Errors generated by the use of a linear model of optical diffuse reflectance in coastal waters

1. Errors generated by the use of a linear model of optical diffuse reflectance in coastal waters Naval Research Laboratory, Ocean Optics Section, Code 7333, Stennis Space Center, USA Vladimir I. HALTRIN e-m: <haltrin@nrlssc.navy.mil>; <http://www7333.nrlssc.navy.mil/~haltrin> Introduction Diffuse reflection coefficient or diffuse reflectance of light from water body is an informative part of remote sensing reflectance of light from the ocean. Diffuse reflectance contains information on content of dissolved and suspended substances in seawater. Diffuse reflectance is an apparent optical property that depends not only on inherent optical properties of the seawater, but also on the parameters of illumination. The dependence on inherent optical properties is expressed as a dependence on a ratio of backscattering coefficient bb to absorption coefficient a. In the open ocean under diffuse illumination of the sky diffuse reflectance R is linearly proportional to the ratio of bb to a, i. e.R=kbb /a, with k=0.33 according to Morel and Prieur. The abovementioned linear equation is very good for the Type I open ocean waters. It is also acceptable for certain types of coastal waters. In fact, it is valid for all types of waters when the ratio of bb to a is less than 0.1. From physical considerations R should always lie between zero and one for any ratio bb /a between zero and infinity. The linear equation fails to pass this criterion, i. e. it exceeds unity when bb /a becomes greater than 1/k, or a<kbb (highly scattering water with a lot of very small particles). It means that indiscrete use of the linear equation for coastal waters, when parameter bb /a exceeds limitations of smallness, can cause unacceptable errors in processing of in situ and remote sensing optical information. In order to estimate possible errors in determining diffuse reflectance we used different approaches to generate diffuse reflectance as a function of bb /a, or g=bb /(a+bb). One approach is based on numerical calculations using Monte Carlo simulation, and other approaches were theoretical. The input values of bb /a have been varied from very small to very large numbers. It was found that numerically and theoretically generated results for all varieties of input parameters satisfactory correspond to the available experimental data. It was found both theoretically and using Monte Carlo that diffuse reflectance strongly depends on backscattering coefficient and has very weak dependence on the shape of the phase function used. References Conclusion Linear model: (Morel-Prieur, 1977): 1. G. A. Gamburtsev, “On the problem of the sea color,” Zh. RFKO, Ser. Fiz. (Journal of Russian Physical and Chemical Society, Physics Series),56, 226-234 (1924). 2. P. Kubelka and F. Munk. “Ein Beitrag zur Optik der Farbanstriche,” Zeit. Techn. Phys., 12, 593-607 (1931). 3. C. Sagan and J. B. Pollack, “Anisotropic nonconservative scattering and the clouds of Venus,” J. Geophys. Res., 72, 469-477 (1967). 4. H. R. Gordon, O. B. Brown, and M. M. Jacobs, “Computed relationships between the inherent and apparent optical properties of a flat homogeneous ocean,” Appl. Optics, 14, 417-427 (1975). 5. A. Morel, L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr., 22, 709-722 (1977). 6. V. I. Haltrin (a.k.a. V. I. Khalturin), “Propagation of light in sea depth,” in Remote Sensing of the Sea and the Influence of the Atmosphere (in Russian), V. A. Urdenko and G. Zimmermann, eds. (Academy of Sciences of the German Democratic Republic Institute for Space Research, Moscow-Berlin-Sevastopol, 1985), pp. 20-62. 7. V. I. Haltrin, “Exact solution of the characteristic equation for transfer in the anisotropically scattering and absorbing medium,” Appl. Optics, 27, 599-602 (1988). 8. V. I. Haltrin and G. W. Kattawar “Self-consistent solutions to the equation of transfer with elastic and inelastic scattering in oceanic optics: I. Model,” Appl. Optics, 32, 5356-5367 (1993). 9. V. I. Haltrin, and A. D. Weidemann, “A Method and Algorithm of Computing Apparent Optical Properties of Coastal Sea Waters”, in Remote Sensing for a Sustainable Future: Proceedings of 1996 International Geoscience and Remote Sensing Symposium: IGARSS’96, Vol. 1, Lincoln, Nebraska, USA, p. 305-309, 1996. 10. V. I. Haltrin, “Self-consistent approach to the solution of the light transfer problem for irradiances in marine waters with arbitrary turbidity, depth and surface illumination,” Appl. Optics, 37, 3773-3784 (1998). The widely used linear model is very good for bb /(a+bb) < 0.1 and very satisfactory for bb /(a+bb) ≤ 0.2, it produces wrong results for bb /(a+bb) > 0.2. The majority of coastal water and almost all open ocean water cases fall in the range of applicability of linear model. But the linear model may be very inadequate in some important and interesting coastal water conditions like hazardous blooms, spills, etc. For the reasons to avoid possible unacceptable errors and missing interesting optical events it is advisable to avoid using linear model to process information related to coastal (Type II and III) waters. All presented non-linear equations (except the Kubelka-Munk equation that is not acceptable for seawater at small values of bb /(a+bb) < 0.2, and Gordon’s equations that are not valid at bb /(a+bb) > 0.2, and at bb /(a+bb) < 0.0001) are capable to produce values of R that are correct for all possible values of bb /(a+bb). In order to detect special optical cases the non-linear equations should be used in automatic processing of in-situ and remotely obtained optical information. Non-linear solutions for Diffuse reflectance: Exact (Haltrin, 1988) Self-Consistent Asymptotic (Haltrin, 1985, 1993, 1997) Self-Consistent Diffuse (Haltrin, 1985) Two-Stream: (Gamburtsev, 1924; Kubelka-Munk, 1931; Sagan and Pollack, 1967) Because we do not have reliable in situ measurements of diffuse reflectances that represent the whole range of variability of inherent optical properties, 0 < bb /(a+bb) < 1, we have to choose a dependence which can be regarded as sufficiently “precise one” in order to be a basis for error estimation. Such dependence exists in literature (Haltrin, 1988) and represents an exact solution of radiative transfer for diffuse reflection of light in a medium with delta-hyperbolic phase function. This solution lies exactly in the middle of two Monte Carlo and two theoretical solutions for diffuse reflections for small values of bb /(a+bb) < 0.2, and it gives precise and asymptotically correct values for 1 - bb /(a+bb) < < 1 (see first two figures). Direct Monte Carlo (Gordon, Brown, and Jackobs, 1975) Diffuse Acknowledgment Semi-Empirical (Haltrin and Weidemann, 1996) The author thanks continuing support at the Naval Research Laboratory through the Spectral Signatures 73-5939-A1 program. This article represents NRL contribution AB/7330-01-0168.