Keywords: Aerosol optical depth, Structure function, SPOT

Abstract
In applying the contrast method, such as the structure function method, to estimate the atmospheric aerosol depth with satellite data, a uniform landcover area in image is chosen as a test area to avoid the probable errors induced by the poor structure function patterns. Unfortunately, the uniform area is not easy to pick up or does not exist in some complex landuse regions, such as in Taiwan. In order to pursue the potential application of the structure function method, especially for complex terrain areas, our study extends the original single-directional structure function to multi-directional, and introduces an “optimal number” into our procedure to improve the accuracy of aerosol depth estimation. The comparison between the estimated and sunphotometer-observed aerosol depths shows that the accuracy is improved significantly by our improvements.

Introduction
Basically, satellite remote sensed data are affected mainly by the scattering effects of atmospheric molecular and aerosols in the visible band, Fraser et al., 1984. Some atmospheric correction schemes have been proposed in applying to the satellite visible band data, such as Landsat and SPOT images in the past studies(Griggs, 1975; Mekler et al., 1977; Tanre et al.,1988; Rao et al., 1989; Holben B. N. et al., 1990 and Liu et al., 1997). Some studies showed that the structure function method can be used to estimate accurately the atmospheric aerosol depth, however obvious errors could be induced in some cases by the bad structure function patterns. Further analysis indicates that the abnormal patterns probably are caused by the change of satellite observation geometry, the change or the complexity of landcover. Therefore, the aim of this study is to try to provide a solution to reduce the probable errors in applying the structure method and improve the accuracy of optical depth estimation.

Methodology
Assuming the surface observed by satellite sensor is Lambertian, the apparent reflectance, r*, observed by satellite is (Tanre et al., 1988)

where m_s=cosq_s,q_s is the solar zenith angle, m_v=cosq_v ,q_v is the observed zenith angle, r^a is the atmospheric reflectance,f is the relative azimuth angle between the sun and the satellite, t is the optical depth, r is the surface reflectance, T is the transmittance from the sun to surface, s is the atmospheric albedo, <r> is the mean surface reflectance, and t_d is the diffused transmittance from surface to satellite.

The multi-scattering effect of the surface and atmosphere is small and can be neglected. In other words, <r>S=0. Assuming the <r> values to be the same in local area, the apparent reflectance difference between two neighboring pixels, (i,j) (i,j+d), in distance d can be written as
D r*_(i,j)=D r_(i,j)T(m_s)exp[-t/m_v]              (2)
where i, j are the row and column index of image, respectively. If we assume the D r_(i,j) value being constant in time, the r*(i,j ) will be the function of t, which depends upon atmosphere condition. Tanre et al. (1988) has defined a structure function parameter, M, as

where N is the total pixel number in the test area. In their original method, a single direction difference is calculated in the structure function. However, in our study, we consider a multi-directional(i, j and cross(c) directions shown in Fig.1) structure function, which can be expressed as

and the structure function M * (d) derived by satellite observation can is written as
M*²(d)=M²(d)T²(m_s)exp[-2t /m_v]              (5)
where M is the real surface structuren function.


Figure 1 The directions of i, j and c of the multi-directional structure function.

If assuming the landcover remaining unchanged from t1 to t2 observation time, i.e. M ² (d,t₁)=M² (d,t₂), the relationship of observed structure functions between t1 to t2 is,

Equation (6) shows that if one of the aerosol optical depths in t₁ or t₂ is known, the other optical depth can be derived from satellite observation.

Analysis shows that the estimated optical depth could be different for different d values. In the general cases, the mean value of optical depths from d=1 to d=10 is used, but significant errors could be observed in some cases when the correlation is low for different distances. So, the “optimal number” is adopted to remove the abnormal structure function areas(Liu et al., 1997). The introduction of “optimal number can remove the poor structure functions and improve the accuracy of the optical depth estimation.