4 Error budget

To perform a statistical analysis, we introduce the time interval $T_{\rm Residual}$ which is the difference between the measured time interval $T_{\rm Obs}$ given by Eq. (1) and the computed travel time $T_{\rm Comp}$. The residual precision $\sigma _{\rm Residual}$ is defined by the root mean square of $T_{\rm Residual}$, measured in stable conditions. Since the measurements are corrected by the term $\left\langle t_{\rm Calib}-t_{\rm Start}\right\rangle$ at a period $\tau _{\rm Calib}$, all the instrumental precisions have to be evaluated over a period $\tau _{\rm Calib}$.This condition concerns the start and the return detectors, the laser, and the timers. The accuracy of the measurement $E_{\rm Echoes}$ illustrates the error between the real distance and the measured distance. It depends on the precision of the measurement and on the unknown systematic errors. One obtains only an estimation of the accuracy. The measurements being degraded by some noises, which are time dependant, it is convenient to introduce the time stability $\sigma _{x}(\tau )~$of these measurements over a period $\tau$. This can be computed by the Time variance TVAR (Allan et al. 1991). To evaluate the precision, the accuracy and the stability, we first present the different uncertainty sources involved in the lunar measurement.

4.1 Uncertainty sources

4.1.1 Laser

The laser characteristics are:

- type: YAG

- 4 pulses at a rate of 10 Hz

- $\lambda ~=~532$ nm and 1064 nm

- FWHM = 70 ps

- Energy: 75 mJ @ 532 nm; 75 mJ @ 1064 nm per pulse.

The mean value of the light pulse width is in the range of 70 ps. The shape of the pulse is Gaussian. The pulse width experiences some variations between 50 ps and 90 ps, caused by temperature inhomogeneities in the laser cavity. These variations are only encountered during the warm-up of the laser and a stable regime is obtained after thirty minutes. The start detector response is sensitive to the variation of the pulse width since the detector receives a large amount of photons. The variation $\delta t$ of the laser width introduces a shift equal to $\delta t/2$ between the pulse centre and the photo-detector electrical response. The laser width root mean square of about 50000 laser pulses, measured in stable conditions, is 9 ps. This quantity will introduce in the global error budget the term $\sigma _{\rm Laser\ Edge}=4$ ps rms. The return detector response is sensitive to the width of the laser pulse since it is working in a single photon mode. The term added to the error budget, deduced from the laser FWHM, is $\sigma _{\rm Laser\ Width}=30$ ps rms. Figure 2

  

Figure 2: Time stability of the laser pulse width. The time interval between two laser pulses is 0.2 s

shows the time stability of the pulse width. The time interval $\tau _{0}$ between each measurement is 0.2 s. All these laser width measurements are performed with a streak camera Hamamatsu C1587.

4.1.2 Start detector

The start detector is an InGaAs photodiode of 80 $\mu$m aperture, coupled with a very fast comparator. The laser light is extracted for the start detection at the cavity output with a mono-mode fiber of 15 m length. The detection is performed on the infrared pulses. Since the green pulses are generated from the infrared ones by a second harmonic generation phenomenon in a nonlinear medium, there is no time variation between the infrared and the green pulses. The amount of photons impinging the start detector and the configuration of the device permit to obtain an electrical response which is quite independent of the photon number: lower than 10 ps/octave. Some individual experiments have shown that the time precision $\sigma _{\rm Start\ Detector}$ of the start detector is better than 5 ps.

4.1.3 Return detector

The return detector is an avalanche photodiode built by Silicon Sensor (SSOAD230H) working in Geiger mode (Ekstrom 1981); (Samain 1998). Its aperture is $230~\mu$m. The photodiode break-down voltage is 140 V. The total voltage applied on the photodiode to detect the photons is 290 V. The detector operates at $-40~^{\circ}{\rm C}$. The quantum efficiency in Geiger mode is 0.2. Some workbench experiments have shown that the transit time experiences a variation versus the position of the light spot on the active surface of the detector. With r being the distance between the photodiode centre and the spatial point where the photon is absorbed, and $T_{\rm Return\ Detector}(r)$ the transit time difference between the response obtained with a photon absorbed at the centre and the response obtained at a distance r, we have measured

 

$$T_{\rm Return\ Detector}(r)~=~ar^{2}; a=0.005~{\rm ps}/\mu {\rm m}^{2}.$$ (2)

In the case where the light spot's radius is R, the transit time mean value $<T_{\rm Return\ Detector}\gt$ is a function of the above equation and of the probability P(r) that a photon is absorbed in a ring of width $\delta r$ and of diameter r. We then have  

$$P(r)=\frac{2r\delta r}{R^{2}}$$ (3)

and  

$$\left\langle T_{\rm Return\ Detector}\right\rangle = \int\no... ...s_{0}^{\rm R}\frac{2r{\rm d}r}{R^{2}}ar^{2} =\frac{1}{2}aR^{2}.$$ (4)

Experimentally, we found, with $R=115~\mu$m, $\left\langle T_{\rm Return\ Detector}\right\rangle =39$ ps (the above computation gives 33 ps).

The best time precision $\sigma _{\rm Return\ Detector}$ of the photodiode is obtained with r=0 and with a spot radius R as small as possible. Then, we have $\sigma _{\rm Return\ Detector}$ $(r=0,R=20~\mu{\rm m})=35$ ps. In the case where the whole active area of the photodiode is used we have $\sigma _{\rm Return\ Detector}$ (r=0,R=115 $\mu {\rm m})=50$ ps.

The size of the lunar return spot is determined by the optical configuration and by the atmospheric turbulence. The seeing, which roughly varies from $1^{\prime\prime}$to $7^{\prime\prime}$, modifies the spot size from 50 to 200 $\mu$m and the return detector precision $\sigma _{\rm Return\ Detector}$ lies in an interval of 35-50 ps rms.

4.1.4 Timers

The timers were built by Dassault Electronique, a French company, for the OCA Lunar Laser ranging station. These timers were also designed to perform a Time Transfer by Laser Link experiment (under development at OCA and CNES, French space agency) with some performances improved by two orders of magnitude as compared to the current time transfer techniques (Fridelance et al. 1997). The time base of both timers, generated from a 10 MHz signal, is performed by a single synthesizer also designed by Dassault Electronique. The frequency time base of the timers is 200 MHz. The time precision $\sigma_{\rm Timer}$ of the timer is 5 ps and the resolution is 1.2 ps. The linearity error is of the order of 1 ps without any table correction. The time stability of the timer coupled within the synthesizer is 8 fs over 1000 s $(\tau _{0}=400~\mu{\rm s})$. The time drift in a laboratory environment is below 5 ps over two months. The sensitivity in temperature is 0.5 ps per degree.

4.1.5 Retroreflector array orientation

Since the return detection operates in a single photon mode, an orientation difference between the normal axis of a retroreflector array n and the axis defined by the direction (retroreflector array, telescope) will introduce a dispersion in the measurements. This orientation difference depends on the lunar libration and on the initial orientation of the panel as compared to the mean orientation of the Earth centre as seen from the Moon (O x ) (see Fig. 3).

  

Figure 3: Retroreflector orientation. The reflection point precision depends on the angle between n and Ot

This latter is estimated at $\pm 1{{}^{\circ }}$ for the Apollo retroreflectors and at $\pm 5{{}^{\circ }}$ for the Lunakhod ones. The reflection point will be known with a certain precision which we expressed in time as $\sigma_{\rm Retroreflectors}$. For simplification, one considers that the retroreflector array is located at the centre of the Moon. The vector O t defines the instantaneous direction of the Earth telescope as seen from the Moon's centre. At a given date, the orientation O t, as compared to the mean direction O x, is given by the libration longitude $\lambda$ and the libration latitude $\beta$. The dispersion would be proportional to the angle ($\widehat{{\vec{ Ot}},{\vec{ Ox}}}$) if the panel were symmetrical. This can be applied to the retroreflectors Apollo XI and XIV which are square. Then,

 

$$\sigma _{\rm Retroreflector}=\frac{2D\sin (a\cos (\cos \beta \cos \lambda ))}{c}\frac{1}{2\sqrt{3}}$$ (5)

where D is the equivalent size of the panel. The term $\frac{1}{2\sqrt{3}}$is introduced to obtain a rms precision, since the probability to obtain a return from a given corner cube of the panel does not depend on the location of the panel. For the Lunakhod and Apollo XV panels, the direction of the panel major axis p has to be taken into account in the dispersion computation. One defines L as the largest size of the panel (in the direction p) and l as the smallest. Approximating that the equivalent size D in Eq. (5) varies between L and l according to an ellipse, we get  

$$D\approx \sqrt{L^{2}\cos ^{2}\theta +l^{2}\sin ^{2}\theta }$$ (6)

where $\theta$ is the sum of the angles between the projection of t on the planes (yOz) and (Oy), and the angle a between p and the plane (x Oy). One has  

$$\theta =\frac{\sin \beta }{\cos \beta \sin \lambda }+\alpha.$$ (7)

The panel sizes are (Faller 1972); (Fournet 1972)

$1040\times 630$ mm² for Apollo XV

$450\times 450$ mm² for Apollo IX and XIV

$440\times 190$ mm² for Lunakhod 1 and 2

$\sigma_{\rm Retroreflectors}$ lies in an interval from 0 to 350 ps rms for the larger retroreflectors (Apollo XV), and from 0 to 150 ps for the smallest. The correlation between the theoretical and the observed precisions on Apollo versus the lunar libration XV will be studied in the following (Sect. 4.2).

4.1.6 Noise

Some noise events can degrade the precision of the measurements. These noise events are introduced by photons coming from the Sun and by thermal triggering of the avalanche photodiode. The latter events can be neglected. When the retroreflector is in the lunar day, the measured noise frequency $f _{\rm Noise}$ is of about 2 MHz. On the contrary, in the lunar night the noise frequency is a few kHz. This noise depends on the field of view of the optics, the spectral width of the interferential filter placed in front of the detector, and the telescope aperture. The light events outside a temporal gate $\Delta t=6~\sigma _{\rm Residual}$ are eliminated. The noise event number accumulated during an observation period $T_{\rm Obs}$ in this gate $\Delta t$ is $f_{\rm Noise}\ f_{\rm Shoot}\ T_{\rm obs}\Delta t$, where $f_{\rm Shoot}$ is the laser rate (10 Hz). The noise events being random, the residual precision $\sigma _{\rm Residual}$ is given by  

$$\sigma_{\rm Residual}^{2}=k\sigma _{\rm Noise}^{2}+(1-k) \sigma_{\rm Residual^{\prime }}^{2}$$ (8)

where k is the fraction between the noise events and the total events number inside the gate $\Delta t$. $\sigma _{\rm Noise}$ is the root mean square of the noise events taken in the gate $\Delta t$, and $\sigma _{\rm Residual^{\prime }}$is the residual precision without any noise events. Since the noise is white, we have $\sigma _{\rm Noise}=\frac{\Delta t}{2\sqrt{3}}$. The filtering at $\pm$ 3 sigma implies $\sigma _{\rm Noise}=\sqrt{3}\sigma _{\rm Residual}$.Finally, one gets

$$\begin{eqnarray} &&\sigma _{\rm Residual}^{3}\cdot \frac{18f_{\rm Noise}f_{\rm S... ...rm Obs}}{ N_{\rm Residual}}+\sigma _{\rm Residual^{\prime }}^{2}=0\end{eqnarray}$$ (9)

where $N_{\rm Residual}$ is the events number inside the gate $\Delta t$ taken in the computation. Figure 4,

  

Figure 4: Precision degradation versus the quantity $f_{\rm Noise} /N_{\rm Residual}$, for different values of $\sigma _{\rm Residual^{\prime }}$. Better is the time precision without any noise $\sigma _{\rm Residual^{\prime }}$, less sensitive to noise is the global precision $\sigma _{\rm Residual}$

which illustrates Eq. (9), allows us, for $T_{\rm obs}~=~600$ s, $f_{\rm Shoot}$ = 40 Hz, to evaluate the precision degradation versus the fraction $f_{\rm Noise} /N_{\rm Residual}$ for different $\sigma _{\rm Residual^{\prime }}$. In the case where $f _{\rm Noise}$ = 2 MHz, $N_{\rm Residual}$ = 100, and $\sigma _{\rm Residual^{\prime }}$ = 200 ps, one gets $\sigma _{\rm Residual}$ = 300 ps. If $f _{\rm Noise}$ is in the kHz range, (lunar night and echoes obtained during the night), the precision degradation due to noise is not significant.

4.1.7 Time base

The time base of the timers is a commercial atomic clock HP 5071A. The frequency accuracy of this clock is better than 10^-12. The temporal error corresponding to the clock frequency error is of the order of $E_{\rm Clock}=3$ ps over 2.5 s. In the time interval 0.2 $<\tau <$ 3 s, the clock signal is perturbed with a flicker frequency modulation. The Allan variance root square $\sigma _{y.{\rm Clock}}$ is  

$$\sigma _{y.{\rm Clock}}^{{}}=5~10^{-12}\tau ^{0};~0.2~<~t~<~3~s.$$ (10)

These stability and accuracy data are provided by the constructor. After conversion into time variance (TVAR) one gets  

$$\sigma _{x.{\rm Clock}}^{2}=0.645\frac{\tau ^{2}}{3}\sigma _{y.{\rm Clock}}^{2}$$ (11)

where the coefficient 0.645 is the ratio of the modified Allan variance to the Allan variance. The temporal stability of the clock over 2.5 s will degrade the global precision of the measurement by $\sigma _{x.{\rm Clock}} (\tau=2.5~{\rm s})=6$ ps. Figure 5

  

Figure 5: HP 5071A - HP 5061A clocks time stability

shows a time stability measurement of the HP 5071A and HP 5061A atomic clocks. The measured stability corresponds to the HP 5061A stability, and the time stability performance of the HP 5071A is probably better. Taking into account the measured global time stability we get $\sigma _{x.{\rm Clock}}(\tau =2.5~{\rm s})=10$ ps.

4.1.8 Calibration

The distance between the corner cube used for the calibration and the LLR spatial reference (cross of the mechanical axis) is measured geometrically. The maximum error of this measurement is estimated at 3 mm. This error will introduce a systematic time error $E_{\rm Geometric}=$ 10 ps. The optical components of the telescope are tied together with some invar rods. Consequently, the variation in distance of the calibration path can be neglected.

The transit time sensitivity of the return detector versus the spatial position of the light spot on the detector will introduce an error in the distance measurement if the light spot of the calibration and the light spot of the lunar returns are located at different places or, if the spot sizes are different. The speed aberration, due to Earth and Moon rotation, combined with an eventual non alignment of the laser axis with the reception axis, can introduce an offset with regard to the centre of the photodiode of the order of 100 $\mu$m. This offset, expressed in time versus Eq. (2), gives $T_{\rm Return\ Detector}=50$ ps. A gelatine polarizer is placed on the calibration path to adjust the calibration level versus the telescope orientation. This polarizer acts also as a diffuser and broadens the calibration spot size which is then limited by the diaphragm. Combining the optical aberration and this diaphragm limitation, the spot radius of the calibration pulse is 100 $\mu$m, and $\left\langle T_{\rm Return\ Detector}\right\rangle$ for the calibration is 25 ps. This magnification of the calibration spot size allows us to obtain a calibration information which is quite independent of the adjustment of the calibration spot position. As compared to $T_{\rm Return\ Detector}$ an error of  25 ps can be introduced in the global error budget by this alignment problem. Another accuracy problem will be encountered if the voltage mean value applied on the return detector for the calibration echoes is different from the one used for the lunar echoes. The observed voltage difference is of the order of 3 V. Converted into time, this will introduce an error in the range of 15 ps. Finally, the phenomena introduced by the return detector degrade the accuracy measurement with a term $E_{\rm Calibration\ Detector}$ = 65 ps.

The echoes detected from the Moon are corrected with the term $\left\langle t_{\rm Calib}-t_{\rm Start}\right\rangle$. This mean value is computed from a set of dates, recorded over a time $\tau _{\rm Calib}$. The period $\tau _{\rm Calib}$should be deduced from the calibration time stability. The typical shape of this time stability is shown in Fig. 6.

  

Figure 6: LLR calibration time stability. Between $\tau _{0}$ and $\tau _{1}$, the calibration is perturbed by some white phase noise

For $\tau$ between $\tau _{0}$ = 0.5 s and $\tau _{1}$ = 1000 s, one has some white phase noise ($\sigma _{x.{\rm Calib}}~=~5.1~10^{-11}~\tau ^{-1/2}$). The determination of the calibration mean value $\left\langle t_{\rm Calib}-t_{\rm Start}\right\rangle$ has to be computed from the larger time interval, $\tau _{\rm Calib}$ less than $\tau _{1}$, that is to say $\tau _{\rm Calib}$ = $\tau _{1}$. If it is computed from $\tau _{\rm Calib}\gt\tau _{1}$, we would loose some information and on the contrary, if $\tau _{\rm Calib}<\tau _{1}$, the residuals will be perturbed by a useless white phase noise. Following this rule, the calibration correction will introduce a dispersion in the global error budget equal to $\sigma_{\left\langle {\rm Calib-Start}\right\rangle }^{{}}$  = 4 ps.

4.1.9 The atmosphere

It is essential to know the air index on the travel path of the laser beam to take into account the delay introduced in the measurements (Herring 1992). This delay is computed from the Marini Murray model (Marini & Murray 1973). The computation uses some atmospheric measurements performed near the station. An error in the evaluation of the real air index value will introduce an accuracy problem. The accuracy $E_{\rm Atmosphere}$ evaluated on the delay is 15 ps when the Moon is observed at an elevation of 90${{}^{\circ }}$ (that is not possible in France) and 70 ps at 15${{}^{\circ }}$. An unknown air index variation on the laser beam path will degrade the precision of the measurement. The short term perturbation 0.5 < $\tau$ < 1000 s of the residual is a white phase noise (see Fig. 8). If the atmosphere has an effect on the measurement, this effect has a time constant lower than 0.5 s or greater than 1000 s. Some stellar interferometry experiments demonstrate that a phase correlation exists between two beacons a few tens of meters apart. The evolution of the phase between the beacons is a continuous function. The phase drift is lower than 100 wavelengths over 100 s. The corresponding time propagation difference between the beacons is then smaller than 0.1 ps. This interferometric measurements are relative (between the beacons) and do not give any information on the absolute time propagation of the light. Nevertheless, it seems improbable that the atmosphere exhibits an important air index variation at a spatial scale of a few tens of meters with a time scale lower than 0.5 s. Concerning the air index fluctuation over a period longer than 1000 s, Fig. 8 cannot yield any consistent information, since the observed flicker noise could be generated by some other sources (computation, clock, return detector, ...). Other experiments should be performed to improve our knowledge of the long term atmosphere comportment. In conclusion, the dispersion $\sigma_{\rm Atmosphere}$ introduced by the atmosphere, computed over a period lower than 1000 s, will be small as compared to other dispersions and will be neglected in the global precision error budget.

4.2 Precision

The error budget for the precision, deduced from Eq. (1), is

$$\begin{eqnarray} \sigma _{\rm Residual^{\prime }}^{2} &=&\sigma _{\rm Start}^{2}... ...e {\rm Calib-Start}\right\rangle }^{2}+\sigma _{\rm Geometric}^{2}\end{eqnarray}$$ (12)

where $\sigma _{\rm Residual^{\prime }}$ is the residual precision without any noise. Taking into account all the elements intervening in the residuals, one gets

$$\begin{eqnarray} \sigma _{\rm Residual^{\prime }}^{2} &=&\sigma _{\rm Start\ Det... ...r \\ &&\sigma _{\left\langle {\rm Calib-Start}\right\rangle }^{2}.\end{eqnarray}$$ (13)

As seen before, one has

- $\sigma _{\rm Laser\ Edge}^{{}}$  = 4 ps: leading edge variation of the laser pulse

- $\sigma _{\rm Laser\ Width}^{{}}$ = 30 ps: width of the light pulse

- $\sigma _{\rm Start\ Detector}^{{}}$  = 5 ps and $\sigma _{\rm Return\ Detector}^{{}}$ = 50 ps: start and return detector

- $\sigma _{\rm Timer}^{{}}$  = 5 ps: electronic timer

- $\sigma _{x.{\rm Clock}}^{{}}(2.5~{\rm s})$  = 10 ps: time stability of the clock over a period of 2.5 s

- $\sigma _{\rm Retroreflectors}^{{}}$  = 0 - 350 ps: retroreflector orientation

- $\sigma_{\left\langle {\rm Calib-Start}\right\rangle }^{{}}$ = 4 ps: calibration.

Numerically, one gets: $\sigma _{\rm Residual^{\prime }}$ = 60 ps in the case where the retroreflector dispersion is 0. This value corresponds to the better precision obtainable on the travel time of light pulses from Earth to Earth via the Moon. This time precision allows us to measure the Earth-Moon distance with a precision of 9 mm. This computation agrees with the calibration precision computed over $\tau _{\rm Calib}$. Among all the instrumental error sources, the main dispersion comes from the return detector.

  

Figure 7: Residual precision computed from the corner cubes array orientation and the intrinsic LLR performances versus the residual precision measured. Retroreflector: Apollo XV

Figure 7 shows the residual precision $\sigma _{\rm Residual}$ obtained on the Apollo XV retroreflector and the theoretical precision taking into account the dispersion generated by the retroreflector array orientation $\sigma_{\rm Retroreflectors}$. The precision is determined from the residual obtained since 1995 on the Apollo XV retroreflector array. $\sigma _{\rm Residual}$ is computed with the echoes accumulated over several hours in order to improve the error on the precision. To avoid any degradation of the precision due to the orbit determination, the residuals are fitted by a fourth order polynomial. The angle between the vector p and the plane (xOy) is fixed to 0. The normal axe of the retroreflector is supposed parallel to the mean Earth direction (Ox) (see Fig. 3). The good correlation between the theory and the measurements demonstrates that all the elements intervening in the precision error budget have been taken into account. This graph shows also that the precision of the Earth-Moon measurement is almost always limited by the retroreflector orientation. A future precision improvement of the LLR stations would be interesting if the echo numbers on the other retroreflector arrays (which are smaller than the Apollo XV ones) were increased.

4.3 Accuracy

The error sources intervening in the accuracy $E_{\rm Echoes}$ of the light pulses travel time from the Earth to the Earth via the Moon are:

- $E_{\rm Clock}$ = 3 ps: frequency clock accuracy

- $E_{\rm Geometrical}$ = 10 ps: accuracy of the geometrical distance of the calibration corner cube as compared to the reference point of the station

- $E_{\rm Calibration\ Detector}$ = 65 ps: accuracy of the return detector for the calibration

- $E_{\rm Atmosphere}$ = 50 ps: accuracy evaluation of the atmospheric delay

- $2\sqrt{3}$ $\sigma _{\rm Residual}$ = 220-1200 ps: residual precision 65 < $\sigma _{\rm Residual}$ < 350 ps.

One gets

$$\begin{eqnarray} E_{\rm Echoes}~ &=&~E_{\rm Clock}~+~E_{\rm Geometrical}~+ \nonu... ...{\rm Atmosphere}~+ \nonumber \\ &&2\sqrt{3}\sigma _{\rm Residual}.\end{eqnarray}$$ (14)

Numerically, one obtains, in the best libration condition, $E_{\rm Echoes}~=350$ ps, which corresponds to a Earth-Moon distance accuracy equal to 50 mm. The quantity $E_{\rm Echoes}$/2 is an evaluation of the absolute value of the maximum difference between the round trip time deduced from one echo and the real distance. It illustrates the worst possible case to envisage. As we will see in the following chapter, the measurement accuracy can be improved by eliminating a fraction of the noise introduced by the residual precision. This is achieved by computing the mean value of many individual measurements.

4.4 Normal point

A normal point is the round trip time of the light pulse at a given time t from the spatial reference of the station to the retroreflector array computed from many individual echoes. This computation permits to decrease the computational time necessary for the scientific use of these data and to reduce the amount of data. For an integration period $\tau _{\rm Normal}$ we have a set of observed round trip time intervals $T_{\rm Obs}(i)$ at the date t(i). We also have a set of computed round trip time intervals $T_{\rm Comp}(i)$at the same date t(i). We compute the date mean value $\left\langle t\right\rangle$ from t(i) and look for the nearest date t(n) from $\left\langle t\right\rangle$. Corresponding to this date t(n) one has a computed round trip time interval $T_{\rm Comp}(n)$. The normal point $T_{\rm Normal}$, giving the mean round trip time at the date t(n) is  

$$T_{\rm Normal}~=~T_{\rm Comp}(n)~+~\left\langle T_{\rm Obs}~-~T_{\rm Comp}\right\rangle.$$ (15)

This computation method has been introduced by Christian Veillet. Since the beginning of LLR, the determination of $\tau _{\rm Normal}$ has been more or less arbitrarily fixed to ten minutes. To be sure that there is no information lost during the realization of a normal point, a time stability study of the measurement has to be performed. Figure 8

  

Figure 8: Residual time stability computed from echoes obtained on Apollo XV. The normal point precision deduced from the 23/09/96 observation session is 10 ps (1.5 mm), and the best $\tau _{\rm Residual}$ is 1500 s

shows the residual time stability computed from two complete observation nights (23/09/96 and 30/01/97) on the Apollo XV retroreflectors. It appears that two kinds of noise are present in the data. For the best stability curve, one gets:

- 10 < $\tau$ < 1500 s: $\sigma _{x.{\rm Residual}}(\tau )~=~3~10^{-10}~\tau ^{-1/2}$ corresponding to a white phase noise

- 1500 < $\tau$ < 5000 s: $\sigma _{x.{\rm Residual}}(\tau )~=~10~10^{-12}~\tau ^{0}$ corresponding to a flicker phase noise.

In this case, the best choice for $\tau _{\rm Residual}$ is 1500 s. The precision of the normal point computed over this period is $\sigma _{x.{\rm Residual}}(\tau _{\rm Residual}=1500~{\rm s})=10$ ps which corresponds to a distance precision equal to 1.5 mm. The accuracy of the normal point is improved as compared to the accuracy of each individual echo. One gets

$$\begin{eqnarray} E_{\rm Normal}~ &=&~E_{\rm Clock}~+~E_{\rm Geometrical}~+ \nonu... ...er \\ &&2\sqrt{3}\sigma _{x.{\rm Residual}}(\tau _{\rm residual}).\end{eqnarray}$$ (16)

Numerically, for the 29/09/96, $E_{\rm Normal~}$= 160 ps (24 mm).

In practice, the period $\tau _{\rm Residual}$ is not a constant. This way, to preserve all the information in the normal point, a time stability analysis is necessary. Another way is to set $\tau _{\rm Calib}$ small enough so to be sure that no information has been eliminated. In this case, both the computational time and the amount of data will be raised. Furthermore, the accuracy and the precision of the normal point will not illustrate the best accuracy and precision available. In all cases, since information storage is today not a problem, the echoes should be saved without any filtering (one year of LLR echoes would represent only a few Mo).