You are viewing a version of the Ocean Optics Web Book site speciallyformatted for printing.
The purpose of a "lidar equation" is to compute the power returned to a receiver for given transmitted laser power, optical properties of the medium through which the lidar beam passes, and target properties. There are, however, many versions of lidar equations, each of which is tailored to a particular application. For example, Measures (1992)
develops lidar equations for elastic and inelastic backscattering scattering by the medium, for fluorescent targets, for topographical targets, for longpath absorption, for broadband lasers, and so on.
The lidar equation developed here applies to the detection of a scattering layer or inwater target as seen by a narrowbeam laser imaging the ocean from an airborne platform. This equation explicitly shows the effects of atmospheric and seasurface transmission, the water volume scattering function and beam spread function, watercolumn diffuse attenuation, and transmitter and receiver optics. (Acknowledgment: I learned this derivation from Richard C. Honey, one of the pioneering geniuses of optical oceanography, and of many other fields including antenna design and laser eye surgery. Dick Honey is unfortunately little known to the general community because he spent much of his career doing classified work.)
PreliminariesTable lists for reference the variables involved with the derivation of the present form of the lidar equation.
Figure shows the geometry of the lidar system for detection of a scattering layer.
Recall from Eq. (1) of the Volume Scattering Function page that the volume scattering function (VSF) is operationally defined by where is the power incident onto an element of volume defined by a surface area and thickness , and is the power scattered through angle into solid angle . These quantities are illustrated in Fig. . In this derivation, quantities directly proportional to the layer thickness will be labeled with a . Thus is the power scattered by the water layer of thickness . The incident power onto area gives an incident irradiance .
Now consider exact backscatter, which is a scattering angle of , or 180 deg. The backscattered power exits the scattering volume through the same area , so the backscattered irradiance is . The VSF for exact backscatter can then be written as This gives (Do not confuse this backscattered or reflected irradiance with , which is defined for any incident radiance distribution and for an arbitrarily thick layer of water. Here is the irradiance for a collimated incident laser beam and a thin layer of water.)
Suppose a collimated beam is emitting power in direction . Then scattering and absorption in the medium will give some irradiance on the surface of a sphere of radius at an angle of relative to the direction of the emitted beam, as illustrated by the green arrow in Fig. . The beam spread function (BSF) is then defined as
The beam spread function and its equivalent, the point spread function, are discussed in more detail on the Beam and Point Spread Functions page.
Derivation of the Lidar EquationThe derivation of the lidar equation for the stated application now proceeds via the following steps:
Equation () nicely shows the effects of the transmitted power ( ), atmospheric and surface transmission ( ), receiveroptics ( ), watercolumn ( ), and layer thickness ( ). The takehome message from this equation is that in order to understand lidar data, the water inherent optical properties you need to know are the beam spread function and . This observation was in part the incentive for the work of Mertens and Replogle (1977), Voss and Chapin (1990), Voss (1991), McLean and Voss (1991), Maffione and Honey (1992), Gordon (1994b), McLean et al. (1998), Sanchez and McCormick (2002), Dolin (2013), Xu and Yue (2015), and others. These papers present several models for beam/point spread functions in terms of the water absorption and scattering properties. Several of those models are reviewed in Hou et al. (2008). There are various arguments about what to use for , which depends on both the water optical properties and on the imaging system details. It is intuitively expected that , which is defined for a horizontally small patch of upwelling irradiance, will be greater than the diffuse attenuation coefficient for upwelling irradiance, , which is defined for a horizontally infinite light field. Likewise, we expect that will be less than , the beam attenuation coefficient. Thus . Because is an attenuation function for a finite patch of reflected irradiance, computing its value is inherently a threedimensional radiative transfer problem. To pin down the value of more accurately thus requires either actual measurements for a particular system and water body, or threedimensional raditative transfer simulations (usually Monte Carlo simulations) tailored to a given system and water properties.
Example numbersTo develop some intuition about Eq. (), consider the following example application. Suppose a 532 nm laser is being used to look for objects in the water that have an area of . The receiver FOV must be small enough that the object can be distinguished from its surroundings. For and , this requires that For a 15 cm radius receiving telescope, . For normal incidence at the sea surface, , and for a clear atmosphere, . Suppose the water is Jerlov type 1 coastal water for which (Light and Water, page 130), and assume that . Further assume that , since the lidar beam attenuation will be more "beam like" than diffuse attenuation, and . Finally, take (Light and Water, Table 3.10 for "coastal ocean" water). Then for at a depth of 10 m, the water returns the fraction of the transmitted power. Now suppose that the laser beam hits an object at = 10 m whose surface is a Lambertian reflector of 2% (irradiance) reflectance. Then 0.02 of is reflected into . The layer backscatter is then replaced by If all other terms remain the same, the object would return about three times the signal as the water itself.

