Page updated: April 12, 2020
Author: Curtis Mobley
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The Lidar Equation

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 long-path absorption, for broad-band lasers, and so on.

The lidar equation developed here applies to the detection of a scattering layer or in-water target as seen by a narrow-beam laser imaging the ocean from an airborne platform. This equation explicitly shows the effects of atmospheric and sea-surface transmission, the water volume scattering function and beam spread function, water-column 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.)

Preliminaries

Table 1 lists for reference the variables involved with the derivation of the present form of the lidar equation.





variable definition units



HHeight of the airplane above the sea surface m
zDepth of the water layer being imaged m
ΔzThickness of the water layer being imaged m
PtPower transmitted by the laser W
ΔPrPower detected from water layer Δz W
TaTransmittance by the air nondimen
TsTransmittance by the water surface nondimen
ArReceiver aperture area m2
ΩFOV Receiver field of view solid angle sr
AzArea at depth z seen by the receiver m2
ΩSolid angle of the receiver aperture as seen from depthz sr
βπWater VSF for 180-deg backscatter m1sr1
BSFWater beam spread function m2
EiIrradiance incident (downward) onto the water layer at zWm2
ΔErIrradiance reflected (upward) by the water layer Δz Wm2
K̄upDepth-averaged (over 0 to z) attenuation coefficient
for upwelling (returning) irradiance ΔEr m1




Table 1: Variables occurring in the lidar equation.

Figure 1 shows the geometry of the lidar system for detection of a scattering layer.


PIC

Figure 1: Geometry of the lidar system for detection of a scattering layer. The left panel illustrates the physical variables. The right panel illustrates the system optics; this panel ignores refraction at the sea surface.

The volume scattering function (VSF)

Recall from Eq. (1) of the Volume Scattering Function page that the volume scattering function (VSF) is operationally defined by

β(ψ) = ΔΦs(ψ) ΦiΔzΩ ,

where Φi is the power incident onto an element of volume defined by a surface area A and thickness Δz, and ΔΦs(ψ) is the power scattered through angle ψ into solid angle Ω. These quantities are illustrated in Fig. 2. In this derivation, quantities directly proportional to the layer thickness Δz will be labeled with a Δ. Thus ΔΦs(ψ) is the power scattered by the water layer of thickness Δz. The incident power Φi onto area A gives an incident irradiance Ei = ΦiA.


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Figure 2: Quantities associated with the definition of the volume scattering function.

Now consider exact backscatter, which is a scattering angle of ψ = π, or ψ = 180 deg. The backscattered power ΔΦs(π) exits the scattering volume through the same area A, so the backscattered irradiance is ΔEs(π) = ΔΦs(π)A. The VSF for exact backscatter can then be written as

βπ β(ψ = π) = ΔΦs(π)A (ΦiA)ΔzΩ = ΔEs(π) EiΔzΩ .

This gives

ΔEs(π) = EiβπΩΔz. (1)

(Do not confuse this backscattered or reflected irradiance with Eu = REd, which is defined for any incident radiance distribution and for an arbitrarily thick layer of water. Here Ei is the irradiance for a collimated incident laser beam and a thin layer of water.)

The beam spread function (BSF)

Suppose a collimated beam is emitting power P in direction 𝜃 = 0. Then scattering and absorption in the medium will give some irradiance E(r,𝜃) on the surface of a sphere of radius r at an angle of 𝜃 relative to the direction of the emitted beam, as illustrated by the green arrow in Fig. 3. The beam spread function (BSF) is then defined as

BSF(r,𝜃) E(r,𝜃) P (m2). (2)

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Figure 3: Geometry for definition of the beam spread function.

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 Equation

The derivation of the lidar equation for the stated application now proceeds via the following steps:

1.
As illustrated in Fig. 1, the laser pulse has transmitted power Pt to start. After transmission through the atmosphere and sea surface, the pulse has power
Pw(0) = PtTaTs

just below the water surface.

2.
The laser pulse is still a narrow beam when it enters the water, but it then begins to spread out because of scattering, and it is attenuated by absorption. This process is quantified by the beam spread function. We are interested in the ”on-axis” irradiance incident onto a layer of water at depth z. From Eq. (2), this is given by
Ei(z) = Pw(0)BSF(z,𝜃 = 0).
3.
The irradiance that is reflected by a layer of thickness Δz at depth z is given by Eq. (1):
ΔEr(z) = Ei(z)βπ(z)ΩΔz.

The solid angle Ω is determined by the aperture size of the receiver and the distance from the receiver to the layer. This is discussed in step 7 below.

4.
The irradiance ΔEr(z) heading upward from the layer Δz will be attenuated by some diffuse attenuation function K̄up as ΔEr propagates back to the sea surface:
ΔEr(0) = ΔEr(z)exp(K̄upz).

The receiver sees only an area Az depth depth z. It is assumed that Az is less than the total area illuminated by the spreading laser beam at depth z. Thus we can multiply ΔEr(z) by Az to get the power leaving the illuminated area that is seen by the receiver. The fraction of the total power reaching the surface, which is seen by the detector, is then

ΔPr(0) = ΔEr(z)Az exp(K̄upz).

Az is determined by the receiver field of view and the distance from the receiver to the layer.

5.
The power just below the surface is now transmitted through the water surface and through the atmosphere to the receiver. The power detected is then
ΔPr = ΔPr(0)TsTa.
6.
Combining the above results gives
ΔPr = PtTa2T s2A zΩBSF(z,0)βπ exp(K̄upz)Δz.
7.
Az and Ω can be written in terms of known parameters. We are interested only in power that is reflected by the layer Δz into the solid angle Ω that will take the light into the receiver. As shown in the right panel of Fig. 1, the receiver aperture Ar and the range H + z determine Ω = Ar(H + z)2. Only power heading into Ω from area Az, which is seen by the receiver, reaches the receiver. Az is determined by the receiver field of view solid angle ΩFOV and the range: Az = ΩFOV (H + z)2. Thus the AzΩ factor in the preceding equation can be rewritten as
AzΩ = ΩFOV (H + z)2 Ar (H + z)2 = ArΩFOV .

This equation shows how the receiver optics affects the detected power. (This is an application of the ”AΩ” theorem of optical engineering. AΩ is also called the throughput or étendue of the system. Strictly speaking, the in-air solid angle ΩFOV decreases by a factor of 1n2 upon entering the water, where n is the water real index of refraction. However, the in-water Ω increases by a factor of n2 when entering the air. For a round trip, air to water to air, these n2 factors cancel. They are thus ignored here and in Fig. 1.)

8.
Collecting the above results gives the desired lidar equation:
ΔPr = PtTa2T s2A rΩFOV BSF(z,0)βπ exp(K̄upz)Δz. (3)

Equation (3) nicely shows the effects of the transmitted power (Pt), atmospheric and surface transmission (Ta2Ts2), receiver-optics (ArΩFOV ), water-column (BSF(z,0)βπ exp(K̄upz)), and layer thickness (Δz). The take-home 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 βπ. 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 K̄up, which depends on both the water optical properties and on the imaging system details. It is intuitively expected that K̄up, which is defined for a horizontally small patch of upwelling irradiance, will be greater than the diffuse attenuation coefficient for upwelling irradiance, Ku, which is defined for a horizontally infinite light field. Likewise, we expect that K̄up will be less than c, the beam attenuation coefficient. Thus Ku < K̄up < c. Because K̄up is an attenuation function for a finite patch of reflected irradiance, computing its value is inherently a three-dimensional radiative transfer problem. To pin down the value of K̄up more accurately thus requires either actual measurements for a particular system and water body, or three-dimensional raditative transfer simulations (usually Monte Carlo simulations) tailored to a given system and water properties.

Example numbers

To develop some intuition about Eq. (3), 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 1m2. The receiver FOV must be small enough that the object can be distinguished from its surroundings. For H = 100m and z = 10m, this requires that

ΩFOV 1m2 (100 + 10)m2 8 × 105sr.

For a 15 cm radius receiving telescope, Ar = π(0.15m)2 0.07m2. For normal incidence at the sea surface, Ts 0.97, and for a clear atmosphere, Ta 0.98. Suppose the water is Jerlov type 1 coastal water for which Kd(532) 0.15m1 (Light and Water, page 130), and assume that K̄up = Kd. Further assume that BSF exp(0.2z), since the lidar beam attenuation will be more ”beam like” than diffuse attenuation, and c > Kd. Finally, take βπ = 103m1sr1 (Light and Water, Table 3.10 for ”coastal ocean” water). Then for Δz = 1m at a depth of 10 m, the water returns the fraction

ΔPr Pt = (0.98)2(0.97)2(0.07)(8 × 105)e(0.2)10(103)e(0.15)10(1) 1.5 × 1010

of the transmitted power.

Now suppose that the laser beam hits an object at z = 10 m whose surface is a Lambertian reflector of 2% (irradiance) reflectance. Then 0.02 of Ei is reflected into 2πsr. The layer backscatter βπΔz = 0.001sr1 is then replaced by

0.02 2πsr 0.003sr1.

If all other terms remain the same, the object would return about three times the signal as the water itself.

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