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\title{Mars 1064-nm Spectral Radiance Measurements
from the Receiver Noise Response \\ 
of the Mars Orbiter Laser Altimeter  \\

 }



\author{ Xiaoli Sun

     \thanks{NASA Goddard Space Flight Center, Code 694, Laser Remote Sensing Branch,  Greenbelt, MD 20771, USA

            (telephone: 301-614-6732, e-mail: xiaoli.sun-1@nasa.gov) } \\
   Gregory A. Neumann
     \thanks{Department of Earth, Atmospheric and Planetary Sciences, 
Massachusetts Institute of Technology,
54-518, 
Cambridge, MA 02139-4307, USA, also NASA GSFC, Code 697, Greenbelt MD 20771} \\

         James B. Abshire  

     \thanks{NASA Goddard Space Flight Center, Code 690,  Greenbelt, MD 20771, USA } \\
     Maria T. Zuber
     \thanks{Department of Earth, Atmospheric and Planetary Sciences, 
Massachusetts Institute of Technology,
54-518, 
Cambridge, MA 02139-4307, USA }   \\

}


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\date{March 26, 2005}



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\begin{abstract}
The Mars Orbiter Laser Altimeter provides not only Mars surface topography from the laser pulse time-of-flight, but also two radiometric measurements, the active measurement based on the ratio of the reflected to the transmitted laser pulse energy, and the passive measurement based on the ratio of the incident to the reflected sunlight. A novel technique was developed to compute the background light power on the photodetector from the detection threshold and the noise threshold crossing rate.  The passive radiometry measurements provide images of Mars at 1064-nm wavelength over a 2 nm bandwidth with sub-km spatial resolution and with 2\% or better precision under full solar illumination. This paper describes the principle of operation, receiver model, calibration, and performance assessment from sample orbital measurements.
\end{abstract}


\clearpage


\section{Introduction}

The Mars Orbiter Laser Altimeter (MOLA)
\cite{Zuber1992, Abshire2000a, Afzal1994, Ramos1994} on board the Mars Global Surveyor (MGS) spacecraft 
\cite{Albee98, Albee2001} was designed primarily as a laser altimeter that measures range via the laser pulse time-of-flight. MOLA measurements have produced a high precision topographic map of Mars with 670 million individual laser shots over a period of more than one Mars year \cite{Smith2001a}. MOLA also provides data for two Mars radiometric measurements, (a) the ratio of the transmitted to the echo pulse energies and (b) the solar background light onto the photodetector. The former, which we refer to as active radiometry,  measures the reflectance to laser pulses at zero phase angle. The latter, which we refer to as passive radiometry, measures the Mars reflectance to sunlight. The MOLA receiver design and its calibration for the laser pulse time-of-flight and active radiometry measurements are described in \cite{Abshire2000a}. This paper describes the receiver model and performance analysis for passive radiometry measurements. 

The MOLA passive radiometry measurement is accomplished by converting the receiver's false alarm rate and the threshold setting into the background light power onto the detector. The false alarm rate, which is the number of threshold crossings per unit time due to detector dark noise and background light, is registered by a noise counter.  On board instrument algorithm dynamically adjusts the receiver's detection threshold to maintain a nearly constant false alarm rate. The detection threshold level thereby tracks the background light and allows the receiver to operate at maximum sensitivity in altimetry measurement under a prescribed false detection probability. The receiver's false alarm rate and threshold level can also be used to solve for the amount of background light entering the detector. The MOLA receiver thereby provides a passive radiometric measurement in a 2-nm-wide bandwidth centered at 1064-nm. As MGS orbits Mars, MOLA can function as a single pixel camera and provides maps of the apparent surface brightness and seasonal variation at 1064 nm wavelength.

\section{MOLA Receiver and Solar Background Radiance}
The MGS spacecraft orbits Mars in a near-polar, sun-synchronous, near-circular orbit at 400-km mean altitude. MOLA is oriented by the spacecraft toward the planet surface along with other payload instruments. Figure 1 shows a simplified diagram of the MOLA measurement configuration and light paths to the receiver. The MOLA receiver parameters are summarized in Table 1. The MOLA receiver field of view is co-aligned with the laser beam and covers an area larger than the laser beam footprint. The receiver telescope collects photons from the reflected laser light and sunlight scattered from the Mars surface and atmosphere within the receiver field of view (FOV). The laser light is pulsed and may be filtered out temporally. The background light is continuous and causes the detector noise to rise whenever MGS is on the sunlit side of the planet. 

The background light power $P_o$ incident on the MOLA detector can be written as
\begin{equation}
\label{PofO}

P_o =  I_{Mars} \  

      \Delta \lambda \ \eta_{rcvr} \ \pi \left( \frac{ \theta_{FOV} }{2} \right)^2 A_{tel} 

\end{equation}
where $I_{Mars}$ is the Mars radiance in watts per unit area, unit bandwidth, and unit solid angle subtended by the MOLA receiver,
$\Delta \lambda$ is the receiver optical bandwidth, $\eta_{rcvr}$ is the receiver optics transmission, $\theta_{FOV}$ is the receiver field of view, and $A_{tel}$ is the receiver telescope aperture area. Since the receiver parameters are known, $P_o$ can be calculated from the radiance of Mars and vice versa.

The average receiver false alarm rate is a predictable function of the background light and the threshold setting. Figure 2 shows an example of how the MOLA average receiver false alarm rate varied with the detection threshold under different background light levels during a special test on 13 September 2001.  The prelaunch dark noise data are also plotted for comparison. The test was conducted by cycling through a series of threshold settings while monitoring the false alarm rate as MGS orbited from night into martian daylight. These data show that the false alarm rate
varies over several decades at a given threshold setting in response to the changing background light.
At a nearly constant background radiance, the false alarm rate varied over the full dynamic range of the 16-bit counter as the threshold changed. 

Optical radiance measurements are usually expressed in terms of the radiance factor, or `I/F', which is defined as ratio of the measured radiance to that of an ideal diffusive surface in vacuum with 100\% reflectance under the same solar illumination.
An ideal diffusive surface (Lambertian) scatters the incident light uniformly into  $2 \pi$ steradians, and the radiance at any observation angle can be written as $I_0 \,cos( \theta_{s} ) / \pi $, where $I_0$ is the solar irradiance in
W m$^{-2} \mu m^{-1}$ and $\theta_{s}$ is the solar incidence or zenith angle. The radiance factor can be written as
\begin{equation}
I\!/\!F = \frac{\pi I_{Mars} }{I_0 \, cos( \theta_{s} ) } 
\end{equation}
The solar irradiance is well known and has varied by only 0.5\% over the past 100 years \cite{Frohlich1998}. The solar irradiance at 1064-nm wavelength is 647 mW m$^{-2}$ nm$^{-1}$ at 1 AU \cite{Accetta1993}, or 279 mW m$^{-2}$ nm$^{-1}$ at Mars' mean solar distance.
As Mars orbits  the sun with an eccentricity of $e$=0.0934,
the solar irradiance varies by the square of the Mars solar distance, or from 82\% to 122\% of its mean value during a Mars year.
The phase angle, the angle between the light source and the observer line of sight, varies as the spacecraft orbits the planet and can be calculated from the spacecraft attitude and navigation data.

The major differences between MOLA's passive and active radiometry measurements are  
the effects of the atmosphere and the angle of illumination. The passive radiometry measures a combination of sunlight reflected by the surface and scattered by the atmosphere. 
The angle of illumination from the sun varies with MGS orbit position and pointing angle. The active radiometry measures the direct surface reflectance times the two-way atmospheric transmission and the angle of illumination is always the opposite of the observation. The active radiometry measurement is also influenced by the opposition enhancement \cite{Hapke1993}. The ratio of passive and active radiometry measurements gives a measure of the column atmospheric backscatter to extinction ratio. The atmospheric effect on the surface reflectance measurement can be removed if the atmosphere backscattering and extinction coefficients can be determined via other independent measurements, such as those described in \cite{ivanov98}. 

The sampling rate of the passive radiometry measurement is determined by the integration time of the noise counter. The noise counters are read and reset at the end of the integration time and there is little dead time between successive counting intervals. 
During MOLA ranging operations, the integration time was 1 second. MOLA operated in this mode during the Aerobraking and Mapping mission phases spanning two martian years \cite{Albee2001}.
During the MGS Extended Mission, after meeting all its measurement objectives, MOLA's on-board clock oscillator failed, ending ranging operation. MOLA was then configured to an enhanced passive radiometry measurement mode and continues to operate to this date. 
In this mode the noise counter readings are read at 8 Hz, which corresponds to an integration time of 0.125 s.
The threshold levels were also lowered so that the false alarm rate is maintained at a higher value. This improves measurement precision by reducing the random counting error. Table 2 gives a timeline of MGS and MOLA operation along with radiometry measurement configurations.

MOLA has operated nearly continuously in the enhanced passive radiometer mode since 10 October 2001. At 400-km orbital altitude the detector field of view covers a $\sim$340-m-diameter circle on Mars' surface.
At the MGS ground track speed of 3 km s$^{-1}$, a 0.125-second sampling interval corresponds to an along-track distance of 375 m, which gives a spatial resolution of $\sim$0.5 km when convolved with the receiver FOV. The cross-track resolution depends on the total number of tracks used in forming the image. The MGS orbital period is about 1 hour 58 minutes, and there are 8400 tracks over a Martian year with a track spacing of $\sim$3 km at the equator.

\section{Converting the Receiver False Alarm Rate and Threshold Level to Incident Optical Power} 
The key to the passive radiometry measurement is an algorithm to convert the receiver false alarm rate and the threshold level to the power of the background light on the detector. This is achieved by developing a precise receiver model and calibrating it with the prelaunch and in-orbit test data.

A simplified block diagram of the MOLA receiver is shown in Figure 3 with instrument parameters given in Table 1. Details on MOLA laser ranging and active radiometry measurement have been previously described in \cite{Zuber1992, Abshire2000a, Smith2001a}. The subsystems that collectively perform the passive radiometry measurement consist of the detector, postamplifier, lowpass filters, threshold comparators, and the counters that monitor the false alarm rates. Note that the receiver does not directly measure the continuous photocurrent as in a conventional radiometer, because the received signal is AC coupled between the detector and the postamplifier. Instead, it measures rate of the threshold crossings, which is the integral of the probability density function above the threshold. 

The MOLA receiver was designed to dynamically adjust its detection threshold according to Neymann-Pearson detection criterion \cite{Skolnik1962}, in which the detection threshold is set as low as possible while keeping the average false alarm no greater than a predetermined value. False alarms are caused by both the inherent detector dark noise and the quantum noise from the background light onto the detector. The false alarm rate is determined by the probability density function of the detector noise and the detection threshold. The detector dark noise is known from the preflight tests and has changed little on orbit. Therefore,  changes in the false alarm rate are caused solely by the background light power and the detection threshold. The background light power can be uniquely solved from the false alarm rate and detection threshold level based on the receiver model and the noise distribution functions.

The average number false alarms expected from the MOLA receiver within a counting gate interval can be written as 

\begin{equation}

\label{Nfa}

N_{fa} = \frac{T_g}{ T_{fa} } ,

\end{equation}

where $T_g$ is the gate interval and $T_{fa}$ is the average time between adjacent false alarm events.

The average rate of false alarms, $T_{fa}^{-1}$, is related to the probability of false alarm at a given time as \cite[p.31]{Skolnik1962}

\begin{equation}

\label{pfa1}

p_{fa} = \frac{ \tau_W}{ T_{fa}} .

\end{equation}

Here  $ \tau_W$ is the width of a noise pulse above the detection threshold and can be approximated as  $ \tau_W \approx 1/ BW_{3dB}$, where $BW_{3dB}$ is the receiver bandwidth at the -3dB response point ($1/ \sqrt{ 2 } $ in signal amplitude). For 
bandwidth-limited white noise, the occurrences of false alarms are statistically independent and the number of false alarms within a given time interval follows a Poisson distribution.



The probability of a false alarm occurring at a fixed time can be written as

\begin{equation}

\label{pfa2}

p_{fa} = \int_{y_t}^{\infty} p(y) dy ,

\end{equation}

where $p(y)$ is the probability density function (p.d.f.) of the instantaneous noise.


The noise from the MOLA detector is the sum of the shot noise from the photodetector and the electrical noise of the preamplifier, which are statistically independent and have different distributions. The p.d.f. of the total noise can be written as the convolution of the p.d.f.'s of the preamplifier circuit noise, $p_c (x)$, 

and that of the photodetector shot noise, $p_d (x)$, as

\begin{equation}

\label{pdfy1}

p(y) = \int_{-\infty}^{\infty} p_c (y-x) p_d (x) dx .

\end{equation}

For convenience, we define the integration variables, $x,y$, in units of the number of equivalent photoelectrons at the input to the preamplifier
within the receiver integration time. 


MOLA uses a silicon avalanche photodiode (APD) as the photodetector. The p.d.f. of the detector noise can be approximated as \cite{Webb1974}

\begin{equation}

\label{pdx1}

p_d (x) = 

\frac{1}

{ 

( 2 \pi G^2 F \bar{n}_e )^{1/2} 

\left[ 1+ \frac{ (x-G \bar{n}_e )(F-1) }{ GF \bar{n}_e } \right]^{3/2}

} 

\cdot

e^{ \left\{ 

\frac{ - ( x - G \bar{n}_e )^2 }

  { 2G^2 F \bar{n}_e \left[ 1+ \frac{ (x-G \bar{n}_e )(F-1) }{ GF \bar{n}_e } \right] } 

  \right\} } ,

\end{equation}

where $G$ is the average APD gain, $F$ is the excess noise factor, and $\bar{n}_e$ is the average number of the detected photons, or the primary photoelectrons, over the integration interval. The excess noise factor can be calculated as \cite{Webb1974}

\begin{equation}

F = k_{eff} G + \left(2 - \frac{1}{G} \right) (1-k_{eff} ) ,

\end{equation}

with $k_{eff}$ the ratio of the ionization coefficients of holes and electrons in the APD.

The mean and standard deviation of the APD output are given by \cite{Webb1974}

\begin{equation}

\label{xbar}

\bar{x} = G \bar{n}_e

\end{equation}

\begin{equation}

\label{sigmad}

\sigma_d^2 = F G^2 \bar{n}_e .

\end{equation}

The

average number of primary photoelectrons is given by

\begin{equation}

\label{nebar}

\bar{n}_e = \left[ 

\frac{ \eta_{APD} P_o }{hc/ \lambda } + \frac{I_{db}}{q}

      \right] \tau_n ,

\end{equation}

where $\eta_{APD}$ is the APD quantum efficiency, $P_o$ is the received optical power,

$h$ is Planck's constant, $c$ is the speed of light, $\lambda$ is the wavelength,

$I_{db}$ is the APD bulk leakage (dark) current,

$q$ is the electron charge, and 

$\tau_n$ is the equivalent receiver noise integration time. 

The other source of noise in the receiver is the preamplifier noise, which can be modeled as 
a zero-mean, Gaussian random variable. The standard deviation of the preamplifier noise can be estimated more conveniently from the frequency domain using the spectral noise density, which can be directly measured. The total noise variance can be obtained by integrating the noise spectrum over the noise bandwidth. For an ideal

integrator, the integration time is related to the one-sided noise bandwidth, $BW_n$, as

\begin{equation}

\label{taun1}

\tau_n = \frac{1}{2BW_n}  .

\end{equation}


The electronic filters in the MOLA receiver are not ideal integrators and their frequency responses can be approximated as 5-pole Bessel lowpass filters. The noise bandwidth is related to the 3-dB bandwidth by $BW_n = 1.04 \times BW_{3dB}$ using the definition of the noise integration time in Eq.(\ref{taun1}). The standard deviation of the integrated noise can be written as

\begin{equation}

\label{sc2}

\sigma_c^2 = \left( I_c^2 + 2 q I_{ds} \right) BW_n  \left( \frac{ \tau_n }{q} \right)^2
= \left[ \frac{ I_c^2 }{2 q^2 }  + \frac{ I_{ds} }{q} \right]  \tau_n  ,

\end{equation}

with $I_c^2$ the one-sided preamplifier input noise spectral density in A$^2$/Hz

and $I_{ds}$ is the APD surface leakage current in amperes.



Substituting Eq.'s (\ref{pdfy1}), (\ref{pdx1}), (\ref{xbar}), and (\ref{sigmad}) into (\ref{pfa2}),

exchanging the order of the double integrals, and using the standard complementary error function 

$ er\!f\!c(u) =  \frac{2}{ \sqrt{\pi} } \int_u^{\infty} e^{ -t^2 } dt$,
Eq.(\ref{pfa2}) can be rewritten as

\begin{equation}

\label{pfa4}

p_{fa} = \int_{- \infty }^{\infty}

\frac{1}{2} er\!f\!c \left( \frac{ y_t - x }{ \sqrt{2} \sigma_c } \right)

\frac{ exp \left\{ 

    \frac{ - ( x - \bar{x} )^2 }

    { 2 \sigma_d^2 \left[ 1+ \frac{ (x- \bar{x} )(F-1)G }{ \sigma_d^2 } \right] } 

     \right\}

}

{ 

( 2 \pi \sigma_d^2 )^{1/2} 

\left[ 1+ \frac{ (x-\bar{x} )(F-1)G }{ \sigma_d^2 } \right]^{3/2}

}

dx  .

\end{equation}



The average number of false alarms in a given interval can be obtained by substituting

Eq.'s (\ref{pfa1}) and (\ref{pfa4}) into (\ref{Nfa}).

All the parameters in the above equations are constant except for $\bar{x}$ and $\sigma_d$, which

are functions of the received optical power defined in Eq.'s (\ref{xbar}), (\ref{sigmad})

and (\ref{nebar}).
The threshold used in Eq.(\ref{pfa4}) can be scaled from the threshold voltages in the telemetry, $v_{th}$, as,

\begin{equation}
\label{vth}
y_{th} = \left(
\tau_n \frac{ \eta_{d} G }{hc/ \lambda }

\frac{a_{thre}}{ R_{det} } 
\right)
v_{th} ,
\end{equation}

where $R_{det}$ is the detector responsivity in V W$^{-1}$ and $a_{thre}$ is the scaling factor from the detector assembly output to the comparator input of the appropriate channel.
Other MOLA instrument parameters relevant for the passive radiometry measurement are given in Table \ref{molapar}.



The number of false alarms per unit time, the threshold level, and the received

optical power are all monotonic functions of each other. A unique solution can be found for any one of them given the other two.
A numerical method is derived to solve the integral equations, Eq.(\ref{pfa2}) through (\ref{nebar}),  for the background light power as a function of the threshold level and the false alarm rate, which we denote as  $f(v_{th}, N_{fa})$. 
Once the received optical power is found, the Mars radiance factor of Mars can be obtained from Eq.(\ref{PofO}).


\section{Calibration}
The MOLA receiver was calibrated using data from prelaunch testing. In one of these tests,  the false alarm rates were measured as a function of the threshold level under a set of known background light levels which were simulated by the MOLA ground support equipment. 


Figure 4 shows a plot of the calculated noise count rate vs. threshold level for several values of background light level. The measured data from the prelaunch testing are also plotted. Three receiver parameter values were refined from those given in \cite{Abshire2000a} in order to optimize the fit between the theory and the measurements. These parameters were the APD quantum efficiency, from 35\% to 40\%; the preamplifier noise current spectral density, from 2 $pA/ \sqrt{\rm Hz}$  to 1.74 $pA/ \sqrt{\rm Hz}$; and the receiver signal voltage gain (or threshold scaling factor), from 1.32 to 1.28. These parameter values had little effect on MOLA ranging performance estimates in \cite{Abshire2000a} but they had a significant effect in determining the exact shape of the noise probability density function on which the passive radiometry measurement depends. The parameter values given in Table \ref{molapar} are the revised values. The original values of the APD quantum efficiency and preamplifier noise density used in
\cite{Abshire2000a} were based on the manufacturer's minimum acceptable performance specification for the batch of detectors. The revised values are more close to the typical values from various tests. The small adjustment in the receiver gain is well within the tolerance of the circuit components used.

Figure 5 shows a surface plot of the optical power of the background light as a function of the threshold level and the false alarm rate. This surface plot is used as a look-up table for $f(v_{th}, N_{fa})$ in the MOLA
ground data processing to obtain the optical power of the background light from the observed false alarm rate and the threshold level.

\section{Measurement Precision and Accuracy}
Several factors influence the precision and accuracy of the MOLA passive radiometry measurements. These include variations in the detector responsivity  due to temperature, random fluctuations in the false alarm rate, uncertainty in the threshold level, and the slow drift in the receiver dark noise level and threshold voltage.

The receiver model and the parameter values described in the previous section are for a constant room-temperature condition. However, under different operating conditions during flight, the detector temperature can vary between 0$^\circ$C and 40$^\circ$C. 
The Si APD bias voltage is internally temperature compensated to maintain a nearly constant responsivity over this temperature range. As the temperature decreases, the APD quantum efficiency at 1064-nm laser wavelength decreases. The temperature compensation circuit then raises the APD gain to keep the overall responsivity constant to within $\pm 10 \%$.
The increase in APD gain causes the detector noise and the false alarm rate to rise.
From laboratory measurements of similar detectors, we found the temperature effects on the passive radiometry measurement can be corrected by a linear function as
\begin{equation}
\label{pow}
P_0 = ( 0.66 + 0.0097 T_{d} ) f(v_{th}, N_{fa}),

\end{equation}
where $T_d$ is the detector case temperature in $^{\circ} C$ and $f(v_{th}, N_{fa})$ refers to the function of background light power vs. threshold and false alarm rate at a nominal detector operating temperature ($35^{\circ} C $).

Unfortunately, the detector case temperature in MOLA is not directly measured but has to be inferred from the detector assembly interface plate temperatures contained in the telemetry data. The detector assembly is connected to the interface plate through the lens barrel. Because the Si APD responsivity is greater at higher temperature, the detector was warm biased with the use of low thermal conductivity media between the detector case and the lens barrel. The power dissipation of the detector itself keeps the detector warmer than the lens barrel. The detector subsystem can be modeled as a three-node cascade thermal system. Prelaunch thermal-vacuum tests of the detector assembly showed the APD case temperature, $T_d$, was  7.25$^\circ$C higher than that of the lens barrel, $T_{m}$, and 12.5$^\circ$C higher than that of the interface plate, $T_{i}$, at steady state.
Neglecting radiative heat transfer, the steady-state temperatures are related by the following equation 
 \begin{equation}
 \label{Q}
Q = k_1 (T_d-T_m) =  k_2 (T_m-T_i) ,
\end{equation}
where  $Q $ is the heat generated by the detector module ($\approx 1.25 \ W$), and  $k_1$ and $k_2$ represent the thermal conductivities of the detector circuit board and mounting fixture, respectively. 

In flight the actual APD temperature change lags the changes in $T_{i}$ by a significant amount due to the thermal inertias of the detector case, the lens barrel, and the interface plate.
For the three-node thermal system, the instantaneous temperatures are
related by a coupled pair of differential equations:
\begin{eqnarray}
\label{Tddot}
{ \dot{T} _d} & = & \frac{Q - k_1 (T_d-T_m)}{c_d}, \\
\label{Tmdot}
{ \dot{T}_m} & = & \frac{k_1 (T_d-T_m) - k_2 (T_m-T_i)}{c_m},
\end{eqnarray}
where $\dot{T}$ denotes the time-derivative of temperature, and $c_d$ and $c_m$ are constants.
Values of $k_{1}$ and $k_{2}$ may be obtained from Eq.(\ref{Q}), and $c_d$=  103.5 and $c_m$ = 433.5 $J/ ^{\circ} C$ were obtained by fitting the transient temperature data in the prelaunch tests.

The response time delay constants for this three-node system are $\sim$2400 s and $\sim$450 s, comparable to the time scales of orbital changes in thermal environment.
These time constants were also consistent with our observations in flight when MOLA experienced sudden temperature changes due to spacecraft operation.
The instantaneous detector case temperature is obtained by numerically integrating Eq.(\ref{Q}) through Eq.(\ref{Tmdot}). 

The detector dark noise also varies with temperature and is automatically measured when the spacecraft is over the night side of Mars. For small changes in temperature, the effect can be approximated as a small offset in the effective detection threshold given as
a linear function of the detector case temperature. The effective threshold, $v_{eth}$, can be written  as 
\begin{equation}
\label{dark}
v_{eth} =v_{th} - ( a_0 + a_1 T_{d}) ,
\end{equation}
with $v_{th}$ given in Eq.(\ref{vth}). For Channel 2, the offset coefficients are estimated to be $a_0 = 3.60 \times 10^{-3}$ and 
$a_1 = 3.13 \times 10^{-5}$. 

The random measurement error in the false alarm rate can be estimated by assuming that the false alarm events follow a Poisson distribution. The variance of the number of false alarms can be approximated by the number of the observed false alarms within the integration time.
When MOLA was in its altimetric laser ranging mode, the false alarm rate was held at about 100/s so that the probability of a false alarm within the 20-km nominal range gate interval ($133 \ \mu s$) was less than 1\%.
The standard deviation of the measurement error in the false alarm rate was, therefore, about 10\%.
During the enhanced  radiometry mode operation, threshold level was lowered to give about 10,000/s false alarms so that the random counting error was reduced to about 1\%. 

The uncertainty in the threshold level mainly comes from the inherent circuit noise at the comparator inputs, and the quantization error in digital-to-analog converter (ADC) output. 
The standard deviation of the inherent circuit noise was assumed to be 1 mV, which was typical for this type of circuit. The quantization error of the ADC may be modeled as a uniformly distributed random variable over the ADC step size.
The ADC step size for MOLA threshold voltage generation is 1 mV. The standard deviation is $1/ \sqrt{12}$ mV. The circuit noise and the quantization noise are independent of each other and the standard deviation is
$ \sqrt{ 1^2 + (1/ \sqrt{12} )^2 } = 1.04 \ mV $. 

The standard deviation of the error in the background light power measurement due to the false alarm count fluctuation and circuit noise can be approximated as
\begin{equation}
\sigma_P = \left[  \left( \frac{ \partial P_o }{ \partial y_{th} } \sigma_{y_{th} }  \right)^2 +
                           \left( \frac{ \partial P_o }{ \partial N_{fa} } \sigma_{N_{fa} }  \right)^2               
                  \right]^{1/2}  ,
\end{equation}
where 
$\sigma_{th}$ is the standard deviation of the threshold noise, and 
$\sigma_{N_{fa}}$ is the standard deviation of the number of false alarms, equal to the square root of the observed false alarm rate. The partial derivatives, $\partial P_o / \partial y_{th}$ and $\partial P_o / \partial N_{fa}$, can be determined from the slope of the surface plot shown in Figure 5 along the horizontal and vertical axis, respectively. Figure 6 shows the resulting normalized radiometry measurement error, defined as the ratio of the standard deviation to the mean. It shows that the majority of the measurement data have a relative error  $<$5\%. Measurements over low to mid latitude regions with higher than 1 nW background light onto the detector have a relative error  $<$2\%. 

During flight, the measured optical power of the background light onto the detector ranges from 0 to approximately 10 nW.
In darkness, the effective noise of the measurement is about 10 pW in standard deviation per 0.125-second interval, measured over a 100 s baseline. At typical daytime background light power level, the noise level may be estimated from observations over relatively featureless terrain, and is approximately 25 pW at an average incident power of about 5 nW, or 0.5\%, per 0.125 s.
The noise is dominated by the counting statistics and the error may be reduced by averaging several successive measurements at the cost of coarser spatial resolution.

The errors in the radiometry measurement may be further reduced by averaging the results from some or all the four receiver channels. Although all the channels share the same detector as the signal source, the false alarm rates are obtained over different electrical bandwidths and at different effective threshold levels. This means that the threshold crossings for different channels are not entirely correlated, thus in principle the uncertainties in the radiometry measurement may be improved by averaging them.
On the other hand, the channels with lower electrical bandwidth are more susceptible to low-frequency circuit noise, such as $1 \! / \!f$ noise and the spacecraft power supply noise. The overall passive radiometry measurements from MOLA were 
found to improve (i.e, less fluctuation in the along-track measurement) by averaging the results from Channels 1 and 2 but not Channels 3 and 4. Note that the threshold for Channel 1 was fixed near its maximum value during most of the laser ranging phase of the MOLA operation in order to reduce the ranging error and to reduce receiver saturation. Therefore, only Channel 2 passive radiometry measurements were available and no averaging could be performed during that time.

We also considered the
effect of space radiation on the receiver performance. The MGS radiation environment was estimated to be about 7 krad(Si) total dose per year during cruise and 6 krad(Si) per Mars year of operation behind the shielding of 2.54 mm (100 mil) thick aluminum sphere. The total dose at the detector which is the most sensitive to radiation damage was much lower, a few krad(Si) or less, because of the shielding provided by its surrounding materials. The total doses for the sensitive electronics components should be similar or a few times higher, depending on their position. In such a moderate radiation environment, the major effects to be considered are the increase in the detector dark current 
\cite{Sun1997}, the degradation of electrical circuits, and the darkening of the optics. 

The Si APD dark noise increase due to space radiation is estimated to be 30 pA/krad(Si) \cite{Sun1997}, which is comparable to the APD dark current before launch. As mentioned earlier, the total detector dark noise is monitored every orbit when MGS is on the dark side of Mars. Any slow increase in dark current due to radiation damage could be compensated by adjusting the threshold bias as in Eq.(\ref{dark}). The effects of space radiation on the electrical circuits and optics are believed to be negligible based on the instrument design, parts selection, and the preflight test data. The flight data shown in Figure 2 also confirmed that there was little change in the MOLA detector dark noise after five years in space. Therefore, the effect of space radiation appears inconsequential.  

\section{Radiometry Measurements from Orbit}
As an example, a sample of the MOLA threshold levels and the false alarm count rates for Channel 2 are plotted in Figure 7 as a function of the laser shot number for an orbit around Mars on 29 June 2001 (Orbit 20316). For this orbit, MOLA was in its normal laser ranging mode and Mars was near its northern autumnal equinox ($L_s \approx 180^{\circ}$). Figure 8 shows the resultant Mars radiance factor based on these passive radiometry measurements. For comparison, the radiance factor obtained from the active radiometry measurement is also plotted. The atmosphere was dusty during this time from a nascent global dust storm. This caused the active radiometry measurement to be substantially attenuated. On the other hand, the sunlight backscattered from the dust caused the scene to be brighter in the  passive radiometry measurement.

Figure 9 shows a Mars image constructed from the MOLA passive radiometry measurements over a period from March to June 1999. An image taken by the Hubble Space Telescope (HST) during the same period over a 61.1-nm optical bandwidth centered at 1042 nm
\cite{Bell2003a} is also shown for comparison. The MOLA measurements show features similar to those seen by HST. It also reveals more features of the Mars surface,
particularly near the polar regions where the lighting and the observation angles for HST are poor. 
The difference between the absolute values of the two was about 60\%.
The HST measurements were taken at a solar phase angle between $2.7^{\circ}$ and $10^{\circ}$, whereas the MOLA measurements are at phase angles greater than $23^{\circ}$ owing to the 2 PM local time MGS orbit, which accounts in part for the greater reflectivity seen by HST. Another reason was that the HST image was taken at a different time and over a relatively short exposure time while the MOLA image were taken by scanning the planet over an 120-day period. Although the MOLA passive radiometry measurements were not absolutely calibrated in space, the receiver performance has been extremely stable since before launch. 

Figure 10a shows a composite image of the Mars mid latitude regions made solely from three Mars years of  MOLA passive radiometry measurements, from the commencement of the MGS mapping phase to December 2004. To estimate the MOLA measurement stability, the average radiance factors of two exceptionally bright and dark regions at Amazonis and Syrtis,  300x150 km in extent, are trended and plotted in Figure 10b.
The sudden shift in the radiance factor in the Amazonis region midway in the plot coincided with a change in the MGS viewing angle (Table 2) was probably due to the  phase angle effect. There is no apparent long-term downward trend besides a seasonal variation, indicating that the receiver responsivity is stable. 

\section{Summarys}
The technique of measuring radiance from the detector noise density is shown to provide a precise and stable passive radiometric measurement over the spectral bandwidth of the receiver. Over time the MOLA passive radiometric measurement has provided a high-precision, near-infrared map of Mars over the $1064 \pm 1$ nm spectral range.
Monitoring of the temporal variations has the potential to study seasonal changes in the atmosphere and cryosphere of Mars
\cite{Zuber2003a, Byrne2003a}.

\section{Acknowledgments}
We would like to thank the guidance and support from the MOLA science team led by D. E. Smith. 
We would like to thank the MOLA instrument manager, R. B. Follas and the system engineer J. C. Smith for their leadership during the MOLA instrument development.

We would like to thank the members of the MOLA electronics team, including

R. L. Kasa, J. F. Cavanaugh, 

C. A. Long, J. N. Caldwell, A. T. Lukemire,  

R. B. Katz,  

G. C. Elman,

J. R. Baker, J. F. McGarry, and P. L. Jester

for their many 

contributions in the receiver design and testing. In particular, we thank J. B. Blair for the  software design and modification that allowed MOLA to continue passive radiometry measurements without the on board clock oscillator.
Also we thank Dr. J. F. Bell III for providing us the HST data used in Figure 9.


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\clearpage


\begin{table}[ht]

\caption{ MOLA Instrument Parameter Values Relevant for Passive Radiometry Measurements}

\vspace{0.1in}

\label{molapar}

\begin{tabular}{l l l} \hline

Symbol & Value & Description   \\ \hline \hline

$\lambda$ & 1064-nm & laser wavelength \\ 

$A_{tel}$ & 0.170 $m^2$ & receiver telescope entrance aperture area \\

$\theta_{FOV}$ & 0.850 mrad & receiver field of view (FWHM))\\

$ \tau_{rcvr} $ & 56.5\% & receiver optics transmission \\

$ \Delta \lambda $ & 2.0 nm & receiver optical bandwidth (FWHM) \\

$ \eta_d $ & 40\% & APD quantum efficiency at 1064-nm wavelength \\

$ G $ & 120 & average APD gain \\

$k_{eff} $ & 0.008 & APD ionization coefficient ratio \\

$I_{ds}$ & 15 nA & APD surface dark current \\

$I_{db}$ & 80 pA & APD bulk dark current \\

$N_{amp}$ & $(2 pA/Hz^{1/2} )^2 $ & preamplifier input noise density \\

$R_{det}$ & $1.26 \times 10^8$ V/W & detector assembly responsivity \\

$a_{thre}$ & 1.28 & threshold scaling factor for Channel 2  \\
$ BW_{3dB} $ & 5.54 MHz & receiver noise bandwidth for Channels 2 \\
                  &         & (scaling factors and bandwidth for other channels  \\
                  &         &  can be found in \cite{Abshire2000a}; those scaling factors \\
                  &         &  may requires minor adjustments to give \\
                  &         &  consistent results as those from Ch.2). \\
\hline

\end{tabular}


\vspace{0.5in}


\caption{MGS Chronology and MOLA Measurement Configurations}

\vspace{0.1in}

\label{molachron}

\begin{tabular}{l l l l l l} \hline

Beginning & MGS & MOLA measurements  & Viewing & Radiometry  \\
Date	& Phase  &   & angle  & resolution \\ 
	&   &   &  & (along track) \\ \hline \hline



7 Nov, 1996 & Launch    \\ 
11 Sept. 1997 & \multicolumn{2}{l}{Orbit Insertion} \\
15 Sept. 1997 & \multicolumn{2}{l}{Aerobraking} \\

3 Mar. 1999 & \multicolumn{2}{l}{Mapping}  \\

3 Mar. 1999 & & Laser ranging & 0$^\circ$ (Nadir) & 0.15 km dia spot  \\
  &  & \& active radiometry & 0$^\circ$ (Nadir)  & at 0.3 km spacing \\
    & &  passive radiometry & 0$^\circ$ (Nadir) & 0.34x3 km footprint \\

31 Jan. 2001 & \multicolumn{2}{l}{Extended Mission}  \\
30 June 2001 &   & \multicolumn{4}{l}{(MOLA clock oscillator anomaly \& investigations)} \\
10 Oct. 2001 & & Enhanced passive &   &   \\
  &   & radiometry & 18$^\circ$ & 0.34x0.5 km \\

  & \multicolumn{5}{l}{Enhanced passive radiometry measurement continues to this date.}  \\ \hline
\end{tabular}
\end{table}



\end{document}



