## Abstract

A method, based on the reciprocity principle of radiative transfer, for using routinely collected field measurements of apparent optical properties in a water body to estimate the total return (time integrated) to an airborne or space borne lidar is presented. It will allow prediction of lidar returns using the databases of apparent optical properties assembled in support of ocean color remote sensing.

©2009 Optical Society of America

## 1. Introduction

In engineering a space-borne lidar system for operation over natural waters, it is important to understand the number of photons expected to be backscattered from the water for a given laser pulse energy. In this note I provide a method based on a reciprocity principle for estimating the photon return (in any marine environment) using water properties that are typically measured in support of passive ocean color remote sensing (apparent optical properties) or computed in various radiative transfer codes [1,2]. The only restriction is that the receiver field of view must be somewhat larger than the excitation spot size at the sea surface.

Zaneveld, Boss, and Behrenfeld [3] reported computation of the lidar return from space using a two-flow model for radiative transfer (RT) in the water and the inherent optical properties of the water and its constituents. Near the end of that report, they discussed the influence of source/receiver spot size on the water surface. In that discussion, they gave a heuristic reciprocity argument to relate active and passive remote sensing. I decided to see if the heuristic argument could be deduced rigorously, and if so, to follow the consequences.

We begin by stating a rigorous form of the reciprocity principle derived from the steady-state radiative transfer equation. Then we apply the reciprocity principle to lidar, starting with a continuous wave (CW) laser and then progressing to pulsed illumination. Next, we consider the time-resolved return from a single pulse. Finally, we examine the return for oligotrophic waters and compare the results with an earlier formulation.

## 2. The reciprocity principle in radiative transfer

The reciprocity principle in RT follows directly from the fact that in a single scattering experiment, the source and receiver can be interchanged with no change in the measured volume scattering function (VSF), i.e., if *β*(*ξ*̂_{1} → *ξ*̂_{2}) is the VSF for scattering from *ξ*̂_{1} to *ξ*̂_{2}, then *β*(*ξ*̂_{1} → *ξ*̂_{2}) = *β*(*ξ*̂_{2} → *ξ*̂_{1}).

Consider two steady-state RT problems involving the same medium without internal sources. The medium is bounded by a surface *S*. (In our application *S* will be the sea surface.) In problem (1), the boundary of the medium is illuminated with a radiance *L*
_{1}(ρ⃗, *ξ*̂), where ρ⃗ indicates position on the boundary, and *ξ*̂ is the direction of propagation. The unit vector *n*̂ is the *outward* normal to the boundary (in our application, *n*̂ points toward the sky). Note that when ,*ξ*̂ ’ ∙ *n*̂<0, *L*
_{1}(ρ⃗,*ξ*̂) is the radiance is *into* the medium. Similarly, in problem (2), the boundary of the medium is illuminated with a radiance *L*
_{2}(ρ⃗,*ξ*̂), etc. Then, it can be shown [4,5] directly from the steady-state radiative transfer equation that

where the integration over *S* is over the *whole* bounding surface of the medium. This is one form of the reciprocity principle of radiative transfer. Note that *L*
_{1}(ρ⃗, -*ξ*̂) is the radiance *exiting* (or reflected from) the medium in the direction - *ξ*̂ at the point ρ⃗ on the surface in problem (1). Variations in refractive index of the medium can be incorporated into the reciprocity principle [5], but this is not necessary for our purposes.

## 3. Application of the reciprocity relationship to lidar

I shall apply the reciprocity relationship to the oceanic lidar problem in two steps: first, the sea surface illuminated by a CW laser; and then the sea surface illuminated by a short (in time) laser pulse.

#### 3.1. CW laser illumination

For application to lidar, we consider two separate RT problems, each with a radiance incident on the sea surface in the form of a parallel beam (e.g., from the sun) of finite extent: *L _{W}*(

*x,y,ξ*̂) and

*L*(

_{N}*x,y,ξ*̂), propagating in the direction

*ξ*̂

_{0}. Mathematically, these are

*L*(

_{w}*x,y,ξ*̂) =

*χ*(

_{W}*x,y*)

*δ*(

*ξ*̂ −

*ξ*̂

_{0})

*F*and

_{W}*L*(

_{N}*x,y,ξ*̂) =

*χ*(

_{N}*x,y*)

*δ*(

*ξ*̂ −

*ξ*̂

_{0})

*F*, where

_{N}*χ*(

_{i}*x,y*) is a function that is zero

*outside*the illuminated surface area on the water surface, ∣

*ξ*̂

_{0}•

*n*̂ ∣

*χ*(

_{i}*x,y*)

*F*is the irradiance distribution on the water surface, and

_{i}*δ*is the Dirac delta function. In the application of interest,

*χ*(

_{N}*x,y*) will be of small (“narrow”extent on the sea surface and

*χ*(

_{W}*x,y*) will be a much larger (“wide” and of essentially infinite extent) on the sea surface. Inserting these into Eq. (1), we find

If the wide beam is essentially of infinite extent *L _{W}*(

*x,y, -ξ*̂

_{0}) =

*L*(-

_{W}*ξ*̂

_{0}), independent of

*x*and

*y*, and if

*χ*(

_{W}*x*,

*y*) = 1 for all

*x*and

*y*,

where the ∞ symbol on the integral means that it is too be taken over the *entire* sea surface. On the left-hand-side of E. (2), *L _{N}*(

*x,y,ξ*̂

_{0}) is the radiance exiting the water in an area

*dx*×

*dy*in the direction -

*ξ*̂, i.e., reflected 180° from the incident beam, and is given by

where *d*
^{2}
*P _{N}*(

*x*,

*y*,-

*ξ*̂

_{0}) is the radiant power leaving

*dx*×

*dy*, in a small solid angle

*d*Ω(-

*ξ*̂

_{0}) around the direction -

*ξ*̂

_{0}. Thus,

where *P _{N}*( -

*ξ*̂

_{0}) is the

*total*radiant power exiting within a solid angle ΔΩ(-

*ξ*̂

_{0}). On the right-hand side of Eq. (2) the integral is simply the total narrow-beam power incident on the surface:

*P*. Therefore,

^{Inc}_{N}or

where

*L _{W}*( -

*ξ*̂

_{0}) is the water-leaving radiance in a direction -

*ξ*̂

_{0}when the water is illuminated by the

*solar*beam of irradiance

*F*propagating in the direction +

_{W}*ξ*̂

_{0}. We will call

*I*the

_{W}*irradiance-normalized*water-leaving radiance. (It is proportional to as the

*remote sensing reflectance*that is associated with the radiance exiting the water in the direction -

*ξ*̂

_{0}.) Thus, if a very narrow beam, e.g., from a continuous wave (CW) laser illuminates the sea surface in a direction +

*ξ*̂

_{0}, and the reflected radiance in the direction -

*ξ*̂

_{0}is collected using a sensor with a wide field of view (FOV), the reflected fraction of the incident power that is collected by the sensor is ΔΩ(-

*ξ*̂

_{0})

*I*(-

_{W}*ξ*̂

_{0}), where ΔΩ(-

*ξ*̂

_{0}) is the solid angle subtended by the sensor at the sea surface. This relates the problem of the reflection of solar radiation from the water to that of the reflection of CW-laser light from the water received by a sensor with a large footprint on the surface.

#### 3.2 Pulsed laser illumination

In order to apply reciprocity to a pulsed lidar system, we need to understand how our basic result, *P _{N}* =

*P*ΔΩ(-

^{Inc}_{N}*ξ*̂

_{0})

*I*(-

_{W}*ξ*̂

_{0}), applies to a pulsed lidar. To effect this, consider a short laser pulse containing

*N*(

_{in}*t*

_{0}) photons centered on time t0 and directed toward the sea surface. This pulse results in

photons returned to the receiver within a time interval *t* and *t* + *dt* at a later time *t* > *t*
_{0}, where *f* is some function that is zero for *t* < *t*
_{0}, and goes to zero as *t* → ∞. The total number of photons that emerge is

Letting *τ*= *t*-*t*
_{0},

Now consider a sequence of identical pulses, each of width Δ*t*
_{0}, to form a continuous input of photons, with each pulse containing Δ*N _{in}* photons. Then the return at time

*t*is

$$\phantom{\rule{1em}{0ex}}=\Delta {N}_{\mathrm{in}}\mathrm{dt}\sum _{i=0}^{\infty}f\left(t-{t}_{0}-i\Delta {t}_{0}\right)=\frac{\Delta {N}_{\mathrm{in}}}{\Delta {t}_{0}}\mathrm{dt}\sum _{i=0}^{\infty}f\left(t-{t}_{0}-i\Delta {t}_{0}\right)\Delta {t}_{0}$$

$$\phantom{\rule{3em}{0ex}}\underset{\Delta {t}_{0}\to 0}{\to}\frac{d{N}_{\mathrm{in}}}{d{t}_{0}}\mathrm{dt}\underset{-\infty}{\overset{t}{\int}}f\left(t-{t}_{0}\right)d{t}_{0}$$

$$\phantom{\rule{3em}{0ex}}=\frac{d{N}_{\mathrm{in}}}{d{t}_{0}}\mathrm{dt}\underset{0}{\overset{\infty}{\int}}f\left(\tau \right)\mathrm{d\tau}=\frac{d{N}_{\mathrm{in}}}{d{t}_{0}}\mathrm{dt}\phantom{\rule{.2em}{0ex}}F.$$

Thus

for the continuous excitation compared with

for the pulsed excitation. Now energy is proportional to the photon number, while power is proportional to the time derivative of the photon number. Thus, in the pulsed case, the CW reciprocity relationship,

becomes

where *E _{N}^{Total}* is the total energy collected at the receiver from the water given an energy

*E*incident in the form of a short pulse.

_{N}^{Inc}#### 3.3 Time development of the reflected pulse (homogeneous medium)

Gordon [6,7] showed that (ignoring the atmosphere) the lidar return from a single *δ*-function pulse sent into a homogeneous water body normal to the surface is

where *C* is a constant dependent on the properties of the lidar system and the scattering properties of the medium, *ν* is the speed of light in *vacuo, m* is the refractive index of water, and κ is the appropriate attenuation coefficient. Multiple scattering computations showed that for a receiver with a field of view (FOV) as large as only a few times the attenuation length (1/c, where c is the beam attenuation coefficient) κ ≈ *K _{d}*, the attenuation coefficient of downwelling irradiance. Now,

therefore

$$=\frac{\nu}{m}\kappa {E}^{\mathrm{Inc}}\mathrm{\Delta \Omega}\left(-{\hat{\xi}}_{0}\right){I}_{W}\left(-{\hat{\xi}}_{0}\right)\mathrm{exp}\left[-\kappa \nu \left(t-{t}_{0}\right)/m\right],$$

by the reciprocity principle. At time *t* the return comes from the top of a slab at depth *z* = *ν*(*t* - *t*
_{0})/(2*m* ∣*ξ*̂_{0}′• *n*̂∣)*z*, where *ξ*̂_{0}′ is the propagation direction in the water, and during the time interval *dt*, the physical width of the slab is *dz* = *νdt*/(2*m* ∣*ξ*̂_{0}′• *n*̂∣) . Thus, in terms of *z*,

$$\phantom{\rule{5em}{0ex}}\approx 2{K}_{d}{E}^{\mathrm{Inc}}\mathrm{\Delta \Omega}\left(-{\hat{\xi}}_{0}\right){I}_{W}\left(-{\hat{\xi}}_{0}\right)\mathrm{exp}\left[-2{K}_{d}z\right],$$

where *K _{d}* is the actual attenuation coefficient for irradiance on a horizontal surface when the incident irradiance is propagating in the

*ξ*̂

_{0}′ direction (In Ref. [8], it is shown that

*K*(

_{d}*ξ*

_{0}′) ≅

*K*(

_{d}*ξ*̂

_{0}′ = -

*n*̂)/∣

*ξ*̂

_{0}′ •

*n*̂∣). This equation allows the computation of the lidar return given experimental measurements (or theoretical calculation) of

*I*and

_{W}*K*.

_{d}The above development rigorously applies to two situations: (1) in the absence of the atmosphere, in which case *I _{W}* is irradiance normalized water-leaving radiance; and (2) in the presence of the atmosphere, in which case

*I*is irradiance normalized, top-of-atmosphere-leaving, radiance. The time resolving capability of the lidar allows backscattering from the atmosphere to be distinguished from the return from the water surface and from within the water body. Our goal is to use day-time measurements of the radiative transfer of solar energy within the water to predict the lidar return, i.e., day-time measurements of water-leaving radiance (at the sea surface or within the water-body). As such, we adopt situation (1), i.e., ignore the atmosphere, but note that any measurement of

_{W}*L*will have already been included the propagation effects of the one-way trip through the atmosphere into the water. Thus, we compute the lidar return in the following manner. First, we assume that the source is sending the radiation into the water such that

_{W}*ξ*̂

_{0}•

*n*̂ = -1, so -

*ξ*̂

_{0}is in the direction of the outward surface normal. Next, we measure

*L*, the upward radiance just beneath the surface propagating toward the zenith,

_{u}*E*, the downward irradiance just beneath the surface, and

_{d}*K*, the attenuation coefficient for downward irradiance. Then,

_{d}*I*is taken to be

_{W}where *T _{f}* is the Fresnel transmittance of the water surface for normal incidence. Using experimentally-obtained

*E*accounts for the effects of the one-way trip through the atmosphere. Finally, we introduce the atmospheric attenuation of the signal from the sea surface to the lidar receiver through an atmospheric transmission factor

_{d}*T*. Thus, Eq. (4) becomes

_{A}and

#### 3.4. Sample computation of the lidar return for oligotrophic waters

Here I use the above equations to compute the return expected over oligotrophic waters for a lidar operating at a wavelength of 532 nm. The altitude is assumed to be 600 km, and the receiver is assumed to have a collecting aperture of 1 m. The solid angle ΔΩ is then 2.2 × 10^{-12} Sr. Measurements in oligotrophic waters near the MOBY [9] site off Lanai, Hawaii [10] yield *L _{u}* ~ 0.53 mW/cm

^{2}μmSr,

*E*~ 130 mW/cm

_{d}^{2}μm, and

*K*~ 0.055 m

_{d}^{-1}. It is important to note that

*L*contains a contribution due to Raman scattering of solar irradiance from ~ 451 nm. Since this excitation is absent in the lidar illumination, it must be removed from

_{u}*L*. For oligotrophic waters at the same solar zenith angle as the measurements described above (36°), the estimated contribution from Raman scattering is ~ 0.13 mW/cm

_{u}^{2}μm Sr [11]. Thus, the purely elastic component of

*L*(532) is ~ 0.40 mW/cm

_{u}^{2}μm Sr. Taking

*T*= 0.9, we find

_{A}Thus, 1 photon will be received for each 3×10^{14} emitted from the laser. Now,

therefore, the fractional return from a slab extending from *z* to *z* + Δ*z* is

Assuming that the laser pulse has an energy of 0.05 J (or contains ~ 1.34 × 10^{17} photons), then *E ^{Total}* ≈ 441 photons and ΔE(

*z*)≈49exp[-0.11

*z*] Δ

*z*photons. Table 1 provides the number of photons at the receiver from various depth ranges. Note that these are the number

photons at the receiver from various depth ranges. Note also that these are the number of available photons at the receiver aperture per pulse. The actual number detected will be the numbers in the table multiplied by the quantum efficiency of the detector times the collection efficiency of the receiver optics, neither of which has been included in this estimate.

#### 4. Comparison with Gordon [6]

From Gordon [6,7], after replacing *P* with *E*, the total return is given by

where *T*
^{2}
_{A} has been added to account for two-way atmospheric losses, *H* is the receiver altitude, *A* is the receiver aperture area, and *β*′(180°) is very close to *β*(180°), the VSF evaluated for 180° scattering. As earlier, using *κ*= *K _{d}*, and the smallest value of

*β*(180°) reported by Petzold [12] in the oligotrophic waters in the Bahamas (Tongue of the Ocean):

*β*(180°) = 5 × 10

^{-4}m-1Sr

^{-1}; the result is

which is in reasonably close agreement with the reciprocity principle, considering that the results are directly proportional to the estimate of *β*(180°).

## 5. Concluding remarks

We have shown how to use the results of passive measurements of apparent optical properties resulting from solar radiative transfer in the ocean-atmosphere system to estimate the total lidar return to space with a receiver having a field of view that is large compared to the spot size of the incident laser beam on the sea surface. This allows the large database of such passive measurements in a wide variety of water types (the SeaWiFS Bio-optical Archive and Storage System, SeaBASS), assembled for ocean color remote sensing [13], to be used in the support of design requirements for oceanic lidar.

For a lidar operating at night, the receiver FOV can be as large as desired; however, in general one would like the FOV of the receiver to be close to that of the laser spot size to reduce the steady-state background of backscattered solar photons that will introduce photon noise into the received subsurface signal. How much larger than the laser spot size does the FOV have to be for the development here to be useful? For an infinitely thin pulse, i.e., *χ*(*x*,*y*) ∝ *δ*(*x*)*δ*(*y*), Gordon [6] shows that

is valid with *β*(180°) ≈(180°)as long as *cD* ≥ 10, where *c* is the beam attenuation coefficient and *D* is the diameter of the FOV at the water surface. For oligotrophic waters, *c*(532)~ 0.1 m^{-1}, so this criterion would require *D* ~ 100 m. This suggests that for a circular laser spot on the surface of radius R, the FOV need only have a radius of *R* + 50 m for the above equation to be valid. Since this equation agrees reasonably well with the result from the reciprocity principle, it seems reasonable to conclude that the receiver spot size need only be a few 10’s of meters larger than the laser spot size.

## Acknowledgments

The author is indebted to C.R. McClain (NASA/GSFC) of pointing out the existence of Ref. [3], Scott McLean (Satlantic) for providing the data used in Section 3.4, J.R.V. Zaneveld for commenting on a draft version of the manuscript, and KJ. Voss for fruitful discussions.

## References and links

**1. **C D. Mobley, B. Gentili, H. R. Gordon, Z. Jin, G. W. Kattawar, A. Morel, P. Reinersman, K. Stamnes, and R. H. Stavn, “Comparison of Numerical Models for Computing Underwater Light Fields,” Appl. Opt, **32**, 7484–7504 (1993). [CrossRef]

**2. **C. D. Mobley, *Light and Water*, (Academic Press, San Diego, CA, 1994) pp. 592.

**3. **J. R. V. Zaneveld, E. Boss, and M. Behrenfeld, “LIDAR photon return calculation in a homogeneous optical medium,” (Unpublished Report)

**4. **K. M. Case, “Transfer Problems and the Reciprocity Principle,” Rev. Mod. Phys. **29**, 651–663 (1957). [CrossRef]

**5. **H. Yang and H. R. Gordon, “Remote sensing of ocean color: Assessment of the water-leaving radiance bidirectional effects on the atmospheric diffuse transmittance,” Appl. Opt. **36**, 7887–7897 (1997). [CrossRef]

**6. **H. R. Gordon, “Interpretation of airborne oceanic lidar: effects of multiple scattering,” Appl. Opt. **21**2996–3001 (1982). [CrossRef] [PubMed]

**07. **In Ref. [6], E (energy) is replaced by *P* (power), and *E _{N}^{Inc}* is replaced by

*P*

_{0}, the laser power; however, that was incorrect because the laser power was taken as

*P*(

*t*) =

*P*

_{0}

*δ*(

*t*-

*t*

_{0}) rather than the correct*P*(*t*) = E_{0}*δ*(*t*-*t*_{0}) so ∫*P*(*t*)*dt*= E_{0}, not*P*_{0}. In the Monte Carlo simulations that were carried out in [6], detected photon numbers were placed in time bins. Photon numbers correspond to energy not power.**8. **H. R. Gordon, “Can the Lambert-Beer Law Be Applied to the Diffuse Attenuation Coefficient of Ocean Water?,” Limnol. Oceanogr. **34**, 1389–1409 (1989). [CrossRef]

**9. **D. K. Clark, H. R. Gordon, K. J. Voss, Y. Ge, W. Broenkow, and C. Trees, Validation of Atmospheric Correction over the Oceans, J. Geophys. Res. **102D**, 17209–17217 (1997). [CrossRef]

**10. **
Scott McLean (Personal commun.)

**11. **H. R. Gordon, “Contribution of Raman scattering to water-leaving radiance: A reexamination,” Appl. Opt, **38**, 3166–3174 (1999). [CrossRef]

**12. **T. J. Petzold, “Volume scattering functions for selected ocean waters,” SIO Ref. **72**–78. (1972).