Optical Energy Variability Induced by Speckle: The Cases of MERLIN and CHARM-F IPDA Lidar

Abstract

In the context of the FrenchGerman space lidar mission MERLIN (MEthane Remote LIdar missioN) dedicated to the determination of the atmospheric methane content, an end-to-end mission simulator is being developed. In order to check whether the instrument design meets the performance requirements, simulations have to count all the sources of noise on the measurements like the optical energy variability induced by speckle. Speckle is due to interference as the lidar beam is quasi monochromatic. Speckle contribution to the error budget has to be estimated but also simulated. In this paper, the speckle theory is revisited and applied to MERLIN lidar and also to the DLR (Deutsches Zentrum für Luft und Raumfahrt) demonstrator lidar CHARM-F. Results show: on the signal path, speckle noise depends mainly on the size of the illuminated area on ground; on the solar flux, speckle is fully negligible both because of the pixel size and the optical filter spectral width; on the energy monitoring path a decorrelation mechanism is needed to reduce speckle noise on averaged data. Speckle noises for MERLIN and CHARM-F can be simulated by Gaussian noises with only one random draw by shot separately for energy monitoring and signal paths.

1. Introduction

Atmospheric methane is a greenhouse gas responsible for about 20% of the additional radiative forcing due to human activities since the industrial revolution [1]. In order to increase knowledge on atmospheric methane burden, the MEthane Remote LIdar missioN (MERLIN) is being developed jointly by CNES (Centre National d’Études Spatiales) and DLR (Deutsches Zentrum für Luft und Raumfahrt) [2,3,4] for a scheduled launch in 2024. It plans to deliver as level 2 products: atmospheric methane total weighted columns (XCH₄) with their associated weighting functions describing the vertical sensitivity of the measurement to CH₄ variation along the atmospheric column. It targets to achieve on XCH₄ systematic (deterministic or correlated) errors less than 3 ppb and random (stochastic or uncorrelated) errors less than 22 ppb for 50 km average when the mean methane content value is 1780 ppb. That corresponds to a signal-to-noise ratio (SNR) value (the inverse of the relative random error) of 81. This challenging accuracy is needed to improve the estimates of methane emissions and sinks [5] and implies to track all the sources of variability, uncertainties and biases in the measurement.

The MERLIN instrument is a double-pulsed IPDA (Integrated Path Differential Absorption) lidar [6]. The methane measurement thus relays on the Differential Atmospheric Optical Depth (DAOD) between two wavelengths. The lidar CHARM-F is an airborne demonstrator of the MERLIN instrument developed by DLR [7] that provides XCH₄ averaged over 2 km (corresponding to 7 s time average as for Merlin).

In the last decade, many studies have been done on the use of doubled-pulsed IPDA lidar to measure CO₂ or CH₄ atmospheric content from space [8,9,10] and many IPDA ground based and airborne demonstrators are been developed around the word by DLR [11,12], NASA Langley [13,14,15,16,17,18,19,20] and JPL [21,22], NIST [23,24], Tokyo Metropolitan University [25,26], Chinese Academy of Sciences [27], or an Italian group [28,29].

Speckle appears when coherent light is scattered by heterogeneities at the scale of its wavelength like variations in the surface roughness or in the refractive index. Scattered waves propagate along different optical paths and interfere in any observation plan showing patterns with granular structure of alternately light and dark spots [30,31]. For IPDA lidar speckle is a serious contributor to the SNR as it induces energy fluctuations on the detector [32,33,34,35,36,37,38,39,40,41].

In Section 2 of this paper, speckle contribution to SNR on XCH₄ is addressed and with MERLIN and CHARM-F specifications the corresponding speckle characteristics are determined. In Section 3, for the two instruments, speckle noise level and shot noise level are compared for one shot and after averaging. And in Section 4, a way to simulate energy fluctuations on a lidar detector due to speckle with some hypotheses on the speckle pattern decorrelation in time is described.

2. The Parameters Which Determines the Speckle Contribution to XCH₄ SNR and Their Values for Merlin and Charm-F

2.1. How Does Speckle Contribute to Signal-to-Noise Ratio on XCH₄ Measurements?

Double-pulsed IPDA lidar emits coherent light pulses and collect the energy scattered back by the ground surface. The atmospheric methane total weighted column (XCH₄) is estimated from the difference in atmospheric transmission between two laser pulses: one (ON) at a wavelength selected to have a high methane absorption, the other (OFF) as reference at a wavelength with significantly less methane absorption. The two pulses are close enough in wavelength and emitted with a small enough time interval in order to consider atmospheric, ground and instrumental optical properties to be identical except for CH₄ absorption. Nevertheless, differences in H₂O and CO₂ absorption between the two beams are accounted. In a typical case, these differences are in the order of a few 10⁻⁴ to compare with the DAOD due to methane in the order of 0.4 to 0.6.

From the vertical DAOD due to methane (DAOD_CH4), XCH₄ is obtained as an averaged value of methane dry-air volume mixing ratio with associated weighting function WF(p):

$$\begin{array}{c}XC{H}_4=\frac{{{\displaystyle \int }}_0^{P_{surf}}XC{H}_4\left(p\right)WF\left(p\right)dp}{{{\displaystyle \int }}_0^{P_{surf}}WF\left(p\right)dp}=\frac{DAO{D}_{C{H}_4}}{{{\displaystyle \int }}_0^{P_{surf}}WF\left(p\right)dp}\\ =\frac{DAO{D}_{slant}cos\mu -DAO{D}_{H_{2}O}-DAO{D}_{C{O}_2}}{{{\displaystyle \int }}_0^{P_{surf}}WF\left(p\right)dp},\end{array}$$

(1)

where μ refers to the incident angle departure from the nadir, P_surf is the surface pressure, and

$$WF\left(p\right)=\frac{{\sigma }_{C{H}_4}\left({\lambda }_{on},p,T_{\left(p\right)}\right)-{\sigma }_{C{H}_4}\left({\lambda }_{off},p,T_{\left(p\right)}\right)}{g\left(p\right)\left(M_{dry}+M_{H_{2}O}{X}_{H_{2}O}\left(p\right)\right)},$$

(2)

where σ_CH4 is the absorption cross sections of a mole of methane depending on wavelength, pressure p and temperature T, g is the acceleration due to gravity, M_dry is the average molar mass of dry-air, M_H2O is the molar mass of water vapor and X_H2O is water vapor dry-air volume mixing ratio. WF(p), DAOD_H2O and DAOD_CO2 are computed from data provided by meteorological centers and spectroscopic data found in GEISA database [42] with specific improvements [43,44].

Slant DAOD is determined as the ratio of the backscattered energy measurements for pulses ON and OFF (P_on and P_off) normalized by emitted energy measurements for each pulse (E_on and E_off) to deal with the fluctuations of the energy delivered by the laser source [40,41].

$$DAO{D}_{slant}=\frac{-1}{2}ln\left(\frac{P_{on}{E}_{off}}{P_{off}{E}_{on}}\right),$$

(3)

The detection chain acquires these energies together with the solar flux energy P_sun which is also acquired on its own. The actual measurements add random noises: electronic noise due to electric fluctuations of intensity or voltage caused by electrons movements in the electronics, shot noise due to fluctuations of time occurrence for electrons release from incident photons caused by the corpuscular nature of light and speckle noise due to fluctuations of the optical energy on detector caused by interference reflecting the wave nature of light. In this study, the focus is put on the speckle, the major contribution on noise coming from the electronic chain is not analyzed. Comprehensive analyses of IPDA lidar noises are developed in [11,33,38].

Mean energy of scattered light by ground is space distributed according to the ground Bidirectional Reflectance Distribution Function [45]. However, for quasi monochromatic light, it fluctuates because of interference. If the light is not quasi monochromatic, interference exists for each wavelength but with a negligible effect on the spatial and temporal distribution of the total energy. For geophysical surfaces speckle pattern is fully developed. For quasi monochromatic light, relative energy fluctuations through a given aperture S during a given time T can thus be linked to the observation geometry and the wavelength through the coherence area S_c and time τ_c. Signal-to-noise ratio due to speckle (SNR^sp) is then given, with P the polarization index of the light, by (cf. Appendix A)

$$SN{R}^{sp}=\sqrt{\frac{2}{1+P²}\left(1+\frac{S}{S_c}\right)\left(1+\frac{T}{{\tau }_c}\right)},$$

(4)

In this analysis speckle due to scattering from aerosol and turbulence is not taken into account [35,37]. For an instrument, SNR^sp can be compared with the shot noise SNR due to the random noise related to the photoelectric effect and to its amplification in the avalanche photo-diode used as a detector. This noise (SNR^sn) can be expressed as a variation of the incoming flux [11]:

$$SN{R}^{sn}=\sqrt{\frac{\eta N}{F}},$$

(5)

where N is the number of photons constituting the optical energy flux, η is the detector quantum efficiency, F is the noise factor of the avalanche photo-diode, it refers to the noise due to the electrons multiplication process (It is one plus the ratio between the variance and the square of the mean of the probability for a primary electron to give secondary electrons). From speckle and shot noise variability, assuming that the uncertainties are not correlated, slant DAOD uncertainty is given by:

$${\sigma }_{DAO{D}_{slant}}^2=\frac{1}{4}\left({\left(\frac{{\sigma }_{P_{on}}}{P_{on}}\right)}^2+{\left(\frac{{\sigma }_{P_{off}}}{P_{off}}\right)}^2+{\left(\frac{{\sigma }_{E_{on}}}{E_{on}}\right)}^2+{\left(\frac{{\sigma }_{E_{off}}}{E_{off}}\right)}^2+2\alpha {\sigma }_{P_{sun}}^2{\left(P_{on}^{-1}-P_{off}^{-1}+E_{off}^{-1}-E_{on}^{-1}\right)}^2\right),$$

(6)

where, for each flux, the variance ${\sigma }_{Xxx}^2$ is ${\sigma }_{Xxx}^{sp}{}^2+{\sigma }_{Xxx}^{sn}{}^2$ and (1−α) is the correlation between solar flux contribution to the other measurements and its own measurement. Neglecting uncertainties coming from the integrated weighting function WF or from DAOD_H2O and DAOD_CO2 estimates and without taking into account the full relation involving the electronic detection response function, the data sampling for ground transmission and the ground processing, the SNR for XCH₄ due to uncertainties of optical fluxes on the detector (SNR^of) is given by:

$$SN{R}_{X_{CH4}}^{of}=\frac{2DAO{D}_{C{H}_4}}{\sqrt{SN{R}_{P_{on}}^{-2}+SN{R}_{P_{off}}^{-2}+SN{R}_{E_{on}}^{-2}+SN{R}_{E_{off}}^{-2}+SN{R}_{P_{sun}}^{-2}2\left(1-\alpha \right)P_{sun}^2{\left(P_{on}^{-1}-P_{off}^{-1}+E_{off}^{-1}-E_{on}^{-1}\right)}^2}},$$

(7)

2.2. MERLIN and CHARM-F Data Sets

For satellite or airborne IPDA lidar observation, the ground is counted as a plane rough surface illuminated by a coherent source and it is modelled as an extended source of chaotic light of intensity I_G (Δx, Δy, t) producing a fully-developed speckle pattern resulting from interference in the propagating space (Figure 1). The field of scattered light is observed at distance z of the rough surface and the area of coherence at this location is determined using results from Appendix A.

The instrument has relative speed to ground v. It emits wavelengths in (λ_on) and near (λ_off) a methane multiplet. The emitted beam is polarized and its divergence is div_beam. The receiver telescope is an afocal design with huge magnification. It consists of two conical mirrors M1 and M2 plus an achromatised ocular lens which generates an image of the entrance pupil. The collecting mirror M1 is elliptical. To obtain a compact design, the secondary mirror M2 is close to M1, and M2 partly vignettes the entrance pupil. Up to the detector the focal length of the receiver chain is f_rec. A band pass filter is incorporated on the optical path to block the light outside a narrow window with size L_filter. The incoming signal light scattered back by the earth is focused on the detector of diameter d_APD which collect also the calibration light from the energy monitoring path [12]. The use of the same detector for signal estimate and energy monitoring avoids variations in the optical-to-electrical response between the two, but the energy extracted from the emitted beam has to be reduced by several orders of magnitude to match the energy level of the lidar returns as the detector performances are for a limited energy range. For this purpose, integrating-spheres with fiber coupling are used [40]. The cut off frequency of the electronic detection is smaller than the sampling frequency ν_sample.

Table 1. MERLIN and CHARM-F lidars parameter values needed to compute speckle impacts on signal.

Parameter	Symbol	MERLIN	CHARM-F
Distance between ground and receiver	z	506.3 km	8.5 km
Instrument speed relative from Earth	v	7.6 km/s	0.2 km/s
ON wavelength	λon	1645.5518 nm	1645.555 nm
OFF wavelength	λoff	1645.8460 nm	1645.860 nm
Polarization of the emitted beam	P	1	1
FWHM laser pulse energy density spectrum	dνl	60 MHz	50 MHz
Beam full divergence at e−2 at transmitter telescope output	divbeam	0.18125 mrad	3 to 6 mrad
Length of elliptical entrance pupil	LM1	0.7325 m	0.06 m
Width of elliptical entrance pupil	lM1	0.69 m	0.06 m
Obscuration of M1 by M2	OM2	0.03%	0.00%
Focal length of reception optics	frec	0.4704 m	0.0303 m
Avalanche photo-diode diameter	dAPD	200 μm	200 μm
Spectral filter width	Lfilter	2 nm	2 nm
Sampling frequency	νsample	75 MHz = 1/13.3 ns	100 MHz = 1/10.0 ns

gives the values provided by manufacturers [6,14] for parameters which allow to compute speckle impact.

Table 2. Some geometric quantities computed from MERLIN and CHARM-F lidars parameters.

Quantity	Symbol	MERLIN	CHARM-F
Diameter at e−2 for energy distribution on ground	de2G = z divbeam	91.8 m	25.5–51.0 m
Diameter of the FOV on ground	dfov = z dAPD/frec	215.3 m	56.1 m
Effective size of the entrance pupil	SEP = π/4 LM1 lM1 (1-OM2)	3850.5 cm2	28.2 cm2

provides some auxiliary values computed from Table 1 data.

3. Speckle Contributions to Signal-to-Noise Ratio on MERLIN or CHARM-F XCH₄ Measurements

Laser pulses energy are Gaussian distributed on the ground. The energy of scattered light is thus modeled as:

$$I_{G\left(\Delta x,\Delta y\right)}=\frac{I_0}{2\pi {\sigma }_R^2}{e}^{\left(\frac{-1}{2}\frac{\Delta x^2+\Delta y^2}{{\sigma }_R{}^2}\right)},$$

(8)

where I₀ the mean intensity and σ_R the spatial standard deviation. That gives by integration in the field of view (FOV) of the receiver with Equation (A16) the effective area of the footprint looked as a secondary source for laser returns:

$$\begin{array}{c}{S}_{eff}=\frac{{\left[{{\displaystyle \iint }}_{FOV}{I}_G\left(\Delta x,\Delta y\right)d\Delta xd\Delta y\right]}^2}{{{\displaystyle \iint }}_{FOV}{I}_G^2\left(\Delta x,\Delta y\right)d\Delta xd\Delta y}=\frac{{\left(2\pi {\sigma }_R^2\right)}^2}{2\pi {\left(\frac{{\sigma }_R}{\sqrt{2}}\right)}^2}\frac{{\left(1-e^{\frac{-1}{2}{\left(\frac{d_{fov}}{2{\sigma }_R}\right)}^2}\right)}^2}{1-e^{-{\left(\frac{d_{fov}}{2{\sigma }_R}\right)}^2}}=4\pi {\sigma }_R^2\frac{e^{\frac{1}{2}{\left(\frac{d_{fov}}{2{\sigma }_R}\right)}^2}-1}{e^{\frac{1}{2}{\left(\frac{d_{fov}}{2{\sigma }_R}\right)}^2}+1}\\ \sim 4\pi {\sigma }_R^2=\frac{\pi }{4}{d}_{e2G}^2.\end{array}$$

(9)

Solar energy is uniform in the FOV. The energy of scattered solar flux is thus modeled by a top hat distribution:

$$I_{G\left(\Delta x,\Delta y\right)}^{sun}=I_0^{sun}\mathrm{inside}\mathrm{the}\mathrm{FOV},0\mathrm{elsewhere},$$

(10)

so, (still using Equation (A16)) the effective area for sun light is given by:

$$S_{eff}=\frac{\pi }{4}{d}_{fov}^2,$$

(11)

Because of the movement of the satellite (or of the aircraft) during the signal energy estimation there is time speckle renewal due to changes in the illuminating of the ground at the wavelength scale [36]. With d_eff the characteristic size of the coherent area (the diameter for a circular area), the characteristic time of this renewal is d_eff/v [32] which remains considerable even for space lidar (in the order of 0.1 ms). Moreover, as pulsed lidar are used, regardless of the laser beam characteristic time, there is a full coherence inside one pulse. There is no averaging of the interference pattern with increasing integration time as there is no more signal.

Table 3. Speckle characteristics for MERLIN and CHARM-F IPDA lidars (λ = 1645.7 nm).

Speckle Characteristics	Symbol	MERLIN	CHARM-F
Effective surface on ground for laser fluxes	Sefflas ~ π/4 de2G2	6618.7 m2	510.7–2042.8 m2
Effective surface on ground for solar flux	Seffsun = π/4 dfov2	36406.4 m2	2471.8 m2
Coherence surface for laser fluxes	Sclas = 4/π (λ/divbeam)2	105 mm2	0.38–0.096 mm2
Coherence surface for solar flux	Scsun = 4/π (λ × frec/dAPD)2	19 mm2	0.079 mm2
Characteristic time for sun light	τcsun = 1/(Lfilter × c/λ2)	4.52 10−3 ns	4.52 10−3 ns
Number of spatial speckles for laser fluxes	Mslas = 1 + SEP/Sclas	3668	7440–29449
Number of spatial speckles for solar flux	Mssun = 1 + SEP/Scsun	20267	35786
Number of temporal speckles for laser fluxes	Mtlas = 1	1	1
Number of temporal speckles for solar flux	Mtsun = 1 + δtdis/τcsun	296–2951	222–2213

provides both for MERLIN and CHARM-F the effective surface for incoherent source on ground, the coherence area, the coherence time and the number of speckles.

The number of temporal speckle for solar flux is estimated for a discretisation time δt_dis settled between a tenth of the sampling time to a sampling time. Even if for simulation purposes the time signal is discretised at a tenth of the sampling time, for each discretisation interval there are several hundreds of temporal speckles for solar flux.

Subjective speckle for laser and solar fluxes are not to be counted as the detection optics is built in such a way that all the photons collected by the entrance pupil reach the detector. So, speckle only modulates the spatial distribution of the energy on the detector and not its integrated value.

Along the optic calibration path (see Figure 2), speckle induces fluctuations of the energy amount at each location where there is energy dilution, specifically the output of the integrated spheres and the entrance of the optical fibers.

For CHARM-F it is assumed that the detector does not truncate the fiber output [14], but for MERLIN in order to avoid alignment issues, the illuminating area is larger than the detector size. Therefore, subjective speckle also occurs. For MERLIN, AIRBUS has done a dedicated study on preliminary design for this calibration path and estimated SNR_Eon/Eoff around 43 mainly from detector and fiber entrance face [personal communication]. For CHARM-F, the SNR_Eon/Eoff could be estimated from [40] around 59 mainly from fiber entrance face.

Table 4. SNR (signal to noise ratio) and RRE (relative random error) from speckle for different in the case of MERLIN and CHARM-F lidars.

	MERLIN	CHARM-F		MERLIN	CHARM-F
SNRPon/Poff	61	86	RREPon/Poff	1.6%	1.2%
SNRPsun	3470–10948	3986–12585	RREPsun	0.0%	0.0%
SNREon/Poff	43	59	RREEon/Eoff	2.3%	1.7%
SNRsn	<49	about 3000	RREsn	>2.0%	0.0%

summarizes SNR estimates obtained with Equation (4) and data from Table 3 for the different fluxes in the case of MERLIN and CHARM-F. P_on and P_off are polarized, P_sun is not, E_on and E_off neither because integrated-spheres depolarize [46]. The maximum number of backscattered photons for MERLIN has been estimated to less than 18000 and for CHARM-F it is about 3550 time more (because the distance between ground and the receiver is 8.5 km instead of 506.3 km). The relative random error (RRE) is the inverse of the SNR and can be expressed in percentage form.

Speckle noise is of the same order but smaller than shot noise for MERLIN lidar. Conversely, for CHARM-F lidar, speckle noise is dominant compared to shot noise which becomes negligible with the smaller distance between the receiver and the ground.

Using a mean value of 0.53 for DAOD and a mean value of 1780 ppb for methane content the random uncertainties coming from speckle on XCH₄ estimated with value of Table 4 and Equation (7) are summarized in

Table 5. Random noise on XCH₄ due to speckle for MERLIN and CHARM-F lidar.

	MERLIN	CHARM-F
Noise for one shot	60 ppb	41 ppb
Noise for a 7s average	5 ppb	3 ppb

. Noise for 7 s average corresponds to 50 km average for MERLIN and to 2 km average for CHARM-F.

The random error of 5 ppb from speckle is compatible with the specification of MERLIN random error less than 22 ppb [2] but shows how difficult it will be to reach the target user requirement for random error estimated at 8 ppb.

4. Discussion and Conclusions

For any satellite or airborne IPDA lidar, speckle induces variability of incident energy on the detector both for atmospheric and energy monitoring branches. Subjective speckle has no impact on the energy absorbed by the detector as long as it does not truncate the energy collected. SNR from speckle can be estimated as above (Table 4) on a shot by shot basis. The associated noises are smaller with respect to the electronic noise, for which averaging is needed. There is no difficulty in calculating such an average for simulated data even if geophysical variability should be taken into account to be realistic [47]. But the noise correlation between the various samples describing one shot and between different shots must be included.

For signal path the speckle pattern changes like the target from one shot to another. If the speckle pattern changes there is no correlation between the noises. But for energy monitoring path, the speckle pattern may be highly correlated from one shot to another. And to limit speckle impact on random noise for XCH₄ spatially averaged, it is necessary to either fully stabilize the speckle pattern, or to deliberately change it over time [48,49]. This second solution is easier to do on mobile lidar and the first one may induce systematic bias on the methane estimation. Using moving parts, specific systems have been designed to change speckle pattern between successive pulses. Tests with CHARM-F showed resulting noise exhibiting pure white noise behavior which can be reduced by averaging [40]. The introduction of such a mechanism prevents any correlation between speckle simulated noises for one shot and another [41].

Simulations of IPDA data with realistic speckle noise may be performed after computations similar to those in Section 3, and with the two hypotheses of full correlation for the samples of one shot (because the noise is fully correlated at this time scale as the speckle pattern is constant during the full registration of one pulse) and of full decorrelation between shots.

In summary, nothing has to be done for sun flux. Only Gaussian random noise has to be added, both on each sampling of energy monitoring fluxes and on each sampling of signal fluxes. Only one random draw is needed for energy monitoring path and signal path per shot. However, a new random draw is necessary for each shot because the speckle pattern vary from one pulse to the other according to the use of the decorrelation mechanism acting between two pulses.

For the MERLIN mission, the SNR on optical energy of backscattered signals due to speckle is around 60, always higher than the shot noise SNR which is less than 50. On the contrary for CHARM-F, speckle SNR is smaller than the shot noise SNR. Speckle occurring on the energy monitoring path is associated to pulse by pulse SNR about 40 for MERLIN (60 for CHARM-F) and so it is the dominant speckle impact even with a decorrelation mechanism. A Gaussian model of these noises has to be included in the Merlin simulator.