Uncertainty analysis for the miniaturized laser heterodyne radiometer (mini-LHR) for the measurement of carbon dioxide in the atmospheric column

Presented here is a sensitivity analysis for the miniaturized laser heterodyne radiometer. This passive, ground-based instrument measures carbon dioxide (CO2) in the atmospheric column and has been under development at NASA/GSFC since 2009. The goal of this development is to produce a low-cost, easily-deployable instrument that can extend current ground measurement networks in order to (1) validate column satellite observations, (2) provide coverage in regions of limited satellite observations, (3) target regions of interest such as thawing permafrost, and (4) support the continuity of a long-term climate record. In this paper an uncertainty analysis of the instrument performance is presented and compared with results from three sets of field measurements. The signal-to-noise ratio (SNR) and corresponding maximum uncertainty for a single scan are calculated to be 329.4 ± 1.3 by deploying error propagation through the equation governing the SNR. Reported is an absorbance noise of 0.0024 for six averaged scans of field data, for an instrument precision of 0.14 ppmv for CO2.


Introduction
The miniaturized laser heterodyne radiometer (mini-LHR) [1][2][3] is a portable ground instrument that measures greenhouse gases in the atmospheric column that is under development at NASA Goddard Space Flight Center. While LHRs have been in common use at NASA for several decades [4,5], development of the miniaturized version began in 2009 and regular measurements in the atmospheric column have Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. been ongoing since 2012. The miniaturization of the LHR instrument was made possible by recent developments in the telecommunication industry, including the implementation of low cost distributive feedback (DFB) lasers, and the use of fiber optics to eliminate free space optical pathways. In addition to testing at NASA, the instrument has been fieldtested at three sites to establish instrument performance:  Collimating optics, attached to an AERONET suntracker, collect sunlight that has undergone absorption by CO 2 . The sunlight is mixed with a DFB laser (local oscillator) in a fast InGaAs detector. The resultant beat signal is amplified by a series of amplifiers, and then detected by a square law detector. The CO 2 absorption feature is measured by scanning the local oscillator across the feature and measuring the changes in amplitude of the beat signal.
The mini-LHR operates in tandem with an AERONET [6] sun photometer-using its sun tracker as a platform for the mini-LHR light collection optics. AERONET is an established global network of more than 450 instruments [6][7][8] and provides a path for future mini-LHR deployments. This partnership offers a simultaneous measurement of aerosols, known to be important modulators in regional carbon cycles, and includes coverage in key artic regions.
In this paper, an estimation of absorbance noise for the mini-LHR is presented to validate the accuracy of the measurements by analyzing component-level sources of error. The analysis resulted in a detailed characterization of a signal-to-noise and uncertainty in measurement. By adding together the instrument noise (found through the signal-tonoise ratio (SNR) calculation) and the experimental error (found through the uncertainty of the measurement) a total value for absorbance noise was calculated.

Description of the mini-LHR instrument technique
The mini-LHR, shown in figure 1, is a passive instrument that measures the concentration of CO 2 in the atmosphere through its absorption of sunlight in the infrared. Sunlight is collected using collimating optics (0.2 • field of view) attached to the side of an AERONET suntracker with a pointing stability of 0.1 • , and then modulated with an optical chopper. The sunlight is then superimposed with light from a DFB laser (centered at 1611.51 nm) in a 50/50 single-mode fiber coupler, and then mixed in a fast (10 GHz) InGaAs photodiode. The resultant beat signal, or intermediate frequency signal, is amplified with a stage of RF (radio frequency) amplifiers and detected with a square-law detector that outputs a dc voltage proportional to the square of the input voltage. The RF output is further amplified and low-pass filtered through a video amplifier circuit. The final output is processed with a lock-in amplifier to measure the magnitude of the beat signal. As the laser is scanned across the CO 2 absorption feature, the magnitude of the beat signal is monitored [1].
The retrieval algorithm, developed at the George Washington University [1,9], converts the scanned absorption feature to a column mole fraction. In this algorithm, the atmosphere is modeled as a series of spherically symmetric shells with boundaries specified at defined altitudes. Profiles of atmospheric temperature, humidity, and pressure used in the retrieval algorithm are taken from the Modern-Era Retrospective Analysis for Research and Applications [10]. Spectra are then calculated for 100 m steps from ground level to 60 km. The resultant spectra are then fit using a Nelder-Mead simplex algorithm [11] iterating the CO 2 concentration, wavelength adjustment (to account for slight dither in laser calibration), and instrument bandwidth. A multi-linear regression is then used to determine concentrations.
In the current instrument configuration column CO 2 is measured at 8 min intervals throughout the day during sunlight hours when clouds are not present. Figure 2 shows an example of a typical single data scan collected in May 2013 at the NOAA Mauna Loa Observatory in Hawaii. The data is shown with a modeled fit of the absorption feature produced by the algorithm. The data was collected with an integration time of 500 ms for each data point. The scan stepped across the absorption feature at 0.01 nm increments for the wings, and 0.002 nm increments for the feature. The increased step magnitude on the wings was done to increase scan speed, within AERONET observation window of less than 2 min.

Uncertainty analysis
The following analysis calculates a value and uncertainty for the SNR for the mini-LHR in order to calculate the absorbance noise. The absorbance noise ( abs ) (fluctuations in the absorbance signal) is the combination of the instrument (systematic) noise ( sys ) and the experimental (random) noise ( ran ): abs = sys + ran (1) where sys = 1/SNR and ran = SNR /SNR and SNR is the uncertainty in the SNR. The uncertainty in the SNR ( SNR ) was found by measuring uncertainties for each component in the instrument and tracking the propagation of the error. Subsections 3.1-3.5 provide detail for each of the component-level measurements.
For a laser heterodyning radiometer implementing a distributive feedback laser, the SNR is given as [12]: where T BB is the temperature of the blackbody (the sun), η e is the effective quantum efficiency of the InGaAs photodiode, B is the bandwidth of the intermediate frequency (or IF), ν is the frequency, h is Planck's constant, κ B is Boltzmann's constant, τ is the equivalent integration time and T 0 is the total transmission factor [12]. In this system the signal is proportional to the blackbody radiation (calculated via Planck's law, and degraded by transmission loss and quantum efficiency of the detector), and the noise is proportional to the 1/ √ Bτ [13].With values discussed below, and assuming a blackbody temperature of 5777 ± 10 K [14], a SNR of 329.4 is calculated, correlating to an instrument noise of 0.0030.
The uncertainty in the SNR, which is responsible for the random error for this instrument, is dependent on the component-level uncertainties. The error propagation is then calculated through: where SNR is the uncertainty of the SNR, dq/dx is the partial derivative of the SNR equation with respect to individual components of the equation, and x is the error that is associated with each component of the SNR equation [15]. The uncertainty in the SNR calculated through the error propagation is ± 1.3. This corresponds to a fractional uncertainty of the calculated signal, or random noise component of the instrument, of 0.038. The component level values and uncertainties are listed in table 1.

IF bandwidth (B)
The IF bandwidth of the instrument is the range of beat frequencies that can be measured by the mini-LHR. The

Integration time (τ )
The uncertainty in the integration time was determined by analyzing an electric signal with a low duty cycle. A pulse generator was configured to send an electric pulse, at a relatively slow frequency (200 Hz) and a relatively small pulse width (5 μs), into the input of the lock-in amplifier. The lock-in amplifier could not 'lock' onto the reference signal due to the small duty cycle (long time between pulses), and thus the 'integration window' could be measured by digitizing the output of the lock-in amplifier. The resulting 'integration window' was measured on an oscilloscope and was equal to twice the integration time (due to the lock-in amplifier sampling both positive and negative phases). Using an 80 μs integration time, a histogram of the 'integration window' width was complied. The resulting error measurement from a Gaussian fit analysis produced associated uncertainty of 0.024%. For an integration time of 500 ms (consistent with the operation of the mini-LHR) there is an uncertainty in the measurement of 500 ± 0.120 ms.

Total transmission factor (T 0 )
The total transmission factor was determined by isolating the optical components of the mini-LHR, as shown in figure 4. Light from a DFB laser, centered at 1611.51 nm (a) was first sent into a 50/50 fiber coupler/splitter (b). One end of the output was sent to a power meter (c) to monitor input power. The other end of the fiber coupler was sent into a collimator, to make it a compatible source for input into the optical components of the mini-LHR. Components (a)-(c) represent the input source, of known intensity, for the mini-LHR. Components (d)-( f ) are the isolated optical components of the mini-LHR. The optical loss was determined by measuring the power at the exit point of this isolated system (g). Component (h) was a terminated fiber that was not in use.
The optical loss was determined by the ratio of the power measured at (c) (input power) to the power measured at (g) (output power).
In addition, optical fluctuations were measured in the input of the mini-LHR when in operation tandem to an AERONET sun photometer. The addition of this uncertainty yielded in a total transmission factor of 0.022 ± 0.001.

Frequency (ν)
The uncertainty in the frequency of the local oscillator of the mini-LHR was measured by recording the wavelength stability using an optical spectrum analyzer. The central wavelength was set and measured for an extended period of time to simulate a full day of operation. This helps to include any long term changes in the laser operation into the uncertainty measurement. This measured value was converted to frequency with a final value of (1.860 322 ± 0.000 0004) × 10 14 Hz.

Quantum efficiency (η e )
The photodiode sensitivity was measured by sending a 1611.51 nm laser into a 50/50 fiber splitter. One side of the splitter sent light to a power meter while the other side was directed into the photodiode. The photodiode was then connected to an oscilloscope to monitor the amplitude of the signal. Data was collected for an extended period of time to simulate a full day of data collection, and thus any drift or degradation would be incorporated into the error measurement. This value for photodiode sensitivity was then converted into quantum efficiency using the relationship R = ηe/hν (4) where R is the measured responsivity, η is the quantum efficiency term that is in question, ν is the frequency, and h is Planck's constant [13]. This equation was rearranged such that it was solving for the quantum efficiency and then error was propagated through the equation using the general error propagation equation to arrive at a final uncertainty value for quantum efficiency of 0.5124 ± 0.0001.

Comparison of analytical analysis with field data
Using the previously discussed component errors (section 3) and implementing them into the general error propagation equation, the SNR and error were calculated as 329.4 ± 1.3. This corresponds to an absorbance noise calculation of 0.0068, through the addition of instrument and experimental noise. These values are comparable to the current performance of the mini-LHR. Since its initial development in 2009, the mini-LHR has undergone significant changes to reducing error. The absorbance noise for a single random scan was seen to decrease from 0.0216 to 0.0089 to 0.0054 for field campaigns in Park Falls, Atwater, Mauna Loa, respectively. The absorbance noise was decreased mostly through the implementation of a shorter scan time. Other factors that helped to reduce noise were an upgrade in the quality of InGaAs photodiode, and the replacement of the DAQ with a lock-in amplifier. Data sets from Mauna Loa show that the single scan performance is within 0.0014 of calculated theoretical performance.
To correlate the noise in a typical scan to the detection limit in the column, a Monte Carlo simulation was used to calculate the CO 2 concentration given varying levels of random noise. The concentration of CO 2 in the atmosphere was treated as a constant (at 390 ppmv) and the zenith angle was assumed to be 25 • . The results of these simulations are shown in figure 5 where the standard deviation in the calculated CO 2 concentration is plotted as a function of absorbance noise. Figure 6. The mini-LHR absorbance noise has improved from 0.0202 to 0.0024 for lowest absorbance noise achieved by averaging scans in a given hour throughout the three field campaigns. Here the absorbance noise is compared to the number of averaged scans.
An absorbance noise of 0.0068 corresponds to a sensitivity of 0.4 ppmv.
For the modeling efforts that this instrument supports, only hourly reports of trace gas levels are required. Thus, it was possible to reduce the error further by averaging consecutive scans. Figure 6 shows the number of scans averaged in a given hour and the reduction of absorbance noise. During the first field campaign at the Park Falls, the scan time was 20 min and the lowest absorbance noise reached, in a given hour, was 0.0202 for an average of three scans. At the second field campaign in Atwater, the scan time was reduced to less than 3 min by transitioning from LabView to Python software for data acquisition, resulting in an absorbance noise of 0.0027, when an hours' worth of data was averaged. During the third field campaign at Mauna Loa Observatory, the scan time was reduced to less than a minute and an absorbance noise of 0.0024 was reached in six scans. Only six scans could be collected within a given hour because the AERONET suntracker was operated in an automated mode which was set up to make a measurement every 8 min. Once again referring to figure 5 an estimated precision of 0.14 ppmv is determined.

Conclusions
In this paper a characterization and quantification of errors and uncertainties for the miniaturized laser heterodyne radiometer (mini-LHR) instrument has been presented through analytical analysis, yielding a signal-to-noise ratio of 329.4 ± 1.3. The calculated absorbance noise from this analysis (0.0068) is within 0.0014 of the absorbance noise of a field collected scan (0.0054). The minor discrepancy in these values is due to an overestimation of the random error component of the calculated absorbance noise ( abs = sys + ran )specifically in the IF bandwidth. By averaging multiple scans, a column precision 0.14 ppmv in molar column concentration of CO 2 was achieved.