Passive remote sensing of the Earth system has used spatial resolutions ranging from meters to kilometers. It is thus necessary to understand how data products with different spatial resolutions can be compared with each other, and how sub-pixel variations may affect data comparison. This is particularly important for ocean color remote sensing where the measured signal (water-leaving radiance or remote sensing reflectance) is a non-linear function of sub-surface constituents. As a result, products at low resolution are not necessarily arithmetic or geometric means of those at higher resolution. Here, we developed analytical expressions to link ocean color properties derived from high- and low-resolution data, and the proof-of-concept is further demonstrated with both simple examples and measurements of MERIS full-resolution (FR) and reduced resolution (RR). These results suggest that current global chlorophyll concentration is likely underestimated due to the coarse spatial resolutions. Application of the expressions will facilitate cross-sensor comparisons and may also reduce uncertainties.
©2012 Optical Society of America
Remote sensing plays an indispensable role for the acquisition of geophysical information of large areas. These missions employ various platforms including aircraft and satellites, with sensors measuring the Earth’s surface at resolutions ranging from sub-meters to kilometers. For instance, an image pixel from the WorldView 2 satellite measurement has a size of ~1.5 x 1.5 meter (high spatial resolution), while a pixel from Hyperion or Landsat measurement is ~30 x 30 m, and a pixel from the Moderate-resolution Imaging Spectroradiometer (MODIS) or the Sea-viewing Wide Field-of-view Sensor (SeaWiFS) measurement is ~1000 x 1000 m (medium spatial resolution). Thus, one MODIS pixel encompasses ~33 x 33 Hyperion or Landsat pixels and ~660 x 660 WorldView 2 pixels. The high-resolution measurements provide detailed information for a targeted area, while the low-resolution measurements provide synoptic information over larger areas. On the other hand, this spatial mismatch produces difficulties in comparing products measured with different spatial resolutions when the target is inhomogeneous [1, 2]. For a spatially homogeneous environment, it is easy to visualize that the product of a unit area (e.g., absorption coefficient, concentration of chlorophyll) of a low-resolution pixel is equivalent to the product of any pixel of high-resolution pixels encompassed by the low-resolution pixel. For a spatially inhomogeneous environment, the situation becomes complex. For information mainly generated from a 2-dimensional (horizontal) domain such as surface temperature or Earth surface albedo , normally an arithmetic mean of the high-resolution measurements can represent the low-resolution measurements. For ocean color remote sensing, however, because of its 3-dimensional (horizontal and vertical) characteristics, it is not known if this arithmetic mean is still representative. For instance, for vertically stratified waters, the remotely observed property is not a simple arithmetic mean of its vertical profile, but a weighted mean with the weight decreased exponentially with increasing depth [4–6]. The same argument also applies to horizontally inhomogeneous waters (see Fig. 1 for instance): although the water-leaving radiance from a low-resolution pixel is an arithmetic mean of those from the corresponding higher-resolution pixels, the non-linear nature of the dependence between water-leaving radiance and sub-surface constituents makes it non-straightforward to extend the “averaging” concept to the retrieved data products. This raises the question of how best to compare products measured with different spatial resolutions? The same question is also raised in planning field measurements for the validation of remote sensing products , where discrete point measurements are used to compare with the synoptic data from the remote sensing estimates.
Based on established radiative transfer models for ocean color remote sensing, this short letter addresses these questions for water and bottom properties, with the ultimate objective of providing an analytical solution for comparing products at different resolutions. In addition, it tries to provide an easy understanding of low-resolution product for spatially inhomogeneous environments. Because of the particular importance of sub-pixel variations in optically shallow waters where the bottom contributes to a significant amount of satellite signal, we address these questions for two distinctive environments: optically deep and optically shallow waters.
Optically Deep waters
After radiometric calibration and atmospheric correction, the fundamental property of ocean color remote sensing is the remote sensing reflectance (Rrs, sr−1), which is used for the derivation of subsurface properties [7, 8]. Rrs is defined as the ratio of water-leaving radiance (Lw, mW cm−2 µm−1 sr−1) to downwelling irradiance just above the surface (Ed, mW cm−2 µm−1). For optically deep waters, Rrs can be expressed as  (a more detailed review is provided in Lee et al )11, 12], g0,1 (sr−1) are model coefficients and they are constants for a given water body and sensor-sun geometry; a (m−1) and bb (m−1) are the absorption and backscattering coefficients, respectively. a is composed of the contribution of pure seawater, phytoplankton pigments, colored dissolved organic matter (CDOM), and detrital particles; while bb is composed of contribution of pure seawater and suspended particulates .
For Rrs of a low-resolution pixel (RrsLo), Eq. (1) becomes
Now consider n high-resolution pixels that make up the low-resolution pixel. For each RrsLo, because the downwelling irradiance on each high-resolution pixel is virtually the same, RrsLo is the arithmetic mean of RrsHi from the n high-resolution pixels,Eqs. (1-3) indicate the following is generally true:13–16] or when satellite products with different spatial resolutions are compared with each other.
For easier derivation and explanation, we use the following simplified model  to describe the relationship between Rrs and a and bb,Eq. (3), there is9, 11] and considered not to vary significantly within the high-resolution pixels, the above relationship is reduced to
Further, because Rrs is generally linearly proportional to particle backscattering coefficient (bb, see Eq. (5)) and because absorption at longer wavelengths (except absorption peaks of various phytoplankton pigments) is nearly stable [12, 17], bbLo in the longer wavelengths can then be derived following Eq. (6) as7, 18]. Thus, Eqs. (8) and (9) show how the effective bbLo and aLo can be derived from their high-resolution measurements. Equation (8) further indicates that, because RrsLo represents the arithmetic mean of RrsHi (see Eq. (3)), if the high-resolution pixels are not homogeneous (e.g., in frontal zones, coastal waters, etc.), aLo derived from RrsLo is the backscattering-weighted mean of aHi instead of an arithmetic mean of aHi.
Figure 2 shows an example to demonstrate this concept. Rrs spectra of 4 high-resolution pixels were simulated with the Lee et al  IOP-Rrs model. In this simple illustration, the 4 pixels had the same phytoplankton absorption and particle backscattering coefficients while the absorption coefficient of colored dissolved organic matter (CDOM) was varied from 0.1 to 0.6 m−1. The Rrs spectrum perceived from the low-resolution pixel encompassing all 4 high-resolution pixels was calculated as an arithmetic mean (Eq. (3)), as shown in Fig. 2(a) (black line). From a spectral optimization inversion algorithm , a was derived from RrsLo and shown in Fig. 2(b) as aLo (see Table 1 for definition of symbols). The arithmetic (and geometric) mean of absorption from the 4 high-resolution pixels showed much higher values (about 29% and 22% at 410 nm and 440 nm, respectively) in the blue-green wavelengths than aLo (see Fig. 2(b)). In contrast, the weighted mean, aweit, calculated with Eqs. (8) and (9) from RrsLo, closely (average absolute percentage difference is ~1.1%) follows aLo.
These results clearly showed that the absorption coefficients in the shorter wavelengths perceived by a low-resolution pixel is much lower than the arithmetic mean of that from the high-resolution pixels, but it could be well represented by the backscattering-weighted mean. This result further suggests that the chlorophyll concentrations derived from low-resolution measurements are likely underestimated compared to that measured by high-resolution measurements, as the empirically derived chlorophyll concentration closely follows the total absorption coefficient [21–23]. Specifically for the examples shown in Fig. 2, using the same OC4 algorithm , the arithmetic mean chlorophyll concentration of the 4 high-resolution pixels are 5.4 mg/m3, but it is 4.7 mg/m3 from RrsLo, which is reduced by ~13%. Note that empirical band-ratio algorithms do not distinguish the impacts of phytoplankton and CDOM, so the variation of CDOM is perceived as variation of phytoplankton (Chl-a).
Optically shallow bottom
It has long been demonstrated that the depth of optically shallow bottom could be retrieved from spectral measurements of water color [25–28]. On the other hand, bottom depth (H, m) in shallow waters may change substantially in a small region, creating a similar problem to the above when comparing bottom depths derived from remote sensing measurements at different spatial resolutions. To address this problem, analytical expressions to relate low-resolution and high-resolution bottom depth products, such as Eq. (8) for optically deep waters, are investigated here.
To focus on the bottom depth retrievals, water properties (i.e., rrsC) are assumed homogenous across multiple high-resolution pixels (but within a low-resolution pixel) due to current and tidal mixing [30, 31]. Then, the following equations can be derived from Eq. (3) and Eq. (9):
To help visualize this non-linear impact, consider that e-x is proportional to 1/x for x in any narrow range, Eq. (11) can then be simplified asEquation (14) shows that in order to interpret HLo, information of the low-resolution perceived bottom albedo (ρLo) and the high-resolution bottom properties is required. In addition, unlike the case of backscattering coefficient of deep waters, ρLo in general is not an arithmetic mean of ρHi(i) because the contribution of substrate from the multiple high-resolution pixels to a low-resolution pixel depends on both bottom albedo and depth. For a special case when the bottom albedo is uniform, Eq. (14) reduces to
Similar to the deep-water example in Fig. 2, an example is shown in Fig. 3 to illustrate the non-linear effect of bottom depth on high- and low-resolution pixels. The example used homogenous water properties and bottom albedo but varying bottom depths (2 – 10 m) across 4 high-resolution pixels, and simulated their effect on the perceived bottom depth from a low-resolution pixel encompassing all 4 high-resolution pixels. The 4 high-resolution Rrs spectra (see Fig. 3(a)) were generated from a shallow-water model (see Eq. (10)), and the arithmetic mean of these Rrs, as measured from a low-resolution pixel, were fed back to the simulation model with HLo derived via spectral optimization . The perceived HLo from RrsLo is again much lower (~30% and 17%, respectively) than the arithmetic and geometric means of the four bottom depths, but agrees very well (within ~1% for this example) with that predicted by Eq. (15). This is consistent with radiative transfer as shallower bottom (assuming all other properties remain the same) results in greater radiance at sea surface and then contributes more to the total measured signal.
Tests on MERIS measurements
The analyses and figures above demonstrate that water column and bottom depth properties derived from low-resolution data are lower than their arithmetic or geometric means derived from high-resolution data, and the former can be better described using a combination of high-resolution data and analytical equations similar to Eqs. (8), (14) and (15). To further prove this concept, satellite measurements from the Medium Resolution Imaging Spectrometer (MERIS) were used. MERIS measures the oceans at a nominal resolution of 300-m (full resolution or FR), while aggregated MERIS data have 1.2-km resolution (reduced resolution or RR), with 1 RR pixel corresponding to 16 FR pixels.
MERIS Level-2 data over the southern Ocean east of Argentina on 26 December 2011 were obtained from the U.S. NASA Goddard Flight Space Center (GSFC). The data contained spectral Rrs derived from the most recent updates in sensor calibration and processing algorithms embedded in the software package SeaDAS6.4. The FR Rrs (RrsFR) data for a meso-scale eddy were used with the quasi-analytical algorithm  to derive total absorption and backscattering coefficients at 440 nm, aFR(440) (m−1) and BFR(440) (m−1), respectively, at 300-m resolution. Then, aFR(440) was aggregated to 1.2-km resolution to obtain arithmetic mean (), geometric mean (), and backscattering-weighted mean () following Eq. (8). Further, aRR(440) was derived from RR Rrs (RrsRR) data using the same algorithm and compared with the various means of aFR(440) data.
Figures 4(a) and 4(b) show aFR(440) and aRR(440), respectively; while the ratios of aRR(440) to and , respectively, are calculated and shown in Figs. 4(c) and 4(d). Ratio of aRR(440) to is also calculated but not shown (except its histogram, see Fig. 5 ). Compared with , aRR(440) is biased low for the patchy eddy (see Fig. 4(c)) where the ratio is often < 0.9. This trend is consistent with the simulation results in Fig. 2(b), although there are 16 FR pixels corresponding to 1 RR pixel here. In contrast, agrees much better (see Fig. 5) with aRR(440), also consistent with the simulation results in Fig. 2(b). These contrasting results are further demonstrated in Fig. 5 where the histogram distributions of the a(440) ratios clearly show the improvement in the backscattering-weighted mean when using high-resolution products to compare with low-resolution products.
Implications for satellite remote sensing
The simulations and MERIS examples clearly show that when the water is patchy, low-resolution products are likely underestimated when compared to the “truth” (i.e., arithmetic mean of products from high-resolution measurements). This is simply because that the result of arithmetic mean favors higher values, and higher remote-sensing reflectance in the blue-green domain (the spectral window for sensing subsurface constituents or bottom depth) is likely associated with lower absorption or shallower bottom depth, although the actual situation will depend on the detailed spatial distribution of water and/or bottom properties. There are many cases where ocean waters are patchy [33, 34], for example in eddies, fronts, river plumes, or coastal blooms. Then, when high-resolution measurements are not available and there is substantial spatial variability, one can infer that the low-resolution products are possibly biased low. The same argument can also be applied for the global biomass standing stocks and for validation of satellite data using discrete field data. However, it is yet to be investigated to what degree the global biomass is underestimated due to a combination of ocean patchiness and low-spatial resolution measurement.
Analytical expressions are developed to provide an explicit interpretation of low resolution measurements and an easy way to remove uncertainties when comparing high-resolution and low-resolution data. It is found that the low-resolution product can be well represented using weighted means of high-resolution data. The concept has been demonstrated through simulations and satellite data analysis. These relationships provide insights into the spatial meaning of remote sensing products, especially for spatially inhomogeneous environments. Application of these relationships can reduce uncertainties when comparing measurements from platforms with different spatial resolutions. For spatially inhomogeneous environments, because remote-sensing reflectance (or water-leaving radiance) is a non-linear function of the water column constituents and bathymetry, products (e.g., chlorophyll-a concentration, CDOM absorption) derived from low-resolution measurements are likely biased low compared to their arithmetic means from the corresponding high-resolution measurements. Thus, to overcome the uncertainty resulted in low- or medium-resolution measurements by ocean color sensors (e.g., SeaWiFS, MODIS), it is necessary to have concurrent high-resolution measurements (e.g., measurement with a few wide-bandwidth, high-spatial resolution, bands).
Financial support provided by the U.S. NASA Applied Sciences, Ocean Biology and Biogeochemistry (OBB) and Water and Energy Cycle programs, and the Naval Research Laboratory is greatly appreciated. We are extremely grateful to the comments and suggestions from three anonymous reviewers that greatly improved this manuscript.
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