3.1 Derivation of D(L) from R-T Model Calculation
Several investigators have given approximate expressions of D(L), [5,6]. All these expressions didn't consider function of surface reflectance. In fact, different surface reflectance result not only the different magnitude of adjacency effect but also variation of the form of D(L). Here we derive a new expression for D(L) by the following integration (refer to Fig.2).


Fig.2. Schematic diagram of adjacency effect.
D(L) =D₁(L) + fd₂(L) (10)

D₂(L) is same as D₁(L) except in the integration the scattering phase function P(z,q)=1 for characterizing multiple scattering as isotropic. f is the fraction determined by RT calculation. For homogeneous surface, the adjacency effect can be accurately estimated effect can be accurately estimated. Thus



Fig.3 Relative significances of atmospheric path radiance, target direct reflection, and adjacency for homogeneous surface with different surface reflectance.
For given atmospheric and surface condition, all of r, r_o, T(q₀),d(q_v), D₁(L),D₂(L) can be obtained as the function of t and A, thus value of f can be determined. In fact, We have made numerical calculation for same atmospheric model s used in section2 with different t and A. fig 3. shows the percentage of r₀, r₁ (target reflection), r₂(adjacency effect) as function of t and A. The expression of D(L) is also valid for airborne remote when the sensor height is within the atmosphere. In this case, t is the optical depth between Z and Zs, the height of airplane. It is shown that for different t and A, path radiance and adjacency effect have different fraction.

Fig. 4 gives the example of D(L) (relative unit), it is shown that D(L) is also depending on surface reflectance. In this Figure, expression of D(L) by [6] is also given. Comparison with our results revels that for different D(L) should be used. Based on above model calculation, we can derive empirical fitting of D(L) for operational atmospheric correction. This will be done in our next paper.


Surface reflectance with the same line style (downward) is 0.8, 0.5, 0.2,0.02 respectively, symbol results are Kaufman(6) with Ha=1.0km

Fig.4D(L) as derived by Eqs. (10)-(12) for different surface reflectance.
3.2 procedure for Atmospheric correction of Inhomogeneous Surface.
Based on discussions and derivation of previous section, we suggest the following procedure of atmospheric correction.

3.2.1 When atmospheric parameters are available

1. Calculation of r₀ and reducing r₀ from observed apparent reflectance r.
2. Derivation of average A_b(see.2.2)
3. Derive D(L) for average Ab centered at each target pixel centered at (x,y) with Eq (10)-(12) of its empirical formulae;
4. Reduction of adjacency effect by Eq(8) and to obtain high resolution surface reflection A⁽¹⁾(x,y) with averaged A_b(x,y) as first guess of A(x,y)
5. Repeat d, until|A⁽ⁿ⁺¹⁾(x,y)-A⁽ⁿ⁾(x,y)|x,y)|< e, e is a predetermined error. Then taking final solution A(x,y) = A⁽ⁿ⁺¹⁾(x,y).

3.2.2. When atmosphere parameters are not available but with observed spectral but with observed spectral apparent reflectance image.

1. Follow procedures of section 2.3 to retrieve atmospheric spectral optical reflectance.
2. Repeat procedures of section 3.2.1.

4. Sumary
In previous sections, we briefly discussed the strategy of atmospheric correction and the simultaneous remote sensing of atmospheric aerosol optical depth and surface reflectance. This strategy is based on the radiative transfer of intrinsic atmospheric scattering, target surface direct reflectance, as well as adjacency effect, respectively. A new expression for atmospheric spread function D(L) is derived. Based on above analysis, we suggested the procedures to retrieve atmospheric aerosol spectral optical depth and surface spectral reflectance by using space-borne or airborne spectral images. We are using this strategy for TM's atmospheric correction.

Acknowledgment
This work was supported by National natural Science Foundation of China under the Major Project "IMGRASS" (No. 49790020) and also supported by China National High -Tech Program on Space Technology (863-2)

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