Upward irradiance and radiance

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 environment (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 variation 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 consider 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.

Quantum irradiance ( ^einsteins m 2 s 1) 1 10 100 1000 10 000

Quantum irradiance ( ^einsteins m 2 s 1) 1 10 100 1000 10 000

1 10 100 1000 Quantum irradiance (1018 quanta m-2 s-1)

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

1 10 100 1000 Quantum irradiance (1018 quanta m-2 s-1)

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

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°.636 For 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 1m, by which depth in this turbid water the light field was well on the way to reaching the asymptotic state (see §6.6), R increased 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 expect R 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 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 mathematical modelling of the underwater light field. A simplified version of radiative transfer theory339,956 leads to the conclusion that R is proportional to bb/(a + bb) 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 than a, we might expect that to a reasonable approximation, R should be proportional simply to bb/a. Numerical modelling of the underwater light field for waters of various optical types, by Monte Carlo and other methods,484,702,1091 reveals that this is indeed the case and we can write

where R(0) is the irradiance reflectance just below the surface. The constant 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 m0 decreases, and indeed can be expressed as an approximately linear function of (1- m0),706,712 1/m0,640 or [(1/m0) - 1)],478 in, for example, a relationship such as

where C(1.0) is the value of C(m0) for zenith sun (m0 = 1.0) and M is a coefficient whose value is determined by the shape of the scattering phase

AfJQ 710

function. , It 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 6.3 is usually given the symbol, f, it being recognized that f is itself a function of solar altitude and the scattering phase function. Thus

For any given oceanic location, an approximate but realistic estimate of f (= C[m0]) can be obtained from eqn 6.4, using the solar zenith angle, and selecting a plausible value of M from those previously published for a range of water types.712 Using radiative transfer theory, and applying certain approximations, Hirata and Hojerslev (2008) have shown that for waters and wavebands for which the absorption coefficient is numerically much larger than the backscattering coefficient (i.e. excluding 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), , and the average cosine of backscattering, gb, in accordance with gb - 0.0849

Morel and Prieur (1977) compared their measurements of R across the photosynthetic spectrum with the values calculated using eqn 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 satisfactory 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 as Lu. 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(0) within this range, or an average value over this range, can be taken as a reasonable estimate of Lu. Lu, like Eu, varies for a given water, in parallel with Ed, and we shall refer to the ratio Lu/Ed as the radiance reflectance. In the context of remote sensing it is often referred to as the subsurface remote sensing reflectance, rrs.

where Q is the ratio of upward irradiance to nadir radiance

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 recognition of this fact, Morel and Gentili (1993, 1996) define another version of Q, as a function of zenith solar angle (00), the nadir angle (0') of the radiance direction under consideration, and the azimuth difference (f) between the vertical plane of the radiance and the plane of the Sun.

For the subsurface irradiance reflectance, they use the symbol, R(00), 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/Ed and inherent optical properties, such as that embodied in eqns 6.3 and 6.4, if we know Q, the ratio of Eu to Lu, we can relate Lu/Ed (i.e. rrs), to bb and a. Substituting for R(0) in eqn 6.8, we obtain f bb tr 1 ^

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 to p. In fact, the radiance distribution is not Lambertian (see §6.6 and Fig. 6.13) and measurements in Lake Pend Oreille at a solar altitude of 57°1380 showed that Q was equal to 5.08, near the surface in this water body.42 Monte Carlo modelling calculations (Kirk, unpublished)

have yielded values of Q, just below the surface, of about 4.9 for waters with b/a values 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 with b/a values in the range 1.0 to 5.0 give a value off/Q of ^0.083, i.e.

for a solar altitude of 45°.

Aas and Hojerslev (1999) present data showing Q as a function of solar elevation (hs °) at 5 m depth, based on measurements of angular radiance distribution at 70 stations in the western Mediterranean. Q decreased with increasing solar elevation, falling from ^5.2 at hs ~ 0° to ^3.4 at hs ~ 90°. The relationship could approximately be represented by

Q = (5.33 ± 0.30) exp[—(0.45 ± 0.08) sin hs] (6.13)

in agreement with one proposed earlier by Siegel (1984)

for clear water in the central Atlantic Ocean. Aas and Hojerslev suggest that this relationship (eqns 6.13 and 6.14 being 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 §3.4 for 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, Q increased 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°, corresponding 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 is Rrs, the above-surface radiance reflectance -the vertically upward water-leaving radiance (Lw) divided by Ed (0+), the downward irradiance above the surface

which is in the first instance determined (we here adopt the convention that 0+ 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 n2

where 0 is the above-surface zenith angle of the radiance, 0' is the corresponding refracted nadir angle in the water, f is the azimuth angle, n is the refractive index of sea water and [1 - p(0', 0)] is the Fresnel reflection at the water-air interface for radiance at nadir angle 0'491,951. Because of surface reflection, the incident solar flux gives rise to a slightly reduced downward irradiance just beneath the surface

where Ed (0-)0 is the downward irradiance at zero depth which is created by that solar radiation which has just penetrated the surface, and p is the Fresnel reflection at the water surface for the whole, Sun + sky, incident solar flux.

The total downward irradiance at zero depth, Ed (0-), is slightly greater than Ed(0-)0 because 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-)0 R, where R is irradiance reflectance, R = Eu/Ed. Part of this upward flux undergoes downward reflection at the water-air interface, giving rise to a new downward flux with irradiance, Ed(0)0Rr, where r is the water-air Fresnel reflection for the whole diffuse upwelling radiation stream, and has a value of-0.48. This new downward flux in turn, as a result of upward scattering followed by internal surface reflection, gives rise to a second additional downward flux with irradiance Ed(0)0(Rr)2. An infinite series of diminishing downward fluxes is generated in this way

Ed (0-) = Ed (0-)0[1 + Rr + (Rr)2 + (Rr)3 + ... + (Rr) + ...]

which, when summed gives us

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

In eqn 6.20, the water-leaving radiance, Lw(6, f), and the surface-incident irradiance, Ed(0+), are the experimentally determined input parameters; refractive index, n, is known (—1.34); the Fresnel reflectances, p and p, can be calculated; since R in ocean waters is usually <0.1 and r is —0.48, the (1 — Rr) term is close to 1.0 anyway, and can be estimated if a plausible value of R 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 (see Chapter 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, R can 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 impoundment. 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 rrs

Florida Irradiance
Fig. 6.7 Spectral distribution of upward irradiance and irradiance reflectance in an oceanic and an inland water (plotted from data of Tyler and Smith, 1970). (a) Gulf Stream (Atlantic Ocean) off Bahama Islands, 5m depth. (b) San Vicente Reservoir, San Diego, California, USA, 1 m depth.

occurring between 480 and 650 nm is observed: the chlorophyll fluorescence 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 distribution peaked in the green-yellow region at about 575 nm, as a consequence 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, resulting from strong absorption in the blue and green regions by high concentrations of soluble and particulate humic substances. Rivers

550 600 Wavelength (nm)

Fig. 6.8 Spectral distribution of subsurface upward irradiance in Lake Gin-ninderra, ACT, Australia. (a) 20 April 1983. Appearance - clear, green. b — 3.2 m . a440 — 1.22 m . (b) 15 August 1984. Appearance - turbid, brown. b — 28.2 m-1. a440 — 23.1m-1.

550 600 Wavelength (nm)

Fig. 6.8 Spectral distribution of subsurface upward irradiance in Lake Gin-ninderra, ACT, Australia. (a) 20 April 1983. Appearance - clear, green. b — 3.2 m . a440 — 1.22 m . (b) 15 August 1984. Appearance - turbid, brown. b — 28.2 m-1. a440 — 23.1m-1.

derived from glaciers are characteristically milky white or grey in appearance, due to the presence of high concentrations of mineral particles ('glacial flour'), but little organic material.

The apparent colour of a water body is determined by the chromaticity coordinates of the flux received by the observer, and these can be calculated from the spectral distribution using the CIE (International Commission on Illumination) standard colorimetric system (see Jerlov, 1976, for further details). Davies-Colley et al. (1988) have carried out such calculations using upwelling spectral distribution data, for 14 New Zealand lakes, and suggest that this is a potentially useful tool for water resource managers with a concern for the aesthetic quality of the water bodies for which they are responsible. A comprehensive treatment of colour and clarity in natural water bodies in the context of human use of such waters may be found in the book by Davies-Colley, Vant and Smith (2003).

In the oceanic waters studied by Tyler and Smith (1970), irradiance reflectance for total PAR at 5 m depth varied from about 2 to 5%. For slightly to very turbid inland water bodies in southeast Australia, irradi-ance reflectance values for PAR just below the surface were usually between 4 and 10%, but values as low as 2% and as high as 19% were observed, the higher values being associated with higher turbidity.697,700 The reflectance values increased somewhat with depth (see §6.6), the maximum value observed so far being about 24%. In inland waters with low scattering, but intense colour due to high concentrations of soluble yellow substances, reflectance of PAR can be very low. In a series of lakes of this type in Tasmania, irradiance reflectance for PAR just below the surface ranged from about 1.2% down to 0.14%.152

Where there is a bloom of coccolithophores - haptophycean algae whose cells are covered with highly scattering calcareous scales (coccoliths, Fig. 4.10) - the reflectance of the ocean is greatly increased. In a coccolithophore bloom in the Gulf of Maine, Balch et al. (1991) measured subsurface reflectance values in the blue-green waveband of 5 to 7% at one station, and 22 to 39% at another. The high reflectance values appeared to be due primarily to large numbers of detached cocco-liths suspended in the water, rather than to the whole cells.

In clear ocean waters with little colour, Raman scattering of the predominantly blue-green downwelling light stream gives rise, because of the associated shift to longer wavelengths (see §§4.2, 7.5), to a faint diffuse scattered light field in the 520 to 700 nm range.869,1291,1321 While this is of little importance for photosynthetic primary production, and makes only a small contribution to the downwelling light field, it can contribute significantly to the upwelling light stream in the lower region of the euphotic zone, and is the probable cause of anomalous increases in reflectance at greater depths, which have sometimes been observed.

In water bodies that are sufficiently shallow for significant light to reach the bottom, then unless the bottom is very dark in colour there is an increase in Eu near the bottom due to reflection from it.

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  • nasih
    Why radiance value are smaller than irradiance value?
    2 years ago

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