Page updated: April 9, 2020
Author: Curtis Mobley

# K functions

We now further examine the diﬀuse attenuation functions, which are often called ”K functions.” Under typical oceanic conditions, for which the incident lighting is provided by the sun and sky, the various radiances and irradiances all decrease approximately exponentially with depth in homogeneous water, when far enough below the surface (and far enough above the bottom, in optically shallow water) to be free of boundary eﬀects. It is therefore convenient to write the depth dependence of ${E}_{d}\left(z,\lambda \right)$, for example, as

${E}_{d}\left(z,\lambda \right)\phantom{\rule{1em}{0ex}}\equiv \phantom{\rule{1em}{0ex}}{E}_{d}\left(0,\lambda \right)\phantom{\rule{0.3em}{0ex}}exp\left[-{\int }_{0}^{z}\phantom{\rule{1em}{0ex}}{K}_{d}\left({z}^{\prime },\lambda \right)\phantom{\rule{1em}{0ex}}d{z}^{\prime }\phantom{\rule{0.3em}{0ex}}\right]\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}},$

where ${K}_{d}\left(z,\lambda \right)$ is the diﬀuse attenuation function for spectral downwelling plane irradiance. Solving for ${K}_{d}\left(z,\lambda \right)$ gives the equation previously used for the deﬁnition of ${K}_{d}\left(z,\lambda \right)$:

${K}_{d}\left(z,\lambda \right)\phantom{\rule{1em}{0ex}}=\phantom{\rule{1em}{0ex}}-\frac{d\phantom{\rule{0.3em}{0ex}}ln\phantom{\rule{0.3em}{0ex}}{E}_{d}\left(z,\lambda \right)}{dz}\phantom{\rule{1em}{0ex}}=\phantom{\rule{1em}{0ex}}-\frac{1}{{E}_{d}\left(z,\lambda \right)}\phantom{\rule{0.3em}{0ex}}\frac{d\phantom{\rule{0.3em}{0ex}}{E}_{d}\left(z,\lambda \right)}{dz}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}.$

Similar equations can be written for other radiometric variables and their corresponding K functions.

As explained in the previous discussion of AOPs, in order to be useful for relating light measurements to water properties, the K functions should depend strongly on water IOPs but only weakly on external environmental conditions like sun location, sky condition, or surface waves. To illustrate these dependencies, various K functions were numerically computed using the HydroLight radiative transfer numerical model. In most situations it is preferable to work with real data. However, use of this model gives us the ability to simulate diﬀerent environmental conditions at will and to control things that cannot be controlled in nature, such as turning chlorophyll ﬂuorescence on or oﬀ. This can be very useful for understanding the interdependence of various quantities.

HydroLight simulations were ﬁrst performed at one wavelength for homogeneous idealized water bodies dominated by either scattering or absorption. For the ”highly scattering” water, the absorption coeﬃcient was set to $a=0.2\phantom{\rule{1em}{0ex}}{m}^{-1}$, and the scattering coeﬃcient was $b=0.8\phantom{\rule{1em}{0ex}}{m}^{-1}$, so that the albedo of single scattering was ${\omega }_{o}=b∕\left(a+b\right)=0.8$. These values correspond roughly to what might be found in Case 1 water with a chlorophyll concentration of $5\phantom{\rule{1em}{0ex}}{mg\phantom{\rule{1em}{0ex}}m}^{-3}$ at blue or green wavelengths. An ”average particle” scattering phase function was used, the sun was placed in a clear sky, and the water was inﬁnitely deep. Note that since $a+b=c=1\phantom{\rule{1em}{0ex}}{m}^{-1}$, the optical depth is numerically equal to the geometric depth z in meters.

Figure 1 shows various K functions as a function of depth for the highly scattering water when the sun was placed at a zenith angle of 40 degrees and the surface was level (wind speed of $U=0$). As is conventional in radiative transfer theory, the $K\left(𝜃,\varphi \right)$ are for radiance propagating in the $\left(𝜃,\varphi \right)$ direction. HydroLight measures depth positive downward from the mean sea surface, and polar angle $𝜃$ is measured from the $+ẑ$ or nadir direction. Thus $𝜃=0$ refers to light heading straight down into the water (corresponding to ${K}_{Ld}$), $𝜃=180$ refers to light heading straight up (corresponding to ${K}_{Lu}\right)$, and $𝜃=90$ refers to light traveling horizontally through the water. Azimuthal angle $\varphi =0$ refers to light heading towards the sun, which was placed at $\varphi =0$; $\varphi =180$ thus refers to light heading away from the sun. In this simulation, ${K}_{sun}\left(30,180\right)$ corresponds to looking into the sun’s refracted beam underwater, which is light heading downward and away from the sun. Figure 1: Fig 1. Computed K functions for ”highly scattering” water. The sun was at a zenith angle of 40 deg and the sea surface was level. The optical depth is numerically equal to the geometric depth.

There are several important features to note in Fig. 1:

• The various K functions can diﬀer greatly near the sea surface. This due to boundary eﬀects on the solution of the radiative transfer equation. The surface boundary aﬀects radiances in diﬀerent directions in diﬀerent ways depending on the relative location of the sun. The large near-surface values of ${K}_{sun}\left(30,180\right)$ indicate that the sun’s direct beam is decreasing rapidly with depth due to absorption and scattering out of the beam. On the other hand, ${K}_{Lu}$, which is looking straight downward at radiance propagating upward, is almost constant with depth.
• K functions can be positive or negative near boundaries. A negative K means that the radiometric variable is increasing with depth. ${K}_{Ld}$, which is looking upward at the zenith at radiance propagating downward, is negative in the ﬁrst couple of meters below the surface. At depth 0 just below the sea surface, the downwelling radiance ${L}_{d}$ comes mostly from the zenith sky radiance transmitted through the level surface. Going deeper into the water, ${L}_{d}$ increases with depth as scattering from the sun’s strong direct beam contributes more path radiance to ${L}_{d}$ than is lost by absorption and scattering out of the downward beam. Eventually the sun’s direct beam becomes weak enough that the path radiance contribution to ${L}_{d}$ is less than the attenuation due to absorption and scattering out of the beam, and ${K}_{Ld}$ becomes positive. The same eﬀect is seen less dramatically in ${K}_{h}\left(90,180\right)$, which corresponds to looking horizontally toward the sun.
• K functions are not constant with depth even in homogeneous water. Again, this is a manifestation of the surface boundary eﬀects. If the water IOPs depend on depth, then the K functions also vary with depth, even far from a boundary. Thus radiometric variables never decrease exactly exponentially with depth, although this is often a good approximation for homogeneous water.
• Far from boundaries (i.e., very deep in the ocean and very far from the bottom), all K functions approach a common value, the ”asymptotic K value” K, that depends only on the IOPs. Its value for the IOPs of this simulation was ${K}_{\infty }=0.3082\phantom{\rule{1em}{0ex}}{m}^{-1}$. Thus at depths great enough for boundary eﬀects to be negligible, all K functions are the same and these AOPs become an IOP. In the present simulation, the K functions are all the same to within 3% by 30 m depth; ${K}_{d}$ is within 0.2% of ${K}_{\infty }$ by 30 m depth. The asymptotic K functions are discussed in detail on the asymptotic radiance distribution page.

Figure 2 shows the K functions corresponding to the same conditions as Fig. 1, except that the wind speed was $U=15\phantom{\rule{1em}{0ex}}{m\phantom{\rule{1em}{0ex}}s}^{-1}$. We see that there is very little diﬀerence between Figs. 1 and 2. Thus, as hoped, the K functions are almost unaﬀected by the surface waves. Figure 2: Fig 2. Computed K functions for ”highly scattering” water. The conditions were the same as for Fig. 1, except that the wind speed was $15\phantom{\rule{1em}{0ex}}{m\phantom{\rule{1em}{0ex}}s}^{-1}$, so that the sea surface was not level.

Figure 3 shows the K functions for a level surface and the sun at the zenith, rather than at 40 deg. Again, the irradiance K functions are almost unchanged. However, the radiance ${K}_{Ld}$ function now corresponds to looking straight upward into the sun’s direct beam. Thus ${K}_{Ld}$ now looks very similar to ${K}_{sun}\left(30,180\right)$ in the previous ﬁgures. Similarly, $K\left(30,180\right)$ now looks much like ${K}_{Ld}$ in the previous ﬁgures. This is because moving the sun from 40 deg in air (28 deg in water) to the zenith gives $K\left(30,180\right)$ almost the same scattering angle relation to the sun’s direct beam as ${K}_{Ld}$ had in the previous ﬁgures. Figure 3: Fig 3. Computed K functions for ”highly scattering” water. The conditions were the same as for Fig. 1, except that the sun was at the zenith.

Figure 4 shows K functions for ”highly absorbing” water: the absorption coeﬃcient was $a=0.8\phantom{\rule{1em}{0ex}}{m}^{-1}$, and the scattering coeﬃcient was $b=0.2\phantom{\rule{1em}{0ex}}{m}^{-1}$, so that the albedo of single scattering was ${\omega }_{o}=b∕\left(a+b\right)=0.2$. These values correspond roughly to what might be found at red wavelengths, where absorption by the water itself begins to dominate the IOPs. Other conditions were the same as for Fig. 1. Figure 4: Fig 4. Computed K functions for ”highly scattering” water. The sun was at a zenith angle of 40 deg and the sea surface was level. The optical depth is numerically equal to the geometric depth.

Comparing Figs. 1 and 4, we note that

• The rate of approach to the asymptotic value depends on the IOPs. In highly scattering water, the approach to ${K}_{\infty }$ is much faster than in highly absorbing water. This is because the near-surface radiance distribution must be ”redistributed” by multiple scattering into the shape of the asymptotic radiance distribution ${L}_{\infty }$ in order for the K’s to approach ${K}_{\infty }$. The more scattering, the faster the initial photon directions are changed by multiple scattering into their asymptotic distribution, which depends only in the IOPs.

Comparing Figs. 1 and 4 also shows that the K’s have changed greatly because of the change in IOPs, which is what is desired in any AOP. For the highly absorbing water, ${K}_{\infty }=0.8681{m}^{-1}$.

The distinction between beam and diﬀuse attenuation is important. The beam attenuation coeﬃcient c is deﬁned in terms of the radiant power lost from a collimated beam of photons. The downwelling diﬀuse attenuation function ${K}_{d}\left(z,\lambda \right)$, for example, is deﬁned in terms of the decrease with depth of the ambient downwelling irradiance ${E}_{d}\left(z,\lambda \right)$, which comprises photons heading in all downward directions (a diﬀuse, or uncollimated, radiance distribution). In the above simulations, $c=a+b=1.0\phantom{\rule{1em}{0ex}}{m}^{-1}$ at all depths, but all K functions except the near-surface ${K}_{sun}\left(30,180\right)$ in the high absorption case are less than c. Radiative transfer theory shows (e.g., Light and Water (1994) Eq. 5.71), for example, that in general $a\le {K}_{d}\phantom{\rule{0.3em}{0ex}}{\stackrel{̄}{\mu }}_{d}\le c$, where ${\stackrel{̄}{\mu }}_{}$ is the mean cosine of the downwelling radiance. These inequalities are seen to hold true in the above simulations.

HydroLight was next used to simulate a homogeneous Case 1 water body with a chlorophyll concentration of $Chl=1\phantom{\rule{1em}{0ex}}{mg\phantom{\rule{1em}{0ex}}m}^{-3}$. As for Figs. 1 and 4, the sun was at 40 deg and the surface was level. Figure 5 plots several quantities as a function of wavelength at 10 m depth for this simulation. We see that between 300 and about 600 nm, the various $K\left(10\phantom{\rule{1em}{0ex}}m,\lambda \right)$ functions are very similar and proportional to the total (including water) absorption coeﬃcient $a\left(10\phantom{\rule{1em}{0ex}}m,\lambda \right)$. However, beyond 600 nm the K functions diﬀer from each other, and they are all much diﬀerent from a. The reason for this behavior is inelastic scattering. Figure 5: Fig. 5. K at 10 m depth for Case 1 water with $Chl=1\phantom{\rule{1em}{0ex}}{mg\phantom{\rule{1em}{0ex}}m}^{-3}$. The HydroLight run included Raman scatter by the water, and chlorophyll and CDOM ﬂuorescence.

At the near-UV to blue to green wavelengths below 600 nm, most of the radiance at 10 m depth (for these IOPs) comes from sunlight being transmitted through the upper 10 m of the water column. Above about 600 nm, absorption by the water itself has removed most of the sunlight. For example, at 700 nm where ${a}_{water}=0.65\phantom{\rule{1em}{0ex}}{m}^{-1}$, we expect roughly $exp\left(-az\right)\approx 0.001$ of the surface light to reach 10 m. However, Raman scatter and CDOM ﬂuorescence inelastically scatter light from shorter wavelengths, where sunlight is present, into the red wavelengths, and thus create additional red light at 10 m. Thus, beyond 600 nm, the various radiances and irradiances no longer decrease with depth in the simple exponential fashion expected. Note that ${K}_{d}$ tracks a longer than do ${K}_{u}$ and ${K}_{Lu}$. This is because ${E}_{d}$ continues to collect whatever downwelling sunlight remains, and thus the inelastic contribution to ${K}_{d}$ is not noticeable until the chlorophyll ﬂuorescence contribution begins near 670 nm. ${K}_{u}$ and ${K}_{Lu}$ on the other hand have only a small amount of backscattered sunlight, so that the inelastic contribution becomes signiﬁcant sooner, at around 600 nm.

It is easy to verify that the peculiar behavior of the K functions beyond 600 nm is due to inelastic scatter. The HydroLight run was repeated with Raman scatter and CDOM and chlorophyll ﬂuorescence ”turned oﬀ.” Figure 6 shows the results. Now, the K functions all track the absorption nicely at all wavelengths. Figure 6: Fig. 6. K at 10 m depth for Case 1 water with $Chl=1\phantom{\rule{1em}{0ex}}{mg\phantom{\rule{1em}{0ex}}m}^{-3}$ as in Fig. 5, except that inelastic scatter was turned oﬀ in the HydroLight run.

As seen in these ﬁgures, radiative transfer theory shows that K functions are very ”absorption like,” meaning that the K functions are strongly correlated with the total absorption coeﬃcient when inelastic scatter eﬀects are negligible. For ${K}_{d}$, the approximate relation ${K}_{d}\approx a∕\phantom{\rule{0.3em}{0ex}}{\stackrel{̄}{\mu }}_{d}$ gives close agreement between the exact (computed by HydroLight) ${K}_{d}$ and the value estimated from the absorption coeﬃcient and the downwelling mean cosine ${\stackrel{̄}{\mu }}_{d}$of the radiance distribution, which was also obtained from the HydroLight simulation.

These few simulations are enough to establish the salient features of diﬀuse attenuation functions. Their use has a venerable history in optical oceanography. Smith and Baker (1978) listed some of their virtues:

• The K’s are deﬁned as ratios and therefore do not require absolute radiometric measurements.
• The K’s are strongly correlated with phytoplankton chlorophyll concentration (via the absorption coeﬃcient). Thus they provide a connection between biology and optics.
• About 90% of the diﬀusely reﬂected light from a water body comes from a surface layer of water of depth $1∕{K}_{d}$; thus ${K}_{d}$ has implications for remote sensing.
• Radiative transfer theory provides several useful relations between the K’s and other quantities of interest, such as the absorption and beam attenuation coeﬃcients and other AOP’s.

#### Gordon’s normalization of Kd

Gordon (1989) developed a simple way to normalize measured ${K}_{d}$ values. His normalization for all practical purposes removes the eﬀects of the sea state and incident sky radiance distribution from ${K}_{d}$, so that the normalized ${K}_{d}$ can be regarded as an IOP. The theory behind the normalization is given in his paper; the mechanics of the normalizing process are as follows.

Let ${E}_{d}\left(sun\right)$ be the irradiance incident onto the sea surface due to the sun’s direct beam, and let ${E}_{d}\left(sky\right)$ be the irradiance due to diﬀuse sky radiance. Then the fraction f of the direct sunlight in the incident irradiance that is transmitted through the surface into the water is

$f\phantom{\rule{1em}{0ex}}=\phantom{\rule{1em}{0ex}}\frac{t\left(sun\right)\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{E}_{d}\left(sun\right)}{t\left(sun\right)\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{E}_{d}\left(sun\right)\phantom{\rule{1em}{0ex}}+\phantom{\rule{1em}{0ex}}t\left(sky\right)\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{E}_{d}\left(sky\right)}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}.$

Here t(sun) and t(sky) are respectively the fractions of the direct beam and of the diﬀuse irradiance transmitted through the surface; these quantities can be computed using methods described in Light and Water (1994) Chapter 4 [where they are denoted by t(a,w)]. However, if the solar zenith angle in air, ${𝜃}_{}$, is less than 45 degrees , then $t\left(sun\right)\approx 0.97$. If the sky radiance distribution is nearly uniform (as it is for a clear sky), then $t\left(sky\right)\approx 0.94$. In this case, we can accurately estimate f from measurements made just above the sea surface:

$f\phantom{\rule{1em}{0ex}}\approx \phantom{\rule{1em}{0ex}}\frac{{E}_{d}\left(sun\right)}{{E}_{d}\left(sun\right)\phantom{\rule{1em}{0ex}}+\phantom{\rule{1em}{0ex}}{E}_{d}\left(sky\right)}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}.$

The sun and sky irradiances are easily obtained from an instrument on the deck of a ship. When both direct and diﬀuse light fall onto the instrument, it records ${E}_{d}\left(sun\right)+{E}_{d}\left(sky\right)$. When the direct solar beam is blocked, the instrument records ${E}_{d}\left(sky\right)$. (Advanced technology is not required here: just hold your hat so that its shadow falls on the instrument.)

Next compute the nadir angle of the transmitted solar beam in water, ${𝜃}_{sw}$, using Snell’s law:

${𝜃}_{sw}\phantom{\rule{1em}{0ex}}=\phantom{\rule{1em}{0ex}}{sin}^{-1}\left(\frac{sin{𝜃}_{sa}}{1.34}\right)\phantom{\rule{0.3em}{0ex}}.$

Finally, compute the quantity

${D}_{o}\phantom{\rule{1em}{0ex}}=\frac{\phantom{\rule{1em}{0ex}}f}{cos{𝜃}_{sw}}\phantom{\rule{1em}{0ex}}+\phantom{\rule{1em}{0ex}}1.197\phantom{\rule{0.3em}{0ex}}\left(1\phantom{\rule{1em}{0ex}}-\phantom{\rule{1em}{0ex}}f\phantom{\rule{0.3em}{0ex}}\right)\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}.$

This value of ${D}_{o}$ is valid for ﬂat or rough sea surfaces as long as ${𝜃}_{sa}\le 50$. For larger values of ${𝜃}_{sa}$, or for an overcast sky, a correction must be applied to ${D}_{o}$ to account for surface wave eﬀects on the transmitted light; the correction factors are given in Gordon (1989, his Fig. 6). Gordon’s normalization then consists simply of dividing the measured ${K}_{d}$ by ${D}_{o}$:

${K}_{d}\left(normalized\right)\phantom{\rule{1em}{0ex}}=\phantom{\rule{1em}{0ex}}\frac{{K}_{d}\left(measured\right)}{{D}_{o}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}.$

Physically, ${D}_{o}$ is a function (essentially $1∕\stackrel{̄}{{\mu }_{d}}$) that reduces ${K}_{d}$ values to the values that would be measured if the sun were at the zenith, if the sea surface were level, and if the sky were black (i.e., if there were no atmosphere). The zenith-sun, level-surface, black-sky case is the only physical situation for which ${D}_{o}=1$. In other words, normalization by ${D}_{o}$ removes the inﬂuence of incident lighting and sea state on ${K}_{d}$. The same normalization can be applied to depth-averaged values of ${K}_{d}$.

We recommend that experimentalists routinely make the simple measurements necessary to determine ${D}_{o}$, because normalization of ${K}_{d}$ enhances its value in the recovery of IOP’s from irradiance measurements.