Introduction
Broadband solar and terrestrial irradiance are key parameters in the
atmospheric energy budget. Classical instruments measuring broadband
irradiance are pyranometers and pyrgeometers based on thermophile
sensors .
The pyrgeometer performance is quantified by the absolute calibration
accuracy, measurement stability, angular response, dome spectral
transmissivity, direct solar heating and dome temperature effects
e.g.,.
For a modified Eppley Precision Infrared Radiometer (PIR),
quantified daytime and nighttime precision with
0.4 % (±1 Wm-2) at night and 1 %
(±2 Wm-2) at day. The effect of differential
temperatures between the dome and the thermophile sensor, as discussed
by , was minimized by adding two additional
temperature sensors in the dome . For an unshaded
pyrgeometer, solar radiation may cause nonhomogeneous dome temperatures
in cloud-free conditions. showed that this may
result in an overestimation of up to 10 Wm-2 for PIR
pyrgeometer while for CGR-4 pyrgeometer by Kipp and Zonen this
temperature effect ranges within the measurements' uncertainties.
demonstrated that the World Infrared Standard Group
(WISG) of pyrgeometer (including PIR and CGR-4) underestimates downward
irradiance in clear sky conditions by 2 to 6 Wm-2 compared to
independent radiometers traceable to International System (IS) of Units.
However, internal consistency and stability of the WISG pyrgeometers ranges
within ±1 Wm-2.
Further uncertainties of pyrgeometer measurements result from the
spectral variability of the dome transmissivity
. This mostly affects clear sky measurements with
low values of integrated precipitable water vapor (IWV) causing a bias
of about -0.5 Wm-2 when IWV is less than
10 mm. However, showed that the spectral
dome transmissivity of CGR-4 pyrgeometer is more uniform over the
entire wavelength range, and IWV effects can be neglected. Similarly,
effects caused by a nonuniform angular response of the
pyrgeometer dome are not crucial for the overall measurement
uncertainties .
One main drawback of pyranometer and pyrgeometer is their relative
slow response to changes in the incident irradiance. Sensor,
ambient air and dome have to be in thermal equilibrium which can
take up to 6 s (1 e-1 response time) as
specified for the CGR-4. While the response of the thermophile
sensor itself is faster with response times of about 2 s
. Pyranometer response times are in the same
order ranging between 2 s for the CMP-22 by Kipp and Zonen
and 15 s for the Eppley Standard Precision Pyranometer
(SPP). For ground based measurements of the energy budget, the slow
response of pyranometer and pyrgeometer is not critical. However,
airborne measurements of broadband irradiance require a high
temporal resolution as changes in irradiance appear more rapidly
and can be faster than the response time of pyrgeometer and
pyranometer. For example, vertical profiles of net broadband
radiation and derived vertical heating rates depend on a dense
sampling in particular near cloud top
e.g.,. As shown
by the terrestrial cooling, a crucial measure
influencing cloud dynamics, increases sharply within a few meters
at cloud top. Therefore, analyzing the terrestrial cloud top
cooling and solar heating by measurements requires a sufficient
temporal resolution of pyrgeometer and pyranometer measurements.
For downward solar irradiance measured by airborne pyranometer an
attitude correction is needed. If the pyranometer measurements are
lagged compared to the aircraft attitude angles due to the slow
response time of the pyranometer, the correction can fail.
introduced a time shift equal to the
pyranometer response time in the irradiance time series to
compensate this effect in the attitude correction. However, the
reduced temporal resolution of the slow-response pyranometer data are not corrected by adding such a time shift.
The time response of pyrgeometer measurements was already
addressed by and . For the
PIR they investigated the consequences of different response times
of the thermophile, the dome temperature and the thermophile sink.
Usually the thermophile response (about 2 s) is of one
magnitude faster than the time the dome needs to establish
equilibrium with the ambient air temperature (about 5 min
response time). For aircraft measurements with sufficient air flow
around the dome, the response time is reduced. However,
as well as concluded that
during ascents, descents or short horizontal flight legs the
pyrgeometer data are systematically biased as the stabilization of
the pyrgeometer temperature equilibrium is in the order of
several minutes. showed that after rapid
descents the equilibrium of dome and thermophile temperature of
the PIR is reached after about 4 min. Similar
investigation for the CGR-4 are not available. The response time
of 6 s specified for the CGR-4 preliminary accounts for
the thermal equilibrium of sensor, dome and ambient temperature,
when only the terrestrial irradiance changes. This is valid as a
first approximation for flights in constant altitude (constant
ambient temperature).
As a consequence previous airborne measurements focused on sampling
during horizontal flight legs and integrated over several minutes to
allow the sensor to establish thermodynamic equilibrium
e.g.,. Measurements during
ascents and descents by showed that the observed
vertical profiles of heating rates in Arctic stratus clouds could not
represent the vertical variability of heating rates derived by
a radiative transfer model.
To circumvent these limitations, the spatial resolution of airborne
pyrgeometer measurements can be improved by using slowly moving
platforms or decreasing the ascent and descent rates in the flight
pattern. In this way, derived continuous
profiles of radiative heating within a 1 km thick cirrus layer
by extending the descent for 10 min with a rate of
100 mmin-1. For marine stratocumulus,
measured vertical profiles of net irradiance using a tethered
balloon. This platform allows slow ascent and descent rates down to
20 mmin-1. With a sampling time of about 6 s,
a vertical resolution of about 2 m is obtained. However, an
entire profile of a 500 m thick cloud needs about
25 min, which creates problems in the data
interpretation. Stable horizontal homogenous conditions have to be
assumed which does not hold for all cases .
Airborne pyrgeometer measurements of high temporal resolution are also
needed to study horizontal variations of the terrestrial radiation
budget. For example, above stratocumulus differences of the cloud top
cooling induce vertical motion and entrainment . In
the Arctic, open leads in sea ice cover may lead to enhanced sensible
and latent heat fluxes and emitted terrestrial radiation
. As shown by this effect
can be observed for even small leads with a diameter down to about
100 m. For typical aircraft velocities of at least
50 ms-1, this corresponds to a measurement time of only
2 s, which is less than the response time of pyrgeometer.
To improve the temporal resolution of pyrgeometer with slow
response time a method is introduced to reconstruct the
fluctuations in the measured time series. The approach is based on the
deconvolution theorem of Fourier transform, which is popular in
different scientific fields such as the reconstruction of climate
proxy time series e.g.,.
In Sect. the theory of the deconvolution theorem is
described. It is exemplarily applied to pyrgeometer measurements in
this paper but can also be adapted to slow-response pyranometer. The
potential of the method is discussed by analyzing the minimum scales
of fluctuation which can be reconstructed for different sample
frequencies assuming a perfect measurement with a defined noise
level. Laboratory measurements are discussed in Sect.
using two types of predefined time series to investigate the impact of
instrument noise on the reconstruction. In Sect. , the
reconstruction is applied to two exemplary measurement time series
(leads in sea ice and broken clouds) of upward terrestrial irradiance
from airborne observations. For these measurements, the limitations of
the method in case of a digital data acquisition are discussed.
Deconvolution of time series
Theory
An instantaneous change of a given quantity can be described with the
Heaviside step function H(t), defined by
H(t≤0)=X0 and H(t>0)=X0+ΔX. In
that case, the response of pyrgeometer and pyranometer follows an
exponential decay with time t characterized by the response time
τ. The temporal instrument readout X(t) as a response of the
instantaneous change of the measured quantity is described by
X(t)=X0+ΔX⋅exp-tτ,
with the response time τ defining the time when the difference
between readout and actual value reaches 1/e (37 %) of the
initial difference ΔX. The function given by
Eq. () represents a convolution H∗g(t) of the step function H(t) with the
convolution kernel g(t),
X(t)=H∗g(t)=∫-∞∞H(t′)⋅g(t-t′)dt′.
For an arbitrary change of the measured quantity x(t) the
instrument readout is similarly described by
X(t)=x∗g(t)=∫-∞∞x(t′)⋅g(t-t′)dt′.
In case of an exponential response characteristic of the instrument it
holds:
g(t)=1/τ⋅exp(t/τ),t≤00,t>0.
applied an approximation to correct for the slow
time response of pyranometer and pyrgeometer by using the Newton's
formula including the first derivative of the measurement time series
by,
x(t)=X(t)+τ⋅dX(t)dt.
However, an exact analytical solution for the convolution is given by
the convolution theorem of Fourier transform:
FX=Fx∗g=Fx⋅Fg,
with FX, Fx,
and Fg the Fourier transform of the
instrument readout, the actual measured time series and the
convolution kernel, respectively. If the convolution kernel g is
known, Eq. () can easily be inverted and used to
calculate the Fourier transform of the actual time series not
smoothed by the delayed instrument response,
Fx=FXFg.
By inverse Fourier transformation of Fx
the original time series x(t) can be derived,
x(t)=∫-∞∞Fx⋅ei⋅2πf⋅tdf.
In practice, the reconstruction may fail when instrument noise is
artificially amplified. To suppress these effects, an additional
low pass filter (moving average) is applied and the inverse
Fourier transformation calculated only from frequencies below a
specified cut-off frequency fc. The Fourier transform
of a moving average (boxcar function) with window length of
Tm in units of seconds is given by the sinc function
what finally results in the inverse transformation,
x(t)=∫-fcfcsincTm⋅f⋅Fx⋅ei⋅2πf⋅tdf.
The deconvolution of long time series of high sampling frequency
is demanding. Fortunately, due to the exponential kernel function
a single measurement of a time series is affected only by
measurements within a certain time span. The fast decay of the
exponential kernel function is used to accelerate the
computational speed by splitting the time series into smaller
sections. A criteria of 99.99 % decay and corresponding time
segments of tsplit=τ⋅ln(104⋅τ-1)
should be sufficient. A section length of (10+2)⋅tsplit including an overlap of overlap 2⋅tsplit to glue the single sections, was chosen for
further investigations in this paper. For example, in case of a
response time of τ=4 s the section length is about
6 min, the overlap 1 min. The overlap is used for
the computation of the deconvolution only and rejected when gluing
the section into the final entire time series.
Reconstruction of synthetic time series
To demonstrate the potential of the deconvolution method to
reconstruct pyrgeometer measurements synthetic time series of
convoluted data are analyzed. The convoluted time series represent
ideal measurements without any noise. In this way, it is studied how
large the amplitude of fluctuations is in the measured time series and
if this damped amplitudes range above or below the noise level of the
instrument. For the original time series a periodic function with
symmetric rise and decline was chosen. The frequency of the
oscillations fp was varied between 0.1 and
4 Hz. The corresponding theoretically measured time series
were calculated by convolution taking into account the response time
τ in the range of 0.1–10 s. The convoluted time series
were calculated for a sampling frequency of
fs=20 Hz, which is typical for airborne pyrgeometer
and pyranometer measurements.
In Fig. three exemplary time series with
frequencies fp= 0.25/1/2 Hz are analyzed. A hypothetical
noise level of 0.25 Wm-2 is illustrated by the gray area
(typical for CGR-4 pyrgeometer used in this paper). The original time
series (red line) with amplitude ±5 Wm-2 is
convoluted for different instrument response times. In all cases,
a decrease of measured amplitude and an increasing of the phase shift
with increasing response time is obvious. With increasing frequency of
the fluctuations the phase shift and the amplitude of the measured
time series become smaller. For τ=4 s the maximum amplitude is
1.8 Wm-2 for fp=0.25 Hz, while for
fp=2 Hz the maximum amplitude of
0.25 Wm-2 is close to the noise level of pyrgeometer
measurements. In this case, the small damped fluctuations registered in
the measurements might be superimposed by noise and the reconstruction
of the given fluctuations by deconvolution is not meaningful.
Original synthetic time series (red) and convoluted time
series (black) of irradiance for three different frequencies
fp= 0.25/1/2 Hz (a–c). The different
black lines show convoluted time series for different response times
τ between 0.1 and 12 s. The gray area indicates
a hypothetical noise level of 0.25 Wm-2.
A signal to noise ratio of ΔFc/ΔFn≥1 between the amplitude of the convoluted data
ΔFc and the noise amplitude ΔFn
was defined as a criteria of the ability to reconstruct a specific
time series. We assumed a noise amplitude of ΔFn=0.25 Wm-2 and translated the results of the
calculations into a specific minimum amplitude ΔFmin
representing the smallest periodic fluctuations which from theory can
be reconstructed from the measurements. Taking into account the
amplitude of the original time series of ΔF=10 Wm-2 it is
ΔFmin=ΔFΔFc⋅ΔFn.
These minimum amplitudes are presented in
Fig. a as function of response time τ of the
sensor and frequency of the fluctuations fp showing an increase
of ΔFmin with increasing τ and fp. For slow
response times and slow oscillations, the fluctuations with amplitudes
below 0.5 Wm-2 can still be resolved while ΔFmin can exceed 10 Wm-2 for frequencies above
1 Hz and response times larger than
2 s. Figure a suggests a linear dependence
between ΔFmin and the product (τ⋅fp). The
product of τ and fp can be interpreted as ratio of the
characteristic times of sensor (τ) and oscillations (1/fp)
as it is shown in Fig. b. For low values of
(τ⋅fp), the minimum amplitudes are close to the predefined
noise criteria what means that almost all fluctuations above the noise
criteria can be reconstructed. For (τ⋅fp)>0.5 the plot
follows an almost perfect linear behavior given by
ΔFmin=1.3⋅(τ⋅fp).
All fluctuations above this line can potentially be reconstructed.
(a) Minimum amplitude ΔFmin of
fluctuations which theoretically can be resolved by deconvolution in
dependence of the sensor response time τ and the oscillation
frequency fp. (b) Similar to (a)
ΔFmin is given as a function of the product
(fp⋅τ). Symbols illustrate the results of the
calculation. The red line indicates a linear fit for large
(fp⋅τ).
In general, this linear relation depends on the defined noise amplitude
ΔFn. The lower the noise of the instrument, the lower
the amplitudes that can be resolved by deconvolution. This relation
is linear meaning that reducing the noise level by a factor of 2
reduces the minimum amplitudes by a factor of 2. Based on the fit
given in Eq. () (Fig. b) the
following relation is postulated:
ΔFmin=A⋅ΔFn⋅(τ⋅fp).
The empirical parameter A depends on the shape of the original
periodic time series. For the shape of the oscillations used here
A=5.2 was determined. Calculations, not shown here, using a square
wave (boxcar) function give a smaller coefficient with A=4 while for
a triangular wave function A is larger with A=8. This illustrates
that the theoretical capability of deconvolution increases for sharper
fluctuations. For fluctuations with a continuous increase and decrease
(triangular wave) ΔFmin is lower by a factor of 2
compared to an instantaneous increase and decrease of the oscillations
(square wave). Other shapes of fluctuations will likely vary between
these two extremes. For a periodic but not symmetric oscillation based
on the laboratory measurements presented in Sect. ,
A=6.4 was determined, which is slightly closer to the triangle
function than the more smooth oscillation presented here.
Laboratory measurements
Instrumentation
A CGR-4 pyrgeometer manufactured by Kipp and Zonen was used in
combination with the analog signal amplifier CT 24 also provided
by Kipp and Zonen. The amplified thermoelectric voltage was
digitized by an analog-to-digital (AD) converter USB-6009 by
National Instruments. The AD converter sampled with up to
48 kHz sampling frequency and 14 bit resolution.
Three AD channels were utilized, one for the pyrgeometer voltage
and two for the thermistor (input and output voltage) measuring
the internal sensor temperature inside the CGR-4. This reduced the
sampling frequency fs to 16 kHz. To
investigate the net terrestrial irradiance two pyrgeometer (upward
and downward facing) are needed. In the case of two CGR-4 being logged with
the same AD converter, the sampling frequency is reduced to
fs=8 kHz. If additionally solar pyrgeometer
data are logged, a maximum sampling frequency of
fs=4 kHz can be obtained for such kind of AD
converter.
The full time series sampled at 48 kHz can not be saved.
Therefore, a recording frequency of 20 Hz was selected. All
samples in between are averaged reducing the noise in the measurements
significantly.
As discussed by the pyrgeometer thermophile
and the dome temperature exhibit different response times.
Therefore, in this section only net irradiance measured by the
pyrgeometer thermophile were analyzed, which excludes additional
effects due to the time response of the internal thermistor. For the sake of comparison, all irradiance time series were subtracted by the mean
irradiance measured by the pyrgeometer highlighting the amplitude
of the fluctuations in the time series.
Measured (black) and reconstructed irradiance without filter (blue) and with filter
(red) for an original boxcar function (gray) for three different sampling
frequencies fs= 16/8/4 kHz (a,
c and d). The corresponding power spectra for the
measured (black), deconvoluted (green) and reconstructed time series
(blue and red) as well as the convolution kernel (dark blue) are given in
(b, d, and f).
Boxcar function
Irradiance time series with shape of a boxcar function were
generated to determine the response time of the pyrgeometer. The
pyrgeometer dome was covered for 40 s by a plate with
higher temperature compared to the ambient air. The data were
recorded with fr=20 Hz and different sampling
frequencies of fs=16/8/4 kHz. Examples of the
measured time series are given in Fig. .
Within the 40 s, the measured irradiance (black line)
approaches the emitted irradiance by the plate (gray line).
After the plate was removed the temporal decay of the measured
irradiance was analyzed as an alternative measure of the response
time. For both increase and decrease τ was calculated,
estimating the time of 63 % increase or decay. For the CGR-4
used here, a mean value from increase and decrease of
τ=3.3 s was calculated and used to reconstruct the
original boxcar function.
The time series was reconstructed for all three measurements with
different sampling frequencies (compare
Fig. a, c and e). The corresponding power
spectra of measured and deconvoluted time series are given in the
right panels of Fig. . Additionally, the
Fourier transform of the kernel (exponential decay,
Eq. ) and the frequencies used for the inverse
transformation are illustrated. The low-pass filter characteristic
of the kernel (slow sensor response) is obvious in the decay of
the kernel Fourier transform for high frequencies. The convolution
damps the Fourier coefficients, and thus the amplitudes of
fluctuations with high frequencies. However, the scale break of
the decay at high frequencies in the power spectra of the
measurements shows that the data are contaminated by instrument
noise. The noise significantly differs for the three measurement
examples as the scale break is shifted to smaller frequencies for
lower sampling frequency fs.
By deconvolution as given in Eq. () these frequencies
are artificially amplified as indicated by the increasing power
spectra of the deconvoluted time series for high frequencies. To
suppress this noise in the reconstructed time series different
cut-off frequencies fc were applied when calculating
the inverse transformation using Eq. (). For
a sampling frequency of fs=16 kHz the noise of
the measurements is lowest with an amplitude of ΔFn=0.08 Wm-2, which allows for the use a high
cut-off frequency of fc=1.5 Hz. With a reduced
sampling frequency (fs=8/4 kHz) the noise
amplitude of the measurements increases
(0.12/0.25 Wm-2) and lower cut-off frequencies were
feasible (fc=1.2/1.0 Hz).
The general structure of the original boxcar function could be
reconstructed in all three examples, indicating that the response time
was correctly determined. The steps in the boxcar function were
reproduced with the accuracy expected from the chosen cut-off
frequency. A closer look into the data showed that the step takes
about 0.65 s in the reconstructed time series for
fs=16 kHz, which is close to the inverse of the
cut-off frequency of fc=1.5 Hz. Additionally, no
differences for the positive and negative step of the boxcar function
are found, which is in agreement with the symmetry of the pyrgeometer
response discussed by .
In all cases small oscillations are present in the reconstructed
data. This still might be a result of the instrument noise although
the cut-off frequencies were chosen carefully to reduce noise. Using
higher fc, increased the oscillations, but for lower
fc the oscillations could not be removed completely. This
indicates that the remaining oscillations result primarily from the
Gibbs phenomenon. At a jump discontinuity as represented by the boxcar
function, the application of the Fourier transform is limited, causing
ringing artifacts in the inverse transformation if the number of
coefficients in the Fourier transform is not infinite. Therefore, the
boxcar function is an extreme to illustrate how the deconvolution acts
for different instrument noise levels. In real pyrgeometer
measurements sharp changes are not typical.
The oscillations can be reduced by applying an additional moving
average filter as described in Eq. () which acts
similar to the sigma-approximation often used to eliminate the
Gibbs phenomenon. Applying filter with window length Tm=1/fc did significantly improve the results shown in Fig. (red lines). The magnitude of the
oscillations is reduced but consequently, the width of the step
slightly increases to about 1 s what is still in the range
of the inverse of the cut-off frequency. As this might be
different in other cases, the use of a moving average filter has
always to be weighted in each individual case with the negative
consequence of a reduced temporal resolution caused by the
smoothing. Similar to the results without filter, the remaining
oscillations again become larger when the sensor noise is
increased due to a reduction of the sampling frequency.
Periodic function
Continuous oscillating time series of terrestrial irradiance were
generated using a chopper wheel. The chopper was operated in front of
the pyrgeometer sensor head and rotated with frequencies
fp in the range between 0.1 and 2 Hz. Behind the
chopper a plate with a temperature higher than the chopper wheel was
placed. The exact irradiance time series generated by the chopper was
determined by measurements at fixed chopper positions. For this case,
(fixed chopper) each measurement was taken long after the
slow-response pyrgeometer showed stable values (after about
20 s). During the measurements the temperatures of plate and
chopper wheel were additionally observed by an infrared
thermometer. Changing the temperature of the plate allowed for the study of
the response of the pyrgeometer to periodic functions with different
amplitudes between 1 and 8 Wm-2.
Influence of sampling frequency fs
To illustrate the influence of different sampling frequencies (similar
to Sect. ), first measurements of slow
oscillations fp=0.1 Hz were investigated. Three
sampling frequencies, fs=16/8/4 kHz, were
applied. For all sampling frequencies 20 s time periods of
measured and reconstructed terrestrial irradiance are shown in
Fig. . The mean value of the period was
subtracted in order to provide time series oscillating around
zero. Two different amplitudes of oscillations were illustrated
representative for high (ΔF=8 Wm-2, left panels)
and low (ΔF=2 Wm-2, center panels) fluctuations.
The corresponding Fourier power spectra are shown in the right panels.
Measured (black) and reconstructed irradiance (red) in case
of a periodic oscillating time series (gray) for three different
sampling frequencies fs= 16/8/4 kHz.
Oscillations with two different amplitude were generated,
8 Wm-2 amplitude in the left panels (a,
d and g) and 2 Wm-2 amplitude in the
center (b, e and h). The corresponding
power spectra for the measured (black), deconvoluted (green) and
reconstructed time series (red) as well as the convolution kernel
(blue) are given in (c, f and i).
Depending on the sampling frequency, the recording produced
different noise levels which are obvious in the originally measured
irradiance (black lines) and in the flattening of the power
spectra, which are shifted to lower frequencies for low sampling
frequencies. From the 16 kHz data the noise amplitude of
the measurements was about ΔFn=0.08 Wm-2. For 8 and 4 kHz
higher noise levels with ΔFn=0.12 Wm-2 and ΔFn=0.25 Wm-2 were found. From
Eq. () (with A=6.4) minimum amplitudes of ΔFmin=0.5 Wm-2 for the noise level are derived
indicating that reconstruction works in all cases investigated
here using the full deconvoluted spectra.
In practice a cut-off frequency of fc=1 Hz was
chosen for the reconstruction, which is sufficient to reduce noise
and adequately recover 0.1 Hz oscillations. In this case – a relativly slow oscillation compared to the response time and
sampling frequency – an additional low pass filter (moving average)
with window length Tm=1 s could be applied to
further reduce the impact of noise on the deconvolution. The
chosen window length is short enough to avoid a reduction of the
amplitude of the reconstructed data. The reduction of noise is
shown in the decrease of the power spectra used for the
reconstruction (red lines) close to the cut-off frequency. These
power spectra are significantly lower than the power spectra
obtained after the deconvolution (green lines).
The comparison of reconstructed and original irradiance measurements
showed in general differences below 0.5 Wm-2 for both low
and high amplitude and all sampling frequencies. The general shape of
the time series generated by the chopper was reconstructed by
deconvolution and the phase shift was removed. However, small scale
fluctuations are obvious in the reconstructions of the low amplitude
oscillations using measurements with fs=8 kHz and
fs=4 kHz. This illustrates that even when applying
a conservative cut-off frequency and an additional low pass filter
high sampling frequencies (low noise) improve the quality of
deconvolution.
The differences between original and reconstructed irradiance were
quantified by their normalized percentaged standard deviation (SD)
cv=σ/ΔF⋅100%. The results given in
Table 1 show that for the oscillation with high amplitude ΔF=8 Wm-2 the differences are almost equal for all
sampling frequencies fs=16/8/4 kHz with
cv ranging between 2 and 3 %. For the chopping with lower
amplitude ΔF=2 Wm-2 the relative differences are
higher ranging between 4 and 8 %. A tendency of higher cv
for lower fs is found. Both is closely related to the
signal to noise ratio, which is lower for the lower amplitudes, and
decreases with sampling frequency. Not observing this pattern for the
oscillations with high amplitude indicated that here the sampling
frequency is sufficient in all cases.
Influence of oscillation frequency fp
Figure shows reconstructed time series for
different oscillations with fp=0.2/0.5/2 Hz. For the sake of comparison, a low pass filter was not applied because the filter
would cause a reduction of the oscillation amplitudes if the window
length is chosen in a similar range as in
Sect. . For fp=0.2 Hz and
fp=0.5 Hz, a cut-off frequency of 1 Hz was chosen
for the reconstruction, while for fp=2 Hz a cut-off
frequencies of 4 Hz, higher than the oscillation frequency,
was applied.
Applying Eq. () to calculate the potential of
deconvolution for these oscillations (A=6.4) gives ΔFmin=0.34 Wm-2 for fp=0.2 Hz and
ΔFmin=0.85 Wm-2 for
fp=0.5 Hz. Both minimum amplitudes range below the
actual amplitude of the oscillating irradiance time series. However,
for fp=2 Hz a minimum amplitude of
3.4 Wm-2 is derived for which at least the oscillation
with low amplitude ΔF=2 Wm-2 should disappear in
the sensor noise.
Figure shows that for fp=0.2 Hz
and fp=0.5 Hz the reconstructed and theoretical
irradiance again agree. For the oscillations with high amplitude
(Fig. a and d), the corresponding normalized
percentaged SD given in Table 1 are in the range observed for the slow
oscillations of Sect. with
cv=3.4 % and cv=7.2 %. For the
oscillations with low amplitude (Fig. b and e)
the deviations are larger and apparent in a ripple structure of the
reconstructed irradiance. Here cv increased up to
10 %, which indicates that the signal to noise ratio is lower in
these cases and moderately affects the accuracy of the
reconstruction. Especially for fp=0.5 Hz the
amplification by deconvolution of high frequencies in the power
spectra affects also frequencies used for the reconstruction.
Normalized percentaged SD cv between original and
reconstructed irradiance for measurements of different oscillations, as
illustrated in Figs. and .
fp (Hz)
fs (kHz)
cv(%)
cv (%)
ΔF=8 Wm-2
ΔF=2 Wm-2
0.1
16
2.5
4.0
0.1
8
2.0
6.9
0.1
4
2.8
8.3
0.2
16
3.4
9.1
0.5
16
7.2
10.2
2.0
16
29.3
63.8
Measured (black) and reconstructed irradiance (red) in case
of a periodic oscillating time series (gray) for three different
frequencies of the oscillation fp= 0.2/0.5/2 Hz.
Oscillations with two different amplitude were generated,
8 Wm-2 amplitude in the left panels (a,
d and g) and 2 Wm-2 amplitude in the
center panels (b, e and h). The
corresponding power spectra for the measured (black), deconvoluted
(green) and reconstructed time series (red) as well as the
convolution kernel (blue) are given in (c, f, and
i).
Figure g, h and i show the results for an
oscillation with fp=2.0 Hz. For both low and high
amplitudes, the reconstruction by deconvolution fails. Although the
oscillation is in phase and the amplitude is partly correct, in this
case the results of the reconstruction are not reliable.
A cv of 29 and 64 % is given in Table 1. The main
reason for these differences is the use of a higher cut-off frequency
of 4 Hz to cover the fast oscillations. This did increase the
influence of noise as obvious in the power spectra. Here the frequency
range above 1 Hz where the power spectra increases with
frequency, indicating an amplification of noise, dominates the
reconstructed time series. This illustrates that such oscillations of
high frequencies can not be reconstructed with the instrument setup
used here.
Application
Airborne measurements of broadband terrestrial radiation from the
international field campaign Vertical Distribution of Ice in
Arctic Clouds (VERDI) were used to demonstrate the application of
the deconvolution method. VERDI was based in Inuvik, Northwest
Territories, Canada, and took place in April and May 2012. It
included 13 research flights over the Canadian Beaufort Sea. The
research aircraft Polar 5 of the Alfred Wegener Institute for
Polar and Marine Research (AWI) was equipped with remote sensing
and in situ instruments with the purpose to investigate Arctic
clouds . In this paper,
upward terrestrial irradiance measured by a Kipp and Zonen CGR-4
pyrgeometer and upward nadir brightness temperatures from
a Heitronics KT19.85 II are analyzed. Both instruments were
operated with 20 Hz sampling frequency. The CGR-4 was
amplified with the digital signal amplifier AMPBOX by Kipp and
Zonen providing a digital resolution of about
0.2 Wm-2. For noise reduction, the AMPBOX applies
a low pass filter averaging the signal over 440 ms. The
response time of the KT19 is specified by an internal bessel
filter of second order with τ=0.43 s while for the
CGR-4 the response time τ=3.3 s of the laboratory
measurements presented in Sect. was assumed.
The KT19 collects nadir radiance with a field of view of 2∘
while the CGR-4 measures hemispheric irradiance. The KT19 is sensitive
to wavelengths in the atmospheric window (9.6–11.5 µm)
and gives a measure of the surface temperature while the CGR-4 covers
almost the entire terrestrial wavelength range (4.5–42.0 µm) and is also sensitive to atmospheric
properties (temperature and humidity profile). Therefore, a direct
quantitative comparison of measurements from both instruments is not
trivial. However, leads (surface temperature differences) and clouds
(altitude of the emitting surface) will have similar effects on the
measurements of both instruments allowing a qualitative
comparison. Furthermore, the KT19 provides a higher spatial resolution
due to its geometry and, therefore, serves as a reference to
investigate the reconstruction of CGR-4 time series with respect to
small scale fluctuations.
For the reconstruction of the time series presented in this
section, the thermophile output of the CGR-4 and the sensor
temperature measured by an internal thermistor were treated
separately. The deconvolution was only applied to the thermophile
output as the thermistor has a much slower time response and
temperature changes in constant flight altitude are negligible.
Upward irradiance over leads
On 3 May 2012 Polar 5, operating on a north–south transect at
134.5∘ W longitude between 70.5 and 72.0∘ N
latitude, crossed a cloud free area with dense sea ice. Several
open leads with maximum dimensions of about 1 km have
formed in the sea ice. Polar 5 flew at the low flight altitude of
about 150 m above ground, crossing the leads. In
Fig. , an exemplary time series of the
measurements of KT19 and CGR-4 collected over leads are shown. As a comparison, the irradiance of the CGR-4 is translated into
brightness temperatures using an emissivity of ϵ=1. Three
open leads with duration of 9, 11 and 25 s (550, 700 and
1600 m length for ground speed of 63 ms-2)
were observed within the 3 min of measurements. Mean
surface temperatures of -2.1 ∘C over open water and
-7.6 ∘C over the ice were observed by the KT19. The
CGR-4 time series was reconstructed (red line) using a response
time τ=3.3 s for the deconvolution and a cut-off
frequency of fc=2 Hz. Due to the digitization
of the CGR-4 AMPBOX, a moving average filter (2 s window
length) was applied to avoid overshootings by the deconvolution.
Comparing measured brightness temperature from CGR-4 and KT19 shows
principal deviations due to the differences in the measured quantities
as discussed above. The CGR-4 time series also shows the three leads
but significantly lags in time due to the lagged time response of the
sensor. The center time of the three leads is shifted about
2.9 s compared to the KT19. Similarly, the magnitude of the
brightness temperatures fluctuations is affected be the slow sensor
response. In the area between the first two leads, clearly identified
as ice covered by the KT19, the lagged response hinders the CGR-4
signal from reaching the temperature levels measured before and
afterwards above ice. In the center between the two leads, the distance
of Polar 5 to the lead edges was about 250 m. In this distance
and at 150 m altitude the irradiance should only slightly be
influenced by the leads.
These differences are partly removed in the reconstructed CGR-4 time
series (red line in Fig. ). The edges of the
leads match with the KT19 data and also in the gap between the two
first leads, the brightness temperatures reach the level of the sea
ice area. At least over the larger third lead, the reconstructed data
shortly reach an almost constant level of about -5.2 ∘C
(292 Wm-2), which can be used as a representative
brightness temperature of the leads. Above the first two leads
brightness temperatures increase to -5.8 ∘C
(290 Wm-2) still not reaching the value of the third
lead. However, the upward terrestrial irradiance did significantly
increase after reconstruction. The maximum irradiance in the two
smaller leads is at least 2.5 Wm-2 higher than indicated
by the raw data.
Brightness temperatures of upward terrestrial radiation from
KT19 (black) and CGR-4 (blue) on 3 May 2012 above sea ice with
leads. The red line shows the reconstructed CGR-4
data.
Upward irradiance over broken clouds
On small scales clouds are highly variable especially in case of
broken clouds. On 15 May 2012, Polar 5 crossed the edge of a cloud
field at about 69.4∘ N and 136.6∘ W where the
closed boundary layer cloud cover transformed into scatter clouds.
These clouds were located over sea ice providing a high contrast
between surface and cloud brightness temperatures. The
corresponding measured time series of brightness temperature of
CGR-4 and KT19 are shown in Fig. . Within the
2 min the KT19 brightness temperature (black line) dropped
by about 12 K when no clouds have been below the aircraft.
Clouds and cloud gaps were in the range of 1–10 s flight
time corresponding to sizes of 80–800 m at
80 ms-1 ground speed of Polar 5.
The CGR-4 (blue line) does not show similar strong and fine scale
variations as the flight altitude was about 3 km with cloud
top altitudes at about 0.9 km. At this distance the hemispheric
integrating of the CGR-4 does not allow for the sampling of individual clouds of
this size. However, by reconstructing the measured irradiance with the
deconvolution method (red line) some features observed by the KT19 are
also obvious in the CGR-4 time series. Even small clouds and cloud
gaps such as around 2670 or 2700 s flight time become obvious
in the reconstructed data.
Using the raw CGR-4 data, the period between 2660 and 2675 s
would be identified as cloud free, and with an irradiance above sea ice of
258.5 Wm-2 (13.3 ∘C brightness temperature)
would be estimated. However, the reconstructed irradiance, resolving
the small clouds, shows values of 257.0 Wm-2 in the
remaining cloud free areas. This is 1.5 Wm-2
(0.4 K) lower and illustrates that using raw CGR-4
measurements can overestimate the upward irradiance of ice areas close
to clouds. Similarly, the maximum brightness temperatures and
irradiance above small clouds such those at 2655 or 2680 s is
significantly underestimated by the raw pyrgeometer time series. The
reconstruction results in about 0.8 K and
3.3 Wm-2 higher values. Especially for investigations of
the cloud top cooling driving cloud dynamics and entrainment at cloud
top such bias might be of significance.
The reconstruction by the deconvolution method was compared to the
fast approximation presented by .
Figure shows CGR-4 data from
Fig. (red line) and results using
Eq. () (green line). The method by
is very sensitive to noise in the measurements
because the time derivative is calculated for each data point. We
applied a moving average filter with 2.5 s window length
to reduce noise in the CGR-4 raw data before calculating the
derivative. The smoothed CGR-4 raw data are already shown in
Fig. (blue line). However, still the
reconstruction using the approximation of
Eq. () reveals a significantly higher noise
level compared to the reconstruction using the Fourier theorem of
deconvolution. A further reduction of this noise can only be
achieved by smoothing the raw data a with larger window length.
However, that would cause the reconstruction to fail at high
frequency fluctuations. This is obvious between
2668 and 2675 s in Fig. , where the
result of the approximation does not exactly follow the brightness
temperatures reconstructed by the deconvolution method.
Brightness temperatures of upward terrestrial radiation from
KT19 (black) and CGR-4 (blue) on 15 May 2015 above broken
clouds. The red line shows the reconstructed CGR-4
data.
Limitations
The relative variations in upward terrestrial irradiance analyzed in
the last sections are large. For investigations of small scale
fluctuations of upward irradiance, e.g., to analyze the variability of
cloud top cooling of homogeneous stratus, the resolution of the
measurements has to be increased both in time and dynamic range. This
is only possible if a high cut-off frequency can be selected for the
inverse Fourier transformation.
For the airborne measurements presented here,
fc=2 Hz was chosen and an additional low pass
filter (moving average with window length of 2 s) had to
be applied. Less smoothing would lead to significant overshooting
by the deconvolution in the reconstructed data. This is
illustrated in Fig. a by an exemplary 40 s
measurement with relative low change of the terrestrial irradiance
of 1 Wm-2 in 20 s which has been
reconstructed using two different configurations A and B. For the
data of reconstruction A fc=1.3 Hz and
a moving average of 1 s length was used while for
reconstruction B a moving average of 5 s window length was
applied. For reconstruction A, overshootings of about
±0.5 Wm-2 are obvious in Fig. a.
These overshootings correspond to steps in the raw data which can
be identified by smoothing the raw data (blue line). The
deconvolution transforms these steps into single positive or
negative peaks.
Comparison of reconstruction using the deconvolution method
as described here and the approximation presented by
. The data are shown for the broken clouds case of
15 May 2015. The measured CGR-4 data are smoothed by a moving average
filter with 2.5 s window length as used in the
approximation.
Comparison of the reconstruction using different low pass
filter of 1 s window length (A) and 5 s window
length (B). In (a) the airborne instrumentation using
the digital AMPBOX are analyzed while in (b) the
laboratory instrumentation with the analog signal amplifier CT 24
are shown.
These steps are artificial and originate from the digitalization
of the amplified thermoelectric voltage by the AMPBOX amplifier.
Compared to an analog signal which allows continuous values, the
digital output of AMPBOX for the setup used on Polar 5 has
a minimum resolution of 0.2 Wm-2. To avoid
overshooting due to the digitalization, the window length of the
moving average filter had been increased to smooth the steps.
Results using a 5 s window length, reconstruction B, are
shown in Fig. a as red line. However, by the
smoothing real small scale fluctuations are removed; thus, the
capability of the reconstruction is reduced. The parameters of the
reconstruction used for the airborne measurements represents
a compromise between avoiding overshootings and resolving small
fluctuations.
To show the potential of an analog amplifier, similar data (same
change of terrestrial irradiance) from the CGR-4 pyrgeometer
configuration used in the laboratory measurements of
Sect. are plotted in Fig. b. The
maximum sampling frequency of 16 kHz was used to demonstrate the
maximal improvement. In the results using a 1 s averaging
window length, the overshooting with about
±0.1 Wm-2 amplitude is significantly lower
compared to the airborne measurements and can not be related to
a specific pattern in the raw data. Therefore, these
reconstructed fluctuation may either originate from the electronic
noise or they are real.
Conclusions
As shown here by calculation, laboratory and field measurements
the deconvolution is a powerful tool to reconstruct high-resolution time series of terrestrial irradiance measurements of
pyrgeometer. By characterizing the response function of the sensor
the smoothing due to the slow-response of the sensor can be
removed efficiently using the convolution theorem of Fourier
transform.
An empirical equation which can be used to estimate the capability of
the reconstruction for measurements with 20 Hz recording frequency is
derived. Depending on the assumed noise criteria, Eq. ()
gives the minimum amplitudes which might be reconstructed. However,
comparison with laboratory measurements revealed that the estimates
are optimistic. This is probably caused by the noise criteria of
a signal to noise ratio of 1 which is assumed. Using a higher signal
to noise ratio may provide a more conservative estimate of the
amplitudes that may be resolved. Eq. () and the
coefficients A have to be considered as an upper limit of the
deconvolution method.
In laboratory measurements different synthetic irradiance time series
(boxcar function and periodic oscillation) have been measured by
a CGR-4 pyrgeometer (τ=3.3 s) and reconstructed using the
deconvolution method. The results followed the theory showing that
a low noise level of the measurements, achieved by a high sampling
frequency, favors the reconstruction of time series contaminated by
the slow response of the pyrgeometer. The higher the frequency and the
lower the noise, the smaller the fluctuations which can be
reconstructed because more information on the original time series is
included in the measurements which can be used for reconstruction.
In case of sharp drops of the irradiance (boxcar function) the method
has certain limits related to the Gibbs phenomenon. Close the edge of
the shifts, ripples remain in the reconstructed time series even if
a high cut off frequency is chosen. However, most applications are
characterized by smooth transitions of irradiance. For idealized
oscillations, the deconvolution method agreed with the original time
series for most of the investigated sampling frequencies, oscillation
frequencies and amplitudes. The normalized SD remained below 10 %
for oscillation frequencies of up to 0.5 Hz and amplitudes
down to 2 Wm-2. A tendency of better agreements for the
oscillations with higher amplitude and high sampling frequencies was
found. Oscillations with fp=2 Hz could not be
reconstructed accurately as the impact of the sensor noise is too
high.
In general, the capability of the reconstruction depends on the
sampling frequency and noise level of the sensor. Even time series
from sensors with slow response times can be successfully
reconstructed if the measured time series is provided with high
temporal and dynamic resolution, high sampling frequency and low
noise. In this respect, the digitalization (AD converter) of the
measured signal is of high importance. To obtain accurate results
from the reconstruction, a digital resolution of 14 bit
and better are beneficial. To resolve fast changes of the measured
irradiance, recording frequencies of 20 Hz or more should
be applied. A focus should be put on the noise level of the
sensors. The instruments used in this study still do not apply
high end AD converter. The USB-6009 by National
Instruments is only at the lower end of available models with a total sampling
frequency of 48 kHz and a 14 bit resolution.
Some of these instrumental limitations have been investigated by
two exemplary measurements of upward terrestrial irradiance
obtained during research flights of VERDI. Over sea ice the
increased surface temperature of open leads was reconstructed and
compared with a KT19 infrared thermometer. The comparison
illustrated that the deconvolution method is capable to correct
phase shifts and reduced amplitudes of the measured fluctuations.
It was shown that upward irradiance over small leads with size of
about 600 m (10 s duration of measurement) may be
underestimated by 2.5 Wm-2 or more when using
uncorrected measurements. Similar results were observed for the
upward irradiance above broken clouds of 80–800 m size
(1–10 s flight time) where cloud top temperatures could be
improved by up to 1 K. These examples revealed only
a small range of applications where the reconstruction method may
improve the measurement.
The reconstruction method presented here for pyrgeometer can be
adapted to pyranometer and other sensors with slow response times
such as contact thermometer or capacitive hygrometers. These
sensors are often used in radio soundings, which also require
a high temporal resolution to resolve temperature inversions or
cloud layers. For example, presented a widely used
method to detect clouds from radio sounding, in which one test is
based on the decrease of relative humidity at cloud top. However,
the humidity sensor of the often used Vaisala RS92 has a response
time of τ= 0.5–20 s depending on temperature. To
avoid this issue, developed a cloud detection
algorithm based on the derivatives of temperature and humidity.
However, a correct estimation of the cloud top temperature may
still be biased because the RS92 temperature sensor also shows
a response time of τ= 0.4–2.5 s depending on air
pressure. These values of hygrometer and thermometer are in the
range of response times discussed here. Based on the theory of an
exponential time response as discussed here,
developed an algorithm to reconstruct
humidity soundings by successive correction the time response in
each data point. They showed that especially a coarse temporal and
dynamic resolution of the measurements complicate the
reconstruction. Increasing the sampling frequency of these
instruments and reconstructing the measurements using the
deconvolution method might significantly improve the data.