As part of the Worldwide Ocean Optics Database (WOOD) Project, The Johns Hopkins University Applied Physics Laboratory has developed and evaluated a variety of empirical models that can predict ocean optical properties, such as profiles of the beam attenuation coefficient computed from profiles of the diffuse attenuation coefficient. In this paper, we briefly summarize published empirical optical algorithms and assess their accuracy for estimating derived profiles. We also provide new algorithms and discuss their applicability for deriving optical profiles based on data collected from a variety of locations, including the Yellow Sea, the Sea of Japan, and the North Atlantic Ocean. We show that the scattering coefficient (b) can be computed from the beam attenuation coefficient (c) to about 10% accuracy. The availability of such relatively accurate predictions is important in the many situations where the set of data is incomplete.
© 2007 Optical Society of America
Numerous ocean optics papers have been written that describe empirical algorithms or equations for (1) converting one wavelength to another for a single parameter or (2) deriving one parameter from another. There are many reasons for wanting to compute new results via such conversions and derivations. For example, in naval applications that involve predictions of optical detection of submerged objects, one typically runs a model that requires inputs of a full set of inherent optical properties (IOPs) at a particular wavelength of interest. IOPs are optical properties whose values do not change as a function of solar elevation, viewing angle, cloud cover, etc. IOPs include the beam attenuation coefficient (c), the scattering coefficient (b), and the absorption coefficient (a). An important “quasi-inherent” optical property is the diffuse attenuation coefficient for downwelling irradiance (K or Kd). A full set of IOPs is almost never available, and the existing data may not have been collected at the desired wavelength. Therefore, it is usually necessary to make wavelength conversions and to estimate the values of unknown parameters from one or more measured parameters.
Austin and Petzold [1,2] provide the most commonly accepted set of equations for converting wavelengths among K profiles at wavelengths ranging from 410 nm (blue light) to 670 nm (red light). Their results are in the form of linear equations given by:
where Kw is the pure water value of K, and M is from a lookup table of empirical slopes of the least squares fit between data at various wavelengths referenced to results obtained at 490 nm. Their results are based on data collected from about latitude 24° to 77° N (i.e., spanning tropical to Arctic conditions) and depths from the surface down to 200 m, but the equations are based solely on depths with light levels within 10% of the surface values to avoid contamination by Raman trans-spectral scattering. Their accuracy is good to about 8% for wavelengths ≤ 590 nm, but the accuracy degrades to about 31% at 670 nm.
To convert a c profile from one wavelength to another, Voss’s  empirical equation:
provides the simplest conversion method, where cw denotes the pure water value of c. This equation is accurate to about 5% for wavelengths from 440 to 670 nm when c(490) values range from cw up to about 1.0/m. Barnard et al.  also provide empirical equations for wavelength conversion of beam attenuation; their method, which is referenced to 488 nm, results in a unique linear fit for specific wavelengths from 412 to 676 nm. The results of Barnard et al., which come from analysis of a broader set of data, are similar to those of Voss and slightly more accurate.
Thus, wavelength conversions exist and are relatively accurate. The more difficult task is to estimate an unmeasured parameter from a measured one. Numerous published papers report empirical relationships among various optical parameters. The most comprehensive public archive for ocean optics data is the Worldwide Ocean Optics Database (WOOD; see http://wood.jhuapl.edu), which has > 242,000 K profiles but only about 18,000 c profiles, 10,000 a profiles, 1,000 b profiles, and 1,200 bb profiles (bb is the backward component of the scattering coefficient and is known as the backscatter coefficient). Therefore, validated, accurate empirical algorithms to convert from K (the most commonly available parameter) to these IOPs would be especially useful.
Shannon  published the following relationship between c and K at 535 nm:
This equation was based on data from a fairly wide range of ocean environments (including clear, open-ocean waters and turbid coastal waters) with c values from 0.11 to 1.6/m; the linear fit was quite accurate (R2 = 0.96). When Eq. (3) is inverted to compute c from K, it is invalid if K < 0.04/m. An examination of Shannon’s original data shows that a logical “revision” to his fit is to use K = 0.57*c for K < 0.06/m (which ensures that K goes to zero when c goes to zero). Equivalently, c can be given in terms of K at 535 nm:
If K is not measured at 535 nm, we can use the Austin-Petzold relationships to convert to 535 nm from other wavelengths. (If the desired c value is at another wavelength, the Voss or Barnard equations also need to be applied after Eq. (4) to obtain c at 535 nm.) One goal of this paper is to report on the empirical accuracy of Eqs. (3) and (4) and on the accuracy of other equations (derived below) using various datasets found in the WOOD.
Assuming we can accurately obtain c, either by measurement with a beam transmissometer or by computing it from K, the next step is to compute b from either K or c. Models also exist that estimate b from chlorophyll, but they apply only in Case 1 waters, and even then, their accuracy can be quite poor. Because the vast majority of existing b data come from WETLabs AC9  instruments that measure c and a (the definition c = a + b is used to compute b), it makes more sense to examine relationships between b and c than between b and K. A number of authors have reported such relationships. For example, Smart  showed that:
Eq. (5) is based on 490-nm data collected at several sites in the Gulf of Alaska, the Sargasso Sea, and the Bahamas; the R2 value was 0.99. Smart also showed that data obtained by Jerlov  at 632 and 655 nm generally followed this same fit.
The result shown in Eq. (5) is similar to those summarized by Levin  for data collected at 550 nm. In that paper, Levin states, “…on the basis of numerous measurements in the open and coastal waters in the range of c = 0.08 −2.5 m-1”:
where ω0 = b/c ranges from 0.3 to 0.9. Levin also presents results from Morel  and Schoonmaker  that have an average value of 0.93 for (b – bw)/(c – cw). For those same data, b/c ranges from 0.67 to 0.94, whereas (b – bw)/(c – cw) ranges only from 0.85 to 1.0. The point is that removing the pure water components from b and c significantly decreases the range in their relative ratio—by about a factor of two for the Morel and Schoonmaker data. The change is even larger when smaller values of ω0 are included.
2. Empirical variability in derived optical properties (a, b, and c)
Several empirical relationships can be used to compute b from c, c from K, and a from K. The three primary IOPs (a, b, and c) can be obtained from K (or from c and K if both are available) with a combination of these methods. We also provide uncertainty estimates for these derived variables.
2.1 Computing b from c
The relationship (b – bw)/(c – cw) = 0.86, as given in Eq. 5, is based primarily on open-ocean data; the only littoral data come from very clear subtropical waters in the Tongue of the Ocean, Bahamas. Also, those data were all collected at 490 nm. Many new datasets spanning a wide range of wavelengths have now been collected and added to the WOOD. Among those data are WETLabs AC9  time series or profiles acquired off the U.S. continental shelf, in the Sea of Japan, and in the Yellow Sea. The least squares fits of (b – bw)/(c – cw) for those data are summarized in Table 1.
The average Coastal Mixing and Optics (CMO) data are from mooring deployments #1 (depths of 13 and 68 m), #2 (depths of 12 and 68 m), and #4 (depths of 12 and 68 m), located at approximately 40.5° N, 70.5° W . Time periods at each depth varied because periods with evidence of biofouling were not included. As an example of the degree of variability observed within the CMO data, the c values at 488 nm (hereafter denoted as c488) in the first deployment ranged from 0.27 to 0.37/m at 13 m depth and from 0.57 to 16.7/m at 68 m depth.
The Chesapeake Bay Outflow Plume Experiment (COPE) data  consist of 359 AC9 profiles (2,872 if the multiple wavelengths are considered) from September 1996. Those c488 data, which range in value from about 0.16 to 6.5/m, come from quite close to the Virginia shoreline and in waters less than 20 m deep.
The Sea of Japan data come from 38 stations occupied in the Sea of Japan during the June to July 1999 Japan/East Sea (JES) project funded by the Office of Naval Research (ONR) . The total c488 values range from about 0.05/m (i.e., close to cw) to 1.6/m. The Yellow Sea data consist of AC9 profiles: 110 from June 2001 and 105 from July 2001. Both the June and July c488 values range from 0.2/m in the clear surface layers (upper several tens of meters) to over 10.0/m near the highly turbid seafloor. Near-bottom values vary by an order of magnitude, depending on location within the Yellow Sea; for further details see Smart .
Figure 1 summarizes statistics on the ratio (b – bw)/(c – cw) for the data given in Table 1. The primary result is that the mean (b – bw)/(c – cw) ratio has a nearly constant value of 0.9 between wavelengths of 530 to 630 nm, increases to about 0.95 from 630 to 715 nm, and decreases to 0.7 below 530 nm. The results at 488 and 550 nm are consistent with those given by Eq. (5) and by Schoonmaker , respectively. What is surprising is the similarity in results for clear, deep waters (from the Sea of Japan) and turbid, shallow waters laden with large amounts of sediment (as in the Chesapeake Bay outflow region and the Yellow Sea). We also observed these kinds of similarities when we compared the near-bottom, high c data to the shallower, lower c data within a given dataset. For example, (b – bw)/(c – cw) values for the CMO 488-nm data at 12–13 m from Deployments 1, 2, and 4 were 0.80, 0.75, and 0.70, while values from the same deployments at 68-m depth were 0.75, 0.72, and 0.77, respectively. In other words, the values were within about 10% of one another despite the differences in water clarity (average c488 values during the CMO Deployment #4 for Julian Days 272 to 285 were about 0.4/m at the shallow depths versus 1.6/m at the near-bottom depths, indicating a significant difference in sediment content.)
The Yellow Sea is a more extreme case in terms of the overall range in c values and the vertical variability of the c profiles. The (b – bw)/(c – cw) ratios were analyzed as a function of altitude off the seafloor and whether the maximum c488 value exceeded a large value (3.0/m in this case). The justification for these distinctions becomes clearer from examination of the actual profile data (Fig. 2). Almost all the profiles show a distinct, almost steplike, change in c488 within about 20 m of the seafloor. Also, most of the profiles have a maximum near-bottom value of < 3.0/m, but a fair number have larger values. These characteristics are evident in both the June and the July data. Therefore, the (b – bw)/(c – cw) ratios were computed separately for the deepest 20 m of each profile and for the shallower data. The ratios were also computed for profiles with very large maximum c values (> 3.0/m) and for the remaining profiles. As summarized in Table 2 and depicted graphically in Fig. 3, the largest (b – bw)/(c – cw) ratio differences occurred between the bottom, sediment-rich layers (deepest 20 m) and the much clearer overlying water layers. However, the shapes of the (b – bw)/(c – cw) ratios were quite similar in both layers, and the actual range of the ratios varied at most by a few percent at wavelengths > 488 nm. Similarly, the profiles with large (> 3.0/m) near-bottom c values consistently had higher (b – bw)/(c – cw) ratios (the effect was most pronounced for wavelengths < 510 nm) than the profiles without high near-bottom c values. However, the percentage differences were no more than about 6% at 412 nm, and the differences decreased rapidly at the longer wavelengths.
The consistent dependence of the (b – bw)/(c – cw) ratio on wavelength indicates that the mean values given in Table 1 can be used to reliably estimate b from c. The “tightness” of the relationship, indicated by the minimum and maximum values in Table 1, suggests that the resulting overall average estimates should generally be accurate to within about 10% of the correct value at all wavelengths between 412 and 715 nm. If the water is relatively turbid (e.g., if c488 > 1.0/m), using the upper bound (b – bw)/(c – cw) ratios to compute b from c might produce better accuracy, while using the lower bound might be more accurate for clear waters.
2.2 Computing c from K
The discussion in the previous section assumes that c has been provided. In cases where only the K profile is available, a method is required for estimating c from K. Three methods have been tested:
- Author’s “c:K” ratio method
Shannon’s results (given in Eq. (3) and revised in Eq. (4) have been tested against data from the U.S. continental shelf  and the Sea of Japan  and also against a large open-ocean (transatlantic) dataset that spanned regions north and south of the Gulf Stream and consisted of 2,793 summer profiles of c and K at 490 nm. More details about these transatlantic data are provided in Smart . Our c:K ratio approach is based on our empirical observations, which indicated that the c:K ratio is not constant with depth. (Excluding nepheloid layers, we consistently observed higher c:K values near the surface than at depth.) The main problem with the Morel and Shannon approaches is that they assume that the relationship between c and K is independent of depth. In our c:K method, we converted c and K versus depth profiles into c and K versus optical attenuation length (AL) profiles and computed the median c:K ratio versus AL. This ratio was then used as an empirical lookup table for converting from K to c (or vice versa) when only one of these two variables was available at a particular station. AL was chosen as the independent variable instead of depth because the vertical variability in K is usually more universally dependent on the available light level than on absolute depth. As shown in Smart , the North Atlantic median c:K ratios computed in this way were quite similar for diverse regions [the oligiotrophic region south of the Gulf Stream, the eutrophic region north of the Gulf Stream, and the Sea of Japan (June–July 1999 data)].
To test the accuracy of these three methods, we used datasets where both c and K were computed from independent measurements. The Morel, Shannon, and c:K ratio algorithms were applied to the K data to compute a “derived” c, which was then compared with the c profile computed from the original beam transmissometer data. Some of the statistical results from these various methods are summarized in Fig. 4. As expected, the depth-independent method based on Morel  gave significant errors in both the open-ocean datasets [Fig. 4(a)] and in the continental shelf data [Fig. 4(b)]. The original Shannon method  (i.e., without the revision given in Eq. (4) gave poor results for the open-ocean data (especially south of the Gulf Stream) but did fairly well when the revision to handle K values below 0.06/m was included. (The improvement for the data south of the Gulf Stream was due largely to the large fraction of low K data found in that region.) In the U.S. continental shelf data, the K values were large enough that the Eq. (4) revision did not apply. The c:K ratio results looked especially encouraging, but because the c:K lookup table was computed from mean c:K lookup tables acquired north and south of the Gulf Stream in the large transatlantic dataset, those results are predisposed to being accurate. The more stringent test is to apply the open-ocean lookup table to the continental shelf data (especially to the data from the nepheloid layer). The results, shown in Fig. 4(b), are quite surprising: the median error of the derived c profiles was less than 25% in both the “clear” water domain and in the nepheloid layer domain. The same lookup table was then applied to the June 1999 Sea of Japan data, and the median errors were even lower (17.7%). The Shannon method gave almost identical results (median error of 18.5%) in the Sea of Japan.
An even more stringent test of these empirical methods for computing c from K is using them on the Yellow Sea data, which contain a very wide range of suspended sediments. The results from application of the Shannon method to the “south-central” region of the Yellow Sea (the region having the widest range in surface-to-seafloor variability) are summarized in Fig. 5. Very similar results were found for the “north-central” region, even though the range of c and K values was much smaller there. As indicated in Fig. 5(c), the median errors were about 15% and distributed approximately equally around zero. The R2 value for the least squares fit between measured and calculated c532 values was 0.94, indicating a high degree of accuracy. In contrast, applying the c:K ratio method with the lookup table computed from either the open ocean or the Sea of Japan gave errors exceeding 100%. The reason is found in an examination of the c:K ratio profiles shown in Fig. 6. The Yellow Sea c:K ratio has a maximum value at depth and a minimum at the surface, which is the opposite of the open-ocean and Sea of Japan c:K ratios.
To assess the best possible result from the c:K ratio method, we computed Yellow Sea region-specific (south-central and north-central) c:K ratio plots for a specific dataset (e.g., July 2001) and then computed c profiles from the K profiles for that dataset and region. The resulting c profiles were accurate to 15%. However, when the c:K ratio from the adjacent region was used (i.e., south-central c:K ratio applied to north-central K profiles or north-central c:K ratio applied to south-central K profiles), errors in c jumped to 24 to 29%, or double the error obtained with the revised Shannon method.
Based on the results summarized in Fig. 4, and especially on these Yellow Sea results, it appears that the most robust method for estimating c from K is the revised Shannon method. In some cases, using a region-specific c:K ratio method could provide a small margin of improvement, but the extra complications involved make that method less attractive than the much simpler Shannon method.
2.3 Computing Absorption Coefficient (a) from K
In theory, one could compute c from K, b from c, and a from c – b. However, using so many empirical relationships results in rather large errors. A more accurate and straightforward approach is:
where the coefficient (relating a and K) is known as the “average cosine” of light scattering and has a nominal value of 0.8 based on data at various wavelengths and in varied environments . For example, in clear open-ocean waters off Sardinia, reported average cosine values range from 0.62 at 427 nm to 0.93 at 633 nm, and in the more turbid waters of the Baltic Sea, the range is from only 0.7 at 427 nm to 0.77 at 633 nm. For the wavelengths between 477 and 535 nm, the open-ocean values are 0.73 and 0.84, and the Baltic values are 0.70 and 0.67. The oceanic range for the average cosine is 0.6 to 1.0 , so choosing a value of 0.8 guarantees about a 20% accuracy. The limited data between 477 and 535 nm suggest that using this value within this wavelength domain will produce even better accuracies.
An example of measurements that have been used to test Eq. (7) is shown in Fig. 7. Figure 7(a) shows the ratio a/K computed as a function of depth for the June/July 1999 Sea of Japan data. The spectral absorption data come from a WETlabs AC9 device, and the spectral K data come from a Biospherical Instruments MER2048 radiometer. The data were provided to the WOOD by Greg Mitchell at Scripps Institution of Oceanography. As shown in Fig. 7(a), the surface values of a/K range from about 0.5 at 490 nm to about 0.95 at 675 nm. The data at 490 nm are likely to be the least contaminated by Raman scattering and sensor noise levels; at depths below about 40 m, they fall within about 10% of the “nominal” average cosine value of 0.8. The subsurface results for the other wavelengths between 412 and 555 nm generally fall between 0.8 and 1.0. Values at or above 1.0 are spurious and can be attributed to a combination of one or more sources of contamination, such as:
- Overly large a(λ) values caused by unusually large (and not properly corrected for) scattering by near-bottom sediments
- Raman contamination, which decreases K values (especially at 675 nm)
- Inaccurate K values as the irradiance sensor approaches its noise floor
The Raman contamination problem has been well-documented; for example, radiative transfer modeling simulations show that Raman scattering strongly influences the asymptotic average cosine at wavelengths greater than about 500 nm in clear waters and 600 nm in more turbid waters . As a result, it is virtually impossible to make an uncontaminated estimate of a/K at 675 nm, and, because most of these Sea of Japan data come from “clear waters,” even the results at 555 nm are increasingly affected with increasing depth. However, it is rather surprising that the asymptotic 490-nm a/K values fall so close to 0.8 while those at 412 and 440 nm appear to be contaminated. One explanation may be that the absolute calibration of the AC9 sensor is biased high for those wavelengths.
Pelevin and Rostovtseva , using data collected over many years at a wide range of oceanographic locations, provide an alternative empirical equation for computing a/K:
where R = Eu/Ed, the ratio of upwelling to downwelling solar irradiance. An advantage of using this equation is that Eu and Ed are fairly common measurements. Also, if time-varying fluctuations are occurring during the data collection, this ratio tends to cancel out those fluctuations. The disadvantage with this equation is that Eu is ~ 50 times smaller than Ed and tends to hit a sensor noise floor at a relatively shallow depth. Thus, a:K can be obtained only down to the valid depth of the Eu data. Figure 7(b) shows the results of using Eq. (8) with Eu and Ed profiles at three of the same wavelengths shown in Fig. 7(a) [and for the same profiles used to create Fig. 7(a)]. The Eq. (8) results were cut off (in depth) when the Eu data approached their sensor noise floor. For all three wavelengths, Eq. (8) consistently gives values higher than 0.8 (mean values of 0.88, 0.89, and 0.91 at 412, 488, and 555 nm, respectively) that show less of a depth dependence than the “measured” results in Fig. 7(a). Based on this limited comparison, it appears that Eq. (8) gave results that were ~ 12% higher than Eq. (7); however, because Eq. (7) was expected to be accurate to only about 20%, the differences were within the confidence bounds associated with Eq. (7). More comparisons could be performed to decide if one of these two equations should be preferred over the other, but the available data may also dictate which one can be used.
Several caveats need to be noted for the claim that we can predict b from c [using the (b – bw)/(c – cw) data from Table 1] to 10% accuracy. That statement assumes perfect knowledge of c and the pure water values for bw and cw. In practice, 10% errors in c are not uncommon (see, for example, Voss  and Voss and Austin ) due in large part to forward-scattered light that reaches the device’s detector. Errors can also arise if the data are not properly calibrated or processed for instrument drift (for a good discussion on how to process data from the instruments used to compute c, see Bishop ). The errors in the values for bw and cw are of less concern because they essentially cancel one another. (To solve for unknown b values, we subtract cw from c, divide by the (b – bw)/(c – cw) ratio, and then add in bw. Also, a common method to obtain cw is to add bw to aw so that errors in bw and cw are not independent.) Another caveat is that none of the datasets in Table 1 is significantly affected by colored dissolved organic material (CDOM; also known as “yellow substance”). The (b – bw)/(c – cw) ratios for waters that are laden with CDOM, such as near the mouth of a river, will be lower than those shown in Table 1, especially at wavelengths below about 488 nm (i.e., in the blue part of the spectrum).
Historical ocean optical measurements, such as those found in the WOOD, were collected for a limited set of properties and wavelengths. Users of those data frequently need to convert those results to another wavelength and/or to other parameters. Empirical wavelength conversion algorithms exist to convert a measurement of c(λ1) to c(λ2), or K(λ1) to K(λ2). Their accuracy is about 5% and 8% for c and K, respectively, for wavelengths less than 590 nm. (The accuracy of K predictions degrades to about 31% at 670 nm.)
Existing empirical algorithms for conversions from c to b (or vice versa) are accurate (except in CDOM-laden waters) to about 10%. Conversions from K to c are less accurate; among the algorithms tested in this work, the most robust approach was the revised Shannon method, whose accuracy was about 25% in the Sargasso Sea, 40% in the waters of the U.S. continental shelf, 18% in the Sea of Japan, and about 15% in the Yellow Sea. A caveat associated with these results is that, except for the CMO data used to relate b to c, most of these test datasets came from summer or early fall periods (e.g., the Yellow Sea data came from June, July, and October). A more complete test of the robustness of these relationships, especially c to K, would include data from spring and winter months.
5. Future research
The hundreds of thousands of bio-optical profile data found in the WOOD provide a resource for building empirical algorithms to fit a particular geographic region and/or season. As is discussed in Smart , the empirical fit, e.g., of c:K versus optical depth, can be totally different in one location than another, so it makes sense to develop region-specific (and/or season-specific) fits provided that adequate data exist within that region/season.
An alternative approach is to derive one parameter from another by an iterative approach involving inverse radiative transfer techniques. For example, we are collaborating with Dr. Dale Kiefer (Professor of Marine Biology and Biological Oceanography at USC) and Dr. Curt Mobley (Chief Scientist at Sequoia Scientific, Inc.), to develop and test a software package with embedded radiative transfer models that computes an entire suite of IOPs and apparent optical properties (e.g., diffuse attenuation coefficient) from a limited set of optical profile data (such as a downwelling irradiance and an upwelling radiance profile).
The advantage of this approach is that it does not rely on historical relationships to predict profiles from current data. Instead, as is discussed further in Kiefer , the Hydro-OPtical Analysis System (HOPAS) predicts virtually all the unknown variables from current data. HOPAS is a software package that enables users to import and process oceanic optical data using exact radiative transfer code to derive physical, chemical, and biological descriptions of the water column. The HOPAS inverse scheme uses an iterative loop where water column characteristics are treated as “unknowns,” and the nonlinear programming search routine is used to continuously “adjust” their values during the solution processing to provide the best match between optical measurements and Ecolight calculations of these same optical variables. (Ecolight is a special version of Hydrolight, the well-known high-fidelity software package/physics model that uses the full radiative transfer equation (RTE). Ecolight solves the azimuthally averaged RTE and provides only the irradiances, nadir and zenith radiances, and various reflectances and diffuse attenuation functions as output.) HOPAS also uses the Environmental Analysis System (EASy©), an advanced geographical information system designed for the storage, analysis, and presentation of diverse spatially and temporally referenced information .
We are providing Dr. Kiefer with sample datasets from the WOOD to evaluate the HOPAS-derived parameters. The results of those evaluations will be the topic of a future paper.
This work was supported in part by ONR Grant # N000149810773, “World-Wide Ocean Optics Database (WOOD)” funded by the Office of Naval Research, Code 322OP.
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