To recover the actual responsivity for the Ultraviolet Multi-Filter Rotating
Shadowband Radiometer (UV-MFRSR), the complex (e.g., unstable, noisy, and
with gaps) time series of its in situ calibration factors (

While many instruments generate relatively stable data time series over
short time windows, dynamic uncertainty levels, variable sampling densities,
and/or different lengths of gaps with missing data can complicate the
analysis of long-term datasets. For example, the 5-year time series of a
solar variability indicator (Mg II core to wing index) shows consistency on
the order of days but increasing noise level and gaps are observed at the
month scale (Cebula et al., 1992). The time series of the
geopotential scale factor, a function of the geoidal potential, is also
relatively stable on shorter timescales but demonstrates a slowly
increasing long-term pattern (Burša et al., 1997). Additionally,
the time series of a ratio (

Long-term measurements of irradiance by Multi-Filter Rotating Shadowband Radiometers (MFRSRs) are also subject to errors imposed by the factors mentioned above. The MFRSR measures direct normal, diffuse horizontal, and total horizontal irradiances at seven visible channels with a roughly 10 nm full half maximum width (FHMW) (Harrison and Michalsky, 1994). The Ultraviolet (UV) version of MFRSR measures the same three irradiance components at seven UV channels (i.e., 300, 305, 311, 317, 325, 332, and 368 nm) with a 2 nm FHMW (Gao et al., 2010). Currently, the US Department of Energy (DOE) Atmospheric Radiation Measurement (ARM) Climate Research Facility (Mather and Voyles, 2013), the NOAA Surface Radiation (SURFRAD) (Augustine et al., 2005), and the US Department of Agriculture (USDA) UV-B Monitoring and Research Program (UVMRP) (Gao et al., 2010) maintain their own MFRSR and/or UV-MFRSR at multiple sites across the US. To capture immediate instrument responsivity variation, the UVMRP performs in situ calibrations using the Langley method (Slusser et al., 2000; Harrison and Michalsky, 1994) or derived approaches (e.g., Chen et al., 2013, 2015, 2016) on (UV-)MFRSR direct beam measurements on days with extended clear-sky periods (Gao et al., 2010).

Many factors contribute to the error or uncertainty of the Langley method, including variations in aerosol and/or other atmospheric constituents over the course of the calibration period (Augustine et al., 2003; Chen et al., 2015; Zhang et al., 2016), the presence of thin cirrus (Shaw, 1976), and instrument errors (e.g., instrument tilt and misalignment, incorrect nighttime offset and angular corrections) (Alexandrov et al., 2007). Thus, the sequence of original UVMRP (UV-)MFRSR in situ calibration factors exhibits certain levels of noise. Among these uncertainties, variable AOD is considered the major contributor to the variability in the Langley calibration factors obtained in typical atmospheric conditions over the continental United States (Alexandrov et al., 2008), even with careful cloud screening (e.g., Chen et al., 2014; Alexandrov et al., 2004). In addition, extended cloudy periods and low solar zenith angles during winter months further reduce the sequence quality and appear as large time gaps in the datasets. Since the in situ calibration factor represents the instrument's responsivity, which is assumed to be relatively stable, it has been suggested that one applies some smoothing methods (e.g., averaging or fitting a smooth curve) to the daily calibration time series (Alexandrov et al., 2008) to reduce the issue. Currently, UVMRP implements an outlier detection and moving smoothing technique to overcome these issues. However, the process involves manual interaction, performs unreliably during sparse and gapped periods, and lacks the uncertainty estimation.

Analyses of complex long-term time series, such as those of (UV-)MFRSR

For problem (ii), the input error statistics (e.g., input uncertainty) is often assumed to be known or roughly estimated in advance. In practice, a typical approach may use some predetermined constant (e.g., the nominal uncertainty of an instrument, or the standard deviation of its observation) to estimate input uncertainty for the entire dataset. However, this kind of approach omits the information of the possible time-varying observation error, leading to over- or underestimation of the input uncertainty at a given (temporal) location (Chandorkar et al., 2017). A sophisticated approach may treat the dynamic input uncertainty as additional parameters and solve them together with other model parameters through optimization under the Bayesian framework (Kavetski et al., 2006a, b). However, this method requires the specification of valid error and uncertainty models, which are normally poorly understood in practice (Kavetski et al., 2006a, b).

In this study, we developed and validated a generic solution that combines GP regression with a new dynamic input uncertainty estimation method to determine the underlying continuous trend and the corresponding uncertainty for the given time series. In Sect. 2, we briefly summarize the basics of the GP regression and develop the dynamic input uncertainty estimation method. We also describe a complex (noisy, gapped, etc.) synthetic time series and real UV-MFRSR in situ calibration factor time series used in the analysis. In Sect. 3, we present and discuss the performance of the GP method on the test data, in comparison with the UVMRP current operational method and a moving average (MA) technique. Validation of the calibration factors determined with the GP method via the comparison of AODs calculated with these factors and those reported by the AErosol RObotic NETwork (AERONET) (Holben et al., 1998) is also discussed in Sect. 3.

A GP is a technique used in the analysis of a finite number of
random variables with a joint Gaussian distribution (Rasmussen and
Williams, 2006). The following briefly introduces the theory of GP
regression. An observed dataset,

Based on the (optimized) joint distribution Eq. (1), the theorem that
derives the conditional distribution from the joint Gaussian distribution
(Eaton, 1983), and the inversion equations of a partitioned matrix
(Press, 1992), the GP regression predicts

As mentioned before, the statistical properties of the noise

The flowchart of the proposed dynamic input uncertainty estimation method and the complete GP procedure of estimating the mean and confidence interval functions of a given time series is presented in Fig. 1.

Main procedure for deriving the mean and confidence interval
functions using Gaussian process regression

MA is a simple smoothing technique. To assess the
performance of the GP regression with other methods, this study implements
MA for a one-dimensional case as follows. For a given

UVMRP operational algorithm (OPER) was specially designed for smoothing its
in situ calibration factor sequences
(

Ideally, to avoid additional uncertainties caused by the interpolation among
wavelengths, the calibration factors should be validated via a direct
comparison of direct sun signals from the to-be-calibrated UV-MFRSR and a
reference instrument measuring at the 368 nm channel (e.g., the standard
precision filter radiometer (PFR) operated by the
Physikalisches-Meteorologisches Observatorium Davos, World Optical Depth
Research Calibration Center (WORCC)). However, such reference measurements
are not available at most UVMRP stations. Therefore, the estimated mean
normalized

In this study, for the UV-MFRSR at the 368 nm channel, AOD
(AOD

To obtain reliable AOD values, UV-MFRSR measurements with quality concerns or cloud contamination are excluded in the following comparison. More specifically, (1) any measurements with UVMRP-provided quality control flag(s) relevant to the data quality of the direct beam at the 368 nm channel are excluded; (2) data with small (direct beam) measurements at 368 nm are also excluded because they are more sensitive to noise or errors introduced during various calibration steps; and (3) a simple variation check is performed to reduce the potential of mixing cloud and AOD. If the ratio between the standard deviation of TODs and the mean TOD value in the 15 min time window exceeds 0.05, they are excluded from further analyses.

AERONET (v2.0) provides AOD at the 340 and 380 nm channels. These values are
interpolated to the effective wavelength of the UV-MFRSR 368 nm channel for
comparison using the Ångström exponent as follows. Note that in the
log-transformed coordinate system (i.e., log(AOD) vs. log(wavelength)),
log(AOD) is generally linear between 340 and 380 nm (Krotkov
et al., 2005a). First, the AERONET AOD spectrum between the two wavelengths
is derived by linear interpolation of AERONET AODs at 340 and 380 nm in the
log-transformed coordinate system. Next, since the UV-MFRSR AOD at 368 nm is
a band-pass value over a narrow band (i.e 2 nm FHMW), the equivalent AERONET
AOD at that channel is derived by

Since AERONET and UV-MFRSR AOD values at 368 nm are derived from
measurements involving different instruments and wavelengths, the
uncertainties when comparing these AOD values should be noted. Some
important sources of uncertainties include the following.

We generate a synthetic time series that is composed of six segments with a
varying base function and noise levels (Fig. 2a). The base function (Eq. 10),
including linear, quadratic, and cubic functions, simulates a wide variety of
functions for which the proposed technique is applicable. The noise levels
are the same within each segment but different across segments. The noise at
segment

In this study, the in situ calibration factors of UVMRP UV-MFRSRs are used
as application cases to test the performance of the three smoothing methods
(i.e., GP, MA, and OPER). These UV-MFRSR in situ calibration factors over
several months or years are obtained through the Langley method on clear
days. Their varying uncertainties are mainly attributed to two aspects. One
is the optical stability of atmospheric constituents (e.g., the aerosol,
ozone, and thin clouds) when the in situ calibration factor is derived
(Chen et al., 2015), and the other is the aging status of the
radiometer. UVMRP publish its in situ calibration factors on their website
(

The three UVMRP 368 nm UV-MFRSR in situ calibration factor time series for test.

The proposed dynamic input uncertainty estimation method is first applied to the synthetic case. To observe the statistical properties and characteristics of the estimated input uncertainty, this procedure was applied to 200 synthetic time series, each of which is generated by adding random noise into the base function (Eq. 10) following the procedures discussed in Sect. 2.5.1.

Figure 2b shows the means (dark blue circles) and
confidence intervals (light blue area) of estimated uncertainty of the 200
estimated input uncertainty sequences. The mean of the estimated uncertainty
is close to the true uncertainty (RMSE

Validation of the input uncertainty and mean of GP prediction using
four input uncertainties: the input uncertainty estimated by the proposed
method (Sect. 2.1.2), overall standard deviation of the synthetic time
series (30.95), minimum true uncertainty of the synthetic time series
(2.00), and maximum true uncertainty of the synthetic time series (15.00).
RMSE stands for root-mean-square error. LR stands for linear regression.

Note:

To demonstrate the improvements in the GP resulting from the dynamic input
uncertainty estimation, the GP is also run with three typical constant input
uncertainties: overall standard deviation of the synthetic time series
(30.95), minimum true uncertainty of the synthetic time series (2.00), and
maximum true uncertainty of the synthetic time series (15.00). The results
from all three constant input uncertainties are less accurate than the
estimated input uncertainty generated by the proposed method
(Table 2). The proposed method has a significantly
smaller RMSE (i.e., 0.6321) compared with the three constant input
uncertainties (i.e., 24.1152, 6.5226, and 8.7921, respectively). Similarly,
the LR between the estimated and true uncertainties shows
that the proposed method has slope and

The kernel function in the GP regression used in this study is the RQ kernel, with two parameters: length scale and alpha (Eq. 2). To use RQ with GP regression, we need to provide the initial (estimated) values for these two parameters. First, we round the original data points (red points in Fig. 2a) to the nearest 0.25 interval grids. Then, we calculate the autocorrelation on these rounded data points from lags of 0.25 to 22.25 (approximately equivalent to lags of 1 to 90 points). Next, we perform curve fitting on autocorrelation results and obtain 9.80 and 1.05 as initial length scale and alpha estimates, respectively. With these initial RQ parameters and the estimated input uncertainty (from the proposed method or using three representative constant input uncertainties), GP regression predicts the mean and uncertainty functions. Figure 2c shows the GP results for the proposed method: dark blue line for the mean function and the light blue area for the confidence intervals (4.42 times of the GP-predicted uncertainty function).

In terms of the GP-predicted mean function vs. the base function (Eq. 10), the proposed input uncertainty estimation method shows a 12.0 % to 15.7 % improvement on RMSE over the three constant input uncertainties (i.e., 1.1785 vs. 1.3146, 1.3976, and 1.3146) (Table 2). Similarly, the slope of the LR between the two functions is closer to 1 for the proposed uncertainty estimation method (i.e., 1.0082) than the three constant uncertainties (i.e., 1.0228). In addition, the predicted mean function from the proposed method is close to the base function even near the gaps (G1, G2, and G3 in Fig. 2a, c). Additionally, the proposed method's predicted uncertainty function (or confidence intervals) shows better agreement with the true uncertainty of the synthetic time series (Fig. 2c) while the three constant input uncertainties' results show a consistent over- or underestimated pattern over the entire time series (figures not shown). It is noted that the predicted confidence intervals from the proposed method are wider near the three gaps (G1, G2, and G3 in Fig. 2a) than nearby locations with similar uncertainty. This is anticipated because the constraint in the gaps are from distant points at which the RQ kernel gives low correlation.

The results of the three smoothing methods (i.e., GP – Gaussian
process; MA – moving average; OPER – UVMRP operational algorithm) for the
three UVMRP in situ calibration factor sequences:

The same GP procedure is applied to three in situ calibration factor
(

Figure 3a3, b3, and c3 show the estimated means (dark blue line) and confidence intervals (light blue area) after the initial pass through GP. The length scale parameters of the RQ kernel for the HI02, IL02, and OK02 sites are 6.091, 6.369, and 6.228 (days), respectively. Their corresponding alpha parameters of the RQ kernel function are all close to 1.0 (i.e., 0.948, 0.862, and 0.944, respectively). As expected, the confidence interval is narrower near time windows with more data points, and the confidence intervals are wider near gaps (Fig. 3b3).

As depicted in Fig. 1, the outlier removal and GP are repeated following the initial GP regression, giving the final GP results shown in Fig. 3a4, b4, and c4. After this final pass, the length scale parameters of the RQ kernel function for the HI02, IL02, and OK02 sites are 6.091, 11.149, and 6.907 (days), respectively. Compared with the first round, all length scale parameters increase as more outliers are removed (except for HI02). At HI02, the average ratio between GP means and standard deviations is lower than the threshold (i.e., 0.01) after the first round and the iteration stops. The corresponding alpha parameters of the RQ kernel function are still all close to 1.0 (i.e., 0.948, 1.010, and 1.110, respectively). Because of outlier removal, compared with the first-round results, GP generates smoother mean functions and narrower confidence intervals at the last round.

The 368 nm AOD scatter plots between UVMRP (

The other two methods (i.e., MA and OPER) are applied to the same in situ
calibration time series. They can provide mean functions but not confidence
intervals. The MA (win_size

Time series of UVMRP and AERONET 368 nm daily average AOD at the
HI02, IL02, and OK02 sites. The daily AOD mean values derived from the GP
mean in situ calibration factor (

Following the procedures described in Sect. 2.4, the UVMRP AODs at the 368 nm
channel generated by GP, MA, and OPER are validated against the
corresponding AERONET AODs at the three collocated sites (i.e., HI02 –
Mauna_Loa, IL02 – BONDVILLE, OK02 – Cart_
Site). The scatter plots between these UVMRP and AERONET AODs are displayed
in Fig. 4. The performance of all three methods
at HI02 (Fig. 4a, d, g) are similar. For
example, the average bias “

Statistical metrics (average absolute difference, average absolute
relative difference, and linear regression) on comparing 368 nm AOD between
UVMRP (AOD

Table 3 shows two additional statistical metrics for validation:
“Avg(

Overall, the 368 nm AODs by GP show higher correlation, closer to 1 slopes, and lower absolute and relative biases compared to AERONET AODs than MA and OPER at all three sites. The improvement of GP over MA and OPER at IL02 and OK02 is more significant than at HI02. The main reason may be that HI02 is the least polluted site among the three sites. Both of its maximum and mean 368 nm AOD values are low: 0.35 and 0.016, respectively. As a result, higher accuracy of Rayleigh and other optical depth components is required to discern small improvement in AOD for HI02. Since AERONET's sun photometer is routinely calibrated, the agreement on AOD values suggests that the calibration factor mean functions generated by GP are more accurate than those of MA and OPER.

In addition, Fig. 5 shows the 368 nm AOD time
series calculated using GP-generated in situ calibration factors at the
three UVMRP sites. The blue solid line represents the AODs calculated using
the GP means, and the green and red dotted lines represent the AODs
calculated using the GP confidence intervals. It is seen that the AOD
confidence intervals are approximately

A new dynamic uncertainty estimation method for noisy time series is
developed in this study. Combining this method with Gaussian process
regression, we provide a solution to estimate the underlying mean and
uncertainty functions of time series with variable mean, noise, sampling
density, and length of gaps. For the synthetic case with linear, quadratic,
and cubic base functions; a noise level varying from 2 to 15; and noticeable
gaps, the proposed solution returns a mean function with the RMSE of 1.1785
(linear regression

Given a time series

Since the true 368 nm in situ calibration factors are not available, their distribution is derived using the AERONET 368 nm AOD distribution via Beer's law (transformed Langley regression).

Beer's law links the irradiance (or voltage,

Box-and-whisker plots of

Figure C1 showed that GP had narrower error ranges compared with the other two methods (i.e., MA and OPER) at all three test sites (i.e., HI02, IL02, and OK02). The median values (the short black lines in blue boxes) of GP are closer to zero at IL02 and OK02 sites, especially for lower air masses. However, regardless of site, air mass, and method, the difference between AERONET and UV-MFRSR AODs still exceeds the WMO AOD U95 criterion for a number of instances.

The in situ calibration factors (sun–earth distance
normalized) used in this study are available from the UVMRP website:

MC and ZS are equally significant contributors to the research. The methodology was developed by MC and ZS. The software was developed by MC. The results were analyzed by MC, ZS, CAC, and JMD and were validated by MC, ZS, and YAL. The original draft was written by MC and ZS and was reviewed and edited by MC, ZS, CAC, JMD, YAL, and WG. The project was supervised and administered by JMD and WG. The funding for the project was acquired by WG.

The authors declare that they have no conflict of interest.

This work is supported by the US Department of Agriculture (USDA) UV-B Monitoring and Research Program, Colorado State University, under USDA National Institute of Food and Agriculture grant 2016-34263-25763. We thank Rick Wagener (PI) and the team for the effort in establishing and maintaining the US Southern Great Plains (SGP) Cloud and Radiation Testbed (CART) site. We thank Brent Holben (PI) and the team for the effort in establishing and maintaining the AERONET Mauna_Loa site. We thank Brent Holben, Christopher M. B. Lehmann, and their team in establishing and maintaining the AERONET BONDVILLE site. Edited by: Andrew Sayer Reviewed by: two anonymous referees