

( 10 )
D. Iteration Method for SST, Precipitable Water, and Representative RadianceThe unknown factors are as follows, (11) Namely, sea surface temperature, precipitable water, and representative radiance.The following cost function is introduced, (12) Then iteration is stopped when the cost function is below the designated value, (13) Radiance of the spectral band i can be rewritten as follows, (14) www.ijarai.thesai.orgThen the following updating equation is introduced, (15) It is rewritten in matrix and vector as follows, (24) Also the second derivatives are represented as follows, (24)The first derivatives of radiance are expressed as follows, www.ijarai.thesai.org(25) Also the second derivatives of radiance is represented as follows,

Fig. 1 .
Fig.1.Example of solution behavior in solution space

TABLE II .
COEFFICIENTS OF EQUATION (8) OBTAINED WITH MODTRAN OF ATMOSPHERIC SOFTWARE CODE