Evaluación

As a result of scattering within the water, at any depth where there is a downward flux there is also an upward flux. This is always smaller, usually much smaller, than the downward flux but at high ratios of scattering to absorption can contribute significantly to the total light available for photosynthesis. Furthermore, in any water the upwelling light is of crucial importance for the remote sensing of the aquatic envir- onment (Chapter 7), since it is that fraction of the upward flux which penetrates the surface that is detected by the remote sensors. At any depth, the upward flux can be regarded as that fraction of the downward flux at the same depth which at any point below is scattered upwards and succeeds in penetrating up to that depth again before being absorbed or scattered downwards. Thus we might expect the irradiance of the upward flux to be linked closely to the irradiance of the downward flux; this is found to be the case. Figure 6.6 shows the upward and downward irradiance of PAR diminishing with depth in parallel together in an Australian lake. Changes in downward irradiance associated with vari- ation in solar altitude, or cloud cover, would also be accompanied by corresponding changes in upward irradiance. Given the close dependence of upward irradiance on downward irradiance, it is convenient to con- sider any effects that the optical properties of the water may have on the upwelling flux in terms of their influence on the ratio of upwelling to downwelling irradiance,Eu/Ed, i.e. irradiance reflectance,R.

A small proportion of the upward flux originates in forward scattering, at large angles, of downwelling light that is already travelling at some angular distance from the vertical. As solar altitude decreases and the solar beam within the water becomes less vertical, there is an increase in that part of the upward flux which originates in forward scattering. The shape of the volume scattering function is such that this more than counterbalances the fact that an increased proportion of back-scattered flux is now directed downwards. The net result is that irradiance reflectance increases as solar altitude decreases, but the effect is not very large. In the Indian Ocean,R for 450 nm light at 10 m depth increased from 5.2 to 7.0% as the solar altitude decreased from 80 to 31.636For Lake Burley Griffin in Australia, irradiance reflectance of the whole photosynthetic waveband (400–700 nm) just below the surface increased from 4.6% at a solar altitude of 75 to 7.9% at a solar altitude of 27. However, at 1 m, by which depth in this turbid water the light field was well on the way to reaching the asymptotic state (see}6.6),Rincreased only from 7.4 to 8.6%.697

Although, as we have seen, there is a contribution from forward scattering, most of the upwelling flux originates in backscattering. Thus we might expectR to be approximately proportional to the back- scattering coefficient, bb, for the water in question. As the upwards- scattered photons travel up from the point of scattering to the point of

0.1 3 2 1 0 1 10 Ed Eu 100 1000 10 000 1 10 100 1000

Quantum irradiance (1018 _{quanta m}–2 _s–1₎

Quantum irradiance ( _meinsteins m–2 _s–1₎

Depth (m)

Fig. 6.6 Parallel diminution of upward (●) and downward (○) irradiance of PAR with depth in an Australian lake (Burley Griffin, ACT) (after Kirk, 1977a).

measurement, their numbers are progressively diminished by absorption and – less frequently – by further backscatterings that redirect them downwards again. We may therefore expect that reflectance will vary inversely with the absorption coefficient,a, of the water. We might also expect that the dominant tendency of reflectance to increase with back- scattering will be somewhat lessened by the contribution of backscattering to the diminution of the upward flux.

The actual manner in which irradiance reflectance varies with the inherent optical properties of the medium has been explored by math- ematical modelling of the underwater light field. A simplified version of radiative transfer theory339,956leads to the conclusion thatRis propor- tional tobb/(a þbb) for media (such as most natural waters) in which bb«a. This is the kind of relation that might be anticipated on the qualitative grounds outlined above. In fact, since bb is generally very much smaller thana, we might expect that to a reasonable approxima- tion, Rshould be proportional simply tobb/a. Numerical modelling of the underwater light field for waters of various optical types, by Monte Carlo and other methods,484,702,1091reveals that this is indeed the case and we can write

Rð0Þ ¼ Cð0Þbb=a ð6:3Þ whereR(0) is the irradiance reflectance just below the surface. The con- stant of proportionality,C(m0), is itself a function of solar altitude, which

we can express in terms of m0, the cosine of the zenith angle of the

refracted solar beam, below the surface. For any given water body it is the case that reflectance increases as solar altitude decreases,478,484,706,949 i.e. C(m0) increases as m0decreases, and indeed can be expressed as an

approximately linear function of (1 –m0),706,7121/m0,640or [(1/m0) – 1)],478

in, for example, a relationship such as

Cð0Þ ¼ Mð10Þ þCð1:0Þ ð6:4Þ where C(1.0) is the value of C(m0) for zenith sun (m0¼1.0) and M is a

coefficient whose value is determined by the shape of the scattering phase function.478,712It turns out to be the case for zenith sun that the constant of proportionality in eqn 6.3, i.e. C(1.0), is approximately equal to 0.33,484,702,1091 and this remains true for waters with a wide range of scattering phase functions.712 In the field of ocean remote sensing, C(m0), the constant of proportionality between irradiance reflectance and bb/a in eqn 6.3is usually given the symbol, f, it being recognized thatf is itself a function of solar altitude and the scattering phase function. Thus

Rð0Þ ¼ fbb

a ð6:5Þ

For any given oceanic location, an approximate but realistic estimate of f(C[m0]) can be obtained fromeqn 6.4, using the solar zenith angle,

and selecting a plausible value ofMfrom those previously published for a range of water types.712Using radiative transfer theory, and applying certain approximations, Hirata and Højerslev (2008) have shown that for waters and wavebands for which the absorption coefficient is numerically much larger than the backscattering coefficient (i.e. exclud- ing violet-blue wavelengths in clear oceanic waters), the constant of proportionality,f, is an approximate function of the average downward cosine of the light field below the surface (largely determined by solar angle),_d, and the average cosine of backscattering,gb, in accord- ance with f ¼ 1 d gb0:0849 gbþ0:8585 ð6:6Þ

Morel and Prieur (1977) compared their measurements of R across the photosynthetic spectrum with the values calculated usingeqn 6.3, for a variety of oceanic and coastal waters. For clear blue oceanic waters, agreement between the observed and the calculated curves of spectral distribution of R was good. For upwelling oceanic waters with high phytoplankton levels and for turbid coastal waters, agreement was satis- factory from 400 to 600 nm, but in some cases not good at longer wavelengths. Part of the problem in productive waters was a chlorophyll fluorescence emission peak at 685 nm (see }7.5) in the upwelling flux, which increased the observed reflectance over the calculated values in this region. At wavelengths greater than 580 nm, spuriously high reflectance values are also caused by Raman emission.

That part of the upwelling flux just below the surface that is directed approximately vertically upwards is of particular significance for remote sensing. The subsurface radiance in a vertically upward direction we shall refer to asLu. The angular distribution of the upwelling flux is such that upward radiance does not change much with nadir angle in the range 0 to 20. Thus, a measured value of Lu(y) within this range, or an average value over this range, can be taken as a reasonable estimate ofLu.Lu, like Eu, varies for a given water, in parallel withEd, and we shall refer to the ratioLu/Edas theradiance reflectance. In the context of remote sensing it is often referred to as thesubsurface remote sensing reflectance, rrs.

rrs ¼ Luð0Þ Edð0Þ ð6:7Þ and rrs ¼ Rð0Þ Q ð6:8Þ

whereQis the ratio of upward irradiance to nadir radiance

Q ¼ Eu

Lu ð

6:9Þ Some remote sensing radiometers have a very wide field of view: in the SeaWiFS scanner for example (}7.1) it is 58.3. While, because of refraction at the surface, the angular range under water is smaller, it is nevertheless the case that some remotely measured radiance values will correspond to underwater radiances well away from the nadir. In recog- nition of this fact, Morel and Gentili (1993, 1996) define another version of Q, as a function of zenith solar angle (y0), the nadir angle (y0) of the

radiance direction under consideration, and the azimuth difference (f) between the vertical plane of the radiance and the plane of the Sun.

Qð0; 0;fÞ ¼ Euð0Þ

Luð0; 0;fÞ ð 6:10Þ For the subsurface irradiance reflectance, they use the symbol,R(y0), to

indicate its dependence on solar zenith angle.

The value of radiance reflectance, like that of irradiance, is a function of the inherent optical properties of the water. Given a relation between Eu/Edand inherent optical properties, such as that embodied ineqns 6.3 and6.4, if we knowQ, the ratio ofEutoLu, we can relateLu/Ed(i.e.rrs), to bbanda. Substituting forR(0) ineqn 6.8, we obtain

rrs ¼ f Q

bb

a ð6:11Þ

The simplifying assumption is often made that the radiance distribution of the upwelling flux is identical to that above a Lambertian reflector (same radiance values at all angles). If this were so, then the ratio,Eu/Lu, would be equal top. In fact, the radiance distribution is not Lambertian (see}6.6andFig. 6.13) and measurements in Lake Pend Oreille at a solar altitude of 571380showed thatQwas equal to 5.08, near the surface in this water body.42 Monte Carlo modelling calculations (Kirk, unpublished)

have yielded values ofQ, just below the surface, of about 4.9 for waters withb/avalues in the range 1.0 to 5.0, at a solar altitude of 45. Thus, for intermediate solar altitudes we may reasonably assume that Eu/Lu 5. Monte Carlo calculations for waters withb/avalues in the range 1.0 to 5.0 give a value off/Qof0.083, i.e.

rrs¼0:083bb=a ð6:12Þ for a solar altitude of 45.

Aas and Højerslev (1999) present data showingQas a function of solar elevation (hs) at 5 m depth, based on measurements of angular radiance

distribution at 70 stations in the western Mediterranean.Qdecreased with increasing solar elevation, falling from5.2 aths0to3.4 aths90.

The relationship could approximately be represented by

Q¼ ð5:330:30Þexp½ð0:450:08Þsinhs ð6:13Þ in agreement with one proposed earlier by Siegel (1984)

Q¼5:3 expð0:5 sinhsÞ ð6:14Þ for clear water in the central Atlantic Ocean. Aas and Højerslev suggest that this relationship (eqns 6.13and6.14being essentially the same) may be generally valid for clear ocean water, close to the surface.

Loisel and Morel (2001) used computer modelling (Hydrolight) to characterize the extent to which the upward radiance field in Case 2 waters (see}3.4for an explanation of this term) departs from isotropy. In turbid, sediment-dominated water, Q increased progressively from 3.53 to 4.09 as solar altitude decreased from 90 to 15. In clearer, yellow colour-dominated water,Qincreased from 3.69 to 5.02, over the same range of solar angle. For nadir radiance the function,f/Q, which prescribes (eqns 6.11, 6.12) the dependence of subsurface remote sensing reflectance on bb/a, was not markedly dependent on solar angle, remaining fairly close to 0.08 over the angular range in both types of water. For radiance at an extreme nadir angle 35, corres- ponding to 50 above the water, however, f/Q was dependent on solar angle, increasing, as solar altitude decreased from 90 to 15, from 0.085 to 0.129 in the turbid water, and from 0.069 to 0.123 in the coloured water.

In oceanic remote sensing it isRrs, the above-surface radiance reflectance – the vertically upward water-leaving radiance (Lw) divided by Ed(0þ), the downward irradiance above the surface

Rrs ¼ Lw

Edð0þÞ ð

6:15Þ which is in the first instance determined (we here adopt the conven- tion that 0þ corresponds to any point just above the surface, and 0 indicates zero depth just beneath the surface). To proceed from this to the subsurface radiance reflectance, we obtain the subsurface radiance from

Luð0; 0;fÞ ¼

n2

½1ð0; ÞLwð;fÞ ð6:16Þ where y is the above-surface zenith angle of the radiance, y0 is the corresponding refracted nadir angle in the water,fis the azimuth angle, n is the refractive index of sea water and [1 – r(y0, y)] is the Fresnel reflection at the water–air interface for radiance at nadir angle y0491,951. Because of surface reflection, the incident solar flux gives rise to a slightly reduced downward irradiance just beneath the surface

Edð0Þ0 ¼ Edð0þÞð1Þ ð6:17Þ whereEd(0)0is the downward irradiance at zero depth which is created

by that solar radiation which has just penetrated the surface, andis the Fresnel reflection at the water surface for the whole, Sunþsky, incident solar flux.

The total downward irradiance at zero depth,Ed(0), is slightly greater than Ed(0)0because to the surface-penetrating flux there is added that

part of the upwelling flux which is reflected downwards again at the water–air interface. To arrive at an estimate of this additional irradiance we can consider the initial downward flux giving rise, as the result of upward scattering within the water column, to an upward flux with irradiance Ed(0)0R, where Ris irradiance reflectance, R ¼Eu/Ed. Part of this upward flux undergoes downward reflection at the water–air interface, giving rise to a new downward flux with irradiance, Edð0Þ0Rr, whereris the water-air Fresnel reflection for the whole diffuse upwelling radiation stream, and has a value of0.48.41This new downward flux in turn, as a result of upward scattering followed by internal surface reflec- tion, gives rise to a second additional downward flux with irradiance Edð0Þ0ðRrÞ

2

. An infinite series of diminishing downward fluxes is gener- ated in this way

Edð0Þ ¼ Edð0Þ0½1þRrþ ðRrÞ

2_{þ ð}

which, when summed gives us Edð0Þ ¼ Edð0Þ0 ð1RrÞ ð6:18Þ and Edð0Þ ¼ Edð 0þÞð1Þ ð1RrÞ ð6:19Þ

For the subsurface remote sensing reflectance, Lu(0)/Ed(0), we can therefore write

rrs ¼ Lwð;fÞn

2_ð1RrÞ

Edð0þÞð1Þ½1ð0;fÞ ð 6:20Þ Ineqn 6.20, the water-leaving radiance,Lw(y,f), and the surface-incident irradiance,Ed(0þ), are the experimentally determined input parameters; refractive index, n, is known (1.34); the Fresnel reflectances, r and, can be calculated; sinceRin ocean waters is usually<0.1 andris0.48, the ð1RrÞ term is close to 1.0 anyway, and can be estimated if a plausible value ofR for the oceanic region is inserted. Thus, subsurface remote sensing reflectance,rrs, a quantity directly related to the inherent optical properties of the water, can be obtained for each pixel of a remotely sensed scene (seeChapter 7).

The spectral distribution of the upwelling flux must depend in part on that of the downwelling flux, but, as eqns 6.3 and 6.5 show, it is also markedly influenced by the variation in the ratio of bb to a across the spectrum. In clear oceanic waters, for example,Rcan be as high as 10% at the blue (400 nm) end of the photosynthetic spectrum where pure water absorbs weakly but backscatters relatively strongly (see }4.3), and as low as 0.1% at the red (700 nm) end where water absorbs strongly.956 Figure 6.7 shows the spectral distributions of upward irradiance and irradiance reflectance in a clear oceanic water and in an inland impound- ment. The upwelling flux in the oceanic water consists mainly of blue light in the 400 to 500 nm band. In productive oceanic waters with high levels of phytoplankton, the photosynthetic pigments absorb much of the upwelling blue light and so the peak of the upwelling flux is shifted to 565 to 570 nm in the green.956 There is also a peak at 685 nm due to fluorescence emission by phytoplankton chlorophyll. In the inland water (Fig. 6.7b), yellow substances and phytoplankton absorb most of the blue light and a broad band, peaking at about 580 nm, with most of the quanta 6.4 Upward irradiance and radiance 175

occurring between 480 and 650 nm is observed: the chlorophyll fluor- escence emission at about 680 nm can be seen in this curve.

Of the upwelling light flux that reaches the surface, about half is reflected downwards again, and the remainder passes through the water–air interface to give rise to the emergent flux(}7.2). It is this flux, combined in varying proportions with incident light reflected at the surface, that is seen by a human observer looking at a water body, and its intensity and spectral distribution largely determine the perceived visual/aesthetic quality of the water body.289,705,709,711 Figure 6.8 shows the spectral distributions of the subsurface upwelling flux in an Australian lake when it had on one occasion a clear, green appearance, and on another, a turbid, brown appearance. In the first case the spectral distri- bution peaked in the green-yellow region at about 575 nm, as a conse- quence of absorption at the blue end of the spectrum by moderate levels of humic substances, and at the red end by water itself. In the second case the upwelling flux had a greater total irradiance, due to intense scattering by suspended soil particles, and a peak in the red region at 675 to 700 nm,