Satellite remote sensing of the Earth's atmospheric composition usually
samples irregularly in space and time, and many applications require
spatially and temporally averaging the satellite observations (level 2) to a
regular grid (level 3). When averaging level 2 data over a long period to a
target level 3 grid that is significantly finer than the sizes of level 2
pixels, this process is referred to as “oversampling”. An agile,
physics-based oversampling approach is developed to represent each satellite
observation as a sensitivity distribution on the ground, instead of a point
or a polygon as assumed in previous methods. This sensitivity distribution
can be determined by the spatial response function of each satellite sensor.
A generalized 2-D super Gaussian function is proposed to characterize the
spatial response functions of both imaging grating spectrometers (e.g., OMI,
OMPS, and TROPOMI) and scanning Fourier transform spectrometers (e.g., GOSAT,
IASI, and CrIS). Synthetic OMI and IASI observations were generated to
compare the errors due to simplifying satellite fields of view (FOVs) as
polygons (tessellation error) and the errors due to discretizing the smooth
spatial response function on a finite grid (discretization error). The
balance between these two error sources depends on the target grid size, the
ground size of the FOV, and the smoothness of spatial response functions.
Explicit consideration of the spatial response function is favorable for
fine-grid oversampling and smoother spatial response. For OMI, it is
beneficial to oversample using the spatial response functions for grids finer
than

Since the launch of the ESA Global Ozone Monitoring Experiment (GOME) in
1995, satellite observations have tremendously advanced our understanding of
the processes governing the atmospheric composition, greenhouse gas
emissions, and air
quality

The retrieval results from satellite sensors are usually total or partial
(e.g., tropospheric or planetary boundary layer, PBL) column density at
individual satellite pixels, i.e., the level 2 product. However, the pixel
geometry may vary significantly even for the same sensor (see Fig. 1 for
example), and data quality screening (by cloud coverage, solar zenith angle,
surface albedo, thermal contrast, etc.) often leaves only small and patchy
fractions of useful level 2 pixels for any given orbit. As such, the level 2
data over many orbits are often projected to a regular spatial grid to better
represent the spatiotemporal variations of the target species through a
gridding algorithm. These “level 3” products help to average out the
observational noise that can be significant for individual level 2 retrieval
and make satellite data more accessible for scientific studies and the
general public. These products may also lead to additional discoveries, such
as emission and lifetime
estimates

The operational level 3 products are typically provided at grid sizes of

Various gridding algorithms have been developed to generate level 3 maps at
a regional scale with much finer grids (0.05–0.01

The OMI instrument aboard the Aura satellite launched in 2004 is a push-broom
UV–visible imaging grating spectrometer. It has a daytime equatorial crossing
at

Across-track (xtrack in figure) ground pixel geometry for IASI, CrIS, and the UV2 and VIS (visible) bands of OMI.

The IASI instrument is an FTS with an across-track scanning range of 2200

We use the most recent neural network (NN) IASI

The CrIS instrument, which is aboard the Suomi NPP satellite and the series
of JPSS satellites, is a step-scan FTS with 2200

The CrIS fast physical retrieval (CFPR)

This section reviews existing gridding methods that map level 2 pixels to level 3 grids. Oversampling conventionally refers to the cases where level 3 grid is much finer than the level 2 pixel size.

The spatial interpolation methods generate continuous data fields from
observations made at discrete locations. The main difference between
interpolation and the point- and polygon-based oversampling approaches
discussed in Sect.

The simple “drop-in-the-box” gridding method can be classified into this
category, as each satellite observation is assumed to be a point on the
surface. The value for each target grid cell is the average of all screened
satellite observations with the center of the FOV falling inside the grid cell
boundaries. A conventional oversampling approach has been developed based on
the drop-in-the-box method; instead of only averaging “in the box”, it
includes satellite observations within a certain radius (much larger than the
grid size) from the center of each grid cell. This averaging radius is chosen
to balance the smoothing and noise but is also somewhat arbitrary. For
example,

Figure

In reality, the OMI ground pixel footprints are not as sharp as quadrilateral
boundaries

Centers of screened OMI pixels in 2005 over a target grid point (red star) near Denver, CO. Pixels that overlap with the target grid point with the pixel center falling within the averaging radius (dashed circle) are plotted as black points (correct oversampling, 40 %). Pixels that overlap with the target grid point with the pixel center falling outside the averaging radius are plotted as purple squares (false negative, 38 %). Pixels that do not overlap with the target grid point with the pixel center falling in the averaging radius are plotted as red triangles (false positive, 22 %). Extreme cases of false positives or negatives are illustrated by OMI pixel quadrilaterals. The percentages of correct oversampling, false positive, and false negative pixels are labeled in the legend.

This approach assumes that each satellite observation footprint is a polygon
on the surface, and calculates the areal proportions of grid cells inside
each polygon. Because calculating these overlapping areas requires filling
irregular satellite footprint polygons with rectangular grid cells, it is
also known as the “tessellation” approach. The contribution of each
satellite observation to a given grid cell is weighted by the overlapping
area and inversely weighted by the total pixel polygon area and the
observational uncertainty, as shown by the following
equations

It is sometimes convenient to define

Figure

Tessellation process for OMI

The tessellation approach discussed in Sect.

For imaging grating spectrometers like OMI, the spatial response function
depends on the diffraction of the fore optics, the instantaneous field of
view (i.e., the instantaneous projection of the slit on the ground from the
point of view of a native detector pixel), the numbers of across- and
along-track bins, and the along-track movement of subsatellite point during
the integration time. The satellite movement only affects the along-track
direction, generally making the spatial response in the along-track direction
smoother than that in the across-track direction.

For FTS systems with stop-and-stare sampling, like IASI and CrIS, the spatial
response function (also known as point spread function by the community) is
more simply defined by the circular aperture and some diffraction around the
edge. The nadir FOV is circular with no difference between across- and
along-track directions, and hence the spatial response function can be
characterized by a 1-D super Gaussian function rotating around the nadir
point. This rotating super Gaussian function is another special case of the
generalized 2-D super Gaussian (Eq.

In the generalized 2-D super Gaussian function (Eq.

The projection of a rectangular FOV for imaging grating spectrometers like OMI
on the surface at large viewing angles leads to distorted quadrilateral
footprints, as shown by the polygon ABCD in Fig.

The proposed oversampling approach represents each satellite observation as a
sensitive distribution, instead of a point or a polygon. If the true
satellite spatial response function is used as the sensitive distribution,
this approach is the theoretically optimal solution to the oversampling
problem, and is hence referred to as “physical oversampling” hereafter. It
follows the same equations as the tessellation approach as in
Eqs. (

This physical oversampling approach can also be considered as a spatial
interpolation method as discussed in Sect.

The tessellation approach is perfect if the spatial response of satellite
observation is a boxcar, but otherwise it will introduce some error in the
oversampled results (referred to as “tessellation error” hereafter). When
the satellite spatial response function is smooth (instead of a boxcar), the
exact solution is to calculate

The satellite sensors have very different spatial responses. The target grid
size for level 3 data ranges from 0.25

Oversampling a synthetic checkerboard pattern, shown in panel

Figure

The analysis for Fig.

Similarly, Fig.

As shown by Fig.

The difference between resolution and sampling density for 1-D spectral data
has been thoroughly discussed in the
literature

First column: spatial response function of an imaginary sensor
discretized at 0.2

Figure

The drop-in-the-box approach shows significant data gaps (5-day averaging)
and high level of noise (5-month averaging), even when its target grid is 10
times coarser than the other oversampling approaches. There are two gaps
where no observation is available for point oversampling over the 5 days
(column 2, rows 1–2 in Fig.

Level 3 results using the drop-in-the-box method (10

Although the physical oversampling using the true satellite spatial response
functions produces the optimal estimation, the result is sometimes noisy and
even unphysical, especially when the observations are noisy and sparse. In
these cases, some spatial interpolation or smoothing methods are often
needed. In addition to the specialized interpolation and smoothing methods
discussed in Sect.

Similar to Fig.

Figure

Physical oversampling results using IASI-A

Oversampling based on Eqs. (

A physics-based approach is developed to oversample diverse satellite observational products to high-resolution destination grids. It represents each FOV as a sensitivity distribution on the ground, which is physically a more realistic representation of satellite observations. This sensitivity distribution can be determined by the spatial response function of each satellite sensor. We propose a generalized 2-D super Gaussian function that can standardize the spatial response functions of many satellite sensors with distinct observation mechanisms and viewing geometries. This generalized 2-D super Gaussian function can be reduced to a rotating super Gaussian to characterize the circular FOV of IASI and CrIS or a 2-D super Gaussian to characterize the quadrilateral FOV of OMI and its successors. It can also represent hybrid cases where the FOV is quadrilateral but with rounded corners. When the shape-determining exponents in the generalized 2-D super Gaussian function approach infinity, the FOV is equivalent to a polygon, as assumed in the tessellation approach.

Synthetic OMI and IASI observations were generated assuming the spatial
response functions are perfectly known to compare the tessellation error and
the discretization error. The balance between these two error sources depends
on the target grid size, the ground size of FOV, and the smoothness of
spatial response functions. The proposed oversampling approach is generally
more accurate for fine-grid oversampling of satellite observations with
smooth spatial responses, whereas tessellation is more accurate for coarse
grids and sharper spatial responses. For OMI, CrIS, and IASI, the threshold
target grid size where both errors are equal are at

The new physical oversampling approach is applied to OMI

A MATLAB implementation of the physical oversampling is
available at

The spatial response functions of IASI are tabulated at

IASI spatial response functions (also known as point spread
functions) defined at the viewing angular space. The corresponding ground
distance at nadir is shown in the axis on the right. The IASI orbit height is
assumed to be 817

Figure

Fitting a tabulated IASI spatial response function for pixel no. 2 using rotating super Gaussian. The fitted exponent is 18.5.

Figure

Slices of spatial response functions for IASI

To compare different discretization schemes, we first construct an ideal
spatial response function using OMI pixel boundaries but sharper edges (

In Fig.

KS, LZ, KY, and GH developed and implemented the oversampling algorithms. KCP provided expertise on the CrIS instrument and products. CCM, KC, GGA, and XL provided expertise on the OMI instrument and products. LC, PFC, MVD, and MZ provided expertise on the IASI instrument and products. KS collected the data, analyzed the results, and wrote the manuscript. All authors contributed to interpretations and edited the manuscript.

The authors declare that they have no conflict of interest.

We acknowledge support from NASA (sponsor contract numbers NNX14AF16G and NNX14AF56G), the RENEW
Institute and School of Engineering and Applied Science at the University at
Buffalo, and Smithsonian Institution subaward SV8-8802. We thank John Houck
at the SAO; Thomas Kurosu at JPL; Holger Sihler at MPI-C; Glen Jaross at
NASA; Rui Wang, Xuehui Guo, and Da Pan at Princeton University; and Likun Wang
at University of Maryland for helpful discussions. We thank the OMI
science team for making the OMI