where Lpath(4) is the path radiance due to Rayleigh and aerosol scattering, t(0v, 4) is the atmospheric transmittance along the path from the ground (or sea) surface pixel to the sensor, t(00, 4) is the atmospheric transmit-tance along the path from the Sun to the ground (or sea) surface and skyEd(0+, 4) is the downward irradiance at the surface due to scattered solar flux in the atmosphere.936 In the case of Landsat studies on aquatic systems, there is as yet no standard method of correcting for these atmospheric effects.
The use of Landsat for remote sensing of water composition has most commonly involved correlating increases in radiance in particular wavebands with increases in concentration of particular types of suspended particles, the increased radiance being due to increased backscattering. In some cases correlations between the uncorrected radiances and concentration of a particular component have been used. The problem in such a case is that on another occasion, when atmospheric conditions have changed, the previously derived relation will no longer apply. In studies on Lake Superior, Canada, for example, the curve of measured radiance against turbidity shifted up and down and changed somewhat in slope on different days, in accordance with changes in atmospheric conditions.1184
A simple, but very approximate, procedure for reducing atmospheric effects is the dark pixel correction method,1134,1135,1142 sometimes also referred to as the dark object subtraction method. This involves locating the darkest pixel in the scene, and subtracting the radiance value for this from the radiance values for all the other pixels. The rationale is that the darkest pixel will have the same atmospheric contribution as all the other pixels, but must have the smallest contribution from scattering within the water. Subtraction of the darkest pixel radiance from all the other pixel radiances should thus remove their atmospheric contribution, and what is left is a function of the difference in particle scattering between the various parts of the water body, and that part with the darkest pixel. To be useful, this method does, however, require that there should exist somewhere in the scene an area of water with a particles content that is at least relatively low: it also assumes constant atmospheric conditions throughout the scene, a reasonable assumption in the case of lakes, the water bodies to which this procedure has usually been applied.
Chavez (1988, 1996) has further developed the dark object subtraction method to take account of the wavelength dependence of scattering. A histogram is prepared of digital number (corresponding to top-of-atmosphere radiance) for all pixels in the scene in selected wavebands. In this way a suitable dark pixel is found to provide the starting dark-object haze value (i.e. path radiance, Lpath from atmospheric scattering). The next requirement is to select a spectral scattering model that best represents the atmospheric conditions at the time of data collection, and the amplitude of the starting haze can be used as a guide to identify these atmospheric conditions. Chavez uses five categories - very clear, clear, moderate, hazy and very hazy. It is assumed that in every case scattering is, in agreement with the Angstr0m law, proportional to X~n. To each type of atmosphere a different value of n is assigned, these being 4,2,1,0.7 and 0.5, respectively, for the five atmospheric categories. The selected scattering model is then used to predict the haze values for the other spectral bands, from the starting haze value. skyEd(0+, 1) in eqn 7.17 is set to zero, i.e. the contribution to Ed from sky radiation is ignored. Chavez (1996) found that the cosine of the solar zenith angle could be used as an approximate estimate of the atmospheric transmittance along the path from the Sun to the ground, and so could be substituted for t(00, 1) in eqn 7.17. The atmospheric transmittance along the path from surface to sensor, t(0v, 1), is assigned a value of 1.0.
Verdin (1985) attempted to determine the actual reflectance values of a water body from Landsat data, with the help of atmospheric radiative transfer calculations. An essential component of the procedure is that there should be an area within the scene that can confidently be assumed to contain clear oligotrophic water. For each waveband a reflectance value for this area is assumed, on the basis of literature data. A plausible initial assumption concerning the prevailing atmospheric turbidity is made from which, by radiative transfer calculation, a value for path radiance is arrived at and tested to see whether, taken together with the assumed water body reflectance, it accounts for the satellite-measured radiance from the clear water. The calculation is repeated in an iterative manner, progressively altering atmospheric turbidity, until a satisfactory value for path radiance is arrived at, and this is assumed to apply throughout the scene. The radiative transfer calculations also yield values for the transmittance of solar radiation down through the atmosphere, and of radiance from the water surface back to the satellite. Thus, for every pixel in the scene, the value of the water-leaving radiance, Lw, and of the incident solar irradiance, can be calculated, and from these the reflectance can be determined.
Brivio et al. (2001) used the radiative transfer calculation procedure of Gilabert et al. (1994) to correct for atmospheric effects in their Landsat Thematic Mapper (TM) study of Lake Garda (Italy). They assumed a horizontally homogeneous atmosphere so that transmittance and path radiance were constant throughout the scene. The path radiance, Lpath(1), was determined from the radiance, Lt(1), measured by the sensor in TM wavebands 1 (450-520 nm) and 3 (630-690 nm), over dark areas within the scene. The Rayleigh contribution was calculated and subtracted from the total path radiance to give the aerosol path radiance, La(1). Through the wavelength dependence of the aerosol path radiance in these two bands an appropriate aerosol model was identified, and used to retrieve the aerosol path radiances in the other spectral bands. The aerosol optical thickness, ta(l), was calculated for each band assuming a linear relationship with La(1). Rayleigh, tR(1), and ozone, tOz(1), optical thicknesses were calculated by standard procedures. Atmospheric transmittance was then obtained as an exponential function of total optical thickness. Total downward irradiance at the surface, Ed(0+, 1), was obtained by summing both direct and diffuse irradiance, as calculated on the basis of atmospheric scattering. In this way, everything required for calculation of pw(1), in eqn 7.17, was obtained.
With six (or seven) narrow wavebands in the visible region, and two in the near-infrared, the SeaWiFS and MODIS (Terra) spaceborne sensors provide much better data for atmospheric correction than the Landsat satellites. Since all three satellites are Sun-synchronous, are at an altitude of ^705 km, and have orbits not widely separated in time, it is generally the case that for a given Landsat 7 TM scene, a concurrent SeaWiFS or MODIS scene will be available. Hu et al. (2001) propose that for atmospheric correction of Landsat 7 TM radiances over aquatic environments, the Rayleigh and aerosol data estimated with SeaWiFS and/or MODIS should be used. Gong et al. (2008) found that vicarious correction with MODIS data worked very well for Landsat TM images of Taihu Lake in China.
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