Digital Commons @ University of Digital Commons @ University of South Florida South Florida

.


Introduction
Uncertainties in satellite ocean color data products are important parameters to define data quality, thus having direct impact on local, regional, and global studies.High uncertainties at image pixel level lead to low image quality to track ocean features or to study changes at fixed locations.High uncertainties at regional or global scale make it difficult to study ocean changes in response to climate variability because changes in the vast oceans may be very small even at decadal scale (e.g., 5-20%) [Behrenfeld et al., 2001;Antoine et al., 2005;Gregg et al., 2005].Thus, understanding sources of uncertainties is important in order to reduce them to improve data quality for a variety of applications.In the past, requirements on data product uncertainties at pixel level have been generally accepted by the ocean color community to be <5% in the satellite-retrieved remote sensing reflectance (R rs , sr 21 ) in the blue bands over clear waters and <35% in the satellite-retrieved chlorophyll a concentration (Chl, mg m 23 ) also for clear waters [Hooker et al., 1992].This is because the former is the lower boundary from atmospheric correction [Gordon, 1997], while the latter was once regarded as the uncertainties in laboratory Chl estimates.Corresponding to these requirements, uncertainties in data products have been evaluated using different approaches, including (1) direct comparison with field measurement [e.g., Gregg and Casey, 2004;McClain et al., 2004;Bailey and Werdell, 2006;Marrari et al., 2006;M elin et al., 2007;Antoine et al., 2008;Zibordi et al., 2009;Wang et al., 2009;Maritorena et al., 2010;Cannizzaro et al., 2013;Moore et al., 2015;many others] ; (2) statistics using satellite data alone [Hu et al., 2013] or cross-sensor comparisons [Zibordi et al., 2006;Barnes and Hu, 2015;Hu et al., 2013] to avoid uncertainties embedded in field measurements.These assessments have led to different findings depending on locations (i.e., open ocean or coastal water), data ranges, and methods used in the assessments.
Such uncertainties in satellite-derived data products can come from several sources, including radiometric calibration, atmospheric correction, and other correction algorithms to estimate remote sensing reflectance Journal of Geophysical Research: Oceans PUBLICATIONS in the visible (R rs (vis)) from at-sensor (i.e., top-of-atmosphere or TOA) total radiance, and bio-optical inversion algorithms to estimate Chl and other in-water properties from R rs (vis) [Gordon, 1990].Additional uncertainty sources at instrument level include instrument artifact and characterization uncertainty (e.g., stray light, nearfield response, crosstalk, spectral response, linearity response, polarization response along scan direction, temperature response, detector or mirror side response), many of which can change from their prelaunch characteristics over time.However, for a well-characterized sensor where all these uncertainty sources are characterized before launch, they can all be dealt with after the satellite is launched.In contrast, another important factor affecting data product uncertainties is the sensor's signal-to-noise ratio (SNR), which must be set before the satellite is launched for obvious reason.
The question then becomes what is the minimum SNR for an ocean color sensor that can minimize spurious pixel to pixel variation, greatly improving image quality.Higher SNR leads to lower product noise, higher data precision, and lower uncertainties in data products (see Figure 1 for an example).However, increasing SNR requires trade-offs with other radiometric and spectral characteristics of the instrument [Gordon, 1990;Hu et al., 2012a].Increasing SNR requires higher instrument sensitivity, which could lead to a lower dynamic range and lower saturation threshold (easier signal saturation over bright targets).Increasing SNR may also require coarser spatial resolution and/or lower spectral resolution in order to collect a sufficient quantity of photons.In addition, other factors contributing to data product uncertainties, as mentioned above, may overwhelm the influence of SNR once SNR is above a minimal threshold.To address the SNR requirement, Gordon [1990] obtained values of $400:1 to 900:1 at 443 nm for sun at 608 from zenith and for scans from nadir to 458 off nadir.In that study, Gordon used the uncertainty of empirically derived Chl as the criterion and assumed that the uncertainty from sensor's noise matched that from the blue-green ratio algorithm for the estimation of Chl.However, the key product from ocean color measurements is the R rs spectrum, thus it will be more informative if minimal SNR values can be obtained with R rs quality being used as the criterion.Indeed, a focus on the surface radiometry as a criterion for data quality has been the objective of NASA ocean color calibration and validation efforts for about two decades [McClain, 2009].
Further, nearly all previous uncertainty estimates are from the statistic of individual pixels.But in reality, most studies on regional or global processes use spatially and temporally binned data at reduced spatial resolution, for example, 4 km monthly means.Uncertainties in these data products are generally unknown as they have not been studied.
Therefore, the objective of this study is twofold.First, the study evaluates uncertainties in satellite-derived R rs (vis) and Chl as induced by sensor noise (i.e., defined by SNR) and makes recommendations on the minimal SNR requirement for ocean color measurements.Second, the study estimates uncertainties in R rs (vis) and Chl data products when the 1 km data products are binned to 4 km resolution at monthly intervals.Such estimates are based on R rs uncertainties estimated from sensor noise only and from realistic Moderate Resolution Imaging Spectroradiometer (MODIS) measurements over ocean gyres using error propagation theory.SNR requirements on future ocean color missions targeted for both global and regional applications are also discussed.

Data and Methods
2.1.Definitions of Error (d), Noise (r), Uncertainty (D), and SNR Several concepts need to be clarified in order to quantify uncertainties and SNR.Specifically, error (d) is a measurement of departure from the hypothetical ''truth.''Sensor noise is defined as the standard deviation of a univariate Gaussian distribution of random noise errors, denoted as r in units of mW cm 22 lm 21 sr 21 , which is a scalar from error statistics (i.e., many d values).Uncertainty is also a scalar, D, defined as the probability of a measured (or estimated) property in approaching the hypothetical ''truth.''The sample standard deviation of the error distribution from many (N) measurements is regarded as the measurement uncertainty (D).Another way to interpret D is that, from a single measurement, the measured value has a 68% likelihood to have its error < D. In this regard, sensor noise is also the uncertainty in the sensor signal.Most importantly, uncertainty (D) is not an error but a measure of error distribution and probability.
Note that there are three notations used here: d, r, and D, which have specific meanings in this context.They are all used as operators, such that dX is the error in X, and rX is the standard deviation of the error distribution in X. DX has the same meaning of rX, but in this context r is reserved for sensor signal (radiance) but D is reserved for data products (e.g., R rs , Chl).
SNR is defined as the ratio of signal versus noise (r).For an input radiance (L, mW cm 22 lm 21 sr 21 ) and r, we have SNR 5L=r: (1) One objective of this study is to determine a minimal SNR with L from typical measurements over the ocean (L 5 L typ ) [Hu et al., 2012a].

Determination of Minimal SNR
As stated above, for a spectral band in the visible, total uncertainties in R rs (vis) from satellite retrievals, DR rs t 8 tal (vis), come from four sources: (1) radiometric calibration, (2) atmospheric correction, (3) sensor noise (r), and (4) sensor artifacts and characterization uncertainty (e.g., stray light, near-field response, crosstalk, spectral response, linearity response, polarization response along scan direction, temperature response, detector or mirror side response) as well as uncertainties in ancillary data (e.g., wind, ozone, atmospheric pressure, water vapor) used during atmospheric correction.Incorrect radiometric calibration will not contribute to sensor noise but instead lead to a systematic bias (offset) in the satellite-derived R rs although such a bias may vary with solar/viewing geometry and other measurement conditions.Because the focus of this study is on noise-induced DR rs t 8 tal (vis), radiometric calibration will not be discussed, but #2 and #3 factors will be the focus here.The #2 factor is considered here because in addition to r(vis)-induced uncertainty in DR rs t 8 tal (vis), r(NIR) will also contribute to DR rs t 8 tal (vis) through the atmospheric correction process as the near-infrared (NIR) bands are used in the atmospheric correction.The #4 factor may play an important role in affecting atmospheric correction accuracy, but it is from sources other than sensor noise, hence not considered here.
Therefore, in order to determine the minimal SNR, two questions need to be addressed Ignoring the impact of sensor's calibration, the two independent terms contributing to DR rs t 8 tal (vis) are DR rs atm (vis) and DR rs r (vis), where the former is induced by the standard atmospheric correction procedure [Gordon and Wang, 1994;Wang, 2007] that was recently updated in Wang et al. [2012], and the latter is a result of sensor noise r(vis).
Journal of Geophysical Research: Oceans 10.1002/2016JC012558 For independent noise terms and following the theory of statistics and probability [Morgan and Henrion, 1990], DR rs t 8 tal (vis) from these two terms can be calculated as DR rs total ðvisÞ5 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ðDR rs atm ðvisÞÞ 2 1ðDR rs r ðvisÞÞ 2 q : (2) Wang et al. [2012] have demonstrated that with the current atmosphere correction procedure that is based on the original Gordon and Wang [1994] scheme with recent updates in aerosol lookup tables (LUTs), DR rs atm (vis) will approach a theoretical limit and will be reduced only marginally with further increased SNR(NIR) after it reached a certain threshold, and this threshold was determined to be 600:1.SNR(NIR) was therefore set to 600:1 in this study, as this number can be regarded as ''enough'' for the NIR bands.
However, it is difficult to answer the question on how much SNR(vis) is ''enough,'' as the answer will depend completely on the oceanographic applications.For certain applications, an SNR(vis) may be ''enough'' but for other applications the same SNR(vis) may be far from ''enough,'' and in the end for some applications even an SNR(vis) of infinity may still not be ''enough where F 0 is the extraterrestrial solar irradiance, h 0 is the solar zenith angle, t 0 is the atmospheric diffuse transmittance from the sun to the sea surface, and t is the atmospheric diffuse transmittance from the sea surface to the sensor.[F 0 (vis)cos(h 0 )t 0 (vis)] in the above equation is the sea surface downwelling irradiance.
In short, we can define SNR(vis) thresholds using the SNR required to achieve the quality of the best field measurement, represented by DR rs field (vis), and the SNR required to achieve the uncertainty level comparable to the uncertainty inherent to the atmospheric correction algorithm, DR rs atm (vis).These two thresholds, namely SNR 1 and SNR 2 , respectively, can be written as and L typ ðvisÞ tðvisÞ Á R rs atm ðvisÞ Á ½F 0 ðvisÞcos ðh 0 Þt 0 ðvisÞ : (6)

Uncertainties in Chl Data Product
Of all the ocean color products, Chl perhaps is the most often used.Because Chl is derived from spectral R rs , DR rs (vis) will lead to DChl through error propagation.The current NASA standard Chl algorithm of MODIS is based on OCI [Hu et al., 2012b], which is a mixture of R rs three-band subtraction (color index or CI) and R rs ratio among three bands (OC 3m ).In addition to MODIS, NASA has applied this algorithm to all ocean color sensors including the Visible Infrared Imaging Radiometer Suite (VIIRS), Sea-viewing Wide Field-of-view Sensor (SeaWiFS), Coastal Zone Color Scanner (CZCS), and Medium-Resolution Imaging Spectrometer (MERIS).
The NOAA ocean color team has also implemented the OCI algorithm for the ocean color data processing from the VIIRS [Wang and Son, 2016].Here we will briefly explain how DR rs propagates to DChl.As stated earlier, both DR rs and DChl represent standard deviations of error distributions.
According to error propagation theory between independent variables and dependent variables, we have The difficulty is how to estimate DR from DR rs t 8 tal (vis) of the individual bands.For simplicity, assuming R 5 R 1 /R 2 where R 1 is the numerator in equation ( 9) and R 2 is the denominator in equation ( 9), then if errors in R 1 and R 2 are independent, error propagation theory will lead to IOCCG [2010] and Hu et al. [2013] showed that DR rs t 8 tal (vis) is not spectrally independent.Here we generated the same error statistics based on Hu et al. [2013] and then calculated DR.First, 5000 random errors of R rs (d, sr 21 ) were generated for each of the three bands (443, 547, and 667 nm), and these errors were generated in a way to agree with the MODIS R rs error statistics given in Hu et al. [2013, Figure10].There is a spectral relationship (i.e., dependence) among the different bands to account for errors due to atmospheric correction, while there is also a certain degree of randomness (i.e., independence) among different bands due to instrument noise (SNR(vis)).Further, we have Here dR is not just a function of dR 1 and dR 2 , but also a function of R 1 and R 2 , which further depend on Chl.In this study, R 1 and R 2 were selected to represent most oceanic waters with Chl ranging between 0.04 and 10.00 mg m 23 .While in reality the same Chl may correspond to different R 1 or R 2 (and vice versa), for illustration purpose in this study Chl was equally spaced (in log scale) between 0.04 and 10.00 mg m 23 , and for each predefined Chl in this range its corresponding R rs (vis) (and therefore R 1 and R 2 ) was determined as the median spectrum from MODIS global R rs statistics.
The 5000 dR values estimated this way will form an error distribution, and its standard deviation is regarded as DR.Once DR is derived, DChl OC3 is derived through equations (10-12).
The CI algorithm takes the following form: where a 0 and a 1 are algorithm coefficients (constants) and CI5R rs ð547Þ2½R rs ð443Þ1ð5472443Þ=ð6672443Þ3ðR rs ð667Þ2R rs ð443ÞÞ CI % R rs ð547Þ20:5ðR rs ð443Þ1R rs ð667ÞÞ Then, for each of the 5000 dR rs of the three bands, we have Journal of Geophysical Research: Oceans Here R 1 , R 2 , and R 3 represent R rs (443), R rs (547), and R rs (667), respectively.
The standard deviation of the 5000 dCI is DCI.Subsequently, we have where The current NASA standard OCI algorithm is a mixture between CI and OC 3m : In practice, data from individual images at its original spatial resolution (e.g., 1 km for MODIS or 750 m for VIIRS) were rarely used for regional or global studies of ocean science due to significant cloud cover or other nonoptimal measurement conditions [Feng and Hu, 2016].Instead, these data of original spatial resolution were often binned to coarser spatial resolutions in 8 day, monthly, seasonal, or annual composites, projected into a plate carr ee grid.These composites help reduce cloud cover and increase coverage, resulting in N valid measurements for each grid cell.According to error statistics and assuming that each measurement error in either R rs or Chl is independent of others, uncertainty for each grid cell will reduce by 1/sqrt(N): Here as an example the number of N for each grid cell was determined from MODIS global measurements.MODIS L3 4 km binned monthly R rs data products were obtained from NASA.In such products, for each grid cell, all valid 1 km pixels that fall in the grid cell within the given month were used to calculate the mean R rs , while the number of valid pixels in the calculation was also recorded.For illustration purpose, the month of June 2008 (north hemisphere summer) was selected.For the i th grid cell uncertainties were estimated as where DR rs t 8 tal (vis) and DChl OCI were from the estimates in section 2.3.

Minimal SNRs
One of the best (i.e., minimal) DR rs (vis) from field measurements, DR rs (vis) field , was from the skylight-blocked approach (SBA) measurements [Lee et al., 2013] taken in blue oceanic waters, as shown in Figure 2. The SBA Journal of Geophysical Research: Oceans 10.1002/2016JC012558 measurement scheme is similar to a typical above-water reflectance measurement except that a custom-made cone is used to block surface-reflected skylight, therefore minimizing potential errors from incorrect removal of reflected skylight.The highest-quality reflectance data collected by marine buoys designed specifically for measuring reflectance typically had 3-4% of uncertainties in the blue bands for blue waters [Antoine et al., 2008], while the uncertainties due to noise in the best SBA measurements over blue waters are only $1% (0.0001 sr 21 divided by a typical blue-water R rs at 443 nm, 0.01 sr 21 , Figure 2).Therefore, these DR rs field (vis) represent the minimal uncertainties that can be achieved by today's state-of-the-art measurements, which is then used to bound SNR 1 calculations in equation ( 5).
DR rs (vis) from atmospheric correction, DR rs atm (vis), for the four visible bands (443, 488, 547, 667 nm) are shown in Figure 3.These were used as the bound to calculate SNR 2 through equation ( 6).While different aerosol types and thicknesses led to different Figure 4a shows the five DR rs terms for each of the four visible bands.Of these, DR rs field is the lowest (<10 24 sr 21 for all bands), representing the state-of-the-art R rs uncertainty from any field measurements.Corresponding to DR rs field , SNR 1 was determined from equation ( 5) under the assumption of typical radiance input with a solar zenith angle of 458, with results shown in Figure 4b (blue bars).Note that even for such a requirement to make DR rs r equal to DR rs field , SNR 1 of 800:1 is sufficient for all visible bands.DR rs atm is much higher than DR rs field for all bands (Figure 4a).Corresponding to DR rs atm , SNR 2 was determined from equation ( 6), with results shown in Figure 4b (orange bars).Note that SNR 2 is <400:1 for all visible bands, much lower than SNR 1 .This suggests that if one were to make DR rs r comparable to DR rs atm , SNR could be lower than 1000:1, a number that was recommended by the IOCCG [2012] report.
Corresponding to SNR 1 and SNR 2 , DR rs t 8 tal derived from equation (2) are shown in Figure 4a as green and yellow bars, respectively, and annotated as DR rs ideal and DR rs practical , respectively.DR rs ideal is slightly higher than DR rs atm , and DR rs practical is 1.41 times DR rs atm , both indicating the dominant role of DR rs atm in determining DR rs t 8 tal .This observation is consistent with those from image statistics [Hu et al., 2012a] where improving SNR(vis) has been shown far less important than improving SNR(NIR) in terms of reducing data product noise and uncertainties in the visible domain.
From this analysis, one can conclude that once SNR(NIR) is >600:1 and the Gordon and Wang [1994] approach is used for atmospheric correction, SNR(vis) can be 800:1 or even 400:1 without sacrificing image quality in terms of noise and uncertainty.This finding is actually consistent with those reported in Gordon [1990] for Chl retrieval and Hu et al. [2012c] where comparison was made between Geostationary Ocean Color Imager (GOCI), SeaWiFS, and MODIS (see Table 1 and Hu et al. [2012c, Figures 1 and 2]).Although GOCI SNR(vis) is much lower ($600:1) than MODIS SNR(vis) (>2000:1) and GOCI SNR(NIR) is also lower ($600:1) than MODIS SNR(NIR) (800:1 at 869 nm and 1000:1 at 754 nm), because GOCI SNR(NIR) is approaching 600:1, the noise levels in their band-ratio Chl products are similar [Hu et al., 2012c, Figures 1  and 2].For the same reason, because SeaWiFS SNR(NIR) is much lower ($200:1) than GOCI SNR(NIR) ($600:1), even if the SNR(vis) of both sensors is very similar, the noise level in SeaWiFS band-ratio Chl is much higher than that in GOCI band-ratio Chl.
Therefore, without examining details of all possible scenarios, the results in Figure 4 and from the above analyses suggest that even for the ''ideal'' requirement where DR rs r is forced to agree with DR rs field , a SNR(vis) of 800:1 is sufficient.For the ''practical'' requirement where DR rs r is forced to agree with DR rs atm , a SNR(vis) of 400:1 is sufficient.Both SNR(vis) choices led to DR rs t 8 tal lower than MODIS DR rs gyre (Figure 4a, purple bars).DR rs (simulated_ideal) is the combined uncertainty from equations ( 2) and ( 5), and DR rs (simulated_practical) is the combined uncertainty from equations ( 2) and ( 6); DR rs (MODIS_Gyre) is DR rs gyre .(b) Minimal SNR determined from equation ( 5) (SNR 1 ) and equation ( 6) (SNR 2 ), i.e., when DR rs field and DR rs atm in Figure 4a were used to bound the DR rs r term.

Product Uncertainties Due to Sensor Noise
Once DR rs t 8 tal at pixel level is determined for each spectral band, it is straightforward to calculate DR rs and DChl for each 4 km grid cell in the monthly data products using the 1/sqrt(N) rule where N is the number of valid observations in each grid cell.Figure 5 shows the # of valid pixels in each 4 km grid cell in June 2008.Note the uneven distributions in the global oceans, for example, the low values in the equatorial regions and in the Indian Ocean due to persistent cloud cover, sun glint, and stray light.
In this study, three forms of DR rs t 8 tal at pixel level were used: DR rs ideal , DR rs practical , and DR rs gyre in Figure 4a, respectively.DR rs ideal is the minimal uncertainty an ocean color sensor can achieve (corresponding to SNR(vis) of $800:1 and SNR(NIR) of $600:1); DR rs practical is the most cost effective way to design a sensor (corresponding to SNR(vis) of $400:1 and SNR(NIR) of $600:1); DR rs gyre represents the best performance of the current ocean color sensors.
Figure 6 shows DR rs in each 4 km grid cell for June 2008, corresponding to the three DR rs t 8 tal at pixel level.
Figure 7 shows the same DR rs in each 4 km grid cell for June 2008 but in relative units (relative to the monthly mean R rs ).For all the 667 nm visible bands, nearly all grid cells showed <5% uncertainties regardless of the assumptions used at pixel level, suggesting that at 4 km monthly scale, ocean color mission goals of obtaining R rs with <5% uncertainties can be met for nearly all waters instead just blue waters.This is attributed to the N > 1 valid observation in nearly all 4 km grid cells at monthly scale.
DR rs t 8 tal at pixel level would propagate to DChl at pixel level through equations (10-23), where R rs errors in different bands (equation ( 14)) are shown in Figure 8 according to MODIS statistics over ocean gyres [Hu et al., 2013].These errors would add on top of the R rs spectra corresponding to different Chl levels (Figure 9a) to result in Chl errors and Chl uncertainties (i.e., DChl).Such DChl for the three DR rs t 8 tal cases (i.e., DR rs ideal , DR rs practical , and DR rs gyre ) are shown in Figures 9b-9d, respectively.For the OC X algorithm, DChl is high for both clear (Chl < 0.1 mg m 23 ) and turbid (Chl > 0.5 mg m 23 ) waters, with the turning point around Chl $ 0.2 mg m 23 .This is understandable because for Chl < 0.1 mg m 23 , the reflectance ratio (R) is )1 but for Chl > 0.5 mg m 23 the ratio is (1.In contrast, DChl from the OCI algorithm is much lower and is nearly a constant (<5%) for Chl < 0.25 mg m 23 because most of the R rs errors in the three algorithm bands cancelled [Hu et al., 2012b].This is exactly what the algorithm was designed for.
Similar to the reduction of DR rs when applied to monthly 4 km data, DChl at pixel level (Figure 9) also decreased significantly when applied to monthly 4 km data (Figure 10).For illustration purpose, only DChl at pixel level in Figure 9d was applied to June 2008 to show the global distributions of DChl in Figure 10.Most of the OC X DChl values are <5%, with some of them between 5% and 10% over some of the ocean gyres where Chl is extremely low (<0.07 mg m 23 ) and number of valid observations is also low.In contrast, nearly all OCI DChl values are <5%, and most of them are <1%, indicating that noise-induced Chl  2) and ( 5)); practical (equations ( 2) and ( 6)); and MODIS gyre estimates.The map is the same for each scenario but the scales are different as shown in the right.uncertainties are negligible in every 4 km location at monthly scale.Note that such estimates are based on the assumption that DR rs t 8 tal at pixel level is comparable to those of MODIS measurements over the gyres (Figure 9d).If the simulated DR rs t 8 tal using SNR(vis) 5 800:1 (Figure 9b) or SNR(vis) 5 400:1 (Figure 9c) were used, DChl would be much lower.Indeed, even if DR rs t 8 tal were to be doubled from DR rs gyre due to other factors that were not accounted for in the simulations, OCI DChl would still be <5% for nearly all 4 km locations at monthly scale.If time series data were to be derived from an ocean basin where many 4 km grid cells are averaged, both DR rs and DChl would be much lower than shown in Figure 10, indicating that basin-scale Chl monthly data are sufficient to address small changes induced by climate variability.

Journal of Geophysical Research: Oceans
Figure 8. R rs errors in several spectral bands showing both dependence (the fitting lines) and independence (data scatter around the fitting lines).These errors were simulated using MODIS observations over gyres, as reported in Hu et al. [2013].These errors were fed into equations ( 14) and ( 18) to estimate errors and uncertainties in band ratio (R) and band subtraction (CI), which were then used to estimate uncertainties in Chl.
Figure 9. (a, b) Typical R rs corresponding to different Chl levels, estimated using MODIS global data.These were used to estimate how DR rs propagates to DChl using equations ( 14)-( 23

Summary
In short, the following findings can be summarized from the above analyses: 1.As concluded in Wang et al. [2012], SNR(NIR) of 600:1 is sufficient to perform accurate atmospheric correction, where further increases in SNR(NIR) would not lead to significant reductions in DR rs atm (vis) because of inherent uncertainties in the atmospheric correction.2. DR rs r (vis) induced by sensor noise in the visible bands (r) is <DR rs atm (vis) under typical radiance input (L typ ) as long as SNR(vis) is >$400:1.Therefore, for practical considerations (engineering design, cost) a sensor with SNR(vis) $ 400:1 may meet the minimal requirement, as DR rs t 8 tal (vis) would be dominated by DR rs atm (vis) when SNR(vis) is much higher than $400:1.Under nontypical conditions, the number may be relaxed to 500:1 (see section 4). 3.Even when DR rs r (vis) were to be minimized using field measurements instead of atmospheric correction as the bound, SNR(vis) $ 800:1 would still be sufficient to meet the minimal requirement, as such SNR(vis) would lead to DR rs r (vis) comparable to those from the existing best field measurements.
4. Even with SNR(vis) $ 400:1, simulated DR rs t 8 tal is still lower than DR rs gyre estimated from a sensor with much higher SNR (i.e., MODIS, SNR(NIR) $ 800:1 to 1000:1; SNR(vis) > 2000:1), suggesting that factors other than sensor noise play important roles in affecting DR rs t 8 tal , which strengthen the argument that SNR(vis) of $400:1 may meet the minimal requirement under typical conditions. 5. DR rs t 8 tal would propagate to other ocean color products such as Chl through algorithm formulations and error propagation theory.6.The above SNR considerations and product uncertainty estimates are based on individual pixels.When they are binned to 4 km monthly products, uncertainties in both R rs and Chl drop significantly as the number of valid observations in each bin is usually )1.Consequently, DR rs is <5% for all blue-green bands for nearly all waters in the global oceans regardless of SNR(vis) at 400:1 or 800:1 as long as SNR(NIR) meets the requirement of 600:1.Correspondingly, DChl from the OCI algorithm is also <5% for all global waters.Of course, these uncertainty estimates considered sensor noise only; when factors other than noise are considered the uncertainties may be higher.

Discussion
It may be surprising to see that a SNR(vis) of $400:1 (under typical input radiance) can still meet the mission requirement, at least from simulations, in retrieving R rs to within 5% uncertainties for blue bands and blue waters, given the fact that most published documents recommended SNR(vis) of 1000:1 or higher.Indeed, even without the sophisticated calculations a simple estimate perhaps can lead to the same conclusion: if only 10% of the total signal comes from the ocean, a SNR(vis) of 400:1 would lead to R rs signal to noise of 40:1, equivalent to 1:40 (52.5%) of R rs uncertainties.Actually, the current SNR(vis) specifications (SNR 1 and SNR 2 ) are derived not from mission specifications on product uncertainties, but bounded by the best field measurement (DR rs field ) or best atmospheric correction (DR rs atm ), respectively.The reasoning for these SNRs is that if DR rs field and DR rs atm always exist, it is counterproductive to have very high SNR(vis) so that DR rs r will be lower than DR rs field or DR rs atm .
Clearly, such derived SNR 1 and SNR 2 are not intended to meet requirements for all water types and ocean applications.For example, for waters rich in colored dissolved organic matter (CDOM), R rs in the blue bands can be very low ($0.002sr 21 or even lower) [Lee et al., 2014, Figure 2a].Then, much higher SNRs than specified here are required to reduce the relative R rs uncertainties in the blue bands.However, for these cases, even if SNR(vis) is raised to infinity, DR rs atm still exists and DR rs t 8 tal will approach DR rs atm , making it impossible to have relative R rs uncertainties < 5% for CDOM-rich waters.Clearly, more effort is required to further improve the current state-of-the-art atmospheric correction.
Finally, by no means does this work suggest that an ocean color sensor's SNR(vis) should be set to SNR 1 or SNR 2 .Rather, these numbers provide lower bounds for sensor SNR specifications.Higher SNR(vis) would lead to lower DR rs t 8 tal , but once the SNR(vis) is higher than these lower bounds the effect of further increased SNR(vis) on the reduction of DR rs t 8 tal is degraded due to the dominant role of DR rs atm .Also, a requirement of higher SNR(vis) may limit the trade space for spatial or spectral resolutions.Nevertheless, if the trade space allows, it is useful to set SNR(vis) higher than these lower bounds, especially for applications that rely on individual images rather than on spatially/spectrally/temporally binned data.

Applicability and Exceptions
It is worth noting that although the above findings are based on a few assumptions which may vary in reality, such variations are unlikely to change these findings significantly.
First, simulations of atmospheric correction only considered several cases (Figure 3) due to the infinite number of combinations of the variable conditions in solar/viewing geometry, aerosol type, and s a .Of these cases, a maritime aerosol (relative humidity of 80%) with s a 5 0.1 in the NIR was used to represent typical atmospheric conditions over the global oceans, as MODIS statistics showed s a $ 0.1 over most of the open oceans.For maritime aerosol with s a $ 0.2, DR rs atm is higher, but at most 30% higher.For tropospheric aerosols, DR rs atm can be 2-3 times higher, but tropospheric aerosols are rare.Indeed, a higher DR rs atm would argue for a lower SNR(vis) according to equation ( 6).Therefore, the choice of SNR 2 ($500:1) should be reasonable.
Second, DR rs field used in this study represents the best result (i.e., minimal uncertainty) from any field measurements (e.g., 1% in the blue bands for blue waters).Even though, the choice of SNR 1 ($800:1) appears to be sufficient.Considering that most DR rs field can be much higher (3-4%) [Antoine et al., 2008], SNR derived from equation ( 5) can be much lower than SNR 1 presented here.
Third, both sensor noise (r) and sensor SNR in this study were assumed to be associated with L typ over the Atlantic Ocean with a solar zenith angle of 458 [Hu et al., 2012a].In practice, both could be different according to the square root law, where r is proportional to sqrt(L/L typ ).The difference in L may result from variable solar/viewing geometry, variable aerosols, and variable water-leaving radiance.Unless the water is extremely turbid (e.g., when atmospheric correction is switched from the NIR to shortwave infrared (SWIR) approach) [Wang and Shi, 2007], variations in L under most scenarios are at most 30-40% [Hu et al., 2012a].This also includes turbid atmosphere (s a $ 0.2) over Chinese marginal seas, as extremely turbid atmosphere where s a > 0.3 is treated as clouds without ocean color retrievals.Therefore, even for moderately turbid waters with turbid atmosphere, a 20% increase in SNR(vis) from those defined here (i.e., SNR 1 $ 800:1; SNR 2 Journal of Geophysical Research: Oceans 10.1002/2016JC012558 $ 400:1) would be sufficient for the ''ideal'' and ''practical'' situations, respectively, with the former converted to SNR 1 $ 1000:1 and latter converted to SNR 2 $ 500:1.
However, all these arguments above are for conventional atmospheric correction schemes (i.e., extrapolation from the NIR to the visible) and for applications where time series of R rs , Chl, or possibly other products are used to study ocean changes and processes.For other atmospheric correction schemes that do not rely on spectral extrapolation (e.g., using optimization to derive atmosphere and ocean properties simultaneously) [Chomko et al., 2003], DR rs atm may be lower than shown here, arguing for higher SNR(vis).For some applications where individual images at pixel resolution are required, such arguments are no longer valid, and other considerations of SNR requirements must be considered instead.These exceptional cases include all feature extraction applications, for example front detection [e.g., Wall et al., 2008], oil detection [Hu et al., 2009], and Sargassum mapping [Hu et al., 2015;Wang and Hu, 2016].This is because that all these features must be delineated at pixel level and some of the features may also be at subpixel level, and a low SNR would lead to missed features (e.g., compare Figure 1a against Figure 1b) that make delineation difficult.A clear example is given in the case of Sargassum mapping.Using simulations, Hu et al. [2015] argued that a SNR of 200:1 in the red and NIR wavelengths could detect Sargassum of 1% pixel size, while with MODIS statistics Wang and Hu [2016] showed that a SNR of 1000:1 made detection possible at 0.2% pixel size.Clearly, for these nontraditional ocean color applications, higher SNR would lead to better results, and there is perhaps no limit on the upper bound of SNR.On the other hand, if a lower SNR enables higher spatial resolution, the latter is also beneficial for such feature extraction applications.
Finally, while the estimates of DR rs are valid at both pixel level and 4 km monthly product level, estimates of DChl in this study only considered errors induced by sensor noise and atmospheric correction where uncertainties in the Chl algorithms were not included.In other words, DChl presented here assumed error-free Chl algorithms.In reality, this is certainly not possible.When the algorithm uncertainties are also included, because these uncertainties may range between 10 and 50% [e.g., Gordon, 1990], following the square root law (equation ( 2)), DChl at pixel level would be much higher than presented in Figure 9.However, as long as the algorithm uncertainties come from random errors rather than systematic errors, DChl (OCI algorithm) in the 4 km monthly product would still be <5% for nearly all locations.

Implications for Sensor Design
Unlike all previous recommendations for ocean color sensor design, a useful finding from this study is that once SNR(NIR) is at least 600:1, SNR(vis) can be much lower than 1000:1.Even after taking into account of the variable L (as opposed to L typ used here), for practical considerations a SNR(vis) of 500:1 can be set as a minimal bound for traditional applications.Given the fact that future ocean color missions such as The Pre-Aerosols Clouds and ocean Ecosystems (PACE) and Geostationary for Coastal and Air Pollution Events (Geo-CAPE) are already designed to have SNR(vis) > 1000:1, the question then becomes so what-why bother arguing for a lower SNR(vis) as these sensors will have SNR(vis) > 1000:1 anyway?Indeed, there are several implications from the findings here.First, they provide scientific support for a trade between SNR(vis) and other factors (e.g., sensor size, cost).Second, they support the use of nonocean color sensors for ocean color applications.For example, Hyperspectral Imager for the Coastal Ocean (HICO) SNR in the blue-green wavelengths is 300:1 to 500:1 [Lucke et al., 2011], and Landsat-8 OLI SNR in the blue-green wavelengths is 300:1 to 350:1 [Pahlevan et al., 2014].They are not significantly lower than SNR 2 (400:1) defined here.Therefore, once SNR(NIR) is increased to 600:1 through 5 3 5 or 9 3 9 smoothing because atmosphere is generally more homogenous than the ocean [e.g., Barnes et al., 2014], ocean color products can be derived from HICO and OLI at raw pixel resolution.Then, for the same reason, for future sensor design targeted for high spatial resolution applications, SNR(vis) can be set at $500:1 as long as SNR(NIR) approaches 600:1 through either spatial smoothing or spectral binning.

Conclusion
The most significant finding in this study is that, generally for open oceans, SNR(vis) can be lower than 1000:1 for an ocean color sensor.This finding is based on the fact that assuming error-free radiometric calibration, most uncertainties in the satellite-retrieved R rs originate from the imperfect atmospheric correction, and further increases in SNR(vis) would not lead to significant reductions in DR rs .In this work, only these two Journal of Geophysical Research: Oceans 10.1002/2016JC012558 uncertainty sources (i.e., sensor noise and atmospheric correction) were considered, while in reality other sources (e.g., instrument artifacts, errors in ancillary data, imperfect bio-optical inversion algorithms, etc.) also impact data product uncertainty.However, unless these other sources depend on sensor noise, they do not invalidate the argument of the minimal SNR requirement here.Another finding is that the DR rs < 5% requirement can be easily met for most global ocean waters in the 4 km monthly data products, where DChl from the OCI algorithm is also <5%, sufficient for most applications to address ocean changes in response to climate variability.Similar to the argument for a minimal SNR requirement, this finding is based only on noise and atmospheric correction induced errors without considering other error sources (e.g., imperfect bio-optical inversion algorithms), therefore it should not be considered as being applicable everywhere unless uncertainties from other sources are well understood.Finally, although the findings about SNR requirements may not be extended to some nontraditional applications (e.g., feature extraction), they support the argument to trade SNR for other requirements (e.g., spatial and spectral resolutions) of ocean sensor design.However, these results only provide lower bounds of SNR, and higher SNR may still be pursued if trade space allows, especially for coastal CDOM-rich dark waters and nontraditional ocean color applications.
Notations d a single value representing error from the hypothetical ''truth.''r measurement noise, defined as the standard deviation of normal d distribution.NEDL noise-equivalent radiance (NEDL), used interchangeably with r in this study.SNR signal-to-noise ratio, defined as the ratio of sensor signal (L) and r.SNR(NIR) SNR in the near-infrared bands that are used for atmospheric correction.SNR(vis) SNR in the visible bands that have a direct impact on sensor r(vis).L typ typical sensor signal, estimated from MODIS measurements over the Atlantic with a solar zenith angle of 458.D uncertainty in a measured or estimated property, or its probability in approaching the truth.Technically, it is defined as the standard deviation of the error distribution.A single measured or estimated value has a 68% likelihood to have its error < D. q wn normalized water-leaving reflectance, dimensionless.R rs remote sensing reflectance, sr 21 ; R rs 5 q wn /p.DR rs atm DR rs in the visible bands originated from atmospheric correction.DR rs r DR rs in the visible bands originated from r(vis) (i.e., directly related to SNR(vis)

Figure 2 .
Figure 2. DR rs field from the best SBA measurements of Lee et al. [2013].This represents minimal R rs uncertainties from any field measurements.Four spectral bands are annotated (443, 488, 547, 667 nm), whose DR rs field values were used to represent one of the two criteria to determine minimum SNR(vis), i.e., SNR 1 .

Figure 4 .
Figure 4. (a) The five DR rs terms for each of the four visible bands: DR rs field is from the best field measurement (Figure 2); DR rs atm is from atmospheric correction over typical global oceans;

Figure 5 .Figure 6 .
Figure 5. Distribution of # of valid pixels in each 4 km grid cell in the June 2008 monthly binned MODIS product.If a 1 km pixel is visited once per day, in theory, the maximum # of valid pixels from any grid cell is about 4 3 4 3 30 5 480.The low values in the equatorial regions and in the Indian Ocean are due to persistent cloud cover, sun glint, and stray light.

Figure 7 .
Figure 7. Same as in Figure 6, uncertainties in DR rs are expressed in relative terms (%, i.e., divide Figure 6 by the monthly mean R rs ).The map is the same for each scenario but the scales are different as shown in the right.
Figure9.(a, b) Typical R rs corresponding to different Chl levels, estimated using MODIS global data.These were used to estimate how DR rs propagates to DChl using equations (14)-(23); (c-e) DChl (in %) estimated from OC X and OCI algorithms using the corresponding R rs in Figures9a and 9bas input after considering DR rs at pixel level using three forms: (c) DR rs ideal , (d) DR rs practical , and (e) DR rs gyre .Note that OCI switches to OC X for Chl > 0.25 mg m 23 .

Figure 10 .
Figure 10.DChl (in %) from the 4 km monthly Chl product for June 2008, derived from (a) OC X and (b) OCI algorithms, respectively.DChl at pixel level is from Figure 9e, where # of valid observations in each 4 km grid cell is shown in Figure 5.
OCI 5Chl CI ½for Chl CI 0:175 mg m 23 ; 5Chl OC3 ½for Chl CI > 0:25 mg m 23 ; 5a3Chl OC3 1b3Chl CI ½for 0:175 < Chl CI 0:25 mg m 23 ; DChl OCI derived above were presented as the statistics of error distribution from individual pixels.As stated earlier, for repeated measurements, they indicated the standard deviation of error distribution.For a single measurement, they indicate the probability (68%) of the measurement error within DR rs where DChl OC3 and DChl CI are derived from equation (10) and equation (19), respectively.t 8 tal (vis) and DChl OCI .
). DRrs in the visible bands originated from both DR rs atm and DR rs r .DRrs from field measurements, used in this study as a reference to bound the SNR(vis) selection.DR rs gyre DR rs estimated from MODIS/Aqua measurements over ocean gyres, representing realistic DR rs from ocean color measurements.DChl uncertainty in the retrieved chlorophyll a concentrations, in relative (%) term.OC X band-ratio chlorophyll retrieval algorithm.OCI a hybrid chlorophyll retrieval algorithm to combine band subtraction and band ratio.SNR 1 SNR(vis) determined through forcing DR rs r to equal to DRrs field .SNR 2 SNR(vis) determined through forcing DR rs r to equal to DRrs atm .