Alexandre Castagna,^{1,}^{*}
B. Carol Johnson,^{2}
Kenneth Voss,^{3}
Heidi M. Dierssen,^{4}
Heather Patrick,^{2}
Thomas A. Germer,^{2}
Koen Sabbe,^{1}
and Wim Vyverman^{1}

Alexandre Castagna, B. Carol Johnson, Kenneth Voss, Heidi M. Dierssen, Heather Patrick, Thomas A. Germer, Koen Sabbe, and Wim Vyverman, "Uncertainty in global downwelling plane irradiance estimates from sintered polytetrafluoroethylene plaque radiance measurements," Appl. Opt. 58, 4497-4511 (2019)

Global downwelling plane irradiance is a necessary variable to
normalize water-leaving radiance measurements, reducing the magnitude
and spectral variabilities introduced by the incident light field. As
a result, the normalized measurements, known as remote sensing
reflectance, have higher correlation with the inherent optical
properties of the water body and so to the composition of optically
active water components. For in situ measurements,
the global downwelling plane irradiance can be estimated from the
exitant radiance of sintered polytetrafluoroethylene plaques or other
diffuse reflectance standards. This allows use of a single
spectrometer to measure all necessary variables to estimate the remote
sensing reflectance, reducing cost in acquisition and maintenance of
instrumentation. However, despite being in use for more than 30 years,
the uncertainty associated with the method has been only partially
evaluated. In this study, we use a suite of sky radiance distributions
for 24 atmospheres and nine solar zenith angles in combination with
full bidirectional reflectance distribution function determinations of
white and gray plaques to evaluate the uncertainties. The isolated and
interactive effects of bidirectional reflectance distribution,
shadowing, and tilt error sources are evaluated. We find that under
the best-performing geometries of each plaque, and with appropriate
estimation functions, average standard uncertainty ranges from 1% to
6.5%. The simulated errors are found to explain both previous
empirical uncertainty estimates and new data collected during this
study. Those errors are of the same magnitude as uncertainties of
plane irradiance sensors (e.g., cosine collectors) and overlap with
uncertainty requirements for different uses of in
situ data, which supports the continued use of the plaque
method in hydrologic optics research and monitoring. Recommendations
are provided to improve the quality of measurements and assure that
uncertainties will be in the range of those calculated here.

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Code and datset of uncertainty simulations for downwelling
plane irradiance from plaque measurements as normalization factor for
remote sensing reflectance, as used in hydrologic optics field
spectroscopy.

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Mean Absolute Percentage Errors (%) in the Visible Range Due to
BRDF Effects for Two Conversion Schemes from
${\mathsf{L}}_{\mathsf{p}}$ to ${\mathsf{E}}_{\mathsf{g}}$ Represented by Eqs. (6) and (7)^{
a
}

Smallest errors for each sky condition and plaque are
formatted in bold. Numbers in parenthesis are the standard
deviations. Statistics calculated over all simulations
included in each evaluation and over all visible
wavelengths.
Clear skies for $20\xb0\le {\theta}_{\mathrm{s}}\le 60\xb0$.
Standard deviation is zero, since both relative sky
radiance distribution for overcast conditions and the
normalized BRDF of gray plaque are independent of
wavelength.

Table 2.

Mean Absolute Percentage Errors (%) in the Visible Range Due to
Shadow Effects and Combined Shadow and BRDF Effects^{
a
}

Conversion from ${L}_{\mathrm{p}}$ to ${E}_{\mathrm{g}}$ is provided with
Eqs. (6) and (7). Smallest errors for each sky condition and
plaque are formatted in bold. Numbers in parenthesis are
the standard deviations. Statistics calculated over all
simulations included in each evaluation and over all
visible wavelengths.
Clear skies for $20\xb0\le {\theta}_{\mathrm{s}}\le 60\xb0$.
Standard deviation of shadow effects for OC is zero, since
the relative sky radiance distribution for OC is
independent of wavelength. For the combined effects of
BRDF and shadow for the gray plaque under OC, standard
deviation is zero because the normalized BRDF of gray
plaque is assumed to be independent of wavelength in this
study.

Table 3.

Mean Absolute Percentage Errors (%) in the Visible Range Due to
Tilt of the Plaque and Due to Combined Effects (BRDF, Shadow,
and Tilt) for Clear Skies and for Overcast Condition^{
a
}

Statistics calculated from level surface (0° tilt)
to maximum zenith tilt angle as indicated in the table,
including all azimuths. Smallest errors for each sky
condition and plaque are formatted in bold. Numbers in
parenthesis are the standard deviations. Statistics
calculated over all simulations included in each
evaluation and over all visible wavelengths.
Clear skies for $20\xb0\le {\theta}_{\mathrm{s}}\le 60\xb0$.

Tables (3)

Table 1.

Mean Absolute Percentage Errors (%) in the Visible Range Due to
BRDF Effects for Two Conversion Schemes from
${\mathsf{L}}_{\mathsf{p}}$ to ${\mathsf{E}}_{\mathsf{g}}$ Represented by Eqs. (6) and (7)^{
a
}

Smallest errors for each sky condition and plaque are
formatted in bold. Numbers in parenthesis are the standard
deviations. Statistics calculated over all simulations
included in each evaluation and over all visible
wavelengths.
Clear skies for $20\xb0\le {\theta}_{\mathrm{s}}\le 60\xb0$.
Standard deviation is zero, since both relative sky
radiance distribution for overcast conditions and the
normalized BRDF of gray plaque are independent of
wavelength.

Table 2.

Mean Absolute Percentage Errors (%) in the Visible Range Due to
Shadow Effects and Combined Shadow and BRDF Effects^{
a
}

Conversion from ${L}_{\mathrm{p}}$ to ${E}_{\mathrm{g}}$ is provided with
Eqs. (6) and (7). Smallest errors for each sky condition and
plaque are formatted in bold. Numbers in parenthesis are
the standard deviations. Statistics calculated over all
simulations included in each evaluation and over all
visible wavelengths.
Clear skies for $20\xb0\le {\theta}_{\mathrm{s}}\le 60\xb0$.
Standard deviation of shadow effects for OC is zero, since
the relative sky radiance distribution for OC is
independent of wavelength. For the combined effects of
BRDF and shadow for the gray plaque under OC, standard
deviation is zero because the normalized BRDF of gray
plaque is assumed to be independent of wavelength in this
study.

Table 3.

Mean Absolute Percentage Errors (%) in the Visible Range Due to
Tilt of the Plaque and Due to Combined Effects (BRDF, Shadow,
and Tilt) for Clear Skies and for Overcast Condition^{
a
}

Statistics calculated from level surface (0° tilt)
to maximum zenith tilt angle as indicated in the table,
including all azimuths. Smallest errors for each sky
condition and plaque are formatted in bold. Numbers in
parenthesis are the standard deviations. Statistics
calculated over all simulations included in each
evaluation and over all visible wavelengths.
Clear skies for $20\xb0\le {\theta}_{\mathrm{s}}\le 60\xb0$.