1. Introduction

This paper continues our study on lidar retrievals of atmospheric constituents from absorption spectra. In a previous study [1] we established a model for the ideal case where the laser frequency channels are placed symmetrically around the center of the target absorption line profile. The present paper generalizes the model for any distribution of the laser frequency channels across the absorption line profile.

Future lidar missions call for unprecedented high precision for global measurement of atmospheric constituents and parameters [2–4]. For example, the Active Sensing of CO₂ Emissions over Nights, Days, and Seasons (ASCENDS) mission planned by NASA [2,3] aims to measure the global distribution of carbon dioxide (CO₂) mixing ratios (∼400 ppm) to ∼1 ppm precision. To meet such stringent requirements, nadir-viewing, direct-detection, and pulsed integrated-path differential-absorption (IPDA) lidar techniques are being developed to measure the two-way optical absorption spectra of the target species from the spacecraft to the surface and back [5,6]. From the absorption spectra and altimetry measurements and other ancillary data of the atmosphere, the dry mixing ratios of the target species can be retrieved [1,7–10].

Throughout this paper, a candidate ASCENDS IPDA lidar approach [6] being developed at NASA is used as a concrete example. As shown in Fig. 1, this approach uses a pulsed laser to rapidly cycle through multiple laser frequency channels across a single CO₂ line at 1572.335 nm [11]. The returned laser pulses are detected directly with a nearly noiseless HgCdTe avalanche photodiode (APD) detector [12], to maximize the signal to noise ratio [13]. To reduce retrieval errors arising from the laser frequency fluctuation, laser pulses at each fixed wavelength are carved from a frequency stabilized continuous-wave (CW) laser [11,14]. Crosstalk from cloud scattering is eliminated by keeping the pulse rate below ∼8 kHz. A fast pulse rate of 8 kHz is chosen to lower the pulse energy requirement (to ∼2 mJ), to average out speckle noise and fast laser frequency noise across more pulses, and to reduce the spectral distortion from surface reflectance variations. A pulse duration of ∼1 µs is chosen to maintain a narrow laser line-width and adequate laser peak power. This approach has also been adopted to scan an O₂ absorption line doublet for an atmospheric pressure measurement [15], and to measure atmospheric methane concentrations [16]. On Earth, such a column absorption line is typically broadened to a few GHz wide, mostly due to pressure broadening in the lower troposphere. Narrow line-width lasers are preferred to scan such absorption lines, to simplify the retrievals and to improve the vertical resolution of the retrieved dry mixing ratios [17].

 

Fig. 1. The IPDA lidar transmits a wavelength-stepped pulse train (left) to repeatedly measure CO₂ absorption at multiple laser frequency channels across a single absorption line (right).

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The classic case of column-averaged retrieval from atmospheric absorption measured at two (online/offline) wavelengths has been analyzed extensively [5,7–9,18–20]. Our previous study addressed multiple-layer retrievals from the absorption spectra measured at multiple pairs of symmetrical laser frequency channels [1]. Based on established inversion methods [21], closed-form formula have been derived that link both the relative random error (RRE) and the relative systematic error (RSE) of the retrieved mixing ratios, arising from the measurement noise and model bias, respectively, to characteristic parameters of the measurement and ancillary data of the atmosphere. By combining the two optical depths (ODs) measured at each pair of symmetrical channels, errors from a common laser frequency drift and spectral baseline tilt (due to, e.g., etalon fringes and surface reflectance variations) can be eliminated. The model includes laser line-shape factor and thus remains accurate even when the laser line-shape is broadened. In the present paper, we generalize this model to quantify the retrieval errors for any laser frequency channels that can be asymmetrical and are no longer combined. By solving for the frequency shift and spectral baseline tilt, atmospheric retrievals degrade only slightly even when such channels are shifted substantially out of symmetry. An optical etalon can thus be used to stabilize the reference laser wavelength.

Our forward model for arbitrary frequency channels is presented in section 2. The retrieval error analysis methods are presented in section 3 with derivation details provided in Appendix C. A numerical example of error estimation for CO₂ retrievals is presented in section 4. More considerations for the retrievals and the observing system are discussed in section 5, and further details are provided in the Appendices. Throughout this paper, matrices are denoted by bold face upper case, e.g., K, column vectors by bold face lower case, e.g. b, the transpose by superscript T, e.g., K^T. $\hat{{\textbf b}}$ represents an estimate of b, and $\det ({\textbf K})$ the determinant of ${\textbf K}.\;{\mathop{\textrm {Cov}}\nolimits} (x,y)$ represents the covariance between $x$ and $y,\;{\sigma ^2}(x)$ represents the variance, $\sigma (x)$ the standard deviation, $\bar{x}$ the ensemble average of x.

2. Forward model for arbitrary frequency channels

The laser frequency noise contributes to measurement errors through two complementary factors: the energy spectral density (ESD) L(${\nu_{F}}$) of each single laser pulse (as a function of the Fourier frequency ${\nu_{F}}$), and the fluctuation of the line-center frequency ${\nu_{c}}$ of L(${\nu_{F}}$). Both factors are jointly determined by the CW laser frequency noise and the laser pulse amplitude [22]. The present analysis adopts a forward model established previously for uncombined laser frequency channels [1], as summarized in this section.

In this generalized model, the m laser frequency channels are not necessarily distributed symmetrically about the center ${\nu _0}$ of the column absorption line. Let ${\nu _i} \equiv \bar{\nu }_c^i$ denote the ensemble mean of the laser line-center frequency and $\Delta {\nu _i} \equiv {\nu _i} - {\nu _0}$ the frequency offset of channel i (i = 1, 2, …, m). The atmospheric pressure p is conveniently used as the vertical coordinate and $p(r)$ denotes the atmospheric pressure at range r from the spacecraft. The column absorption line is slightly asymmetrical due to the pressure shift and ${\nu _0}$ is taken to be at the absorption peak.

The detected laser pulse energy $W_s^i$ backscattered from the surface is given by [22]

The effective two-way OD is related to the dry mixing ratio ${q_{gas}}(p)$ and the two-way effective weighting function $w({\nu _c},p)$ of the target species by $\tau ({\nu _c},2r)\, = \int_0^{p(r)} {{q_{gas}}(p)w({\nu _c},p)dp}$ (see Appendix A for details). We divide the atmospheric column into n_q layers and attempt to solve for the layer-average ${q_j}$ of ${q_{gas}}(p)$ for each layer (j = 1, 2, …, n_q, with layer 1 at the bottom). The layer-average of a dry mixing ratio profile is the ratio of the total number of target molecules to the total number of dry air molecules in a column within that layer. ${q_{gas}}(p)$ in layer j is taken to be proportional to a given dry mixing ratio profile $q_j^a(p)$ and the layer average of $q_j^a(p)$ is denoted as $< q_j^a > $ (given by Eq. (20) in Appendix A). The effective two-way OD $\tau ({\nu _c},2r)\,$ is then related to ${q_j}$ and $w({\nu _c},p)$ by $\tau ({\nu _c},2r) \cong \sum\nolimits_{j = 1}^{{n_q}} {({q_j}/ < q_j^a > )\int_{{p_j}}^{{p_{j - 1}}} {q_j^a(p)w({\nu _c},p)dp} }$ where ${p_j}$ is the pressure at the top boundary of layer j, and ${p_0} = p(r)$.

Let $r_G^i$ denote a certain mean of $r_s^i(k)$ averaged across multiple laser pulses (k = 1, 2, …, n_p) in channel i. ${\tau _i} \equiv \tau ({\nu _i},2r_G^i)$ can be estimated from the following transmittance ${T_i}$ averaged across the n_p pulses

As needed for the retrievals, the covariance matrix ${{\textbf S}_y}$ of ${\textbf y}$ and the model bias ${\bar{\varepsilon }_i}$ are quantified in [1] and summarized in Appendix B where the OD slope at channel i is denoted as ${\dot{\tau }_i} \equiv \partial {\tau _i}/\partial {\nu _i}$ and $\dot{{\boldsymbol \tau }} \equiv {[{\dot{\tau }_1},{\dot{\tau }_2},\ldots ,{\dot{\tau }_m}]^T}.$ It would be ideal if the slow drifts of all laser frequency channels are the same so the common frequency drift can be solved and corrected for. This can be achieved by phase locking the transmitted laser frequencies to a master laser [11,14]. For this ideal case, ${{\textbf S}_y}$ can be expressed as

On the other extreme, the transmitted laser frequencies are uncorrelated (with uncorrelated slow drifts) among different channels so that ${{\textbf S}_y}$ becomes diagonal. The diagonal element ${\sigma ^2}({{y_i}} )$ includes the contribution from the slow frequency drift (in channel i) that essentially does not decrease from pulse averaging. As illustrated in section 4, the uncorrelated laser frequency drifts can drive up retrieval errors significantly.

3. Retrieval error analysis for arbitrary frequency channels

The matrix formulas for the retrievals have been described in [1], and are briefly summarized herein. To simplify the analysis, we assume no a priori information and consider the maximum likelihood (ML) retrieval approach. Since the forward model is in general nonlinear, an estimate $\hat{{\textbf x}}$ of ${\textbf x}$ can only be retrieved iteratively. Nevertheless, it is adequate to linearize the forward model ${\textbf F}({\textbf x})$ about an estimated state ${{\textbf x}_i}$ for the retrieval error analysis. Using the Newton-Gauss method, the following iteration ${{\textbf x}_{i + 1}}$ converges to $\hat{{\textbf x}}$ [21]

 

Fig. 2. Linear least-square fitting (left) for retrieving q and c₀ from monochromatic OD measurement of atmospheric CO₂ line at 1572.335 nm (right). The blue dots mark four pair of symmetrical channels at their ensemble mean frequencies. The red squares mark the channels when they are shifted by 100 MHz.

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The estimated error covariance matrix $\hat{{\textbf S}}$ of $\hat{{\textbf x}}$ is found to be [21]

The RRE and RSE of ${\hat{q}_j}$ for multiple-layer retrievals are derived from Eqs. (7) and (8) in Appendix C. Here we examine the simple case of column-averaged retrieval. In these derivations, $< {a_i} > \, \equiv {{\sum\nolimits_{i = 1}^m {{a_i}\sigma _u^{ - 2}({{y_i}} )} } \mathord{\left/ {\vphantom {{\sum\nolimits_{i = 1}^m {{a_i}\sigma_u^{ - 2}({{y_i}} )} } {\sum\nolimits_{i = 1}^m {\sigma_u^{ - 2}({{y_i}} )} }}} \right.} {\sum\nolimits_{i = 1}^m {\sigma _u^{ - 2}({{y_i}} )} }}$ denotes an average of the m elements of a vector ${\textbf a}$ weighted by $\sigma _u^{ - 2}({{y_i}} ),$ ${{\mathop{\textrm {Var}}\nolimits} _i}({a_i}) \equiv \, < {({a_i} - < {a_i} > )^2} > $ a variance of ${a_i},$ and ${{\mathop{\textrm {Cov}}\nolimits} _i}({a_i},{b_i}) \equiv \, < ({a_i} - < {a_i} > )({b_i} - < {b_i} > ) > $ a covariance between elements ${a_i}$ and ${b_i}$ across m channels. Also, ${\tau _{ij}} \equiv {[{{\textbf K}_q}]_{i,j}}{\hat{q}_j}$ is the estimated two-way OD of the target species within layer j at channel i, $\Delta {\tau _j} \equiv {[{{\mathop{\textrm {Var}}\nolimits} _i}({\tau _{ij}})]^{1/2}}$ is an effective two-way differential absorption OD (DAOD) within layer j that quantifies the spread of the ODs ${\tau _{ij}}$ across m channels, $\Delta \tau \equiv {[{{\mathop{\textrm {Var}}\nolimits} _i}({\tau _i})]^{1/2}}$ is an effective two-way DAOD for the entire column, and a constant ${c_{\Delta \tau }}$ is defined as $c_{\Delta \tau }^2 \equiv {\sigma ^2}({{\delta_{\nu nslow}}} ){{\mathop{\textrm {Var}}\nolimits} _i}({\dot{\tau }_i})[\sum\nolimits_{i = 1}^m {\sigma _u^{ - 2}({y_i})]} .$ We also define the following correlation coefficients: ${r_{\dot{\tau }}} \equiv {{\mathop{\textrm {Cov}}\nolimits} _i}({\tau _i},{\dot{\tau }_i})/{[{{\mathop{\textrm {Var}}\nolimits} _i}({\tau _i}){{\mathop{\textrm {Var}}\nolimits} _i}({\dot{\tau }_i})]^{1/2}}$, ${r_{\Delta \nu }} \equiv {{\mathop{\textrm {Cov}}\nolimits} _i}({\tau _i},\Delta {\nu _i})/{[{{\mathop{\textrm {Var}}\nolimits} _i}({\tau _i}){{\mathop{\textrm {Var}}\nolimits} _i}(\Delta {\nu _i})]^{1/2}}$, ${r_\varepsilon } \equiv {{\mathop{\textrm {Cov}}\nolimits} _i}({\tau _i},{\bar{\varepsilon }_i})/{[{{\mathop{\textrm {Var}}\nolimits} _i}({\tau _i}){{\mathop{\textrm {Var}}\nolimits} _i}({\bar{\varepsilon }_i})]^{1/2}}$, ${r_{\dot{\tau }\Delta \nu }} \equiv {{\mathop{\textrm {Cov}}\nolimits} _i}({\dot{\tau }_i},\Delta {\nu _i})/{[{{\mathop{\textrm {Var}}\nolimits} _i}({\dot{\tau }_i}){{\mathop{\textrm {Var}}\nolimits} _i}(\Delta {\nu _i})]^{1/2}}$, ${r_{\dot{\tau }\varepsilon }} \equiv {{\mathop{\textrm {Cov}}\nolimits} _i}({\dot{\tau }_i},{\bar{\varepsilon }_i})/{[{{\mathop{\textrm {Var}}\nolimits} _i}({\dot{\tau }_i}){{\mathop{\textrm {Var}}\nolimits} _i}({\bar{\varepsilon }_i})]^{1/2}}.$ They all vary between -1 and 1.

From Eq. (37) in Appendix C, the RRE of $\hat{q}$ for ${\textbf x} = {[q,\;{c_0}]^T}$ is found to be

When $m \ge 3,{\delta _{\nu 0}}$ can be included in ${\textbf x}$ (i.e., ${\textbf x} = {[q,{\delta _{\nu 0}},\;{c_0}]^T}$). The RRE and RSE of $\hat{q}$ and ${\sigma ^2}({\delta _{\nu 0}})$ are found to be

Similarly, the RRE and RSE for ${\textbf x} = {[q,{\delta _{\nu 0}},\;{c_1},{c_0}]^T}$ can also be derived when $m \ge 4.$ When the laser frequency channels are nearly symmetrically distributed so that ${r_{\dot{\tau }}} \cong 0$ and ${r_{\Delta \nu }} \cong 0,$ the RREs and RSEs of $\hat{q}$ for ${\textbf x} = {[q,{\delta _{\nu 0}},\;{c_1},{c_0}]^T},$ ${\textbf x} = {[q,{\delta _{\nu 0}},\;{c_0}]^T},$ and ${\textbf x} = {[q,{c_1},{c_0}]^T}$ are also minimized and become essentially the same as those for ${\textbf x} = {[q,\;{c_0}]^T}$ (${[\sum\nolimits_{i = 1}^m {\sigma _u^{ - 2}({y_i})} ]^{ - 1/2}}/\Delta \tau$ and ${r_\varepsilon }{[{{\mathop{\textrm {Var}}\nolimits} _i}({\bar{\varepsilon }_i})]^{1/2}}/\Delta \tau ,$ respectively). This means that ${\delta _{\nu 0}}$ and ${c_1}$ can be solved and corrected for without degrading the RRE and RSE of $\hat{q}$ when the frequency channels are nearly symmetrical. When the channels are symmetrical, the first term of ${\sigma ^2}({\delta _{\nu 0}})$ for ${\textbf x} = {[q,{\delta _{\nu 0}},\;{c_1},{c_0}]^T}$ equals to that for ${\textbf x} = {[q,{\delta _{\nu 0}},\;{c_0}]^T}$ multiplied $1/(1 - r_{\dot{\tau }\Delta \nu }^2).$ For the symmetrical channels illustrated in Fig. 2, $r_{\dot{\tau }\Delta \nu }^2$ approximates zero (as shown in section 4) so that fitting for OD baseline tilt does not degrade fitting for ${\delta _{\nu 0}}$ either. If the frequency channels are symmetrically combined, the information for ${\delta _{\nu 0}}$ and ${c_1}$ is lost, but the information for q and ${c_0}$ is preserved. As shown in Eqs. (11) and (13), solving for ${\delta _{\nu 0}}$ with asymmetrical frequency channels will increase the RRE of $\hat{q}$ by $1/{(1 - r_{\dot{\tau }}^2)^{1/2}}$ and the first term of ${\sigma ^2}({\delta _{\nu 0}})$ by $1/(1 - r_{\dot{\tau }}^2).$ As shown in section 4, these increases are negligible even when the symmetrical channels are shifted out of symmetry by ${\delta _{\nu 0}}$ within ± 1 GHz.

The noise contributions to ${\sigma ^2}(\Delta \tau )$ from sources other than ${\delta _{\nu slow}}$ (such as the signal shot noise, speckle noise, and fast frequency noise) are uncorrelated among different pulses and can be reduced by pulse averaging. The shot-noise limited $\sigma (\Delta \tau ) = {[\sum\nolimits_{i = 1}^m {\sigma _u^{ - 2}({y_i})} ]^{ - 1/2}}$ (and hence the retrieval RRE) is inversely proportional to the square root of the accumulated photon count of the pulses from all m channels.

To restrain the rotation of the rigid bar (hence the RRE and RSE of $\hat{q}$) as illustrated in Fig. 2, it is desirable to “clamp” the bar at its two ends and keep the two ends far apart (i.e., to increase $\Delta \tau$). To reduce the rotational “play” of the rigid bar (hence the RRE of $\hat{q}$), the error bars of ${y_i}$ need to be reduced, particularly at the two extreme ends. The online and offline channels are equivalent in their contributions to $\Delta \tau$ and $\sigma (\Delta \tau ).$ $\Delta \tau$ can be increased by minimizing the absorption of offline channels and shifting online channels towards the absorption peak. Minimizing the absorption of offline channels also reduces their error bars (and hence $\sigma (\Delta \tau )$). Increasing the absorption of online channels increases their error bars (and hence $\sigma (\Delta \tau )$) and could end up reducing $\sigma (\Delta \tau )/\Delta \tau$ and even $\Delta \tau .$ It is often desirable to place some online points on the sides of the absorption line to uniformly sense concentrations in the lower troposphere. This may decrease $\Delta \tau$ and increase the RRE and RSE of $\hat{q}.$ Further error reduction considerations will be discussed along with the numerical example in the next section.

4. Numerical estimation of retrieval errors

We now estimate the errors of atmospheric CO₂ retrievals from the lidar absorption spectra measured with a narrow line-width laser across the 1572.335 nm CO₂ absorption line. The present calculations are based on the same surface reflectance data [23] and essentially the same realistic parameters (listed in Table 1) employed in [1]. The pulses of all channels are assumed to have the same transmitted pulse energy ${E_s}$ of 2 mJ, smaller than the 4mJ assumed in [1]. The detector specifications listed in Table 1 are rather conservative compared to the nearly noiseless HgCdTe APD detectors employed in NASA's IPDA lidars [12]. The additional model bias and measurement noise due to imperfect ranging are negligible for a ranging bias ≤ 0.66 m and a ranging precision ≤ 20 m [22], and are neglected hereafter. The speckle noise is neglected due to the large telescope diameter of 1.5 m and large laser beam spot size of 50 m on the surface [5,24,25]. Unless noted otherwise, the results illustrated in the following figures are for an IPDA lidar with 8 symmetrical channels (as shown in Fig. 2) having a common slow drift ${\delta _{\nu nslow}}$ with $\sigma ({\delta _{\nu nslow}}) = 3\;\textrm{MHz}$.

Table 1. Parameters used for numerical estimation of retrieval errors

View Table

 

Fig. 3. (left) The measurement noise σ(y_i) for atmospheric CO₂ (solid black) as a function of 2τ₀ (for frequency offset < 0), computed using parameters listed in Table 1. The blue dots mark the channels ν₁ to ν₄. Also plotted are partial contributions to σ(y_i) from the signal shot noise (solid grey), slow laser line-center frequency drift (dashed red), solar background (dotted brown), receiver circuitry noise (dash-dotted green), and detector dark count (long-dashed blue). (right) The RRE of $\hat{q}$ (solid black), ${r_{\dot{\tau }}}$ (dotted red), ${r_{\Delta \nu }}$ (dash-dotted green), and ${r_{\dot{\tau }\Delta \nu }}$ (dashed blue) as functions of a common frequency shift ${\delta _{\nu 0}}.$ The RRE of $\hat{q}$ becomes much larger (solid grey) when the transmitted laser frequencies become uncorrelated.

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The measurement noise $\sigma ({y_i})$ and ${\sigma _u}({y_i})$ are computed from Eq. (24) in Appendix B and plotted as functions of two-way OD $2{\tau _0}$ in Fig. 3 (left). The same figure also shows partial contributions to $\sigma ({y_i})$ from the signal shot noise, slow laser line-center frequency drift ${\delta _{\nu slow}},$ solar background, receiver circuitry noise, and detector dark count. The effective fast laser line-center frequency noise ${\sigma ^2}({\delta _{\nu nfast}})$ is averaged down to 28 kHz and its contribution to $\sigma ({y_i})$ is negligible. On the other hand, the noise contribution from ${\delta _{\nu slow}}$ becomes dominant for online channels. The RRE of $\hat{q}$ is computed from Eq. (11) and plotted in Fig. 3 (right) as a function of a common frequency shift ${\delta _{\nu 0}}$ added to the 8 nominal channels illustrated in Fig. 2. Computing from Eq. (9) yields essentially the same RRE. The correlation coefficients ${r_{\dot{\tau }}},$ ${r_{\Delta \nu }},$ and ${r_{\dot{\tau }\Delta \nu }}$ shown in Fig. 3 (right) are ≤ 0.2 so that their squares approximate zero. Even when the channels are shifted out of symmetry by ± 1 GHz, The RRE of $\hat{q}$ does not increase significantly from its minimum of 0.052% at ${\delta _{\nu 0}} = 0.$ Also the square root of the first term of ${\sigma ^2}({\delta _{\nu 0}})$ in Eq. (13) (i.e., the standard deviation of the frequency bias estimation, not shown in Fig. 3) is ≤ 0.53 MHz when ${\delta _{\nu 0}}$ varies within ± 1 GHz and is minimized to 0.41 MHz at ${\delta _{\nu 0}} = 0.$ When the laser channel frequencies are uncorrelated (while all other conditions being the same), the RRE of $\hat{q}$ (shown in Fig. 3 (right)) increases significantly to 0.126% even at ${\delta _{\nu 0}} = 0$.

Figure 4 (left) shows the RREs of $\hat{q}$ for IPDA lidars using two channels (solid grey) and four symmetrical channels (solid black) as functions of $|\Delta {\nu _2}|,$ the absolute value of the online frequency offset. For two channels, the offline frequency offset is fixed at $\Delta {\nu _1} ={-} 15.6\;\textrm{GHz}$ while $\Delta {\nu _2}$ varies from 0 to -2 GHz. For four channels, $- \Delta {\nu _1} = \Delta {\nu _4} = 15.6\;\textrm{GHz}$ and $- \Delta {\nu _2} = \Delta {\nu _3}$. For fair comparisons, the pulse rate for each channel is 4 kHz for two channels and 2 kHz for four channels (while all other parameters being the same). The RRE of $\hat{q}$ jumps to ≥ 0.19% for the traditional two-wavelength lidar. In comparison, adding two more channels symmetrically reduces the RRE to a minimum of 0.048% when $- \Delta {\nu _2} = \Delta {\nu _3} = 0.78\;\textrm{GHz}\textrm{.}$ Also plotted are the partial RSEs of $\hat{q}$ for the two-channel lidar due to a frequency bias of 3 MHz (dashed red) and a small OD slope of $3.3 \times {10^{ - 4}}/\textrm{GHz}$ (dotted green). When the channels are nearly symmetrical (such as the four symmetrical channels), such partial retrieval biases from the frequency shift and OD baseline tilt are essentially eliminated. Figure 4 (right) shows the RSEs of $\hat{q}$ for 1-s averaging time calculated from the surface reflectance data as functions of the starting position along the path for the 8 symmetrical channels (grey), 4 symmetrical channels (blue), and 2-channels (red). The peak RSE of $\hat{q}$ does not change significantly when the 8 (or four) channels are shifted out of symmetry within ± 1 GHz. To simplify the calculation, each beam spot is in effect taken to be a uniformly illuminated square (aligned to the track), and variations of surface reflectance perpendicular to the track are neglected. The computed peak RSE arising from surface reflectance variations is the smallest (∼8×10⁻⁶) for four-channels, larger (∼2.4×10⁻⁵) for 8 channels, and largest for two-channels (∼1.1×10⁻⁴). In reality, the RSEs will be somewhat higher due to non-uniform illumination of the beam spots and cross-track variations of surface reflectance.

 

Fig. 4. (left) The RREs of $\hat{q}$ for two-channel (solid grey) and four-channel (solid black) lidars as functions of the absolute value of the online frequency offset. Δν₁ is fixed at -15.6 GHz for two channels. For four channels, -Δν₁ = Δν₄ = 15.6 GHz and -Δν₂ = Δν₃. Also plotted for two channels are the partial RSEs of $\hat{q}$ due to a frequency bias of 3 MHz (dashed red) and a small OD slope of 3.3×10⁻⁴/GHz (dotted green). (right) RSEs of $\hat{q}$ for 1-s averaging time calculated from the surface reflectance data as functions of the starting position along the path for the 8 symmetrical channels (grey), four symmetrical channels (blue), and two-channels (red).

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5. Discussions

5.1. Retrievals from transmittance measurement

It should be noted that the same state vector ${\textbf x} = {[{q_1},{q_2},\ldots ,{q_{{n_q}}},\;{\delta _{\nu 0}},\;{c_1},\;{c_0}]^T}$ can also be retrieved from the transmittance measurement vector ${\textbf t} = {[{T_1},{T_2},\ldots ,{T_m}]^T}$ by using the following forward model

When the measurement noise is small so that Eq. (15) is valid, it is essentially equivalent to retrieve ${\textbf x}$ from either the transmittance measurement or the OD measurement. The same error formula derived for retrievals from OD measurement (i.e., Eqs. (9)–(13), (33), (36)–(40)) can also be derived from $\hat{{\textbf S}} = {\textbf K}_t^T{\textbf S}_t^{ - 1}{{\textbf K}_t}$ and $\delta {\textbf x} = {\hat{\textbf{S}}K}_t^T{\textbf S}_t^{ - 1}{\bar{{\boldsymbol \varepsilon }}^t}$ for the transmittance measurement (with proper substitutions such as ${[{{\textbf K}_t}({\textbf x})]_{i,j}} = {T_i}{[{\textbf K}({\textbf x})]_{i,j}}$ and ${[{{\textbf S}_t}]_{i,j}} = {T_i}{T_j}{[{{\textbf S}_y}]_{i,j}}$). It is more efficient to retrieve from OD measurement because the forward model becomes nearly linear with respect to ${\textbf x}$ (except for ${\delta _{\nu 0}}$). When retrieving within a short averaging time so that the integrated photon count becomes insufficient, the OD bias arising from the nonlinearity of the log function becomes significant [1]. In this case, it is more accurate to retrieve from the transmittance measurement.

5.2. Multiple-layer retrievals of atmospheric species

For multiple-layer retrievals, the RRE of ${\hat{q}_j}$ equals to the RRE of the effective DAOD $\Delta {\tau _j}$ for layer j (see Eq. (37) in Appendix C). More layers means smaller $\Delta {\tau _j}$ and larger $\sigma (\Delta {\tau _j}),$ and hence larger RRE of ${\hat{q}_j}$. $\sigma (\Delta {\tau _j})$ becomes larger than $\sigma (\Delta \tau )$ due to strong correlation of ${\tau _{ij}}$ among different layers [1]. When the laser linewidth is broadened, such correlation becomes even stronger, leading to increased $\sigma (\Delta {\tau _j})$ and hence even larger RRE of ${\hat{q}_j}$ [17]. Similarly, RSE of ${\hat{q}_j}$ becomes worse with more layers (see Eq. (39) in Appendix C). Even when the column is divided into just two layers, the RRE of CO₂ mixing ratio in the boundary layer (2 km high) is 15-times higher than that for the entire column [1]. It is therefore impractical to derive the vertical profile $q_j^a(p)$ accurately from the absorption measurement itself. Such vertical profiles need to be provided as ancillary data of the atmosphere for the retrievals from lidar absorption spectra.

5.3. More IPDA lidar considerations

It is highly desirable to probe the target absorption line with more than two wavelengths so that ${\delta _{\nu 0}}$ can also be solved for. It has been shown in Eq. (39) that the second term of $\hat{{\textbf x}} - {\textbf x}$ arising from $\dot{{\boldsymbol \tau }}{\dot{{\boldsymbol \tau }}^T}{\sigma ^2}({{\delta_{\nu nslow}}} )$ vanishes if ${\delta _{\nu 0}}$ is included in ${\textbf x}.$ Since ${{\textbf x}_{i + 1}} - {{\textbf x}_i}$ in Eq. (5) is related to ${\textbf y}({{\textbf x}_i}) - {\textbf F}({{\textbf x}_i})$ by the same matrix $\hat{{\textbf S}}{{\textbf K}^T}{\textbf S}_y^{ - 1}$ as $\hat{{\textbf x}} - {\textbf x}$ is to $\bar{{\boldsymbol \varepsilon }}$ in Eq. (8), ${\textbf x}$ (with ${\delta _{\nu 0}}$ included) can be retrieved iteratively without the knowledge of ${\sigma ^2}({\delta _{\nu nslow}}).$ By solving for ${\delta _{\nu 0}},$ this unknown shift does not degrade the RRE and RSE of ${\hat{q}_j}$ significantly. In other words, an unknown frequency shift can be tolerated without causing significant retrieval degradation, as long as the drift is bound to a small fraction of the target atmospheric absorption line-width. This allows an optical reference (such as an optical etalon or a ring resonator) other than a gas absorption cell to be used to lock the master laser wavelength. Such an optical reference can be tailored to any reference wavelengths, and can be made more compact and reliable than a long path-length gas cell needed for weak gas absorption lines. The optical reference needs to be stabilized to remain virtually stationary within each wavelength sweep cycle (∼1 ms) and the slow drift needs to be bound to less than ± 1 GHz, preferably within ± 200 MHz. Such requirements are not too difficult to be met. As illustrated in section 4, the atmospheric lidar absorption measurement across the target absorption line provides an absolute wavelength calibration with high precision of ∼0.5 MHz. By solving for ${\delta _{\nu 0}},$ the optical reference can be better controlled to reduce its drift.

It is desirable for the IPDA lidar to transmit a single-spatial-mode beam. In contrast, transmitting a multi-mode beam (or multiple beams shooting at the same surface spot) requires larger transmitter aperture to illuminate the same surface spot and may lead to variations of beam intensity distribution on the surface among different laser frequency channels. Such wavelength-dependent variations may introduce unwanted spectral distortion for absorption measurement. On the other hand, the slow detector responsivity drift in either the lidar receiver or the transmitted pulse energy monitor does not degrade the retrievals as long as the detector responsivity remains virtually constant during each wavelength sweep cycle [1].

As quantified in Eq. (24) in Appendix B, the speckle noise contribution to ${\sigma ^2}({y_i})$ is $1/({n_p}{M_{sp}}{M_t}),$ and thus can be reduced by increasing the number of spatial correlation cells M_sp within the receiver aperture, the number of coherent intervals M_t within a pulse duration, and the number of pulses n_p within the averaging time. M_sp can be increased by enlarging the receiver aperture and the divergent angle of the transmitted beam. In order to reduce the speckle noise contribution to a small fraction of the shot noise contribution ${F_e}/({n_p}{K_s}),$ the single pulse photon count K_s needs to be limited to the same small fraction of ${M_{sp}}{M_t}/{F_e}.$ Unlike K_s, M_sp and M_t are independent of the laser power. It is thus desirable to choose a fast pulse rate (e.g., 8 kHz) to reduce K_s and to increase n_p for the same average laser power. On the other hand, the peak power of the laser pulses needs to be kept high enough to overcome the solar background noise and the detector noise. The speckle correlation cells during one pulse are taken to be uncorrelated to those during another pulse because the pulse period is longer than the laser coherent time ${\tau _c}$ in practice. For a Gaussian laser line-shape with a FWHM line-width of $\Delta {\nu _{FWHM}},$ ${\tau _c} = 0.664/\Delta {\nu _{FWHM}}$ [24]. For a typical $\Delta {\nu _{FWHM}}$ of 30 MHz [11], ${\tau _c} \cong 22\;\textrm{ns}$ and ${M_t} \cong 45$ for a pulse duration of 1 µs.

6. Summary

A generalized model is presented for retrieving atmospheric constituents from lidar absorption spectra measured at any laser frequency channels. Random and systematic retrieval errors arising from measurement noise and model bias, respectively, are analyzed parametrically and numerically to provide deeper insight. By placing four or more channels symmetrically around the absorption peak, retrieval errors from a common laser frequency shift and spectral baseline tilt can be eliminated. By solving for the frequency shift and spectral baseline tilt, atmospheric retrievals degrade only slightly even when such channels are shifted substantially out of symmetry. An etalon can thus be used for the wavelength stabilization. If the laser frequencies are uncorrelated, the uncorrelated frequency drifts can drive up retrieval errors significantly. Retrieving from the transmittance measurement becomes more accurate than from the OD measurement only when the integrated photon count becomes insufficient. The vertical profile $q_j^a(p)$ needs to be provided and cannot be derived accurately from the absorption measurement itself. It is desirable to transmit a single-mode laser beam to avoid unwanted measurement errors.