The retrieval algorithm for CO

Accurate remote sensing of atmospheric CO

Several types of dual-wavelength (online and offline) IPDA lidar have been
demonstrated previously for measuring XCO

NASA Goddard Space Flight Center (GSFC) has developed an airborne
multi-wavelength CO

This paper describes the retrieval algorithm for the multi-wavelength
CO

The measurement geometry for the CO

Illustrations of the CO

Block diagram of the airborne CO

The lidar receiver detects and records the received laser pulse waveform
over the entire atmosphere column traveled by the laser pulses. In the
airborne lidar all signals are digitized and recorded and the lidar analysis
is performed on ground after the flight. The received signals from the
scattering surface are used to retrieve XCO

An overview of the retrieval algorithm for lidar data is shown in Fig. 3.
The initial processing consists of (a) processing the stored lidar data to
estimate ranges to the reflecting surfaces and form a series of atmosphere
transmission measurements across the CO

Flowchart of the XCO

The signal waveforms are first corrected for the detector baseline offset
and other instrument characteristics and then scaled to the received optical
signal power. The pulse energies from the scattering surfaces are calculated
by integrating the received pulse waveforms over the pulse width interval.
The relative atmosphere transmittances for all laser wavelengths are
calculated by dividing the received pulse energies by the transmitted ones
and then multiplying by the square of the range from the lidar to the
reflecting surface. The signal-to-noise ratio (SNR) of the atmospheric
transmittances at each wavelength is estimated based on the received signal
energy, the estimated background noise, and the detector noise. Finally, a
least-squares fit of the modeled line shape to the lidar measurements is
used to estimate XCO

The lidar returns from clouds are identified by comparing the elevations of
the lidar returns, namely aircraft altitude minus the lidar range, to the
surface elevation from either the onboard radar measurements or a
digital elevation model (DEM). For dense clouds, the laser energies
reflected from the cloud tops are usually sufficient for XCO

The average signal pulse energy reflected from the scattering surface can be
calculated from the lidar equation (McManamon, 2019), as

The product of the surface reflectance and the two-way atmospheric
transmission can be calculated from the received laser pulse energy after
correcting for the range as

The total atmospheric transmission from the lidar to the surface can be
written as

To compute the transmission line shapes of CO

The atmospheric transmissions of each layer for each of the lidar wavelengths
across the CO

Here we assumed that the laser wavelengths are known precisely and the laser
spectral line width is much narrower than the CO

The modeled optical transmission due to CO

The molecular density of CO

In our XCO

The atmospheric pressure, temperature, and water vapor can cause shifts and
broadening of the CO

Since the power and size of the CO

Using the scale factor, the OD which is attributable to the CO

The column XCO

The least-squares fit may be formulated by expressing the lidar measurement
data in matrix form,

The modeled atmospheric transmission given in Eq. (9) can be expressed as a
single column matrix,

A scalar-valued loss function can be defined as the sum of squared
differences between the lidar measurement data and the model, as

For small changes in XCO

Substituting Eq. (12) into Eq. (11) and defining

The use of the above normalization greatly simplifies the mathematical
derivation as well as the data processing since it cancels out the
exponential terms in the derivatives of

The loss function given in Eq. (15) is of the same form as that of a linear
least-squares fit with measurement data

For measurement noise that is zero mean and follows a Gaussian distribution,
the optimal weighting factors are given by the reciprocal of the variance of
the measurement data (Bevington, 1969). In our case, the optimal weighting
factors can be approximated as

The XCO

The covariance of the parameters can be obtained from Eq. (18), as

The variances of the estimated parameters are given by the diagonal elements
of

The total column averaging kernel can be calculated as (Borsdorff et al.,
2014)

The algorithm described here was used to retrieve XCO

The airborne CO

Sample pulse waveforms of the airborne CO

Figure 4 shows an example of a Level-1 data set from our 2017 airborne
campaign. It shows 30 transmitted pulse waveforms and the corresponding
received pulse waveforms averaged over 32 laser wavelength scans. The
decrease (tilt) of laser pulse amplitudes over the pulse width interval is
caused by the depletion of energy stored in the laser gain media, which does
not affect the IPDA lidar measurements. The energies of the transmitted
laser pulses at different wavelengths fluctuate by a few percent, which is
monitored and corrected for in the signal processing. The tails in the
transmitted pulse waveforms shown in Fig. 4b are caused by an artifact of
the laser monitor detector, which is different from the one used in the
receiver. The amplitudes and energies of the received laser pulse waveform
plotted in Fig. 4c clearly show the CO

For the least-squares fit the weighting factor for each wavelength is the square of the SNR of the lidar-detected signals at that wavelength. The SNRs are estimated from the received lidar signal as (Gagliardi and Karp, 1995)

The laser speckle noise term (Goodman, 1965, 1975) is not included in Eq. (23) since it is not a major noise source for our airborne lidar
measurements at nominal flight altitude. This is because of the large number
of speckle cells in the laser footprint and the numerical averaging of the
30 received laser pulses for each XCO

For the XCO

Figure 5 shows the CO

The averaging kernel from the retrieved data (open circles) and the fourth-order polynomial fit with altitude (solid black curve) from the measurement data shown in Fig. 5.

Figure 7 shows the results of the retrieval using the algorithm described
above from the airborne CO

The results of the retrieval sequence from the airborne CO

Figure 8 shows the retrieved XCO

The column XCO

Although the least-squares-fit method minimizes the sum of squared errors
between the modeled line shape and the lidar measurements, it does not
guarantee minimum biases in the estimated parameters. The variance of the
solutions can approach zero as the SNR increases, as shown in Eq. (18), but
biases remain. For example, if the actual CO

The choice of the lidar laser wavelengths is a trade-off among several
factors. The total number of laser wavelengths has to be greater than the
number of parameters to be solved for in the retrieval; however, the total
average laser output power is fixed. Using fewer wavelength samples allows
improvement of the SNR for each sample but provides fewer constraints to the
curve fit. More wavelength samples lower the SNR at each wavelength but
allow us to solving for more parameters and helps to reduce the bias in the
XCO

The airborne CO

The retrieval algorithm described in this paper could also be used for the
online and offline dual-wavelength IPDA lidar to retrieve XCO

It is possible to use the least-squares fit to solve for more parameters of
the CO

An algorithm to retrieve XCO

An IDL (Interactive Data Language) version of the software code for the least-squares fit will be posted at the same website by 1 July 2021 or contact the author xiaoli.sun-1@nasa.gov.

The retrieved XCO

XS led the writing of the manuscript and provided the mathematical formulation of the retrieval algorithm. JBA was the principal investigator of the CO

The authors declare that they have no conflict of interest.

We thank the CO

This research has been supported by the NASA Earth Sciences Technology Office (ESTO) and the NASA ASCENDS Mission pre-formulation program.

This paper was edited by Markus Rapp and reviewed by two anonymous referees.