**Research article**
05 Jan 2022

**Research article** | 05 Jan 2022

# New sampling strategy mitigates a solar-geometry-induced bias in sub-kilometre vapour scaling statistics derived from imaging spectroscopy

Mark T. Richardson David R. Thompson Marcin J. Kurowski and Matthew D. Lebsock

^{1,2},

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**Mark T. Richardson et al.**Mark T. Richardson David R. Thompson Marcin J. Kurowski and Matthew D. Lebsock

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^{1}Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA^{2}Department of Atmospheric Science, Colorado State University, Fort Collins, CO 90095, USA

^{1}Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA^{2}Department of Atmospheric Science, Colorado State University, Fort Collins, CO 90095, USA

**Correspondence**: Mark T. Richardson (markr@jpl.nasa.gov)

**Correspondence**: Mark T. Richardson (markr@jpl.nasa.gov)

Received: 03 Jun 2021 – Discussion started: 11 Aug 2021 – Revised: 12 Oct 2021 – Accepted: 15 Nov 2021 – Published: 05 Jan 2022

Upcoming spaceborne imaging spectrometers will retrieve
clear-sky total column water vapour (TCWV) over land at a horizontal resolution of 30–80 m. Here we show how to obtain, from these retrievals,
exponents describing the power-law scaling of sub-kilometre horizontal variability in clear-sky bulk planetary boundary layer (PBL) water vapour (*q*) accounting
for realistic non-vertical sunlight paths. We trace direct solar beam paths
through large eddy simulations (LES) of shallow convective PBLs and show that retrieved 2-D water vapour fields are “smeared” in the direction of the solar azimuth. This changes the horizontal spatial scaling of the field
primarily in that direction, and we address this by calculating exponents
perpendicular to the solar azimuth, that is to say flying “across” the
sunlight path rather than “towards” or “away” from the Sun. Across 23
LES snapshots, at solar zenith angle SZA = 60^{∘} the mean bias in calculated exponent is 38 ± 12 % (95 % range) along the solar azimuth, while
following our strategy it is 3 ± 9 % and no longer significant. Both bias and root-mean-square error decrease with lower SZA. We include
retrieval errors from several sources, including (1) the Earth Surface Mineral Dust Source Investigation (EMIT) instrument noise model, (2) requisite assumptions about the atmospheric thermodynamic profile, and (3) spatially nonuniform aerosol distributions. By only considering the direct
beam, we neglect 3-D radiative effects such as light scattered into the field of view by nearby clouds. However, our proposed technique is necessary
to counteract the direct-path effect of solar geometries and obtain unique
information about sub-kilometre PBL *q* scaling from upcoming spaceborne spectrometer missions.

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© 2022 California Institute of Technology. Government sponsorship acknowledged.

Spatial scaling in the variability of atmospheric properties such as water
vapour (*q*) can be characterised via structure functions, with the *n*th-order structure function of a field *f*(*x*), *S*_{n} defined as

where *E*[] is the expected value, *x* a location, and *r* a separation between points. Fields of temperature (*T*), *q*, and wind speed are commonly well modelled by a power law:

such that *ζ*_{n} is the log–log gradient of *S*_{n} as a function of *r*. There is strong motivation to quantify and understand these exponents and
the ranges Δ*r* within which they are valid, and here we
specify second-order structure functions *S*_{2} with exponent *ζ*_{2}, describing variance scaling. This is related to the
commonly referenced Fourier power spectrum exponent *β*:

One motivation for obtaining these exponents is that climate model sub-grid
variability in *q* is strongly linked to cloud formation
(Golaz et al., 2002; Perraud et al.,
2011; Sommeria and Deardorff, 1977). In principle, sub-grid variance can be tuned for each model set-up, but scale-aware variance relationships allow a smooth and consistent transition between low-resolution (order ∼ hundreds
of kilometres) and high-resolution (order ∼ km) models (Arakawa et al., 2011; Schemann et al., 2013).

At scales larger than model grid cells, observational estimates of variance
scaling can also be used to assess model performance, as has been done using
estimates of temperature (*T*) and *q* from Atmospheric Infrared Sounder (AIRS) and airborne campaign data
(Kahn et al., 2011).

Furthermore, the scaling exponents are related to the physical processes
that generate the cascade of turbulent eddies in the atmosphere. For
example, while mean-scale statistics retrieved by AIRS are approximately
isotropic in the horizontal, there are differences in scaling between the
horizontal and vertical (Pressel and Collins, 2012).
Scaling following ${\mathit{\zeta}}_{\mathrm{2}}=\mathrm{2}/\mathrm{3}$ is predicted for a passive tracer in
the inertial range of 3-D locally isotropic turbulence following Kolmogorov theory. Accounting for the buoyancy effects can
strongly modify that scaling (Bolgiano, 1959; Obukhov,
1959), with different exponents expected for the velocity (${\mathit{\zeta}}_{\mathrm{2}}=\mathrm{6}/\mathrm{5}$–$\mathrm{7}/\mathrm{5}$) and scalars (${\mathit{\zeta}}_{\mathrm{2}}=\mathrm{2}/\mathrm{5}$), as shown in e.g. Kunnen et al. (2008), Wroblewski et al. (2010), or Boffetta et al. (2012). Exponents of $\mathrm{7}/\mathrm{5}$ have been commonly measured
for vertical wind profiles from dropsondes (Lovejoy et
al., 2007), and values typically near *ζ*_{2} of $\mathrm{6}/\mathrm{5}$ for horizontal
water vapour in non-convective areas and *ζ*_{2}=0.72 in convective areas have been determined from airborne lidar retrievals over
horizontal ranges of up to 100 km (Fischer et al.,
2012, 2013); meanwhile, the sub-kilometre regime remains undermeasured.

Of particular interest for modellers is the existence of “scale breaks”, distances at which the exponents change such that a smooth transition between model resolutions may not be possible. These have been calculated to occur at distances over a broad range of 10–1000 km (Bacmeister et al., 1996; Gage and Nastrom, 1985; Kahn et al., 2011; Pinel et al., 2012), and the range of scales has been speculated to be related to the size of convective systems (Dorrestijn et al., 2018) or changes in the nature of turbulence (Kurowski et al., 2015; Skamarock et al., 2014).

This study focusses on the estimation of horizontal scaling of clear-sky
TCWV at sub-kilometre scales and specifically on the integrated PBL water vapour which we refer to as the partial column water vapour (PCWV_{PBL}). Recent work using airborne data (Thompson et al., 2021) and large eddy simulation (LES) output (Richardson et al.,
2021a) has provided evidence that sub-kilometre horizontal variability in total column water vapour (TCWV) is almost perfectly correlated with PCWV_{PBL}
variability, such that high-spatial-resolution retrievals of TCWV from
visible and shortwave infrared (VSWIR) imaging spectrometers can provide
unique information about PBL *q* variability. This is not a statement that all
water vapour is inside the PBL but rather that on sub-kilometre spatial scales, the variability in low-altitude water vapour dominates the horizontal column variability. It remains a challenge to disentangle variability from
different heights within the PBL such as the sub-cloud layer or a
conditionally unstable cloud layer, which may have different scaling
properties.

In particular, the PBL depth is typically 1–2.5 km in these simulations, while we obtain variability statistics over horizontal ranges of under 1 km. The vertical averaging over a scale larger than the horizontal calculation means that the interpretation of the physical meaning of the derived exponents is challenging and may not be directly related to the theoretically derived exponents discussed above. This study aims only to determine whether the measurement problem of the solar path can be overcome and leaves the physical interpretation beyond its scope. This point will be revisited in Sect. 4.

Several modern and upcoming missions obtain or will obtain VSWIR spectra that could allow retrieval of TCWV at horizontal resolutions from 20 to 100 m. Current examples include the Multi-Spectral Imager (MSI) on Sentinel-2 (Drusch et al., 2012), the PRecursore IperSpettrale della Missione Applicativa (PRISMA, Candela et al., 2016), and the DLR Earth Sensing Imaging Spectrometer (DESIS, Krutz et al., 2019). Upcoming missions such as NASA's Earth Surface Mineral Dust Source Investigation (EMIT; Green and Thompson, 2020) and ESA's Copernicus Hyperspectral Imaging Mission for the Environment (CHIME; e.g. Rast et al., 2019) will obtain a horizontal resolution of order 30–80 m, and this study will assess performance assuming footprint sizes of 40–50 m, i.e. at the mid-point of that range.

This footprint selection was made based on the resolution of available LES output, and the analysis method is based on the output of retrievals developed specifically for EMIT, which primarily retrieves TCWV to allow atmospheric correction for its surface reflectance target observable. Other instruments such as PRISMA or MSI may have different error characteristics, but we treat retrieval errors in a general manner that could be expanded to these other instruments.

This study attempts to determine whether EMIT will be able to obtain
*ζ*_{2} over 0.5–1 km after accounting for (1) random retrieval
error, (2) systematic biases in retrieval mean and sensitivity, and (3) solar zenith angle.

The PBL is of particular interest since it is the location where reflective
low clouds form, and in the Dorrestijn et al. (2018) analysis, *ζ*_{2} varied more in the 850 hPa layer than the 300 or 500 hPa layers. However, previous analyses have generally been restricted to far
larger spatial ranges, with Dorrestijn et al. (2018) referring to 55–165 km scale variance as occurring at the “tiny scale”. Other
examples at higher resolution generally use airborne measurements, with few
calculations for separations under 1 km using onboard sensors
(Cho et al., 1999), using lidar at 5–100 km
(Fischer et al., 2013), or evaluating simulations with lidar for separations > 11 km (Selz
et al., 2017).

This study uses LES outputs and does not make any statements about the
realism or cause of the LES output *ζ*_{2} values but instead aims solely to identify and quantify retrieval biases and errors. Its greatest
contribution is to demonstrate that directional calculation strategies can
remove biases in estimates of *ζ*_{2} introduced by the direct-beam
component of the solar path through the atmosphere.

We show that, in a set of 23 LES snapshots, the non-vertical direct-beam
path prevents accurate retrieval of sub-kilometre *ζ*_{2} when standard methods are naïvely applied. However, errors in *ζ*_{2} depend
on the direction in which it is calculated relative to the solar azimuth,
and selecting the correct solar-aware direction eliminates the bias and
therefore overcomes a fundamental barrier to VSWIR estimation of sub-kilometre *q* scaling. Diffuse sunlight is handled through a plane-parallel radiative
transfer approximation, which means that complex 3-D radiative effects are
neglected. In clear-sky areas near clouds, 3-D effects can brighten observed
spectra (Várnai and Marshak, 2009), with induced biases of
order ∼ 0.25 % for VSWIR column CO_{2} retrievals
(Massie et al., 2021). The consequences for hyperspectral TCWV
retrievals at 30–80 m horizontal resolution are not currently known,
although the effect on retrieved *ζ*_{2} would depend on the spatial
scaling of these TCWV biases. In Sect. 4 we propose a field experiment
design to test our conclusions. Spaceborne spectrometers measure light that
has passed along some path, ${\mathit{r}}_{\downarrow \uparrow}$,
consisting of downward (Sun-to-surface, *r*_{↓})
and upward (surface-to-sensor, *r*_{↑})
components. Retrievals will respond to the path-integrated water vapour (PIWV) between the surface and top of atmosphere (TOA):

Meanwhile the commonly desired value is the vertically integrated water vapour, TCWV:

For a horizontally uniform *q* field (equivalent to “plane-parallel” in radiative transfer parlance) there is a simple geometric relationship between the two:

where *μ* is the cosine of the sensor-viewing zenith angle and *μ*_{0} the cosine of the solar zenith angle. Despite the fact that real *q* fields vary horizontally such that Eq. (6) is not strictly true, the VSWIR
community commonly assumes a plane-parallel atmosphere and reports the
retrieved value as being TCWV (Carbajal
Henken et al., 2015; Diedrich et al., 2015; Grossi et al., 2015; Nelson et
al., 2016; Noël et al., 2004; Preusker et al., 2021). Here we calculate
PIWV by tracing solar rays through 3-D LES output, but for consistency with VSWIR literature terminology, we relate it to an effective TCWV_{eff}:

Our retrieved values, TCWV_{ret}, refer to estimates of this property. Figure 1 compares the true TCWV with TCWV_{eff}
derived from ray tracing through 3-D LES output when solar zenith angle SZA is 45^{∘} and solar azimuth is at 0^{∘}, meaning that the horizontal component of *r*_{↓} is in the negative *y* direction. There is
an apparent “smearing” from Fig. 1a to b in the *y* direction, so we refer to these
solar-geometry-induced changes in TCWV_{eff} due to the horizontal variability in the *q* field as the “solar smearing” effect (for a simplified
illustration of the physical principles behind why our strategy is
anticipated to reduce biases in *ζ*_{2}, see Figs. S1–S3 in the Supplement).

This consequence of solar and view geometry is well known; for example, Thompson et al. (2021) only used flight lines with
very low SZA to minimise its effect in a study of *ζ*_{2}. However,
this severely limits viable data, so here we suggest a new method that exploits the directionality of the solar-smearing effect in order to perform
such calculations across a wider range of conditions. This paper uses the
solar-path-traced outputs of Richardson et al. (2021a) to
demonstrate that by calculating the structure function in a direction
perpendicular to the solar azimuth, the bias in *ζ*_{2} is removed
within the 23 LES snapshots considered. To our knowledge, this is the first
quantification of such a technique to mitigate solar-geometry-induced errors
in spatial statistics of retrieved atmospheric properties.

If this result can be extended to the real world, then upcoming high-spatial-resolution spaceborne VSWIR spectrometers will provide a breakthrough for analysis of moisture scaling across unprecedented spatial scales. Our retrievals are restricted to clear-sky areas over land, but this still represents a substantial advance in the current capacities of other instrument types. Lidar measurements avoid the solar-path issue and can retrieve vertical information including above clouds, but current and anticipated spaceborne lidars do not offer VSWIR's fine spatial resolution or broad spatial coverage from a wide swath. Sounders such as infrared can profile the atmosphere but have footprints that are too large for sub-kilometre exploration.

Even airborne measurements, which offer far less coverage than spaceborne sensors, may suffer from their own challenges. Evidence suggests that the tendency of flights to follow isobars rather than maintain altitude can introduce height variations that blend vertical variation into horizontal calculations and result in exponents similar to those predicted due to buoyancy's vertical effect (Lovejoy et al., 2004; Pinel et al., 2012).

The VSWIR sampling technique introduced here could greatly expand the range
of conditions under which spatial scaling of water vapour can be quantified.
In Sect. 2 we describe the LES output, simulated retrievals, and how
retrieval errors and solar path are accounted for in calculation of *ζ*_{2}. Section 3 presents the results and Sect. 4 discusses and
concludes.

## 2.1 Large eddy simulation output

We use *q* and cloud water (*q*_{c}) output from the five LES runs of shallow convection as in Richardson et al. (2021a), with four cloudy cases (ARM, ARM_lsconv, BOMEX, RICO)
and one case in which clouds do not form (DRY). Simulations use two models,
EULAG for the ARM cases (Prusa et al., 2008) and
JPL-UCONN LES for the others (Matheou
and Chung, 2014). Simulation set-ups are described in Richardson et al. (2021a) and the associated references (Brown et al., 2002; Kurowski et al., 2020; Matheou and Chung, 2014; Siebesma et
al., 2003; vanZanten et al., 2011), and while some simulations represent
oceanic boundary layers, we simply assume a land surface for the retrievals. Static reanalysis profiles from MERRA-2
(Gelaro et al., 2017) are appended
above the LES domain, but the LES domains were shown to capture horizontal
variability in *q* from analysis of LES output and airborne lidar profiles over the Pacific (Bedka et al., 2021). The 23 selected snapshots
are labelled by their time stamp; e.g. DRY_7200s represents 2 h into the DRY simulation. The LES output horizontal resolution
ranges from Δ*x*=20 to 50 m; here we degrade the Δ*x*=20 m cases to Δ*x*=40 m resolution to make the spatial difference
statistics more consistent.

## 2.2 Generating retrieved TCWV fields

### 2.2.1 Emulator development

The methodology here applies the results of Richardson et al. (2021a), which developed a TCWV retrieval emulator to rapidly generate TCWV_{ret} fields given TCWV_{eff} derived from 3-D LES *q*
fields. This emulator was developed from results of an observing system
simulation experiment (OSSE). Our differential optical absorption
spectroscopy (DOAS) retrieval requires an accurate representation of water
vapour spectroscopy, so we selected the MODTRAN6.0 radiative transfer model
for forward and inverse calculations (Berk et al.,
2014, 2015). The retrievals were performed using the Imaging Spectrometer
Optimal Fitting (ISOFIT) code (Thompson et
al., 2018, 2019) with EMIT instrument characteristics and noise. ISOFIT
simultaneously retrieves the surface reflectance spectrum (*ρ*_{s}(*λ*)), aerosol optical depth (AOD), and TCWV_{ret} from radiance over *λ*=380–2500 nm. The retrieval includes a lookup table (LUT) that relates TCWV and AOD to radiance properties, and this LUT is
generated by radiative transfer using uniformly scaled *q*(*z*) and aerosol
extinction (*β*_{ext}(*z*)) profiles to match desired TCWV_{eff} and
AOD.

Some studies performed radiative transfer over complete LES fields, but we found that full-field simulations were too computationally expensive given our toolkit and requirements. For example, Gristey et al. (2019) performed 3-D radiative transfer simulations over full LES fields, but they were interested in broadband fluxes and so could use lower spectral resolution (370 wavelengths versus > 20 000 here). Furthermore, their LES output was smaller (average of ∼ 60 000 footprints versus ∼ 500 000 here), and this work must consider numerous combinations of properties such as surface type and solar zenith angle.

To reduce computational expense, the Richardson et al. (2021a) OSSE selected 101 footprints from each LES snapshot and used their vertical
profiles as MODTRAN6.0 input to generate the true forward radiance spectra.
MODTRAN6.0 is a plane-parallel radiative transfer model which must assume a
horizontally uniform atmosphere but accounts for solar and view geometry, so
this step applies Eq. (6). From the 101 pairs of TCWV and TCWV_{ret} for a given combination of surface type and SZA, a linear relationship between TCWV_{eff} (in this case, TCWV_{eff} = TCWV) and TCWV_{ret} was found:

where *a*_{1} and *a*_{2} are the slope and intercept and *ε* a random sample from a normal distribution whose standard deviation *σ*_{ε} quantifies the random retrieval error. The parameter
*a*_{1} represents the sensitivity ${\mathrm{dTCWV}}_{\mathrm{ret}}/{\mathrm{dTCWV}}_{\mathrm{eff}}$ and *a*_{2} is related to the bias, although it is only equal to the bias if *a*_{1} = 1.
Fits at different time steps from each LES simulation were not significantly different from each other, but parameters did differ between simulations. To
account for this, emulators were generated separately for each LES
simulation, but footprints from all snapshots in each simulation were combined, resulting in a sample size of 303–707 to estimate *a*_{1},
*a*_{2}, and *σ*_{ε} in each case (parameter estimates stabilised around *N*=50).

Figure 2a shows the linear relationship from all ARM snapshots, while Fig. 2b–e show how the parameters may vary with surface type, SZA, retrieval-assumed atmospheric profile shapes, and AOD.

Figure 2b shows that *a*_{2} varies between a
vegetation surface and a mineral surface. Surface type can vary greatly on sub-kilometre scales, and within-scene transitions between vegetation and mineral surfaces
would introduce artificial variance in TCWV_{ret} differences either side
of transition boundaries and potentially affect the derived *S*_{2} and *ζ*_{2}. Variations in *a*_{2} are small when considering mixtures of
vegetation or mixtures of urban-mineral surfaces, so we limit our structure-function analysis to mixed-vegetation or urban-mineral surfaces.

From Fig. 2c, TCWV_{ret} is only weakly sensitive to SZA from 14 to 60^{∘}, the derived parameters do not differ significantly, and for SZA from 14 to 45^{∘} TCWV_{ret} are extremely similar. From this we argue that we can use a single set of derived parameters for each LES case for SZA up to 60^{∘}.

Next, we note that absorption line broadening depends on thermodynamics and so argue that the structure of *q*(*z*) can change *a*_{1} and *a*_{2}. For example,
changes in *q* at a lower, warmer level will result in stronger changes in
absorption at the edges of bands compared with changes in *q* at a higher,
cooler level. Figure 2d presents results from one
test that support this argument, when the atmosphere used in the retrieval's
LUT is changed from mid-latitude summer to tropical and the parameters change. Similarly, substantial changes in the forward model *q*(*z*) profiles change the derived parameters (not shown). We note that Fig. 2a contains results from different LES time steps in which the atmospheric profiles evolved, but this evolution was too small to generate any
detectable differences in the Eq. (8) parameters.

Finally, we considered the relationship between AOD and the parameters in
Eq. (8). In all other panels AOD varied randomly from 0.1 to 0.2 between footprints, but in Fig. 2e the simulations are
performed with AOD fixed across all footprints at either 0.05, 0.20, or 0.35. Changing AOD by 0.3 results in a change in *a*_{2} equivalent to
approximately 0.3 % of TCWV_{eff}. This is a smaller change than
between common surface types (Fig. 2b), and AOD
varies far more smoothly at sub-kilometre horizontal scales than surface type, so we anticipated that our results would be robust to some types of spatial
aerosol variability. In Sect. 3.2 we show how vertical gradients of up to
0.3 AOD km^{−1} can have a minor effect on derived
exponents.

### 2.2.2 Generating retrieved fields

For SZA from 0 to 60^{∘} in increments of 15^{∘}, the PIWV was calculated by ray tracing from the top of the atmosphere to centre of each surface grid cell and then directly up with a solar azimuth of 0^{∘}. This geometry represents a nadir instrument view angle, and PIWV is then
converted to TCWV_{eff} via Eq. (7), which accounts for SZA. This
TCWV_{eff} is then used as input for Eq. (8), thereby generating
TCWV_{ret} fields for analysis.

As mentioned in Sect. 2.2.1, each separate LES case has a unique set of
Eq. (8) parameters, and the fit is generated from plane-parallel radiative transfer calculations. This means that horizontal variability in *q* is not
explicitly accounted for, but we argue that the derivation includes a range
of *q*(*z*) profiles, and there is no fundamental reason why $q\left({\mathit{r}}_{\downarrow \uparrow}\right)$ should introduce fundamental
differences that should affect our Eq. (8) relationship. Therefore, we
assume that the relationship between TCWV_{ret} and PIWV will be captured by Eq. (4).

The same ray-traced calculation is repeated with cloud water *q*_{c} to
obtain cloud water path (CWP). Footprints are then flagged as cloudy or
shaded when CWP > 1 × 10^{−3} mm for any of the selected
SZAs so as to avoid changes in *ζ*_{2} due to changes in the
footprints considered. This CWP is equivalent to approximately *τ* > 0.3 in a typical sub-adiabatic cloud (Szczodrak et al., 2001).

## 2.3 Calculation of spatial statistics and removal of random error

*S*_{2}(*r*) is calculated from the LES field of TCWV_{ret} in one horizontal
direction at a time by including all pairs of footprints separated by *r* in that direction provided that neither of the footprints in the pair is
flagged as cloudy or shadowed for any SZA. To calculate along the LES *x* axis, each different *y* location is effectively treated as a 1-D field and its set of $f(x+r)-f\left(x\right)$ values is calculated, and then all the 1-D subsets are concatenated and
the reported *S*_{2}(*r*) is the expectation value of this combined dataset.

The horizontal directions are either along the *x* axis (“perpendicular” to
the sunlight) or along the *y* axis (the “parallel” case).
Figure 3 displays example *S*_{2}(*r*) for clear-sky
TCWV and their associated scaling parameters, i.e. $\mathrm{d}\mathrm{ln}\left({S}_{\mathrm{2}}\right)/\mathrm{d}\mathrm{ln}\left(r\right)$, showing that *ζ*_{2} will depend on the range of *r* over which it is calculated. The variation of *ζ*_{2} with Δ*r* can be due to changes in physical processes in addition to
imperfect process representation in the LES, such as non-physical dissipation at separations smaller than several grid cells
(Brown et al., 2002). This study is
not concerned with the interpretation of *ζ*_{2} but rather with the
accuracy with which it can be obtained, so we do not explore this further
and follow Thompson et al. (2021) in calculating
the fit over *r*=0.5–1 km.

We estimate and remove the measurement noise following the method of
Richardson et al. (2021a). They derived a
structure-function-based method to estimate *σ*_{ε} from
the TCWV_{ret} field, thereby allowing better estimation of *S*_{2} by subtracting $\mathrm{2}{\mathit{\sigma}}_{\mathit{\u03f5}}^{\mathrm{2}}$ from Eq. (5). This method involves
calculating *S*_{2} in one horizontal direction at *r*=Δ*x* (i.e.
separation of one grid cell, 40 or 50 m) and then smoothing the field by a factor of 2 in the perpendicular direction and recalculating *S*_{2}, which we label ${S}_{\mathrm{2},\times \mathrm{2}}$. At these small separations, $\mathrm{2}{\mathit{\sigma}}_{\mathit{\u03f5}}^{\mathrm{2}}\gg {a}_{\mathrm{1}}^{\mathrm{2}}{S}_{\mathrm{2}}\left(r\right)$, such that ${S}_{\mathrm{2},\times \mathrm{2}}-{S}_{\mathrm{2}}\approx \mathrm{2}{\mathit{\sigma}}_{\mathit{\u03f5}}^{\mathrm{2}}$.

We evaluate the sensitivity of our derived *ζ*_{2} to uncertainty in
the retrieval emulator; when calculating *S*_{2} from the retrieved fields, the emulator changes the retrieved value *S*_{2,ret} to

and the estimated *ζ*_{2,ret} is

This shows that both biases in *a*_{1} (“retrieval sensitivity”) and the
magnitude of random errors can change the derived *ζ*_{2}. Richardson
et al. (2021a) showed that errors in *a*_{1} were the largest
source of uncertainty in estimating the spatial standard deviation. We will therefore perform sensitivity tests for a range of *a*_{1} values and
show only one *σ*_{ε} case since results were insensitive
to changes in *σ*_{ε} of up to a factor of 4 scaling.

The final question we address is whether the scaling exponent estimated at
the very high spatial resolution of missions such as EMIT will be
fundamentally different from that obtained by current sensors such as MERIS
and MODIS that can provide TCWV at a nominal resolution near Δ*x*≈250 m. Large-scale *σ*_{x} narrows as spatial resolution
coarsens, but it is not clear how this affects the spatial scaling
properties considered here, so for this test we sequentially degrade a TCWV
field from Δ*x*=50 m to Δ*x*=250 m and calculate *S*_{2} and
*ζ*_{2} at each resolution.

## 3.1 Variation of horizontal *ζ*_{2} with height

Figure 4 shows how the *ζ*_{2} calculated for
PCWV integrated up to different heights varies within the PBL but that the value becomes fixed by the PBL top. That is, estimates made from TCWV refer
to the value for PCWV_{PBL}, but this conceals vertical structure within the PBL. We can therefore confirm that our derived values are indeed
representative of *ζ*_{2} derived from PCWV_{PBL}, but further work is needed to determine the precise utility of statistics of PCWV_{PBL} and, furthermore, we note that corresponding estimates of PBL height from other
sources may be necessary to help interpret measurements of PCWV_{PBL} scaling.

## 3.2 Retrieval errors and *ζ*_{2}

The results from all sensitivity tests applied to the clear-sky TCWV in the
ARM_18000s snapshot are shown in Fig. 5; results are similar for other snapshots (not shown). Figure 3a shows that random errors that we estimate will be typical for EMIT and result in substantial changes to calculated
$\mathrm{ln}\left({S}_{\mathrm{2}}\right)/\mathrm{ln}\left(r\right)$, with a notable flattening over much of the *r* range and an unacceptably large error in *ζ*_{2} derived from retrieved TCWV. Our
error correction reduces the *ζ*_{2} error from 53.1 % to 1.4 %.
Figure 5b shows that errors in the sensitivity
${\mathrm{dTCWV}}_{\mathrm{ret}}/{\mathrm{dTCWV}}_{\mathrm{eff}}$, which is the emulator parameter *a*_{1} in Eq. (8), shift *S*_{2} but do not substantially affect the gradient over *r*=0.5–1 km, and this conclusion also applies in Fig. 5c when biases in *a*_{1} are combined with
*σ*_{ε} and the error correction is applied. These
results show that even large biases in *a*_{1}, which contribute
proportionally to estimates of errors in *σ*_{x}, do not have a
substantial effect on derived scaling properties.

Figure 5d confirms that scaling properties are
sensitive to the measurement resolution, with derived exponents varying
unpredictably from 0.61 to 0.73 with resolution. For Δ*x*=250 m there are only five points included in the regression. Furthermore, more footprints
will be partially cloudy, meaning that the samples included will be too
small for robust estimation of *ζ*_{2}. We did not apply a matched
cloud mask for this test and expect that this will lead to more uncertain estimates of *ζ*_{2}. Tests across all 23 snapshots show that most
return a higher, and more uncertain, *ζ*_{2} when spatial resolution
is degraded (not shown). This points to new information being obtained from
finer-spatial-resolution retrievals provided that the slanted solar paths do not destroy the correspondence between true and retrieved *ζ*_{2}.

The final panel, Fig. 5e, shows a sensitivity test to strong spatial gradients in AOD. From Fig. 2e, a change of 0.3 in AOD results in a difference of 0.3 % in
TCWV_{ret} in ARM_18000s, and other tests found a smaller
fractional response in DRY_7200s (not shown). We picked the
ARM_18000s results as the worst case and allowed TCWV_{ret}
to vary sinusoidally in the *x* direction by ±0.15 % with a
wavelength of 2 km. This approximates the effect of a change of approximately 0.3 AOD every 1 km. This relatively extreme AOD gradient
changes *ζ*_{2} by < 4 % in all cases except DRY (not
shown). In the DRY LES run our aerosol gradients induce a factor of 2 change
in time step DRY_7200s and 10 % in other time steps. This larger relative error may be related to their small spatial variability of
TCWV and low values of *ζ*_{2}.

This test represents a very large horizontal change in AOD, and since ISOFIT
simultaneously retrieves AOD, such cases could be identified. Overall, we conclude that for scenes where AOD is of order 0.35 or less and horizontal
gradients are smaller than 0.3 AOD km^{−1}, our results will generally be
robust to typical aerosol variability. In practice, we would recommend
testing each case using coincidentally retrieved AOD fields along with the
estimated sensitivity of TCWV_{ret} to AOD to calculate a likely effect of
special AOD variability. This may identify cases where aerosol variability
affects derived *ζ*_{2}.

## 3.3 Solar zenith angle, calculation direction, and *ζ*_{2}

In our final tests we compare *ζ*_{2} calculated on the
TCWV_{ret} fields with SZA changed from 15 to 60^{∘} as a function of the “true” value obtained from the TCWV field. The TCWV_{ret} values are those using each simulation's emulator parameters and include the random
error correction, while the true values refer to the columns directly over
each footprint with no retrieval error. All use Δ*x*=40–50 m with
fits calculated over *r*=500–1000 m, and calculations are either along the
*y* axis and parallel to the solar azimuth or along the *x* axis and perpendicular to the solar azimuth.

Figure 6 shows that there is substantial spread
introduced for realistic SZA, with a strong tendency for bias in *ζ*_{2} to increase with SZA when calculated parallel to the solar azimuth.
This spread and apparent bias are greatly reduced when the calculation is performed perpendicularly to the solar azimuth. In the parallel case there are
still significant differences (*p* < 0.05) between true and retrieved
*ζ*_{2}. The *p*=0.05 value threshold is calculated as 1.96 times the
standard error of the difference in best-fit parameters derived from the
ln (*S*_{2}) as a function of ln (*r*) gradient.

Figure 7 shows that both bias and error range expand
with SZA when calculating in the solar azimuth direction, but the median
(and mean, not shown) bias is eliminated by calculating *S*_{2} in the
direction perpendicular to the solar azimuth. The spread is also wider in
the parallel case, the 5 %–95 % range of differences relative to the
truth is −0.04–0.49 versus −0.20–0.22 when calculating perpendicular,
while the standard deviation is 0.18 for parallel versus 0.13 for
perpendicular.

These results show that realistic SZAs result in path-integrated water vapour structures that have somewhat different characteristics at a 0.5–1 km
scale than that of the PCWV_{PBL} over each footprint. This results in additional error to estimated *ζ*_{2}, but appropriate calculation
strategies that account for solar azimuth angle can reduce the magnitude of
this error and suppress systematic error.

We have shown that for the *q* fields simulated in five LES runs of shallow
convective PBLs, a novel strategy accounting for solar azimuth eliminates
the SZA-induced bias in calculated *ζ*_{2} over *r* of 0.5–1 km from
high-spatial-resolution VSWIR retrievals. This substantially increases the
range of applicable geometries for which VSWIR retrievals can be
used to estimate spatial scaling statistics. For example, in the airborne
case studies of Thompson et al. (2021), only flight
lines with SZA < 15^{∘} could be used, while our method
promises unbiased estimates of sub-kilometre *ζ*_{2} for SZA up to 60^{∘}. Removal of the bias is particularly important for
applications, since *q*-scaling analyses using spaceborne data typically group or average over many sets of measurements (e.g. Kahn and Teixeira, 2009), and this averaging will not reduce
bias in the way that it reduces RMSE. For the first time it also allows
calculation of high-resolution *ζ*_{2} statistics over mid-latitude and polar areas of the globe that do not experience low solar zenith angles.

This approach should be applicable to any instrument that obtains TCWV from
VSWIR with a horizontal resolution approaching 50 m, not just EMIT. Operationally, this requires sufficiently long, continuous sampling
perpendicular to the solar azimuth. We have not analysed length requirements
here but note that for airborne campaigns this is easily addressed on a flight-to-flight basis. For spaceborne instruments, the sampling will depend
on the swath size and orbital geometry. EMIT's approximately 75 km swath
(Bradley et al., 2020) spans distances larger than
those considered here, and so we expect sampling to be sufficient regardless
of orbital configuration. Similarly, ESA's CHIME contractor notes a 128 km
swath, and similar capacities are expected for the missions that address
NASA's Surface Biology and Geology (SBG) and Aerosols, Clouds, Convection
and Precipitation (ACCP) designated observables. However, we note that it
will be necessary to understand the implications of non-uniform footprint
size- and footprint-dependent errors determined from the on-orbit instrument performance in order to increase the confidence in estimates of *ζ*_{2} from these spaceborne sensors.

Our analysis is based on the retrieval outputs of Richardson et al. (2021a), which showed how random retrieval error *σ*_{ε} can be identified and removed from TCWV_{ret} fields by
exploiting the properties of TCWV *S*_{2} at small scales. It also showed
that errors in the *T* and *q* profiles assumed in the retrieval could affect
retrieval sensitivity ${a}_{\mathrm{1}}={\mathrm{dTCWV}}_{\mathrm{ret}}/{\mathrm{dTCWV}}_{\mathrm{eff}}$. We have shown
here that to obtain *ζ*_{2} it is critical to remove the effect of *σ*_{ε} but that while errors in *a*_{1} are the largest error source for estimates of spatial standard deviation, they do not
greatly affect the derived *ζ*_{2}.

Our results were only determined for areas where the surface type is
composed either of mixed vegetation or mixed urban-mineral surfaces, so
either a simultaneous surface classification (as provided by ISOFIT) or
ancillary surface information would be required. We also identified that
horizontal gradients of 0.3 AOD km^{−1} or less have only a small effect on derived *ζ*_{2} in most cases, and since ISOFIT simultaneously
retrieves AOD, it seems likely that cases where this is not true could be identified. However, we highlight further investigation of the effect of
realistic AOD structure as a useful future investigation. A major limitation
of this study is that 1-D radiative transfer was used to derive the
relationship between TCWV and TCWV_{ret}, while clouds can affect nearby clear-sky scenes through 3-D radiative effects. Future work could address this by using a 3-D radiative transfer code (e.g. Evans, 1998; Emde et al., 2011) as the forward
model in an improved OSSE.

Ideally this sampling strategy could be field tested, and a strategy to do
so would be to perform flights both parallel and perpendicular to the solar
azimuth with collocated retrievals of TCWV from VSWIR and an independent
instrument that is not affected by SZA, such as a differential absorption
lidar (DIAL) or passive sounding instrument. The non-VSWIR instrument would
allow calculation of *ζ*_{2} that is not affected by sunlight path,
and the conclusions of this study would be supported if the VSWIR and
non-VSWIR estimates showed good agreement for the perpendicular but not
parallel flight paths. We highlight the High Altitude Lidar Observatory
(Bedka et al., 2021) as a candidate sensor for such an
experiment.

For science applications, it would also be necessary to identify which
scenes are likely to satisfy our requirements: for example, conditions proximate to deep convection may have more substantial above-PBL variability
at < 1 km scales and result in weaker correspondence between TCWV
and PCWV_{PBL}. In addition, it may be necessary to identify PBL height, which would require independent information from weather forecast models or reanalysis, in situ measurements, or other instruments. Furthermore, users would
also need to consider the effect of sampling biases that may depend on cloud
fraction or on the presence of non-isotropic variability in features such as
horizontal convective rolls (e.g. as explored in Carbajal Henken et al., 2015). If locations commonly experience meteorological
features with a preferential orientation, for example due to orography or a
coastline, then this method may not adequately capture its full structure.

Finally, the interpretation of these exponents has not been considered in detail here, but future work could proceed either observationally or theoretically. For an observational study, the relationship between retrieved exponents and other properties, such as later convective initiation, could be investigated. A theoretical study would need to apply current understanding of turbulent physics to address precisely the problem of the horizontal variability of vertically averaged profiles. Further in the future, perhaps multi-angle imaging spectroscopy could provide profiling from VSWIR measurements via computed tomography, which has been demonstrated to retrieve aerosol profiles in some conditions using the Multi-Angle Imaging SpectroRadiometer (MISR) on Terra (Garay et al., 2016), and upper-tropospheric water vapour profiles from airborne measurements with the Gimballed Limb Observer for Radiance Imaging of the Atmosphere (GLORIA, Ungermann et al., 2015). We are not aware of any likely short-term space missions that would allow water vapour tomography from VSWIR retrievals at EMIT-like horizontal resolution, with the upcoming Multiangle Imager for Aerosols (MAIA, Diner et al., 2018) having a horizontal resolution similar to that of MODIS. If such an instrument were launched, high-spatial-resolution tomography would also be subject to the smearing effect from the non-vertical solar path and so may benefit from our proposal, although a detailed study of the particular instrument and retrieval set-up would be required.

Our main conclusion is that a novel sampling strategy can allow a breakthrough in space-based measurement of PBL water vapour scaling and that while individual uncertainties and geographical sampling may vary with the mission, this principle will apply to the array of upcoming VSWIR instruments whose spectra will allow column water vapour retrievals at resolutions from 30 to 80 m or better. The resulting statistics offer a new check on the validity of high-resolution atmospheric models and inform sub-grid parameterisations of coarser ESMs.

The MODTRAN radiative transfer code is available at http://modtran.spectral.com/, licence required, last access: 19 March 2020, Berk et al., 2014, 2015) and requires a licence. The LES profiles used in the emulator development and the 2-D LES output fields used in this analysis are available at https://doi.org/10.5281/zenodo.5717263 (Richardson et al., 2021b).

The supplement related to this article is available online at: https://doi.org/10.5194/amt-15-117-2022-supplement.

MTR processed the data and generated the figures. DRT developed the retrieval code (Isofit) used to generate the retrieval emulators. MJK generated and provided the large eddy simulation output. MDL initialised the project and worked with MTR on analysis design and data interpretation. All the authors contributed to the conceptualisation, writing, and editing.

The contact author has declared that neither they nor their co-authors have any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, USA, under contract with the National Aeronautics and Space Administration. Mark T. Richardson thanks Amin Nehrir and Brian Carroll of NASA Langley for helpful discussion regarding HALO lidar data.

This research has been supported by the National Aeronautics and Space Administration, Jet Propulsion Laboratory (grant no. 80NM0018F0631).

This paper was edited by Joanna Joiner and reviewed by Christoph Kiemle, Alexander Marshak, and one anonymous referee.

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smears2-D maps of imaging spectroscopy vapour retrievals. In simulations we show how this smearing is

towardsor

away fromthe Sun, so calculating

acrossthe solar direction allows sub-kilometre information about water vapour's spatial scaling to be calculated. This could be tested by airborne campaigns and used to obtain new information from upcoming spaceborne data products.