1. Introduction

1.1 Overview

Lidar remote sensing techniques are powerful tools for global measurement of atmospheric constituents and parameters [1, 2]. The required precision and accuracy of such quantitative measurements have been increased to unprecedented levels for future airborne and space missions [3, 4]. To meet such stringent requirements, nadir-viewing, direct-detection, and pulsed integrated-path differential-absorption (IPDA) lidar techniques are being developed to measure the differential absorption optical depths (DAODs) of the target species in the column to the surface [5, 6]. From the DAOD and altimetry measurements and other ancillary data of the atmosphere (temperature, pressure, water vapor, etc.), the column number densities and dry mixing ratios of the target species can be retrieved [7, 8].

This paper addresses the following common issues with such DAOD measurements that have not been well addressed in previous publications: 1) error contribution from the optical frequency noise of a pulsed laser; 2) incorporation of a bias correction factor to reduce the averaging time for “log after averaging” (as named in [9]), and thereby minimize the impact of Etalon and atmospheric turbulence effects; 3) error contribution from the surface reflectance variations [10] when beam spots shift on the surface; 4) methods to reduce errors due to variations of surface height and reflectance; 5) estimation of the measurement precision of the transmitted laser pulse energy.

We present a statistical model to address these issues, to provide insight and estimation of DAOD measurement errors and to guide lidar system design and development. The model addresses both types of DAOD measurement errors: the relative random error (RRE) and the relative systematic error (RSE) (i.e., relative bias). This paper is organized as follows. After introducing several representative IPDA lidar examples in the next subsection, we quantify the error contributions from the laser frequency noise in section 2. In section 3, additional detection noise sources and a bias correction factor are incorporated into the model. A numerical example of the model is presented in section 4. The impact of surface reflectance variation is evaluated in section 5. A new method to reduce errors due to surface height variation is described in section 6. More considerations for the lidar system are discussed in section 7. Some supporting details are provided in the Appendix.

1.2 IPDA lidar examples

The Active Sensing of CO₂ Emissions over Nights, Days, and Seasons (ASCENDS) mission [3] has been planned by NASA to measure the global distribution of carbon dioxide (CO₂) mixing ratios (~390 ppm in average) to ~1 ppm precision. A candidate IPDA lidar measurement approach for ASCENDS, being developed at NASA Goddard, allows simultaneous measurement of CO₂ and surface height in the same path [6, 7]. As shown in Fig. 1 , a pulsed laser is wavelength-stepped across a single CO₂ line at 1572.335 nm [11] to measure the DAOD. To make the measurements uniformly sensitive to concentrations in the lower troposphere, the online wavelengths (ν₂ to ν₇) are placed at the sides of the line where the absorption is mostly from CO₂ molecules in the lower troposphere [12]. The two-way transmittance of the CO₂ line from a 400-km orbit (Fig. 1, right) has a narrow linewidth (~3.5 GHz) and steep slopes on the sides, making the measurements sensitive to laser frequency uncertainties. The offline points (ν₁ and ν₈) are placed in the adjacent window regions where the extremely low CO₂ absorption is essentially wavelength independent. To meet the mission end goal, the DAOD of the atmospheric CO₂ line needs to be measured with a RRE < 0.1% at multiple wavelengths across the absorption lines, and the variations of RSE need to be kept to a small fraction of the RRE. To reduce the RSE and RRE of the DAOD measurements, the detected photon count for each wavelength is averaged across multiple laser pulses before taking the logarithm of the on/off line ratio (i.e., “log after averaging”). This lidar approach will be used as a concrete example throughout this paper.

 

Fig. 1 The laser transmitter provides the wavelength-stepped pulse train (left) to repeatedly measure at 8 points across the 1572.335 nm CO₂ absorption line (right).

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The partial RRE contribution from the laser frequency noise needs to be < 0.03%, to allow more margins for other error sources. In this paper, we show that the standard deviation of the effective laser frequency noise at online points for CO₂ needs to be < 0.23 MHz to meet this target. To satisfy such stringent requirements, laser pulses at each fixed wavelength are generally carved by amplitude modulating a frequency stabilized continuous-wave (CW) laser [13] (Fig. 1, left). The 1-μs wide pulses need to be at least ~100 μs apart, to completely clear the bottom ~15 km of the atmosphere, which eliminates crosstalk from cloud scattering. Chirp-free amplitude modulation [14] is used so that the laser pulses retain the instantaneous frequency of the CW laser.

A similar CO₂ sounder space mission called A-SCOPE [4] has also been formulated by the European Space Agency. The A-SCOPE transmitter generates a pair of pulses, one at an online and the other at an offline wavelength, at an overall pulse pair repetition rate of 50 Hz, with a pulse energy of ~50 mJ. Comprehensive lidar sensitivity and spectroscopic analyses for the A-SCOPE instrument operating at either 1.57 μm or 2.05 μm CO₂ bands have been recently published [5, 8, 15]. A similar approach has been adopted for a planned methane sounder space mission dubbed MERLIN [16].

2. Measurement errors from laser frequency noise

Throughout this paper, $\overline{x}$ represents the ensemble average, $\sigma (x)$the standard deviation, and ${\sigma }^2(x)$the variance of x.

2.1 Treatment of laser frequency noise

The instantaneous frequency ν(t) of a single frequency CW laser can be modeled as an ergodic random process and the frequency noise is defined as ${\delta }_{\nu }(t)\equiv \nu (t)-\overline{\nu }.$ The electric field of a pulse carved from the CW laser can be represented by the analytic signal of the CW laser field multiplied by the real single pulse amplitude $a(t-t_0)$ centered at time $t_0,$ as given below

As shown in Eq. (2), the fluctuation of the line-center frequency${\delta }_{\nu 1}(t_0)\equiv {\nu }_c(t_0)-\overline{\nu }$ is an average of the CW laser frequency noise ${\delta }_{\nu }(t)$ within a pulse duration Δt weighted by $h(t-t_0).$ For a top-hat pulse shape, ${\delta }_{\nu 1}(t_0)$ is simply a running average of ${\delta }_{\nu }(t)$ within Δt. The variance ${\sigma }^2({\delta }_{\nu 1})$ of ${\delta }_{\nu 1}(t_0)$ can be obtained by integrating its power spectral density (PSD), which in turn can be derived from $S_{\delta \nu }(f),$ the PSD of ${\delta }_{\nu }(t).$ Since ${\delta }_{\nu 1}(t_0)$ is a correlation between ${\delta }_{\nu }(t)$ and $h(t)$ as seen in Eq. (2), its PSD $S_{\delta \nu 1}(f)$ is related to $S_{\delta \nu }(f)$ by

Formulation linking $L({\nu }_F)$ to the frequency noise $S_{\delta \nu }(f)$ and the pulse amplitude $a(t)$ is further derived and discussed in Appendix A. Interestingly, the remaining frequency noise that is filtered out by $|H(f){|}^2$ contributes to the bandwidth of the line shape $L({\nu }_F).$

2.2 Single pulse formulation

The energy absorption of a single laser pulse through a thin layer of target molecules is given by

To account for two-way absorption, the path length l is taken to be the accumulated distance that the laser pulse has traveled, running from 0 to $2r_G$ as the laser pulse travels from the spacecraft to the surface (outgoing path) and back to the spacecraft (return path). Here $r_G$ is the distance from the spacecraft to the surface. Our model uses the effective optical depth to include the line shape factor, which causes only a bias $\tau ({\nu }_c,l)-{\tau }_0({\nu }_c,l)$ to the optical depth. As shown in Appendix A, this bias can be accurately corrected. When $L({\nu }_F)$ is much narrower than the spectral width of the target absorption line, this bias even becomes negligible, as shown in subsection 2.6 for the 1572.335 nm CO₂ line. When $L({\nu }_F)$ is broad, the one-way effective optical depth may no longer be the same for the outgoing and return paths so that $\tau ({\nu }_c,2r_G)=2\tau ({\nu }_c,r_G)$ may become invalid (see Appendix A for more details).

Incorporating other attenuation factors [18], the detected laser pulse energy W_s backscattered from the surface becomes

The lidar receiver can operate in either linear mode with an internal gain or photon counting mode. To cover both cases, the analog detector signal and noise are gain normalized and expressed in detected photon count K. For any type of radiation (including thermal light), the ensemble average and the variance of K and the optical energy W entering the detector area are related by [19, 20]

2.3 Averaging across multiple pulses

We now turn to estimate the two-way DAOD $\Delta \tau \equiv \tau ({\overline{\nu }}_{on},2r_G)-\tau ({\overline{\nu }}_{off},2r_G)$ between online and offline laser frequencies ${\overline{\nu }}_{on}$ and ${\overline{\nu }}_{off}.$ Up to section 6, $\tau ({\nu }_c,2r_G)$ is assumed to be constant over the path and duration of the multiple pulses to be averaged. Although the measurement of E_s is affected by the amplified spontaneous emission (ASE) in the laser amplifiers, the resulting measurement errors are negligible in practice and thus will not be included in the present model (see Appendix B for details).

For each online or offline wavelength, the laser pulses are assumed to have the same h(t) and a period of t_p. From Eq. (10), the following sum of normalized pulse energies (SNE) can be introduced to include the effect of the laser frequency noise

Corresponding to S_NE1, the sum of normalized photon count K_s (SNK) is defined as $S_{NK1}\equiv {\displaystyle {\sum }_{i=0}^{n-1}{K}_s(i)/E_s(i)}$ and its mean is found to be $\overline{S_{NK1}}=\alpha \overline{S_{NE1}}.$ Here S_NK1 is named with the following scheme. S stands for sum, NK for normalized photon count, and the trailing digit 1 is used to distinguish S_NK1 from S_NK2 and S_NK3 to be introduced in sections 3 and 6, respectively. Using $\overline{K_s(i)K_s(i\text{'})}={\alpha }^2\overline{W_s(i)W_s(i\text{'})}$ [19], the variance of S_NK1 is found to be

From Eq. (15), Δτ is related to S_NK1 by

2.4 Frequency noise reduction from pulse averaging

We now turn to link the PSD $S_{\delta \nu n}(f)$ of ${\delta }_{\nu n}(t)$ to $S_{\delta \nu }(f).$ From Eq. (14) we have

The present model reveals relationships not available from previous models [5, 8]. Using Eq. (22), the effective frequency noise can be estimated in terms of $S_{\delta \nu }(f),$ as illustrated in Fig. 2 for a representative frequency stabilized seeder laser. The nominal $S_{\delta \nu }(f)$ can be approximated by $S_{\delta \nu }(f)=h_{-1}/f+h_0,$ where constant $h_{-1}$ and $h_0$ quantify flicker noise and a nearly flat noise floor for f above 100 Hz, respectively. The periodical window function $W_{avn}(f)$ suppresses the frequency noise except in its narrow peaks at $f=m_w/t_p$ $(m_w=0,1,2,3,\mathrm{...}).$ Frequency drifts with periods longer than the averaging time fall in the window peak at $f=0$ and thus are not attenuated from this averaging. Since the PSD is nearly constant over the frequency range of all other window peaks $(m_w=1,2,3,\mathrm{...}),$ the variance of the faster frequency noise components (excluding the slow drifts) is essentially reduced by a factor of n. This frequency noise reduction has also been verified experimentally, as shown in the next subsection.

 

Fig. 2 The nominal laser frequency noise PSD $S_{\delta \nu }(f)=h_{-1}/f+h_0$ (solid red) is suppressed to $S_{\delta \nu }\text{(}f\text{)|}H\text{(}f{\text{)|}}^{\text{2}}$(dotted blue) by averaging within a 1-μs top-hat pulse and further reduced to $S_{\delta \nu }\text{(}f\text{)|}H\text{(}f{\text{)|}}^{\text{2}}{W}_{avn}(f)$ (dashed black) by averaging across 1000 identical pulses separated by a period of 1 ms.

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2.5 Experimental results

Equations (3) and (22) have been verified experimentally with our seed laser described in [13]. Figure 3 (left) shows the standard deviation of the absolute frequency of the slave laser and frequency offset between the slave and master lasers measured within a gating time from 1 μs to 1 ms. The absolute frequency noise was measured from the beatnote between two identical and independent laser units. The noise of the frequency offset was measured from the beatnote between the slave and master lasers. A fast frequency counter (HP 5371A, Agilent Technologies, Inc.) was used to measure the beatnote frequencies. The measured noise decreases as the gating time increases, matching the reducing area under $S_{\delta \nu }\text{(}f\text{)|}H\text{(}f{\text{)|}}^{\text{2}}.$

 

Fig. 3 (left) The standard deviation of the absolute frequency of the slave laser and frequency offset between the slave and master lasers measured within a gating time from 1 μs to 1 ms; (right) Both frequencies were also measured periodically within a 1-µs gating time with a 1-ms period and averaged across multiple measurements. The standard deviation of each averaged frequency, computed from 100 averaged samples, is plotted as a function of the number of measurements being averaged.

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Both frequencies were also measured periodically within a 1-µs gating time with a 1-ms period and averaged across multiple measurements. The standard deviation of each averaged frequency, computed from 100 averaged samples, is plotted in Fig. 3 (right) as a function of n, the number of measurements being averaged. The variance of the averaged frequency decreases as n increases, matching the reducing area under $S_{\delta \nu }\text{(}f\text{)|}H\text{(}f{\text{)|}}^{\text{2}}{W}_{avn}(f).$

2.6 Frequency stability requirements for ASCENDS

Using Eqs. (5) and (6), the relative error $(\Delta \tau -\Delta {\tau }_0)/\Delta {\tau }_0$ due to the finite line shape $L({\nu }_F)$ is numerically evaluated and shown in Fig. 4 (left) as functions of the effective bandwidth 1/Δt of a top-hat laser pulse shape for the CO₂ line. As detailed in Appendix A, the line shape $L({\nu }_F)$ is simply taken to be the ESD of the top-hat pulse shape. $(\Delta \tau -\Delta {\tau }_0)/\Delta {\tau }_0$ is smaller than a 0.02% target [5] and thus does not need to be corrected for all three online points (with frequency offsets + 0.5 GHz, + 1.1 GHz, and + 1.7 GHz) when the pulsewidth Δt is longer than 0.5 μs. τ is taken to be the same as τ₀ for the CO₂ line in our numerical examples where Δt is assumed to be ~1 μs.

 

Fig. 4 Relative errors for atmospheric CO₂ DAOD measurements: (left) (Δτ - Δτ₀)/Δτ₀ due to the finite linewidth 1/Δt of a top-hat pulse shape (calculated from Eqs. (5) and (6)); (right) laser line-center frequency noise contributions to the RRE for σ(δ_νn^on) = 0.1 MHz (dashed red, calculated from Eq. (21)), and RSE ln(R_b)/Δτ for σ(δ_ν1) = 10 MHz (dotted blue, from Eq. (18)) as functions of the online laser frequency offset from the absorption line center. The two-way optical depth τ (solid green) is also plotted, with blue dots marking the same online and absorption line-center frequency points used on the left.

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The partial RRE $[(d\tau /d{\nu }_c{)}_{on}/\Delta \tau ]\sigma ({\delta }_{\nu n}^{on})$ for$\sigma ({\delta }_{\nu n}^{on})=0.1\text{MHz},$ the partial RSE $|\ln (R_b)|/\Delta \tau ,$ and $\tau ({\nu }_c,2r_G)$ are plotted in Fig. 4(right) for atmospheric CO₂, as functions of the online laser frequency offset from the absorption line center. Table 1 summarizes the RRE targets, relevant parameters, and resulting effective laser frequency noise requirement for atmospheric CO₂ DAOD measurement. The frequency offset of the offline laser is set at 15.6 GHz. To bound the RRE to < 0.03% for an online frequency ≥ 0.5 GHz away from the absorption line center, the standard deviation $\sigma ({\delta }_{\nu n}^{on})$ of the effective frequency noise needs to be suppressed down to 0.23 MHz. This frequency stability requirement has been satisfied experimentally (see [13] and subsection 2.5).

Table 1. RRE Targets, Relevant Parameters, and Resulting Effective Laser Frequency Noise Requirement for Atmospheric CO₂ DAOD Measurement

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The partial RSE $|\ln (R_b)|/\Delta \tau$ for $\sigma ({\delta }_{\nu 1})=10\text{MHz}$ is less than 0.002% at such online frequencies, and thus negligible. However, this bias and $(\Delta \tau -\Delta {\tau }_0)/\Delta {\tau }_0$ (Fig. 4, left) become significant near the absorption line center, where the excessive CO₂ absorption also results in higher RRE. It is thus undesirable to place an online frequency point near the absorption line center. Referring to Eq. (20), the last bias term $(d\tau /d{\nu }_c{)}_{on}^2{\sigma }^2({\delta }_{\nu n}^{on})/2$ is $<{10}^{-7}$when $[(d\tau /d{\nu }_c{)}_{on}/\Delta \tau ]\sigma ({\delta }_{\nu n}^{on})$ is bound to 0.03%, and thus negligible. The third bias term can be reduced to a negligible level with photon count integration. As shown in section 5, the remaining bias term $|\ln (R_A)|$ can be reduced to $<1.1\times {10}^{-4}.$ Overall, the RSE $b_{\Delta \tau \_SNK1}/\Delta \tau$ can be reduced to < 0.02%.

3. Incorporating additional detection noise sources

We now incorporate additional detection noise sources arising from the background solar radiation, detector dark count, and receiver circuitry noise for the lidar receiver. The mean and variance of the photon count K_n-n produced by the background solar radiation is given by $\overline{K_{n-n}}=2\alpha N_{bgd}{B}_o\Delta t$ and ${\sigma }^2(K_{n-n})=F_e\overline{K_{n-n}}$, where $2N_{bgd}$ is the PSD of the background solar radiation and B_o the optical filter bandwidth. The equivalent detector dark count K_d (normalized by the mean internal gain M_e of the detector) has a mean and variance given by $\overline{K_d}={\lambda }_d\Delta t$ and ${\sigma }^2(K_d)=F_d\overline{K_d}$, where λ_d is an equivalent dark count rate and F_d an effective dark-count excess noise factor. The equivalent photon count K_T arising from the receiver circuitry noise has a zero mean and a variance ${\sigma }^2(K_T)$given by

The laser signal photon count K_s is estimated as $K{\text{'}}_s=K_{tot}-K{\text{'}}_{bgd},$ where K_tot is the total count $K_{tot}=K_s+K_{n-n}+K_d+K_T$ measured within Δt, and $K{\text{'}}_{bgd}=K{\text{'}}_{n-n}+K{\text{'}}_d+K{\text{'}}_T$ is the background count within Δt scaled from a separate measurement taken between the pulse measurements. This background measurement can be taken in a longer duration βΔt, where β is a background integration time multiplier, to reduce the background variance. Obviously, $\overline{K{\text{'}}_s}=\overline{K_s}$. Referencing Eq. (9), the variance of K'_s is found to be

The transmitted pulse energy E_s can be estimated as $E{\text{'}}_s=E{\text{'}}_{tot}-E{\text{'}}_{bgd}$ where E_tot is the total detected energy and the background energy E'_bgd arises from the circuitry noise, dark current in the detector, and ASE in the laser amplifiers. The mean of E'_s equals E_s and the variance of E'_s can be neglected (see Appendix B). The additional noise sources can be included by replacing S_NK1 with $S_{NK2}\equiv {\displaystyle {\sum }_{i=0}^{n-1}K{\text{'}}_s(i)/E{\text{'}}_s(i)}.$ Similarly, S_K is replaced by $S_{K\text{'}}\equiv {\displaystyle {\sum }_{i=0}^{n-1}K{\text{'}}_s(i)}$. Obviously, $\overline{S_{NK2}}=\overline{S_{NK1}}$. The normalized variance of S_NK2 is found to be

As seen from Eq. (20), the “log after averaging” DAOD estimator in Eq. (19) leads to an additive bias term $(1/2)F_e(\overline{S_{NNK1}^{on}}/{\overline{S_{NK1}^{on}}}^2-\overline{S_{NNK1}^{off}}/{\overline{S_{NK1}^{off}}}^2)\cong (1/2)F_e/\overline{S_K^{on}}$, and thus requires significant photon count integration ($\overline{S_K^{on}}$~10⁴) to bring the bias term down to a tolerable level (~10⁻⁴). When using S_NK2, the additional background variance ${\lambda }_{bgd}\Delta t$ in ${\sigma }^2(K{\text{'}}_s)$ results in an additional DAOD bias term $(1/2){\lambda }_{bgd}\Delta t(\overline{S_{NN2}^{on}}/{\overline{S_{NK2}^{on}}}^2-\overline{S_{NN2}^{off}}/{\overline{S_{NK2}^{off}}}^2)$. These two bias terms can be reduced significantly by incorporating the following bias correction factor $C_{\Delta \tau \_SNK2}$into the DAOD estimator given below

Using the bias-corrected DAOD estimator$\Delta {\tau }_{SNK2}$ given by Eq. (27), the bias $b_{\Delta \tau \_SNK2}\equiv \overline{\Delta {\tau }_{SNK2}}-\Delta \tau$ and variance of $\Delta {\tau }_{SNK2}$ are found to be

To minimize the time-varying Etalon and atmospheric turbulence effects, it is desirable to reduce n, the number of pulses being averaged for $\Delta {\tau }_{SNK2}.$ To do this, the overall averaging time $T_{tot}$ is divided into m slots (each is $T_{tot}/m$ long), and a $\Delta {\tau }_{SNK2}$ (denoted as $\Delta {\tau }_{SNK2}(k)$) is evaluated within each slot $(k=0,1,2,\mathrm{..},m-1).$ Δτ can then be estimated from the average of $\Delta {\tau }_{SNK2}(k)$ given below

The variance of $\Delta {\tau }_{SNK2\_av}$ (excluding contribution from the slow laser frequency drifts) also decreases approximately by the number of participating pulses (i.e., $n\times m$), as in the case for $\Delta {\tau }_{SNK2}.$ However, the averaging in Eq. (30) cannot effectively reduce the bias to Δτ.

4. Numerical estimation of measurement errors

Using realistic parameters listed in Table 2 , $\Delta {\tau }_{SNK2\_av}/\Delta \tau ,$ the RRE for atmospheric CO₂ DAOD measurement, is computed and plotted as a function of τ in Fig. 5 . Partial contributions to the RRE from the signal shot noise $~F_e[1+\exp (\Delta \tau )]/\overline{S_{K\text{'}}^{off}}$, frequency noise, solar background, receiver circuitry noise, and detector dark count are also shown Fig. 5. When there are 3200 detected photons (in average) for each offline pulse, the RRE is ≤ 0.1% for τ between 0.66 and 3.2. This photon count can be achieved with $E_s=\text{4}\text{mJ}$ for an average ρ = 0.17. The detector specifications listed in Table 2 can be met with a state-of-the-art HgCdTe avalanche photodiode (APD) detector [21]. The detected solar radiation count K_n-n is estimated for a zenith angle of 75°. Its noise contribution is larger than that from the receiver transimpedance amplifier (TIA). The noise from the dark count K_d is negligible. It should be noted that the RRE increases sharply when the online point moves further away from the line center, confining the frequency offset of the online points to within ± 1.7 GHz from the CO₂ absorption line center. The standard deviation of the effective frequency noise averaged in the 10-s measurement time is taken to be 0.23 MHz for the above calculations. When it is increased to 1 MHz, the RRE rises above the 0.1% target (also shown in Fig. 5).

Table 2. Parameters Used to Evaluate the RRE of Δτ as a Function of τ

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Fig. 5 The RRE σ(Δτ_{SNK2_av}) /Δτ for atmospheric CO₂ DAOD measurement (solid black) as a function of the two-way optical depth, computed from Eqs. (29) and (30) using parameters listed in Table 2. The blue dots mark the three online frequency points used in Fig. 4. Also plotted are partial contributions to this RRE from the shot noise (solid grey), frequency noise (dashed red), solar background (dotted brown), receiver circuitry noise (dash-dotted green), and detector dark count (long-dashed blue). The standard deviation of the effective frequency noise σ(δ_νn^on) averaged in 10 s is 0.23 MHz for the above calculations. When it is increased to 1 MHz, the RRE (dotted black) rises above the 0.1% target (dashed black).

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It is desirable to squeeze down RRE contributions from noise sources other than the signal shot noise. This will translate into reduction in required laser power and hence the cost. The laser peak power entering the lidar receiver needs to be high enough so that the signal shot noise contribution is well above the noise contributions from the receiver circuitry and the solar background count. This is the case for Goddard’s ASCENDS lidar where online points are placed at least 0.4 GHz away from the CO₂ line center with $\tau \le 2.4$.

5. Impact of surface reflectance variation

As discussed earlier, the surface reflectance variation adds a bias $|\ln (R_A)|$to the DAOD estimation. Since this bias varies from one path to another, it needs to be kept to a small fraction of the 0.1% RRE. Also, the PSD of the effective laser frequency noise ${\delta }_{\nu n}(t)$ is reduced by the periodical window function $W_{avn}$ whose average height $<W_{avn}>$ depends on the variation of A_s(i). Both factors are quantified here using measured surface reflectance data in [10] for realistic error estimation.

The parameters used for the evaluation are listed in Table 3 . The laser beam is assumed to travel at the same speed as the spacecraft. As shown in Fig. 1, the laser pulses are assumed to repeatedly cycle through wavelength channels 1, 2, 3, 4, 8, 7, 6, 5, 1, to minimize the separation between online (channels 2 to 6) and offline (channels 1 and 8) pulses. A set of surface reflectance data that has strong variations is used for this evaluation. The surface reflectance measurement was taken in southern Spain using ~10-m laser spot size and a step size of ~6 m [10]. To convert this data to the reflectance for our beam size, a 1-D running average is taken within our beam size (50 m) and the averaged reflectance is used as A_s(i) in our calculation. The raw and averaged reflectance data are plotted in Fig. 6 (left). For an averaging time of 1 s, there are n = 1000 pulses for each wavelength and the path length is 7 km. In contrast, the separation between online and offline beam spots is only 2.1 m at most. Consequently, the ratio R_A is very close to 1. This point is confirmed by our calculation results shown in Fig. 6 (right). The 108-km total path length is divided into 15 sections (7 km each) and $\ln (R_A)$ is evaluated for each section. For each online wavelength channel, its starting pulse for sum ${\displaystyle {\sum }_{i=0}^{n-1}{A}_s^{on}(i)}$ comes in between two adjacent offline pulses, each serves as the starting pulse for its offline sum ${\displaystyle {\sum }_{i=0}^{n-1}{A}_s^{off}(i)}$. Each ${\displaystyle {\sum }_{i=0}^{n-1}{A}_s^{off}(i)}$ is used to calculate $\ln (R_A)$ and the average of the two is plotted in Fig. 6 (right) for the worst case where the bias $|\ln (R_A)|$is $<1.1\times {10}^{-4}.$ The corresponding RSE is smaller than 0.02% if the two-way DAOD is kept > 0.55.

Table 3. Parameters Used for Evaluation of the Impact of Surface Reflectance Variation

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Fig. 6 (left) Relative surface reflectance: measured data (grey) and its running average over a length of 50 m (blue), courtesy of A. Amediek of Deutsches Zentrum für Luft- und Raumfahrt (DLR); (right) $\ln (R_A)$ calculated from surface reflectance data for the worst case. For each online wavelength channel, two $\ln (R_A)$ are calculated from both offline sum ${\displaystyle {\sum }_{i=0}^{n-1}{A}_s^{off}(i)}$ for 1-s averaging time and the average is plotted (blue dot). An average of 10 consecutive $\ln (R_A)$ values, each computed as above over 0.1 s averaging time, is also plotted (red square).

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$|\ln (R_A)|$ is increased when the averaging time (hence n) is reduced. Figure 6 (right) also shows an average of 10 values of $\ln (R_A),$ each computed as above over 0.1 s averaging time consecutively along the worst path. For some wavelength channels, this averaged bias $|\ln (R_A)|$ becomes larger than 10⁻³ and thus unacceptable. To reduce the bias $|\ln (R_A)|,$ it is crucial to allow a fast pulse repletion rate (≥ 1 kHz for each wavelength) and a sufficient integration time (≥ 1 s), and to properly interleave online and offline pulses to minimize their separation.

The average height $<W_{avn}>$ of $W_{avn}(f)$ is also evaluated for the 7-km averaging paths. The calculated $<W_{avn}>$ is quite close to 1/n, varying between 1.01/n and 1.12/n.

6. Mitigating impact of surface height variation

The DAOD errors due to surface height variation can be reduced to negligible levels if pulse-by-pulse knowledge of surface height is incorporated into the formulation. Let $z_0(i)$ be the surface height at laser beam spot i relative to a certain reference ground surface, and $z(i)$ a measured value of $z_0(i).$ $r_G$ now represents the range from the spacecraft to the reference surface. The distance from the spacecraft to the actual surface is $r_G-z_0(i).$ Since $z_0(i)$ could vary by tens of meters over the path of pulse averaging, $\tau (r_G-z_0(i))$ could fluctuate over a few tenths of a percent over the path, resulting in additional DAOD errors. To mitigate this problem, we estimate $\Delta \tau (2r_G)$ (i.e., the two-way DAOD from the spacecraft to the reference surface and back to the spacecraft) from $S_{NK3}\equiv {\displaystyle {\sum }_{i=0}^{n-1}K{\text{'}}_s(i)/[A_z(i)E{\text{'}}_s(i)]},$ where $A_z(i)\equiv \exp [2{\sigma }_{eff}(\overline{\nu },r_G)N(r_G)z(i)]$ is introduced to account for the fluctuation of the optical depth. The corresponding estimator $\Delta {\tau }_{SNK3}$ of $\Delta \tau (2r_G)$ can be easily adapted from $\Delta {\tau }_{SNK2}$ in Eq. (27) by simply replacing $E{\text{'}}_s(i)$ with $A_z(i)E{\text{'}}_s(i)$ in the relevant terms in $\Delta {\tau }_{SNK2}$ (i.e., S_NK2, S_NNK2, and S_NN2). As shown next, the variation of the surface height $z_0(i)$ does not affect such estimation of $\Delta \tau (2r_G).$ Nevertheless, the imperfect measurement of $z_0(i)$ still leads to additional DAOD errors.

To simplify the treatment, we assume $z(i)=z_0(i)+\Delta z+{\delta }_z(i),$ where $\Delta z$ represents a slowly varying altimetry bias, and ${\delta }_z(i)$ a zero-mean random altimetry error. $\Delta z$ is treated as a constant during the n pulse averaging, and the ${\delta }_z(i)$ values among different pulses are assumed to be uncorrelated and have the same variance ${\sigma }_z^2.$ Expanding $1/A_z(i)$in S_NK3 as