## 1 Introduction  
The application that supplied the primary impetus for the present work was bathymetry and near shore bottom mapping. Both problems require inverting the remote-sensing reflectance profile to simultaneously fit both the inherent optical properties of the water column and the bottom spectral reflectance profile. In order to obtain robust and reliable results, the number of fitted parameters must be kept to a minimum. The parameters required to model the water column are already well known from numerous and extensive remote-sensing reflectance studies and detailed in the semi-empirical algebraic irradiance model developed by [PERSON] and [PERSON] [1] that we use as a basis for our work. The model we are proposing here produces a good approximation to bottom reflectance spectra by using only three parameters. In the near-ultraviolet (UV) and visible, the reflectance from mineral compounds and vegetation is predominantly due to absorption and backscattering in the bulk material. For most inorganic liquids or solids such as minerals the absorption comes from the broadened far wing of electronic transitions in the deep UV [2; 3; 4] and the backscattering is dominated by reflections at the interface between the crystalline grains of the material. For vegetation the absorption is primarily due to the chlorophyll-a and accessory pigments contained in the plant chloroplasts while the backscattering is due to reflections at the membranes of the cells and their inner components. The radiative transfer processes in both minerals and vegetation, even though occurring on a much smaller scale, are very similar to that occurring in murky waters. This prompted us to adapt a semi-empirical model for murky type II waters due to [PERSON] and [PERSON] [1] and generalize its results with another model due to [PERSON] [5]. Except for the obvious size scale factor, the key difference between the radiative transfer that occurs in minerals and vegetation against that found in murky waters is due to the physics of the backscattering term. We assume the backscattering term comes from the reflection of the interfaces between the structural elements of the solid. The surfaces of the interfaces are modeled to be rough and randomly oriented. The formulas for this type of backscattering are identical to those derived for randomly oriented particles with rough surfaces [6; 7]. The formulas scale as a function of the relative index of refraction of the solid grains and the material of the gap. If the original reflectance was measured for dry samples, the gaps contain air. If the sample is immersed the gaps are water filled and the relative index is smaller. This occurs at or just below the surface of solid rocks and depends on the porosity and on the state and time of immersion. The same effect occurs to an even greater depth when the mineral is in powdered form such as sand We have used this effect to predict the immersed reflectance spectra given the reflectance measured in air. This new model allows one to use the vast library of spectral reflectance signatures measured in air to the underwater environment. We have also used the model in our bathymetric work by measuring the hyperspectral signature of the coastline and modifying it to use as bottom reflectance. We have found this approach to be particularly effective with sand beaches. The only parameter that needs to be estimated is the mean index of refraction of the sand grains which is very close to either silica or in some cases calcite. To properly model vegetation absorption several effects must be accounted for. The absorption spectrum of chlorophyll-a and accessory pigments at low concentrations is modified by saturation of the absorption through the chloroplasts as the concentration increases. This is known as the package effect and has been extensively studied for spherical chloroplast by [PERSON] and [PERSON] [8]. We extend this work to include disk-shaped chloroplast.