The goal of the subsampling strategy is to preferentially sample observations in the vicinity of a given estimation grid cell, such that both the characterization of the local spatial covariance structure and the ultimate mapped estimate and its associated uncertainty are representative of local variability. This is accomplished by selecting the total number of observations to be used, N, where N is selected to be large enough to yield a representative sample but small enough to make mapping computationally feasible on a given computational platform. For the applications presented in Sect. 3, N= 500 and N= 1000 for the XCO2 and SIF mapping, respectively.

N observations are selected for each estimation grid cell by assigning a relative selection probability to each observation based on that observation's separation distance from the centroid of the grid cell. This selection probability could be application specific, but for the applications presented here we selected Ps∝1/h2, where Ps is the relative probability of a given observation being selected, and h is the great circle distance between the location xi of an observation and the centroid xj of the estimation grid cell: hxi,xj=rcos-1(sin⁡φisin⁡φj+cos⁡φicos⁡φjcos⁡(λi-λj)), where r is the radius of the Earth and φi and λi are the latitude and longitude of location xi.

The form of Ps in Eq. (1) ensures that a comparable number of observations is selected within any equal-area concentric band around an estimation grid cell, thereby also ensuring that observations that are at close distances to one another are preferentially close to the estimation location. This is a desirable feature because observations that are close to one another define the shape of the variogram at short separation distances (Sect. 2.2), and the variogram should reflect variability in the vicinity of the estimation grid cell. Different forms of Ps could also be used, for example if more/fewer observations along a given direction were desirable in order to better represent expected correlations along a given direction.

In previous work (Alkhaled et al., 2008; Hammerling et al., 2012a, b), a fixed application-specific window size was instead defined within which all available observations were used, together with a user-defined fraction of observations outside of the window. The window size was based in part on expected scales of variability in the satellite observations. The updated approach presented here reduces the number of user-selected parameters and explicitly provides a mechanism for ensuring the computational feasibility of mapping in the case of very large data sets, such as the SIF example examined here.

The characterization of the local covariance structure of the observations around each estimation grid cell, based on the subsampled observations, proceeds as described in Hammerling et al. (2012a, Sect. 2.1), except that (1) all possible pairs of observations are included in the formulation of the raw variogram and the nugget-effect variance, representative of the retrieval/measurement errors, is not spatially uniform. The reader is referred to that earlier publication for additional details.

Briefly, for each estimation grid cell, a raw variogram is calculated based on the subsampled observations: γh=12[yxi-y(xj)]2, where γ is the raw variogram value for a given pair of observations y(xi) and y(xj), and h is the great circle distance between the locations (xi and xj) of these observations, as defined in Eq. (2).

A parametric function, the theoretical variogram, is fitted to the raw variogram using non-linear least squares. For the prototypical applications presented here, an exponential variogram function with a nugget effect was used, because it yields a valid covariance function on a sphere (Huang et al., 2011), provided a good match to the known physical characteristics of the observations, and fit the observed variability well: γh=0,forh=0σ2(1-exp⁡-hl+σnug2,forh>0, where σ2 and l are the variance and correlation length of the quantity being mapped, and σnug2 is the nugget variance, typically representative of measurement and retrieval errors in the case of satellite observations. The nugget component can be either prescribed (as in the XCO2 example in Sect. 3) or estimated (as in the SIF example in Sect. 3), depending on the availability of information about measurement and retrieval errors.

The variogram parameters can be used to define a corresponding local spatial covariance structure for the mapped quantity (XCO2 or SIF, in the prototypical examples presented here). For the variogram function in Eq. (4) this becomes qh=σ2exp-hl. The nugget effect is correspondingly used to define the covariance structure of the measurement and retrieval errors: Rh=σnug2,forh=00,forh>0.

Ordinary kriging, a minimum variance linear unbiased mapping method for spatial data, was used in Hammerling et al. (2012a, b) to create contiguous maps of XCO2. In this approach, the spatial support (i.e., footprint) of the estimates corresponds to that of the observations. Although the mapping can be performed at any spatial interval (e.g., once per 1∘ × 1∘ grid cell), the estimates remain representative of the variability at the scale of the observations.

Here, we instead use block kriging (e.g., Webster, 2000), an approach that yields estimates that represent an average within a specified area. This makes it possible to disassociate the native footprint of the observations from the resolution of the mapped product, thereby making it possible to create contiguous maps at any desired spatial resolution equivalent to or greater than the size of the observation footprints. As with moving window ordinary kriging, block kriging provides an optimal estimate of the quantity being mapped (XCO2 and SIF, in the prototypical examples presented here) for each estimation location, based on the subsampled observations (Sect. 2.1) and the local covariance structure (Sect. 2.2), together with a rigorous assessment of the uncertainty associated with the estimate.

The linear system of equations that is solved to obtain the N weights λ assigned to the subsampled observations for a given estimation grid cell is Q+R11T0λ-ν=qA1, where Q is a N×N covariance matrix among the N observations with individual entries as defined in Eq. (5), R is a N×N diagonal measurement and retrieval error covariance matrix among the N observations as defined in Eq. (6), 1 is a N× 1 unity vector, T denotes the vector transpose operation, and qA is an N× 1 vector of the spatial covariances between the estimation grid cell and the N observation locations, defined as qA,i=1n∑j=1nqhi,j, where qA,i is the covariance between the grid cell and observation i, and qhi,j is defined as in Eq. (5) based on the distance hi,j between observation i and n regularly spaced locations within the grid cell. In general, the larger the n the better the representation of the area (i.e., grid cell) to observation covariance. For practical purposes, in the applications presented here, n is defined based on the relative footprint of the observations compared to that of the estimation grid cells.

The system in Eq. (7) is solved for λ and the Lagrange multiplier ν. These parameters are then used to define the estimate (z^) and estimation uncertainty variance (σz^2) for the grid cell as z^=λTy,σz^2=σAA-λTqA+ν, where y is the N× 1 vector of subsampled observations, and σAA is the variance of the mapped quantity (XCO2 or SIF, in the prototypical examples presented here) at the resolution of the estimation grid cell, defined as σAA=1n2∑j=1n∑k=1nqhj,k, where qhj,k is defined as in Eq. (5) based on the distance hj,k between any combination of the n regularly spaced locations within the grid cell defined previously.

The Japanese GOSAT (e.g., Kuze et al., 2009) was launched in 2009 and is the first satellite dedicated to global greenhouse gas monitoring, including CO2 and CH4. GOSAT flies in a polar, sun-synchronous orbit with a 3-day repeat cycle and an approximately 13:00 LT overpass time. GOSAT XCO2 data are being used to examine a number of questions in carbon cycle science, including comparing observed and modeled XCO2 fields (Hammerling et al., 2012b), quantifying sources and sinks of CO2 (e.g., Deng et al., 2014; Basu et al., 2013, 2014; Chevallier et al., 2014; Takagi et al., 2014), detecting perturbations in the carbon cycle (Guerlet et al., 2013), and interpreting seasonal changes in the carbon balance (Parazoo et al., 2013).

Measurements of XCO2 (a.k.a. “Level 2” data) are derived using a number of retrieval algorithms, among them NASA's Atmospheric CO2 Observations from Space (ACOS) algorithm (e.g., O'Dell et al., 2012; Crisp et al., 2012). Filtered and bias-corrected data from the most up to date version of this algorithm (ACOS v3.4 release 3) are used here to demonstrate the mapping approach presented in Sect. 2. Approximately 900 successful retrievals are available per 3-day repeat cycle, with the majority of observations being over land. These data have substantial retrieval uncertainties (e.g., O'Dell et al., 2012) and include large gaps (e.g., Fig. 1). These features prevent the application of simple spatial and temporal binning techniques for generating XCO2 maps at spatiotemporal scales that are directly useful for addressing existing uncertainties in carbon cycle science.

The approach described in Sect. 2 is used to create continuous maps, a.k.a. Level 3 data, based on XCO2 observations obtained over two repeat cycles, namely 2–7 August 2009 (Fig. 1). A 6-day period is used to balance the competing goals of including as many observations as possible, while avoiding time periods over which the XCO2 field itself would change substantially (see discussion in Hammerling et al., 2012a). Maps of XCO2 and associated uncertainties are created at native (Fig. 2a, b) and 1∘ × 1∘ (Fig. 2c, d) resolutions, in order to examine and demonstrate the impact of resolution on mapping uncertainty. Targeting different resolutions is made possible by the use of the moving window block kriging approach presented here. N= 500 subsampled observations are used per estimation location. These maps can also be compared to those presented for an equivalent period in Hammerling (2012b, Auxiliary Figs. 2 and 3), with methodological differences as described in Sect. 2 and representative of the estimated XCO2 at the native resolution of sounding footprints (nadir footprint ∼ 10.5 km diameter) with estimates at 1∘ × 1.25∘ intervals.

Results show that, because of the information content of the sparse observations, the estimated fields (Fig. 2a, c) are similar at native and 1∘ × 1∘ resolutions; however, estimating directly at the coarser 1∘ × 1∘ resolution yields lower uncertainties as observations become more informative for spatially averaged quantities (Fig. 3).

A series of recent studies has demonstrated the potential use of satellite observations of SIF for understanding and quantifying photosynthetic CO2 uptake at large scales, using data from the GOSAT satellite (e.g., Joiner et al., 2011; Frankenberg et al., 2011; Guanter et al., 2012; Joiner et al., 2012; Lee et al., 2013; Frankenberg et al., 2012), the SCIAMACHY (SCanning Imaging Absorption spectroMeter for Atmospheric CHartographY) instrument on board Envisat (e.g., Joiner et al., 2012), the GOME-2 instrument on board MetOp-A (Meteorological operational satellite-A; e.g., Joiner et al., 2013), and the Orbiting Carbon Observatory (OCO-2) (e.g., Frankenberg et al., 2014). Satellite measurements of fluorescence can be used with land surface models to improve the representation of GPP (gross primary production) and to understand the GPP response to environmental stress (e.g., Lee et al., 2013). Among available data sets, GOME-2 provides the highest spatial and temporal density of data.

Until now, studies of SIF have relied on spatially and temporally binned average observations at monthly or coarser timescales and 1∘ or coarser spatial scales (e.g., Fig. 4). The coarse spatial and temporal scales were used to overcome, through the use of simple averaging, spatial gaps in observations and the relatively high uncertainties associated with individual retrievals. One of the limitations of such an approach is that it inherently discards information about SIF variability at fine spatial and temporal scales, which is important for understanding the impact of transient effects such as changes in phenology and water availability (Lee et al., 2013), and developing biospheric models that can represent these effects correctly. A second limitation is the lack of a direct and robust quantification of the uncertainty associated with the mapped products, complicating uncertainty analysis in subsequent applications using the data.

As a second demonstration of the mapping approach proposed here, we use SIF GOME-2 V.14 data (Joiner et al., 2013) with the approach described in Sect. 2 to create contiguous maps of SIF at a single spatial resolution (1∘ × 1∘) but at multiple temporal resolutions. The examination of shorter time periods was selected in order to more directly respond to scientific opportunities in the use of SIF data and to complement the spatial-resolution-focused demonstration of Sect. 3.1. Maps of SIF and associated uncertainties are created at 1-, 6-, and 31-day temporal resolutions in August, 2009 (Fig. 5), where August 2009 was chosen for convenience to correspond with the XCO2 application presented in Sect. 3.1. N= 1000 subsampled observations are used per estimation location. The monthly map can also be compared to the monthly binned map presented in Fig. 4.

Results show that the proposed approach can leverage nearby observations to create realistic contiguous maps even at 1-day resolution (Fig. 5a, b), although, as expected, uncertainties are reduced (Fig. 5d) at coarse temporal resolutions, just as was seen for coarser spatial resolutions in the XCO2 application. In fact, the regions with little or no SIF data for the 1-day application are clearly visible as high-uncertainty bands in Fig. 5b, and a user could explicitly decide whether such uncertainties are acceptable or too high for a given scientific application. When maps are intended to be used to drive and/or validate biogeochemical models, having the ability to choose a desirable balance between temporal resolution and mapping, uncertainty presents a considerable advantage.

Ideally, the temporal resolution at which maps are obtained is as fine as possible so as to capture the dynamics of the observed physical quantity, in this case SIF. The choice of optimal temporal resolution thus, in general, defines a trade-off between having sufficient observations for adequate spatial coverage while minimizing the impact of temporal variability in the quantity being examined (Hammerling et al., 2012a). In Fig. 6 it is apparent that the presented approach makes it possible to obtain maps at temporal resolutions much higher than the monthly (or coarser) resolution of current binned products. As expected, the more abundant observations available at 6-day temporal resolution (Fig. 6d) lead to decreased estimation uncertainty compared to those at 1-day resolution (Fig. 6b). However, at monthly temporal resolutions (Fig. 6e, f) the temporal variability in SIF over a 31-day period increases the discrepancy among (spatially) nearby observations, leading to increased uncertainties at coarse timescales. This effect is apparent in comparing Fig. 6d and f, as uncertainty increases over, for example, eastern South America. A similar trade-off was also noted in selecting mapping timescales for XCO2 (Hammerling et al., 2012a) and further speaks to the advantage of being able to select a mapping timescale based on scientific need and uncertainty tolerance, as is possible with the approach presented here.