Abstract

This paper provides atmospheric CO2 column abundance measurement results from a summer 2011 series of flights of a 2.05-μm laser absorption spectrometer on the NASA DC-8 research aircraft. The integrated path differential absorption (IPDA) method is used for the CO2 column mole fraction retrievals. This instrument and the data analysis methodology developed to achieve retrievals over complex terrain and variable atmospheric conditions provide insight into the capabilities of the IPDA method for both airborne measurements and future global-scale CO2 measurements from low-Earth orbit pertinent to the proposed NASA Active Sensing of CO2 Emissions over Nights, Days, and Seasons (ASCENDS) mission. Demonstrated in this paper is the capability to measure CO2 drawdown caused by crop activity during a midday flight over the U.S. upper Midwest area. In addition, an example is provided of high spatial resolution measurements of CO2 plumes from individual stack clusters of the Four Corners Power Plant in northwestern New Mexico. Complex terrain, the spectral properties of the aboveground scatterers, and potential cloud contamination are factors that complicate the column abundance retrieval. The impacts of these factors and various means of minimizing these influences in the retrievals are discussed.

1. Introduction

Global-scale observations of CO2 mole fractions from Earth orbit, primarily in the lower and midtroposphere with measurement precision equivalent to 1–2 ppm or better on spatial scales of order 102 km × 102 km, are desired to define spatial gradients of CO2, from which sources and sinks can be estimated with much reduced uncertainty than is presently attainable (Rayner and O’Brien 2001; Miller et al. 2007; European Space Agency 2008). Atmospheric CO2 is a long-lived gas, with sources and sinks primarily at the surface. Consequently, a remote sensing technique that can emphasize the lower-tropospheric component, or provide vertical profiles within the troposphere, is preferred. The Greenhouse Gases Observing Satellite (GOSAT), launched in January 2009, has provided valuable insight into the capabilities of a passive spectrometer-viewing reflected solar radiation (Kuze et al. 2009; Crisp et al. 2012). Cloud and aerosol scattering, terrain complexities within the instantaneous field of view (IFOV), and limited signal-to-noise ratio (SNR) outside of daytime midlatitudes are inherent issues that can be mitigated or eliminated using laser absorption spectrometer techniques as differentiated from passive spectrometer techniques. The European Space Agency (ESA) conducted a study of a laser-based concept using the integrated path differential absorption (IPDA) method as an ESA Earth Explorer candidate, leading to a comprehensive report considering two approaches, utilizing the 2.05- and 1.57-μm CO2 absorption bands (European Space Agency 2008). A more detailed exposition of the system-level characteristics and performance capabilities of the various candidate IPDA lidar approaches can be found in Ehret et al. (2008). The National Research Council 2007 Decadal Survey recommended study of a laser-based Earth-orbiting CO2 measurement named Active Sensing of CO2 Emissions over Nights, Days, and Seasons (ASCENDS; National Research Council 2007). The ASCENDS formulation activity is supporting the NASA DC-8 flights of several lidars (laser absorption spectrometers that utilize IPDA rather than range-gated backscatter) to provide insight into the measurement capabilities from airborne platforms as well as demonstrate the relevant instrument technologies. In addition to the Jet Propulsion Laboratory (JPL) Laser Absorption Spectrometer (LAS) instrument, whose measurements we report, instruments operated by groups from the National Aeronautics and Space Administration’s (NASA) Goddard Space Flight Center (Abshire et al. 2010) and ITT Exelis–NASA Langley Research Center (Dobler et al. 2013) were on board and operating during the summer 2011 campaign.

The LAS approach offers the potential to provide the high-accuracy CO2 mole fraction measurements with the vertical and horizontal spatial resolution that is desired by the carbon cycle research community. The LAS probes a well-characterized pressure-broadened absorption line profile with one or more laser frequencies in order to provide weighting functions suitable for retrieving vertical profile information (Menzies and Chahine 1974). An LAS approach to global-scale CO2 measurement from Earth orbit utilizes differentially attenuated multiwavelength backscatter from the earth surface along the suborbital track, with subsequent analysis of the backscatter intensities at each wavelength to retrieve CO2 mole fractions. This is a form of the technique referred to as IPDA. An airborne LAS system utilizing the IPDA technique can obtain mole fraction information on a regional scale with higher spatial resolution than can be obtained from low-Earth orbiting (LEO).

We provide a brief description of the IPDA measurement principles in section 2, followed by a brief description of the instrument in section 3, an overview of the data processing and analysis methodology in section 4, environmental factors that influence the retrievals in section 5, and examples of CO2 retrievals in section 6. These examples are from two flights on the NASA DC-8 research aircraft during August 2011, during a campaign supported by the NASA ASCENDS mission definition activity.

2. Measurement principles

The IPDA technique is not limited to any particular number of transmitted wavelengths. Here we describe the measurement method considering two transmitted wavelengths. The principles of the IPDA technique for two frequency-tuned transmitted (and detected) signals are as follows. (The “online” signal at λ1 is tuned to a frequency that overlaps the CO2 absorption line of interest, while the “offline” signal at λ2 is tuned to a frequency in the vicinity for which there is minimum overlap with absorption lines.)

The instrument measurement parameter of key importance is the transmittance ratio, τ = τonoff, which relates directly to the differential absorption optical depth, DAOD = AODon − AODoff, referring to the optical depths encountered at the online and offline laser sounding frequencies:

 
formula

Here we consider a double pass through the atmospheric path. Then the relationship between DAOD and the (normalized) received power at the online and offline laser frequency channels is

 
formula

The term online in the context of this paper is not restricted to the line center frequency, since we can take advantage of tuning the transmitter to an optimum frequency within the line profile. We assume for the purpose of this introduction of the basic principles that 1) there is no significant contribution to the absorption at either sounding frequency from absorption lines of other gases and 2) the offset between the two sounding frequencies is sufficiently small such that gas continuum absorption, atmospheric aerosol particle extinction, and surface material reflectance contributors to the optical depth produce insignificant differential optical depth. (In practice the influences on DAOD of weak water vapor absorption lines in the neighborhood of 2.05 μm are considered in the actual CO2 retrievals using our airborne instrument data.)

The expression for differential absorption optical depth between online and offline transmitted signals can be expressed as

 
formula

where z = z2 − z1 is the total pathlength or range, σon − σoff is the local differential cross section at z (cm2 mol−1) that is a function of the pressure and temperature at location z, and n(z) is the local CO2 number density. The quantity n(z) can be expressed as

 
formula

where q(z) is the mole fraction of CO2, (z) is the mole fraction of water vapor, and nair(z) is the air number density. (Knowledge of the water vapor is necessary in order to derive the CO2 mole fraction in dry air.) The DAOD as expressed in Eq. (3) is a function of the number density profile of CO2, with a weighting function that is dependent on the pressure and temperature dependence of the CO2 absorption line profile. The DAOD measured through a thin layer of thickness δz at height z in the atmosphere can be expressed as

 
formula

where WF(z) is the normalized weighting function. The vertical weighting depends on the specific displacement of the online laser frequency from the line center because the absorption line is predominantly pressure broadened in the lower atmosphere, which is the region of interest; moving the online frequency farther away from line center results in relatively more weighting at lower altitudes.

In principle, the column-weighted dry air mixing ratio XCO2 can be derived from a measurement of the DAOD, along with knowledge of the spectroscopic parameters, surface pressure, the atmospheric temperature profile, and the water vapor profile. The presence of optically thick clouds precludes a measurement of the full column to the surface. The impacts of thin cloud layers or thick aerosol layers must be considered. Backscatter signals from the surface can still be detected in these cases; however, if the backscatter from thin cloud or aerosol layers in the atmosphere is not filtered out, then their contribution to the overall values of Pon and Poff will reduce the effective pathlength by some amount, which will potentially cause a significant bias in the column retrieval. The effect of the elevated scattering layer on τ = τonoff can be expressed as

 
formula

where is the round-trip online transmittance above the layer and is the round-trip online transmittance below the layer, and RL and RS are the directional reflectances (backscatter) of the atmospheric layer and the surface, respectively (see the  appendix). The effect of the scattering layer (second term) is to increase the effective transmittance ratio τ, that is, decrease the effective differential absorption; the higher the layer altitude, the lower the value of and the more important the second term becomes. Consequently, high cirrus is more consequential than boundary layer aerosol. This is a source of bias if it 1) is sufficiently large to be a significant contributor to the overall bias budget and 2) is not filtered out or otherwise accounted for. As an example using Eq. (6), if the CO2 DAOD below an optically thin cirrus layer in the upper troposphere is 0.6 (single pass) and the surface directional reflectance is 0.1 sr−1 (a moderately low value), then a layer integrated backscatter of 10−4 sr−1 would produce a bias equivalent to 1 ppmv. For cirrus, an integrated backscatter of 10−4 sr−1 would correspond to an optical depth (OD) in the range of 1–3 × 10−3, a very small optical depth. It is certainly advantageous to have a method of filtering out the effects of elevated scattering layers.

There is an optimum DAOD for IPDA sounding (Remsberg and Gordley 1978; Megie and Menzies 1980; Bruneau et al. 2006). If the DAOD is too large, then the attenuation of the online signal results in insufficient online SNR. If the DAOD is too small, then much higher overall SNR at both channels is required to obtain the necessary CO2 measurement precision. In addition, residual sources of bias caused by instrumental imperfections become relatively more significant. The optimum DAOD is close to unity.

The absorption cross section at the probing online frequency is dependent on altitude caused by the pressure broadening of the line profile. The CO2 2.05-μm band strength is such that this permits some flexibility in achieving an optimum or near-optimum DAOD and at the same time a desirable weighting function. In theory, as stated in Menzies and Chahine (1974) and by other authors more recently, moving the online transmitter frequency from line center to one or more pressure-broadened half-widths displaced from the line center changes the preferential weighting from the upper to lower troposphere. When using the most favorable lines in the 2.05-μm band, optimum DAOD is achieved when detuning the online frequency 1–3 (surface pressure) half-widths from line center frequency. The corresponding weighting favors the lower troposphere and peaks at the surface, enabling the selective probing of the CO2 in the lower troposphere, where the CO2 mixing ratio variability of interest is the highest. [Typical weighting functions for CO2 sounding are found in Menzies and Tratt (2003).] The 2.05-μm band strengths are also sufficiently large to enable ground-based differential absorption lidar (DIAL) observations of CO2 with line-of-sight spatial resolution at the 0.5–1-km scale (Koch et al. 2008; Gibert et al. 2008; Ishii et al. 2012).

3. Airborne CO2 LAS instrument

A detailed description of the 2-μm CO2 LAS instrument can be found in Spiers et al. (2011). The instrument was developed jointly by JPL and Lockheed Martin Coherent Technology (LMCT) in the 2002–05 time period (Spiers et al. 2002). The transceiver approach utilizes heterodyne detection, with narrow-line width, frequency-stabilized laser transmitters and local oscillators. Each channel (online and offline) has a dedicated heterodyne detector, and a continuous-wave (CW) single-frequency laser that acts as both the transmit laser and the local oscillator for heterodyne detection of the return signal. The transceiver also includes a separate low-power CW reference laser that provides a frequency reference for offset locking of the online and offline lasers. The photomixers are indium gallium arsenide (InGaAs) photodiodes. The lasers are diode-laser pumped rare-earth ion doped crystal lasers, specifically yttrium lithium fluoride (YLF) crystal with thulium (Tm) and holmium (Ho) dopants. An internal low pressure CO2 gas absorption cell is used to lock the reference laser to the selected CO2 R(30) absorption line of the (20°1)III ← (00°0) band, with the line center at 4875.749 cm−1. The cell is hermetically sealed and temperature controlled. The transmitter laser frequencies are precisely stabilized at selected offsets with respect to the reference laser using mixers and frequency offset lock loops. The reference laser is stabilized at the line center frequency, the online laser is stabilized at 4875.882 cm−1, and the offline laser is stabilized at 4875.225 cm−1 (see the Fig. 1a spectrum; Voigt profiles were assumed in the calculation of this spectrum.) In addition to the strong lines of the 12C isotope, a few weak, spectrally narrow 13C isotope lines appear in the vicinity as well as the relatively broad water vapor lines (indicative of the water vapor being primarily in the boundary layer). The normalized weighting function, WF(z), corresponding to these online and offline frequencies is plotted in Fig. 1b. [Recently, Caron and Durand (2009) reported the results of a systematic comparison of online frequency choices, conducted for the Advanced Space Carbon and Climate Observation of Planet Earth (A-SCOPE) mission study, a comparison that includes potential errors caused by atmospheric modeling uncertainties. The R(30) was selected, although a different point on the R(30) line profile is preferred for a space mission.] The functional layout of the transceiver optical configuration is depicted in Fig. 2. A summary of the instrument parameters is provided in Table 1.

Fig. 1.

Airborne CO2 LAS methodology: (a) vertical path atmospheric spectra for commonly used atmospheric models in the vicinity of the CO2 R(30) line centered at 4875.749 cm−1 and (b) corresponding (peak normalized) weighting function.

Fig. 1.

Airborne CO2 LAS methodology: (a) vertical path atmospheric spectra for commonly used atmospheric models in the vicinity of the CO2 R(30) line centered at 4875.749 cm−1 and (b) corresponding (peak normalized) weighting function.

Fig. 2.

Airborne LAS transceiver functional layout.

Fig. 2.

Airborne LAS transceiver functional layout.

Table 1.

JPL airborne LAS instrument parameters.

JPL airborne LAS instrument parameters.
JPL airborne LAS instrument parameters.

A frequency offset is required between the return signals and their corresponding local oscillators for low-noise heterodyne detection. By pointing the transmit beams at a known offset from nadir, the return signals experience a nominally fixed Doppler shift for a given aircraft velocity, thereby eliminating the need for an additional frequency shifting device in the receiver. For a platform velocity of 60 m s−1 and a transmit angle close to nadir, the change in Doppler frequency shift with the off-nadir-pointing angle along the flight track, at an operating wavelength of 2051 nm, is about 1.0 MHz degree−1 (i.e., angle offset along the aircraft motion vector). In practice the aircraft pitch angle adds to the fixed off-nadir-pointing angle with respect to the LAS instrument mounting frame. The receiver Intermediate frequency (IF) bandwidth can accommodate the signal frequency shifting because of the aircraft pitch angle variations during normal cruise. The NASA DC-8 ground speed is a function of flight altitude and varies in the range of 150–250 m s−1 for the range of altitudes covered by this flight campaign. However, as the airspeed increases, the aircraft pitch angle decreases such that the heterodyne offset frequency during straight and level flight remains within a less than 10-MHz bandwidth centered about the 15-MHz offset frequency. If not taken into account, the CO2 measurement error associated with this Doppler frequency variation, which occurs on the return path only, is approximately 0.1 ppm MHz−1. Doppler shifts due to pointing angle variation are an important ASCENDS design consideration.

The transceiver assembly is mounted to a two-sided optical bench, with custom-designed optical mounts for each of the components. The optical bench is mounted vertically inside a hermetically sealed container, with a cover fastened to the base plate. An O-ring seal allows the transceiver environment to be held near (sea level) atmospheric pressure, with the internal pressure and temperature continuously monitored. Thus, the transceiver assembly is contained within an enclosure with electrical feedthroughs and optical windows. Vibration mounts inside the container are used to isolate the bench from the aircraft environment so that the container plate can be hard mounted to the aircraft via an interface frame that is specific to the aircraft. Since the transceiver subsystem is located in the rear cargo bay of the DC-8, the ambient temperature range can be quite large, varying by up to 60°C over the course of a flight. A temperature controller is employed to limit the maximum and minimum temperatures of the assembly when powered up.

4. Signal processing and data analysis

Our approach to data analysis and CO2 retrieval for calculating values for various atmospheric layers is first, the LAS online and offline signals are sampled, stored, and processed as described below.

 
formula

where P stands for power. Then these results, derived from the measurements, are compared with forward model predictions of DAOD for a range of column average CO2 mixing ratios. We use the line-by-line radiative transfer model (LBLRTM) provided by Atmospheric and Environmental Research Inc. (AER), modified to include a merged line parameter database. In the 5 cm−1 region centered at 4875.5 cm−1, a set of line parameters for CO2 and H2O provided by G. C. Toon (2009, personal communication) were substituted for the AER version 2.1 line parameter database. Line coupling–mixing coefficients (Niro et al. 2005) were included. The Toon line parameters were compiled using the approach described in Hartmann et al. (2009). The forward model is based on the LBLRTM code with this modified line parameter database, plus the atmospheric characterization data needed to provide the normalized weighting function, WF(z). The atmospheric data consist of the surface pressure, and the profiles of temperature and relative humidity. The onboard GPS system provides the aircraft position knowledge (including altitude with respect to the geoid). We rely on a combination of the DC-8 radar altitude and the Shuttle Radar Topography Mission (SRTM) elevation database (Farr et al. 2007) for surface elevation along the ground track.

The IF photomixer signals from the online and offline channels are amplified and the bandwidth is limited to a nominal 8–21-MHz window. The signals from each channel are digitized with a 50 megasamples s−1, 14-bit digitizer. The samples are transformed into the spectral domain using an FFT operation followed by conversion to periodograms. The commonly used “squarer” estimator (Rye and Hardesty 1997) is used to determine the return power in each channel. The default FFT length is 16 K. The integrated power of the signal in the periodogram is related to the intensity of the return signal incident on the photomixer.

When the spatially coherent laser transmitter radiation is backscattered from a rough surface (i.e., rough compared with the laser wavelength), a speckle pattern of backscattered radiation is produced at the plane of the receiver aperture. When the instrument platform is moving, the backscattered radiation takes the form of a dynamic speckle pattern. Independent (uncorrelated) samples of the speckle are obtained as the aircraft moves along the flight track, with speckle decorrelation occurring on the time scale of τdecorr = d/2υacg (where d is the instrument aperture diameter, υacg is the aircraft ground speed. In our case, on the DC-8 aircraft, the sampling duration (320 μs for the 16 FFT) is comparable to the speckle decorrelation time of the signal τdecorr. The probability density of a collection of independent samples obeys negative exponential statistics (Goodman 1975).

Power estimation requires an approach to managing speckle fluctuation effects. Averaging over speckle fluctuations can occur in both spatial and temporal domains. Direct detection lidar optics can be designed such that spatial averaging of speckle lobes occurs at the receiver telescope aperture. Coherent (heterodyne) detection lidar design requires spatial mode matching at the photomixer, precluding spatial averaging at the telescope aperture (or subaperture if multiple photomixers share a common telescope). Obviously, precision in power estimation demands averaging over multiple temporal realizations of the speckle field.

A preselected number of individual periodograms is summed; and the remainder of the signal processing steps operate on a collection of these sums. The signal power becomes a gamma-distributed random variable. The sum of k independent exponentially distributed random variables, each of which has a mean value θ, can be described by the gamma function, f(x; k, θ), with integer values of k. The equation defining the probability density function of a gamma-distributed random variable x is

 
formula

whose shape approaches a Gaussian with increasing k, in accordance with the central limit theorem. For example, selecting a value g = 122 16-K periodograms in the initial summation corresponds to an integration over 40 ms of data. Since τdecorr is ~0.25 ms, k is approximately 160 in this case. We must distinguish between g and k, since the value of k depends on τdecorr, which is influenced by the aircraft ground speed. Neglecting headwind or tailwind influences, the nominal ground speed is ~200 m s−1. Our algorithm initially detects the return signal in the offline periodogram and estimates its heterodyne frequency over short time periods (e.g., 40 ms). The online signal is expected to be at the same heterodyne frequency. Estimates of offline and online signal power are made after summing over a range of time periods (e.g., starting at 40 ms, extending to ~10 s, depending on the circumstances and the measurement objectives). Laser transmitter range to ground is calculated at a 10-Hz rate, based on SRTM and aircraft GPS and attitude data, and is used to compensate for aircraft attitude variability. Since aircraft ground speed and attitude variability cause shifts in the heterodyne signal frequencies, the algorithm has the capability to shift the periodograms accordingly when summing so that the return signal peaks align prior to power estimation. Sudden shifts caused by, for example, encountering turbulence, are recognized by the algorithm, and those time periods are filtered out prior to power estimation.

Characterization of the in-flight instrument stability is critically important. The instrument includes an onboard “validator”: a modified belt sander that provides a Doppler-shifted backscattering signal for each channel. The validator is inserted into the LAS field of view at convenient intervals, by means of remote commanding of a stepper motor, in order to monitor the overall radiometric stability of the instrument. It intercepts the outgoing (offline and online) laser beams and provides a stable source of backscatter. This methodology enables the monitoring of variations in system gain ratio at the precision level of 0.3% and the compensation for these variations in the retrieval algorithm. Additional validation during the flight campaign is accomplished using the Picarro in situ data taken during a spiral above a selected surface location, and comparing with the retrieved weighted-column CO2 when flying over the selected location at fixed altitudes.

5. Environmental factors

This instrument and the data analysis methodology developed to achieve retrievals over complex terrain and variable atmospheric conditions provide insight into the capabilities of the IPDA method for both airborne measurements and future global-scale CO2 measurements from low-Earth orbit that are pertinent to the proposed NASA ASCENDS mission. Although the ASCENDS mission requirements have not yet been finalized, the goals of the airborne measurement demonstrations are measurement precision levels of 1-ppm column CO2 or better over along-track spatial scales of a few tens of kilometers and an understanding of how to categorize and quantitatively evaluate the sources of measurement bias (e.g., spectroscopic modeling, effects of the instrument environment, weighting function errors) at the 1 ppm level or better. The capability to characterize point sources and regional sources is also important. In these cases, the spatial resolution is increased and the measurement precision is relaxed accordingly.

The JPL airborne LAS instrument was one of four instruments on board the NASA DC-8 research aircraft in early August 2011, assessing the capability to retrieve atmospheric CO2 column mixing ratios with high precision in a variety of circumstances. These included partially cloudy atmospheres and a wide range of physical characteristics of the surface locations that were intercepted by the ground tracks. Under these circumstances, several environmental factors exist that complicate the retrievals. We include a discussion of the following, along with methods for minimizing their effects, particularly as potential sources of bias:

  1. cloud detection/filtering

  2. topography

  3. spectral reflectances of surface and aboveground scatterers

a. Cloud detection and filtering

This pertains to the retrieval of column (weighted) CO2 mole fractions in the presence of cloudy atmospheres, specifically scattered cloud and broken cloud cover cases. Clouds in the field of view (FOV) reduce the pathlength, and if not recognized, bias the CO2 retrieval. In cases of scattered cloud and broken cloud cover, breaks or holes exist that permit soundings down to the surface some fraction of the time. The small transmitter footprint of the lidar provides an inherent capability to acquire retrievals in such circumstances. If the lidar provides time of flight to the backscatter source [e.g., a range-gated pulsed system, or a frequency-modulated (FM)–CW system], then any sources of backscatter other than that which occur at the expected range to the surface can be set aside or filtered out. With the current implementation of our airborne system, we do not have this capability. We do not chirp (frequency modulate) our transmitters. However, we do employ alternative methods to detect and filter out the backscatter signals that are due to clouds in the FOV. On what properties and/or capabilities are these methods based? Consider the following:

  • Heterodyne detection provides the capability to see both intensity and spectral properties of backscatter signal.

  • Cloud motion provides a discriminating tool, both broadening and shifting the backscatter signal in the spectral–frequency domain.

  • Clouds in the FOV also cause shortening of atmospheric sounding pathlength—reduced values of retrieved CO2 column.

The heterodyne signals backscattered from the surface are sufficiently narrow to permit identification of cloud backscatter if the cloud movement relative to the surface, along the line of sight, exceeds 0.5 m s−1. Since the typical point-ahead angle in the DC-8 is ~0.1 rad, this corresponds to a threshold horizontal motion of 5 m s−1. However, in practice, the backscatter signals from cumulus and stratocumulus are spectrally broadened, compared with the ~200-kHz full width at half maximum (FWHM) signals backscattered from the surface in clear-air conditions. This provides another filtering method. An example is shown in Fig. 3. The flight on 10 August 2011 headed eastward from Palmdale, California, across the Rockies, and the states of Colorado, Nebraska, and Iowa. The transit across these states occurred during midday, extending through early afternoon. Fair weather cumulus cloud cover increased as the ground track moved into eastern Colorado and through Nebraska and Iowa. The example in Fig. 3 is in the vicinity of the West Branch Iowa (WBI) tall tower. [The WBI tower is a component of the tall tower network that provides regional measurements of CO2 and related gases in the continental boundary layer to the National Oceanic and Atmospheric Administration (NOAA)’s Earth System Research Laboratory (ESRL) for various studies.] This spectral broadening is typical of backscatter from cumulus and also stratocumulus. Broadened backscatter signals from stratocumulus were observed during a 2 August flight over the Pacific off the coast of Southern California. Periodograms corresponding to 40 ms of integration time (about 8 m of alongtrack averaging for the nominal 200 m s−1 aircraft speed) are filtered out of the CO2 retrieval data if the signal spectral widths exceed the normal value associated with surface returns.

Fig. 3.

Cloud broadening of signal spectral width. (a) Camera image showing fair-weather cumulus in the vicinity of WBI tower during an overflight on 10 Aug 2011. (b) The 40-ms periodogram signal is broadened in comparison with a typical surface backscatter signal due to penetration into the cloud and the relative motions within the cloud.

Fig. 3.

Cloud broadening of signal spectral width. (a) Camera image showing fair-weather cumulus in the vicinity of WBI tower during an overflight on 10 Aug 2011. (b) The 40-ms periodogram signal is broadened in comparison with a typical surface backscatter signal due to penetration into the cloud and the relative motions within the cloud.

Another filtering method depends on the ability to discern an abrupt reduction in measured DAOD that would likely be due to the presence of a cloud in the FOV. Clouds shorten the effective pathlength over which the differential absorption measurement is made, thus causing a decrease in the measured DAOD when present. For a cloud OD above a threshold value, this decrease is evident. An example is shown in Fig. 4, during a traverse–overpass of the WBI tower. The ground-track distance covered by this traverse, from left to right, is approximately 60 km. The sharp boundaries between the relatively stable CO2 column mixing ratio in the 370–375-ppm range and the depressed values are evidence of the cloud impacts. The smaller depressions in retrieved CO2 column in the 21.78–21.79 UTC time period (corresponding to about 7 km of track length) are due to occasions when the FOV passed over segments of thin cloud, barely visible in the nadir camera images. The OD threshold for cloud detectability depends on the instrumental DAOD measurement uncertainty over an appropriate averaging interval, and also on the cloud altitude. For the JPL LAS instrument, with increasing altitude the increased averaging needed in order to achieve the desired DAOD measurement precision in the clear air columns tends to smear the boundaries between cloud and clear air, rendering this method ineffective. The surface reflectance at the 2.05-μm lidar wavelength is also a factor. The fundamental measurement uncertainty is due to speckle fluctuations.

Fig. 4.

Impact of scattered cloud cover on column CO2 retrieval during a WBI tower traverse leg at ~2850-m altitude above ground level on 10 Aug 2011. Each decrease in the effective pathlength with sharp boundaries corresponds to the appearance of cloud cover in the FOV during the traverse. Total time duration of the leg plotted here, 0.08 h, corresponds to approximately 60 km of ground-track length.

Fig. 4.

Impact of scattered cloud cover on column CO2 retrieval during a WBI tower traverse leg at ~2850-m altitude above ground level on 10 Aug 2011. Each decrease in the effective pathlength with sharp boundaries corresponds to the appearance of cloud cover in the FOV during the traverse. Total time duration of the leg plotted here, 0.08 h, corresponds to approximately 60 km of ground-track length.

b. Topography

Knowledge of the surface elevation and the associated surface pressure at the location of the lidar footprint is important for the CO2 retrievals. The accuracy with which surface elevation must be known depends somewhat on the circumstances; however, a ballpark estimate is provided by the relationship between small changes in pressure caused by small changes in elevation, dp/p ~ dz/H, where H is the scale height of the atmosphere, approximately 8 km. Thus, a 0.1% change corresponds to dz ~ 8 m. Near sea level this corresponds to dp ~ 1 hPa.

When the ground track is over hilly or mountainous terrain, the rapid changes in surface elevation require either coaligned laser altimetry or other tools and data sources in order to control the contribution of elevation errors to the overall error budget. We use the SRTM database along with the aircraft inertial navigation system (INS)–GPS data to determine surface elevation along the ground track, with an along-track resolution of approximately 50 m. A two-dimensional smoothing algorithm is used to avoid discontinuities in the elevation versus time as the ground track crosses the SRTM 25-m pixel boundaries. Our online and offline signal power data are archived at a 25-Hz rate (8-m averaging along track at the nominal aircraft 200 m s−1 ground speed), with further averaging prior to calculation of the ln(ratio)—as in Eq. (2). The highest-resolution ln(ratio) data are approximately 200-m averages along the ground track. The limiting term in the range to scattering surface error budget is the scattering surface elevation. SRTM vertical precision is ~8–10 m; however, trees influence in different ways the measurement of scattering surface elevation by the SRTM radar versus the infrared lidar. Sloping surfaces also can deteriorate the estimate of “range to ground” averaged over, for example, 50 m. As can be seen in Fig. 5, the slope statistics over two mountainous terrain ground tracks indicate, at the 50-m resolution scale, that slopes are greater than 10° about half the time; that is, an elevation change of 10 m or more over a 50-m ground-track resolution element is ~50% probable. The ground elevation range (from lowest to highest elevation) was about 3500 m for each of these two tracks (over the Sierra Nevada and British Columbia Coast Mountains). The slope statistics for these two cases are remarkably similar. We update the aircraft position and attitude at a 10-Hz rate and apply those data along with the SRTM data to our range-to-ground computation algorithm. Elevation slope versus time can then be computed, and a filter can be set to exclude data at times for which the slopes are above a threshold value. Obviously, the effects of subgrid-scale roughness (e.g., smaller-scale ground slopes, aboveground flora) are not included in our retrieval algorithms, since we do not have a coaligned laser altimeter; however, some discussion of these effects follows. It is worth noting in general that with lidar technology, high-spatial-resolution capability can be utilized to mitigate potential biases associated with rapidly changing ground elevation.

Fig. 5.

Cumulative distributions of ground elevation slopes over mountainous terrain, based on the SRTM 25-m pixel database and a 2D 50-m smoothing parameter. (a) Sierra Nevada range overpass, 3 Aug 2012. (b) British Columbia Coast Mountains range overpass, 7 Aug 2012.

Fig. 5.

Cumulative distributions of ground elevation slopes over mountainous terrain, based on the SRTM 25-m pixel database and a 2D 50-m smoothing parameter. (a) Sierra Nevada range overpass, 3 Aug 2012. (b) British Columbia Coast Mountains range overpass, 7 Aug 2012.

c. Spectral reflectances of surface materials and aboveground scatterers

The heterogeneity of surface materials and aboveground scatterers, natural and synthetic, can cause sudden changes in reflectance—that is, backscatter or retro-reflectance in this case—and consideration must be given to an understanding of how this affects the column CO2 retrieval when that retrieval is an average over a portion of the ground track that includes large changes in reflectance. Sharp changes in reflectivity (e.g., water–land boundaries, or road crossings) can be problematic for IPDA systems with displacements between the online and offline footprints (Amediek et al. 2009), necessitating careful investigation of potential sources of misalignment. Reflectance weighting is inherent in the lidar measurement; that is, the high-reflectance areas are weighted preferentially in the average. When combined with elevation changes within the along-track average, this has the potential of biasing the resulting retrieval. Elevation weighting must be implemented in the retrieval algorithm to account for this inherent reflectance weighting within the ground-track averaging segment, in order to mitigate biases associated with this effect. A particularly egregious case is shown in Fig. 6, a ground track over a high-elevation region of the British Columbia Coast Mountains during the 7 August 2011 flight, with snow cover being prevalent. The low backscatter segments correspond to the snow-covered regions. Snow reflectance is quite low in the 1.6- and 2.0-μm spectral regions where the CO2 absorption bands are located (see, e.g., Aoki et al. 2000 and references therein).

Fig. 6.

Surface backscatter at 2.05-μm wavelength (sr−1) during a portion of the snowline out flight segment over the British Columbia Coast Mountains, 7 Aug 2011. Snow-covered areas (low backscatter) were mixed with patches of bare rock, dirt, and alpine flora. Time duration from left to right: 0.1 h (6 min). The two panels differ only in scale. (bottom) Plot shows the low reflectance snow sections with more clarity.

Fig. 6.

Surface backscatter at 2.05-μm wavelength (sr−1) during a portion of the snowline out flight segment over the British Columbia Coast Mountains, 7 Aug 2011. Snow-covered areas (low backscatter) were mixed with patches of bare rock, dirt, and alpine flora. Time duration from left to right: 0.1 h (6 min). The two panels differ only in scale. (bottom) Plot shows the low reflectance snow sections with more clarity.

These snow backscatter measurements are the first 2-μm lidar measurements from the air over snow in the natural environment. The basis of the snow reflectance derivation is the linkage that we have to ocean surface reflectance as measured over the clear Pacific Ocean on 2 August 2011 in the 1603–1605 UTC time frame. The ocean surface reflectance depends on the (10-m height) surface wind speed. The Cloud–Aerosol Lidar and Infrared Pathfinder Satellite Observations (CALIPSO) mission provides by far the largest study of lidar backscatter from the ocean surface, and we rely on data from Hu et al. (2008) for determination of the surface directional reflectance (backscatter) over this region of the Pacific Ocean. We are able to take advantage of the fact that at the JPL LAS nominal 5° off-nadir angle, the surface backscatter is only weakly dependent on surface wind speed in the range from ~(2.5–12) m s−1 (see Menzies et al. 1998, their Figs. 1 and 2). Somewhat later in this flight, the data taken during a spiral indicated that wind speeds near the surface (10-m height) were likely in the range of 6–7 m s−1. We use this and Hu et al. (2008) to derive a value for the ocean surface backscatter at near-nadir view angle, 0.042 sr−1 at 1064-nm wavelength. The ocean surface reflectance includes a Fresnel reflectance term that depends on the complex refractive index of water. Using the constants from Irvine and Pollack (1968) and Hale and Querry (1973), the value of this term at 2.05 μm is 0.90 times that at 1.064 μm. The next step is the comparison of range-corrected offline return signal values: 2 August 2011 over the Pacific Ocean compared with 7 August 2011 flight over the snow-capped British Columbia Coast Mountains. The LAS instrument radiometric stability is good to within 10% from flight to flight, and flight-to-flight and in-flight variations are monitored with the internal validator. We also monitor laser output power continuously throughout the flights.

In this case the snow backscatter averaged over the ground track is 0.012 sr−1. Two implications of the low reflectance are 1) interspersed areas that are not snow covered can cause sudden and dramatic changes in the backscatter levels and raise the overall ground-track segment average backscatter, as seen in Fig. 6; and 2) deposition of most other materials on the snow-covered surfaces over time can significantly raise the effective surface backscatter at 2.05 and 1.57 μm (Aoki et al. 2000; Boeggild et al. 2010).

Small-scale ground elevation variability and aboveground scatterers contribute to subgrid-scale surface roughness. Aboveground scatterers such as trees can cause the surface scattering elevation (SSE) to momentarily deviate several meters from the ground elevation, the SSE obviously being the preferred parameter for the CO2 column retrieval algorithm. If the tree density is low, then the average difference between the SSE and the ground elevation over the ground-track segment is likely to be small enough such that an approximation of the difference is adequate. Over a dense forest, tree height data are needed. Laser altimetry with sufficient range determination accuracy and sufficient along-track resolution can provide valuable data. Can laser altimetry at a shorter wavelength (e.g., 1 μm or less) provide the needed data with sufficient accuracy? Not necessarily.

Consider the case of a coniferous–evergreen needleleaf forest in winter or spring, with snow on the ground. Assume the trees are predominately pine. Assume the snow on the trees has been blown off or melted away; that is, there is no snow (or very little snow) on the tree branches–needles. How will the SSE values measured with altimetry at 1.06 μm compare with the SSE values at the two CO2 bands? Consider the backscatter area within the lidar footprint or a resolution element along the suborbital track to be split equally between the trees and the ground. The effective SSE aboveground elevation depends on the average tree height and the relative tree and ground reflectance values. The following are nominal reflectance values for the trees and the snow-covered ground at three wavelengths: the commonly used Nd:YAG altimeter wavelength, and the two CO2 band wavelengths,

  • forest–coniferous: 1) 0.45 at 1.06 μm; 2) 0.2 at 1.57 μm; 3) 0.1 at 2.05 μm

  • snow: 1) 0.9 at 1.06 μm; 2) 0.07 at 1.57 μm; 3) 0.035 at 2.05 μm

Using these numbers, the approximate SSE values at the various lidar wavelengths are

  1. 1.06 μm: HG + 7 m

  2. 1.57 μm: HG + 15 m

  3. 2.05 μm: HG + 15 m, where HG is the ground elevation

The deltas in effective scattering surface elevation between the various wavelength pairs are weakly dependent on the tree coverage at least in the 30%–70% range. The effective SSE as measured by the CO2 lidars can differ from the SSE as measured by a laser altimeter at a shorter wavelength. In the above-mentioned example, δz can be more than 8 m and can result in an equivalent ~0.4 ppm CO2 bias. The preferred approach is to perform the laser ranging–altimetry at the CO2 wavelength(s) in order to eliminate complications due to the spectral reflectance characteristics of various scatterers.

6. Observation of CO2 drawdown

The 10 August 2011 flight’s primary objective was the upper Midwest, arriving over the target area (Iowa) near midday, with the expectation that CO2 drawdown in the boundary layer would be observed because of the photosynthetic assimilation by crops over this large-scale agricultural region. (This flight was nicknamed the corn flight). After arriving in the vicinity of the WBI tall tower (Miles et al. 2012), a spiral was implemented in order to profile the CO2 mole fraction using an onboard cavity ring-down spectroscopy sensor (Picarro, Inc.), and several fixed-altitude “tower transits” were conducted at different altitudes. (The NASA DC-8 also has onboard sensors, providing atmospheric temperature, pressure, and relative humidity data to the investigator teams.)

The flight to the Midwest included a long transit at fixed pressure altitude starting near Denver, Colorado, and continuing to the vicinity of the WBI tower in Iowa. We encountered clear atmosphere over the Denver area, with scattered fair-weather cumulus appearing over the eastern Colorado plains. Cloud fraction steadily increased as the flight ground track moved into Nebraska. The observed weighted-column CO2 mixing ratio decreased during this time period as shown in Fig. 7. The aircraft flew at a constant 15-kft pressure altitude during this transit. The SRTM digital elevation model (DEM) data were used to obtain the along-track elevation during this transit. The atmospheric meteorological data that were incorporated into our retrieval algorithm came from the Modern-Era Retrospective Analysis for Research and Applications (MERRA; http://gmao.gsfc.nasa.gov/research/merra/intro.php) products available from the NASA Goddard Space Flight Center Global Modeling and Assimilation Office (GMAO). For example, the surface pressure from MERRA, interpolated along this ground track and “corrected” using the higher-resolution topographical data along the ground track, is shown in Fig. 8. The plot start corresponds to a location a few kilometers south of the Denver International Airport, and the distance covered from left to right is 340 km. The along-track averaging corresponds to about 4-km along-track resolution for the plotted data. The flight altitude CO2 readings from the in situ Picarro instrument measurements trended lower over a narrow range from approximately 389.5 to 387.5 ppmv during the period of time plotted in Fig. 9. The column is likely sampling urban-influenced regional boundary layer air at the beginning. Nadir camera imagery shows a transition to agricultural activity (occasional crop circles) beginning at 2002 UTC, with increasing land use for agricultural activity occurring as the ground track continues eastward. Crossing into Nebraska occurred near 20 h 13 min UTC. Gaps in the data are due to the presence of fair-weather cumulus. The ground track is in the middle of Nebraska at the end of the plotted data. By this time the cumulus coverage had increased, with a corresponding decrease in the durations of the clear-air gaps between clouds, precluding the continuation of the high-precision retrievals.

Fig. 7.

LAS weighted-column CO2 mole fraction retrievals during the flight segment from the Denver vicinity to the middle of Nebraska, 10 Aug 2011 (locations: 39.80°N, 104.72°W at 19.98 UTC and 40.58°N, 100.75°W at 20.37 UTC; distance traveled: 310 km). The 1σ precision level for this retrieval is equivalent to 1.1 ppm. Steady decrease in column CO2 is due to midday drawdown in the atmospheric boundary layer.

Fig. 7.

LAS weighted-column CO2 mole fraction retrievals during the flight segment from the Denver vicinity to the middle of Nebraska, 10 Aug 2011 (locations: 39.80°N, 104.72°W at 19.98 UTC and 40.58°N, 100.75°W at 20.37 UTC; distance traveled: 310 km). The 1σ precision level for this retrieval is equivalent to 1.1 ppm. Steady decrease in column CO2 is due to midday drawdown in the atmospheric boundary layer.

Fig. 8.

Surface pressure from MERRA interpolated to the ground track during the flight segment corresponding to Fig. 7, and corrected using the higher-resolution finescale topographic database. The steady rise is predominantly due to the elevation change between Denver and midstate Nebraska.

Fig. 8.

Surface pressure from MERRA interpolated to the ground track during the flight segment corresponding to Fig. 7, and corrected using the higher-resolution finescale topographic database. The steady rise is predominantly due to the elevation change between Denver and midstate Nebraska.

Fig. 9.

Four Corners Power Plant, New Mexico. There are three main clusters of stacks: (left to right) the tall stack (cluster 1), clusters 1 and 2 separation ~400 m, and clusters 2 and 3 separation ~150–200 m.

Fig. 9.

Four Corners Power Plant, New Mexico. There are three main clusters of stacks: (left to right) the tall stack (cluster 1), clusters 1 and 2 separation ~400 m, and clusters 2 and 3 separation ~150–200 m.

The conclusion that the observed steady decrease in column CO2 abundance is due to drawdown is supported by later measurements in Iowa during a traverse over the WBI tower at 10-kft altitude (Fig. 4), where in situ vertical profile data obtained near the WBI tower from the onboard Picarro instrument indicated boundary layer CO2 mole fraction values ~365 ppm, and free troposphere values averaging 382 ppm. From the data shown in Fig. 4, during the 5-min traverse, we computed LAS-observed weighted-column CO2 mole fraction in the clear-air regions averaging 371–375 ppm. The magnitude of this midday decrease in the boundary layer mixing ratio is consistent with other reported measurements and simulations (Miles et al. 2012; Denning et al. 1996). Regional-scale simulations of the CO2 exchange between the atmosphere and the terrestrial ecosystems (Denning et al. 1996) and measurements at the U.S. upper Midwest tall towers (Miles et al. 2012) show peak daytime net ecosystem exchange (NEE) flux values of −50–60 μmol m−2 s−1 in the summertime, corresponding to midday boundary layer CO2 mole fractions in the 360–365-ppm range at corn-dominated sites such as the WBI tower site and the Mead tower site in western Nebraska. Midday CO2 levels in this region during early August are among the lowest in North America because of strong uptake by corn and other crops.

7. Observation of power plant CO2 plume

On 9 August 2011, the DC-8 flew a northward flight segment at 15-kft pressure altitude, whose ground track was downwind of the Four Corners Power Plant, located in San Juan County, New Mexico (36.690°N, 108.483°W). The ground track was within a few hundred meters of the plant site. The plant has five coal-fired units, with spacing such that the emissions appear to originate from three sources. The source encountered first during this flight leg (leftmost in Fig. 9) is the tall stack. Approximately 400 m from this source are a pair of stacks, and approximately 200 m from this pair is a third stack cluster, dark in appearance from the camera imagery. Figure 10 is a plot of the column-weighted CO2 mole fraction during the pass, with variable along-track resolution. The along-track resolution is 15 m during the 1-km segment immediately downwind of the plant, which is clearly sufficient to resolve plumes from the various stacks or stack clusters.

Fig. 10.

Column-weighted-CO2 mole fraction retrievals during flyby at 15-kft pressure altitude along a south-to-north track, a few hundred meters downwind of the Four Corners Power Plant. Shading corresponds to three spatial resolution segments: 1) 37.2–38.2 km: 150-m along-track resolution, 2) 38.2–38.5 km: 50-m resolution, and 3) 38.5–39.5 km: 15-m resolution.

Fig. 10.

Column-weighted-CO2 mole fraction retrievals during flyby at 15-kft pressure altitude along a south-to-north track, a few hundred meters downwind of the Four Corners Power Plant. Shading corresponds to three spatial resolution segments: 1) 37.2–38.2 km: 150-m along-track resolution, 2) 38.2–38.5 km: 50-m resolution, and 3) 38.5–39.5 km: 15-m resolution.

A simple box model estimate of the power plant CO2 emission rate during the midday time of this flight leg can be made by calculating the CO2 mass crossing a plane of height equal to the aircraft height above ground (3135 m) and a ground-track segment length of 1.0 km for which the mole fraction is significantly above the background or baseline value. The speed of the wind carrying the CO2 plume across the plane at this time, 2.15 m s−1, is obtained from the MERRA reanalysis. The atmospheric temperature in the lowest MERRA layer at this time was 299 K. Thus, for these conditions a CO2 mole fraction delta of 1 ppm corresponds to a mass density delta of 1.5 × 10−3 g m−3. With this information, an estimate of the total CO2 plume emission rate can be made. Our observation amounts to an average weighted-column increase of 55 ppm (i.e., Δ〈C〉 = 82.5 g m−3) above the background value of 400 ppm across this plane (with a 1.0-km ground-track segment), where 〈C〉 indicates an effective spatial average concentration over the dimensions of the plane. Assume the relationship between emission rate ε and Δ〈C〉 is

 
formula

where h and W are the height and width of the plane, respectively, and υ is the component of wind velocity perpendicular to the plane. A puff of emission during time dt results in a slice of elevated concentration Δ〈C〉 of thickness υdt, spatially averaged across the plane.

If the weighting function were uniform (altitude independent), then this would imply ε = 560 kg s−1—that is, a 570 kg s−1 source of CO2. Since our weighting function is not uniform, peaking at the surface (Fig. 1), the integrated Eq. (5) from surface to aircraft altitude is relatively more sensitive to the CO2 near the surface, such as this plume. Assuming the plume is within the first 200 m above the surface, where the weighting function is nearly constant, a source of 470 kg s−1 emission rate produces a 55-ppm weighted-column increment.

The Four Corners Power Plant complex emits in the neighborhood of 14 × 106 metric tons of CO2 annually, according to a September 2011 study prepared by RMT, Inc. for the California Public Utilities Commission (RMT 2011). This corresponds to an average CO2 emission of 440 kg s−1. Surely there is some temporal variability in the emission rate—on daily, weekly, or monthly time scales. We do not have that information. However, we do have a measurement that corresponds closely with the average emission rate. This demonstrates the potential capability of the IPDA measurement method. The most significant uncertainty in this estimate of the source emission rate is the wind speed, since we do not have high-resolution wind field data for this location and time. The MERRA value is an average 50-m-above surface wind speed within a 50 km × 50 km grid box containing the Four Corners Power Plant.

8. Conclusions

The summer 2011 flights of the CO2 Laser Absorption Spectrometer have enabled us to assess and demonstrate the capability of the IPDA method to obtain CO2 retrievals for a variety of atmospheric and surface conditions. With 75 MW of transmitted power in the online channel and 10-cm receiver aperture (~6 × 10−4 W m−2 power-aperture product), CO2 retrieval capability with parts-per-million-level precision was demonstrated at altitudes as high as 5 km above the surface. In terms of a power-aperture product, this would scale to a 400-km Earth orbit with, for example, a 10-W online transmitter and a 0.6-m-diameter telescope, both technically feasible.

Measurements made during a midday flight over the U.S. upper Midwest clearly indicate that we can observe the CO2 drawdown due to photosynthesis at the surface. Measurements made in the vicinity of the Four Corners Power Plant demonstrate the capability to resolve the plumes with high spatial resolution and estimate the source emission rate.

We demonstrated measurement capability in scattered and broken cloud cover conditions. We pointed out the challenges associated with high accuracy, low bias retrievals over mountainous regions. However, this should not be considered a fundamental limitation. The addition of coaligned laser altimetry at a suitable wavelength, meeting the necessary vertical- and horizontal-scale-resolution requirements, will provide the additional data to enable high-quality retrievals in these challenging situations.

The demonstrated capability to identify and measure CO2 sources and sinks using this airborne instrument adds validity to the capability to implement the proposed NASA ASCENDS mission in the 2020 time frame.

Acknowledgments

The authors wish to thank Randy Kawa of the NASA Goddard Space Flight Center for provision of the onboard in situ CO2 mixing ratio data and Lesley Ott of the NASA Goddard Space Flight Center for making available the MERRA meteorological parameters, interpolated to the DC-8 ground tracks, for portions of the 9 and 10 August flights.

This research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.

APPENDIX

Impact of Elevated Scattering Layer

Assume nadir or near-nadir pointing of the lidar, with an aerosol or thin cloud scattering layer in the atmosphere. Assume first that the absorptance of the layer is the same for the online and offline wavelengths and can be neglected, since the objective here is to investigate the influence of the layer’s integrated scattering characteristic on the differential transmittance. Also assume that the scattering characteristics from the atmospheric layer and from the surface are the same for the online and offline laser transmitter wavelengths, and that the atmosphere absorbs the online wavelength radiation but not the offline wavelength radiation. What is the effect of the scattering layer on the overall differential transmittance at the online and offline wavelengths? Let

  • Re = reflectance from the elevated aerosol or thin cloud layer,

  • a = surface albedo,

  • Fd = downward flux at top of elevated layer.

Then the upwelling flux at the lidar receiver telescope Fu is the infinite sum:

 
formula

where the first term is the backscattering from the layer; the second term is the component that passes through the layer, reflects from the surface, and passes through the layer again in the upward direction, etc. Assuming Re is small compared with unity and neglecting the terms of order and higher,

 
formula

and the change in albedo due to the presence of the elevated scattering layer; that is, the albedo of the combination minus the albedo of the surface is

 
formula

[This method is described in Paltridge and Platt (1976). Mitchell (1971) derives the same result using heating budget arguments.]

Now add atmospheric absorption due to atmospheric CO2 at the online wavelength. Let Ta be the atmospheric transmittance above the elevated scattering layer and let Tb be the atmospheric transmittance below the layer. Then applying the same method as mentioned above, the upwelling online flux in the presence of the scattering layer is

 
formula

and the resulting change in albedo is

 
formula

Since Eq. (A2) applies to the offline wavelength and Eq. (A4) applies to the online wavelength, the online/offline transmittance ratio T = Ton/Toff in the presence of the thin scattering layer can be derived from the ratio (A4)/(A2) and after neglecting higher-order terms is

 
formula

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