###### Abstract

The GRACE Follow-On satellites carry the very first inter-spacecraft Laser
Ranging Interferometer (LRI). After more than four years in orbit, the LRI
outperforms the sensitivity of the conventional Microwave Instrument (MWI).
However, in the current data processing scheme, the LRI product still needs
the MWI data to determine the unknown absolute laser frequency, representing
the “ruler” for converting the raw phase measurements into a physical
displacement in meters. In this paper, we derive formulas for precisely
performing that conversion from the phase measurement into a range, accounting
for a varying carrier frequency. Furthermore, the dominant errors due to
knowledge uncertainty of the carrier frequency as well as uncorrected time
biases are derived. In the second part, we address the dependency of the LRI
on the MWI in the currently employed cross-calibration scheme and present
three different models for the LRI laser frequency, two of which are largely
independent of the MWI. Furthermore, we analyze the contribution of thermal
variations on the scale factor estimates and the LRI-MWI residuals. A linear
model called Thermal Coupling (TC) is derived that significantly reduces the
differences between LRI and MWI to a level where the MWI observations limit
the comparison.

###### keywords:

GRACE Follow-On; Laser Ranging Interferometer; Scale Factor; Tone Error;
Thermal Coupling

1 1 0 https://doi.org/ Scale Factor Determination for the GRACE Follow-On
Laser Ranging Interferometer including Thermal Coupling Scale Factor
Determination for the GRACE Follow-On Laser Ranging Interferometer including
Thermal Coupling Malte Misfeldt 1*, Vitali Müller 1, Laura Müller 1, Henry
Wegener 1 and Gerhard Heinzel 1 Malte Misfeldt, Vitali Müller, Laura Müller,
Henry Wegener and Gerhard Heinzel Misfeldt, M.; Müller, V.; Müller, L.,
Wegener, H., Heinzel, G. Correspondence<EMAIL_ADDRESS>
##  1 Introduction

The joint U.S.-German gravity space mission GRACE Follow-On (GRACE-FO)
continues its successful predecessor mission, the Gravity Recovery And Climate
Experiment (GRACE). The twin satellites were launched on the 22nd of May 2018,
and the Laser Ranging Interferometer (LRI) was successfully commissioned in
mid-June 2018 Abich et al. (2019). The GRACE-FO mission was designed to
provide data continuity and thus follows the basic concept and design of the
predecessor mission. Its main scientific instrument for inter-satellite
distance measurements is the K-band Ranging (KBR) (or Microwave Instrument,
MWI) together with an accelerometer on each spacecraft to determine non-
gravitational accelerations acting on the two spacecraft for later removal in
the data processing. Global observations of Earth’s gravitational potential
and its variations from space allow valuable insights into the hydrological
cycle, including rainfall, droughts, ice-melting, and sea-level rise Landerer
et al. .

New to GRACE-FO is the LRI, a technology demonstrator to prove the feasibility
of laser interferometry for distance measurements between two spacecraft
flying a few hundred kilometers apart. The LRI shows drastically increased
precision compared to the KBR instrument Abich et al. (2019); Ghobadi-Far et
al. . Based on the flawless in-orbit operation for over four years and without
any degradation observed so far, the LRI technology is now being adopted to
serve as the primary instrument in future missions like Next Generation
Gravity Mission (NGGM), and GRACE-I(carus)/Mass Change Mission (MCM) Conklin
et al. (2020); Massotti et al. (2021); Nicklaus et al. (2019, ). Evolving from
a demonstrator to a primary instrument will include changes concerning
reliability and redundancy. Moreover, the success in demonstrating inter-
spacecraft laser interferometry was a milestone for the space-based
gravitational wave observatory LISA Amaro-Seoane et al. .

The LRI has shown very low noise in the inter-satellite ranging measurement of
about $0.3\text{\,}\mathrm{n}\mathrm{m}\mathrm{/}\sqrt{\mathrm{Hz}}$ at
Fourier frequencies of $1\text{\,}\mathrm{Hz}\text{/}$ Abich et al. (2019),
which is about three orders of magnitude below the noise of the MWI. However,
the conversion factor between the raw phase measurement of the heterodyne
interferometer and the desired displacement is needed to form the LRI ranging
signal. This conversion factor is the wavelength
$\lambda=c_{0}/\nu_{\mathrm{R}}$, with $c_{0}$ denoting the speed of light in
vacuum. The LRI laser frequency on the reference satellite $\nu_{\mathrm{R}}$
is actively stabilized to a resonance of an optical reference cavity using the
Pound-Drever-Hall (PDH) technique. The variations in the cavity’s resonance
frequency mainly depend on the resonator’s thermal stability, and the
frequency’s absolute value can not be measured directly in flight. Therefore,
the current data processing scheme foresees a cross-calibration of LRI and KBR
to determine the relative scaling between the KBR range and LRI range, using
an initial estimate $\nu_{0}$ for the laser frequency. Through rescaling the
initial value $\nu_{0}$, the actual laser frequency is approximated as

$\nu_{\mathrm{estim.}}=\frac{1}{1+\varepsilon_{\mathrm{SCF}}}\nu_{0}\ .$ (1)

This paper aims to investigate approaches to decrease the dependency of the
LRI data on KBR data in case the latter is unavailable and to study the
performance of a possible LRI-only NGGM. Therefore, we develop different
models for estimating the LRI laser frequency in-flight. Since these models do
not achieve the same level of residuals as the cross-calibration with KBR, a
potential influence of thermal variations into the ranging data is
investigated and modeled.

In section 2, we briefly cover the working principle of the LRI and introduce
the instrument’s most important optical and radiofrequency observables. The
two dominant error sources in the LRI-derived range, which occur in the actual
data processing, are discussed in section 3. Flight data processing is the
topic of section 4, where we analyze the KBR-LRI cross-calibration method and
discuss an observed frequency change of the cavity resonance onboard GF-1. The
derivation and calibration of a telemetry-based absolute laser frequency model
is presented in sections 5 and 6 and an empirical correction to this model is
derived in section 7. In section 8, we use the frequency models to derive
three independent LRI1B-equivalent ranging data sets, which are then compared
to each other and the KBR. In the end, section 9 focuses on minimizing
variations in the relative scale and timeshift of LRI and KBR by reducing
thermally-induced measurement errors, which predominantly manifest as tone
errors. Coupling factors are derived to model this effect and subtract it from
the ranging data. Available technologies to determine the absolute laser
frequency for future gravity missions are discussed in section 10, and the
findings are summarized in section 11.

##  2 Working Principle of LRI

Figure 1: Simplified light paths and frequencies within the LRI. See the main
text for an explanation.

The LRI is set up in an active-transponder configuration Sheard et al. (2012),
the principle of which is shown in fig. 1. Both spacecraft have identical
hardware, including a laser, and both receive and emit light. They are
equipped with photoreceivers to measure the interference between the incoming
and local light fields (shown in red and blue in fig. 1). The LRI is a
heterodyne Mach-Zehnder type interferometer, meaning that the two interfering
light fields have slightly different optical frequencies, which produces an
interference beatnote at the difference frequency. This beatnote frequency is
roughly $10\text{\,}\mathrm{MHz}\text{/}$ for the LRI.

On the reference side, the laser frequency $\nu_{\mathrm{R}}$ is stabilized
utilizing an optical reference cavity using the PDH technique Drever et al.
(1983). The residual frequency fluctuations $\delta\nu_{\mathrm{R}}$ were
required to be below $30\text{\,}\mathrm{Hz}\mathrm{/}\sqrt{\mathrm{Hz}}$ for
Fourier frequencies above $10\text{\,}\mathrm{mHz}\text{/}$, with a relaxation
towards lower Fourier frequencies Sheard et al. (2012); Thompson et al.
(2011). The actual in-flight noise, expressed as amplitude spectral density
(ASD), is well below the requirement and in the order of

$\mathrm{ASD}[\delta\nu_{\mathrm{R}}](f)\approx\frac{${10}^{-15}$\cdot\nu_{\mathrm{R}}}{\sqrt{$\mathrm{Hz}\text{/}$}}\sqrt{\frac{f}{$1\text{\,}\mathrm{Hz}\text{/}$}}\approx$0.3$\frac{$\mathrm{Hz}\text{/}$}{\sqrt{$\mathrm{Hz}\text{/}$}}\sqrt{\frac{f}{$1\text{\,}\mathrm{Hz}\text{/}$}}$
(2)

at frequencies above $200\text{\,}\mathrm{mHz}\text{/}$ Abich et al. (2019),
where it is dominant and directly apparent in the measured signal due to the
lack of other signals at such high frequencies. Assessing the frequency
stability at lower frequencies is difficult due to the dominant ranging signal
arising from gravitational and non-gravitational differential forces acting on
the satellites (cf. black trace in fig. 2, page 2).

On the reference spacecraft, the laser light is split at the beamsplitter into
a local oscillator (LO) part and into the transmit (TX) beam (cf. fig. 1). The
Triple Mirror Assembly (TMA) routes the TX beam around cold-gas tanks (not
shown) towards the distant spacecraft. The emitted frequency is Doppler-
shifted when received on the transponder due to the relative motion of the two
spacecraft. The relative velocity of the two spacecraft is below $\pm
2.5\text{\,}\mathrm{m}\text{/}\mathrm{s}$, which translates into a one-way
Doppler shift of $\nu_{\mathrm{D}}<$\pm 2.5\text{\,}\mathrm{MHz}\text{/}$$
Sheard et al. (2012).

The transponder unit employs a frequency-locked loop with a
$10\text{\,}\mathrm{MHz}\text{/}$ offset, meaning that the frequency
$\nu_{\mathrm{T}}$ is controlled with high gain and bandwidth such that the
beatnote at the photodetector between the local and received, Doppler-shifted
light stays at $f_{\mathrm{off}}=$10\text{\,}\mathrm{MHz}\text{/}$$. This
enforces that the transponder laser frequency $\nu_{\mathrm{T}}$ is the sum of
received Doppler-shifted reference frequency
$\nu_{\mathrm{R}}^{\prime}=\nu_{\mathrm{R}}+\nu_{\mathrm{D}}$ and
$f_{\mathrm{off}}$. The transponder is to send back amplified laser light to
the reference with a well-defined and known optical frequency (and phase).
Since the transponder is in $220\pm 50.\text{\,}\mathrm{km}\text{/}$ distance,
it only receives a fraction of the initially emitted light power (in the order
of nanowatts). The amplified and offset-locked beam travels back to the
reference side, interferes again, and the beatnote between the (once more)
Doppler-shifted transponder and local reference frequency
$f_{\mathrm{R}}=\nu_{\mathrm{T}}^{\prime}-\nu_{\mathrm{R}}=2\nu_{\mathrm{D}}+f_{\mathrm{off}}$
contains the desired ranging information, encoded in the Doppler shift
$\nu_{\mathrm{D}}$.

##  3 Error Coupling in the Range Measurement

The previous section provided a descriptive picture of the LRI working
principle through the beam’s frequencies. However, the LRI actually measures
the differential phase of the two interfering light beams given by the time
integral of the beatnote frequency. To describe the phase observables in a
relativistic framework, we now follow the approach of Yan et al. Yan et al. to
assess potential relativistic effects on the scale factor. We favor this
description in terms of phase since that is invariant in the context of
general relativity, i. e., independent of the coordinate system, in contrast
to the frequency.

The conversion of the measured differential phase

$\varphi_{\mathrm{LRI}}=\varphi_{\mathrm{T}}-\varphi_{\mathrm{R}}$ (3)

to the range observable in a relativistic framework is given in Yan et al. ;
Müller et al. and is omitted here for brevity. The LRI range reads

$\displaystyle\rho_{\mathrm{LRI}}^{\mathrm{raw}}(t)$
$\displaystyle=\frac{c_{0}}{2}\int_{0}^{t}\frac{\mathrm{d}\varphi_{\mathrm{LRI}}(t^{\prime})}{\mathrm{d}t^{\prime}}\frac{1}{\nu_{\mathrm{R}}^{\mathrm{G}}(t^{\prime}-\Delta
t^{\mathrm{[rt]}}(t^{\prime}))}-\left(\frac{\nu_{\mathrm{R}}^{\mathrm{G}}(t^{\prime})}{\nu_{\mathrm{R}}^{\mathrm{G}}(t^{\prime}-\Delta
t^{\mathrm{[rt]}}(t^{\prime}))}-1\right)\,\mathrm{d}t^{\prime}$ (4)
$\displaystyle=\frac{c_{0}}{2}\left(\Delta t^{\mathrm{[rt]}}(t)-\Delta
t^{\mathrm{[rt]}}(0)\right)+\mathrm{errors}\ .$ (5)

Equation 4 provides a recipe to compute the raw biased range as a function of
the coordinate time $t$, which is available after precise orbit determination.
The roundtrip propagation time is in the order of $\Delta
t^{\mathrm{[rt]}}\approx
2\cdot$220\text{\,}\mathrm{km}\text{/}$/c_{0}\approx$1.46\text{\,}\mathrm{ms}\text{/}$$
with the speed of light $c_{0}$ and the absolute laser frequency in the
coordinate frame is given by $\nu^{\mathrm{G}}_{\mathrm{R}}$. The relation
between the frequency of the laser source $\nu_{\mathrm{R}}$ and the apparent
frequency in the Earth-centered geocentric celestial reference frame (GCRF)
system $\nu_{\mathrm{R}}^{\mathrm{G}}$ is

$\nu_{\mathrm{R}}^{\mathrm{G}}=\nu_{\mathrm{R}}\cdot\frac{\mathrm{d}\tau_{\mathrm{R}}}{\mathrm{d}t}\
,$ (6)

where $\tau_{\mathrm{R}}$ is the proper time of the reference spacecraft and
$t$ is the coordinate time in the GCRF. The distinction between those is
discussed in Müller et al. and a neglection yields a tone error in the order
of
$1\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}\,\mathrm{r}\mathrm{m}\mathrm{s}$
at 1/rev. The first term in the integral resembles the well known relation

$\rho(t)=\frac{c_{0}}{\nu}\varphi(t)$ (7)

and the second term accounts for the effect of a varying frequency
$\nu_{\mathrm{R}}^{\mathrm{G}}(t)$.

Equation 5 in turn provides the physical meaning of
$\rho_{\mathrm{LRI}}^{\mathrm{raw}}$ as a time-of-flight measurement, whereby
the errors include Tilt-To-Length Coupling Wegener et al. , laser frequency
noise Abich et al. (2019) and others.

The representation until here is, however, neglecting some error sources. We
now derive a model for two error sources, namely a mismodeling of the laser
frequency, which can be expressed through a scale factor, and secondly from
clock errors. Equation 4 shows that the inter-satellite biased range can be
reconstructed from phase measurements if the conversion factor, given by the
absolute frequency $\nu_{\mathrm{R}}^{\mathrm{G}}(t)$ or wavelength
$\lambda(t)=c_{0}/\nu_{\mathrm{R}}^{\mathrm{G}}(t)$, is known. If we consider
errors in the knowledge of the frequency, given as the difference between
estimated and true frequency
$\nu_{\mathrm{R,est}}^{\mathrm{G}}(t)-\nu_{\mathrm{R}}^{\mathrm{G}}(t)$, this
error is typically expressed as a scale factor

$\varepsilon_{\mathrm{SCF}}(t)=\frac{\nu_{\mathrm{R,est}}^{\mathrm{G}}(t)-\nu_{\mathrm{R}}^{\mathrm{G}}(t)}{\nu_{\mathrm{R}}^{\mathrm{G}}(t)}\quad\Leftrightarrow\quad\nu_{\mathrm{R}}^{\mathrm{G}}(t)=\frac{\nu_{\mathrm{R,est}}^{\mathrm{G}}(t)}{1+\varepsilon_{\mathrm{SCF}}(t)}\
.$ (8)

By applying the replacement
$\nu_{\mathrm{R}}^{\mathrm{G}}(t)\rightarrow\nu_{\mathrm{R}}^{\mathrm{G}}(t)/(1+\varepsilon_{\mathrm{SCF}}(t))$
to eq. 4, we obtain an expression for the estimated range
$\rho_{\mathrm{LRI}}^{\mathrm{raw,est}}$. In the following, we compute the
error of this estimated range. For better readability, we drop the time
dependency of terms evaluated at the measurement epoch $t^{\prime}$ in the
integral.

$\displaystyle\rho_{\mathrm{LRI}}^{\mathrm{raw,est}}(t)-\rho_{\mathrm{LRI}}^{\mathrm{raw}}(t)$
$\displaystyle\qquad=\frac{c_{0}}{2}\int_{0}^{t}\frac{\nu_{\mathrm{R}}^{\mathrm{G}}+\varepsilon_{\mathrm{SCF}}(t^{\prime}\\!-\\!\Delta
t^{\mathrm{[rt]}})\cdot\dot{\varphi}_{\mathrm{LRI}}}{\nu_{\mathrm{R}}^{\mathrm{G}}(t^{\prime}\\!-\\!\Delta
t^{\mathrm{[rt]}})}-\frac{(1+\varepsilon_{\mathrm{SCF}}(t^{\prime}\\!-\\!\Delta
t^{\mathrm{[rt]}}))\nu_{\mathrm{R}}^{\mathrm{G}}}{(1+\varepsilon_{\mathrm{SCF}})\nu_{\mathrm{R}}^{\mathrm{G}}(t^{\prime}\\!-\\!\Delta
t^{\mathrm{[rt]}})}\ \mathrm{d}t^{\prime}$ (9)
$\displaystyle\qquad\approx\frac{c_{0}}{2}\int_{0}^{t}\varepsilon_{\mathrm{SCF}}(t^{\prime}\\!-\\!\Delta
t^{\mathrm{[rt]}})\,\Delta\dot{t}^{\mathrm{[rt]}}-\varepsilon_{\mathrm{SCF}}(t^{\prime}\\!-\\!\Delta
t^{\mathrm{[rt]}})+\frac{\nu_{\mathrm{R}}^{\mathrm{G}}\varepsilon_{\mathrm{SCF}}}{\nu_{\mathrm{R}}^{\mathrm{G}}(t^{\prime}\\!-\\!\Delta
t^{\mathrm{[rt]}})}\ \mathrm{d}t^{\prime}$ (10)
$\displaystyle\qquad\approx\frac{c_{0}}{2}\int_{0}^{t}\varepsilon_{\mathrm{SCF}}\Delta\dot{t}^{\mathrm{[rt]}}+\dot{\varepsilon}_{\mathrm{SCF}}\Delta
t^{\mathrm{[rt]}}-\varepsilon_{\mathrm{SCF}}+\frac{\nu_{\mathrm{R}}^{\mathrm{G}}\varepsilon_{\mathrm{SCF}}}{\nu_{\mathrm{R}}^{\mathrm{G}}(t^{\prime}\\!-\\!\Delta
t^{\mathrm{[rt]}})}\ \mathrm{d}t^{\prime}$ (11)
$\displaystyle\qquad\approx\varepsilon_{\mathrm{SCF}}(t)\cdot\frac{c_{0}}{2}\Delta
t^{\mathrm{[rt]}}(t)+\frac{c_{0}}{2}\int_{0}^{t}\varepsilon_{\mathrm{SCF}}\left(\frac{\dot{\nu}_{\mathrm{R}}^{\mathrm{G}}}{\nu_{\mathrm{R}}^{\mathrm{G}}}\Delta
t^{\mathrm{[rt]}}+1\right)-\varepsilon_{\mathrm{SCF}}\ \mathrm{d}t^{\prime}$
(12) $\displaystyle\qquad\approx\varepsilon_{\mathrm{SCF}}(t)\cdot L(t)$ (13)

where we used some approximations justified below to obtain the simple result.
The approximation in eq. 10 is based on the relation
$1/(1+\varepsilon_{\mathrm{SCF}})\approx 1-\varepsilon_{\mathrm{SCF}}$
together with the definition of the phase derivative in (Müller et al., , eq.
27) and we dropped a second order term in $(\varepsilon_{\mathrm{SCF}})^{2}$.
Equation 11 uses $\varepsilon_{\mathrm{SCF}}(t-\Delta
t^{\mathrm{[rt]}})\approx\varepsilon_{\mathrm{SCF}}(t)-\Delta
t^{\mathrm{[rt]}}\dot{\varepsilon}_{\mathrm{SCF}}(t)$ and neglects product
terms of $\Delta t^{\mathrm{[rt]}}\cdot\Delta\dot{t}^{\mathrm{[rt]}}$. The
result of eq. 12 employed the same type of approximations, namely
$1/\nu_{\mathrm{R}}^{\mathrm{G}}(t^{\prime}\\!-\\!\Delta
t^{\mathrm{[rt]}})\approx(1+\dot{\nu}_{\mathrm{R}}^{\mathrm{G}}/\nu_{\mathrm{R}}^{\mathrm{G}}\cdot\Delta
t^{\mathrm{[rt]}})/\nu_{\mathrm{R}}^{\mathrm{G}}$. To solve the integral, we
omitted product terms of
$\dot{\nu}_{\mathrm{R}}^{\mathrm{G}}/\nu_{\mathrm{R}}^{\mathrm{G}}$ with
$\varepsilon_{\mathrm{SCF}}$ or $\dot{\varepsilon}_{\mathrm{SCF}}$, because
these describe a second order cross-coupling between scale error and
fractional true frequency change that is expected to be negligible.
$L(t)=c_{0}\Delta t^{\mathrm{[rt]}}(t)/2$ denotes the absolute distance
between the spacecraft and the error coupling
$\varepsilon_{\mathrm{SCF}}(t)\cdot L(t)$ resembles the well-known influence
of (fractional) laser frequency variations into the range measurement Sheard
et al. (2012), which can be regarded as a scale factor error.

The second error contributor that we address is a potential timeshift $\zeta$
of the measured LRI data, arising from unmodeled internal delays of the LRI.
At startup, the LRI time is initialized via the Onboard Computer (OBC), which
introduces a delay of $1.5\text{\,}\mathrm{s}\text{/}$ at maximum Wen et al.
(2019), although we only observed values below
$1.0\text{\,}\mathrm{s}\text{/}$. To compensate for this delay, the
differences of LRI time and MWI Instrument Processing Unit (IPU) time are
measured regularly (called the datation report) and are used to correct the
LRI time tags. However, a small deviation $\zeta$ may remain, even after this
subtraction. We linearize the effect of this potential timeshift to first
order as

$\rho_{\mathrm{LRI}}^{\mathrm{inst}}(t+\zeta)\approx\rho_{\mathrm{LRI}}^{\mathrm{inst}}(t)+\zeta\cdot\dot{\rho}(t)\
,$ (14)

where we use the approximate range rate
$\dot{\rho}\approx\dot{\rho}_{\textrm{LRI}}^{\mathrm{raw}}\approx\dot{\rho}_{\mathrm{LRI}}^{\mathrm{inst}}$
for terms that describe a small error coupling and where the highest precision
in $\dot{\rho}$ is not required.

Figure 2: Typical amplitude spectral density of the LRI ranging signal (black)
and effective errors arising from a static scale factor error
$\varepsilon_{\mathrm{SCF}}=10^{-6}$ (green) and timeshift
$\zeta=$50\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$$ (red, cf. eq. 14).
Also shown is the noise requirement of LRI (blue), which is strictly
applicable only for frequencies above $2\text{\,}\mathrm{mHz}\text{/}$, but it
was extrapolated towards lower frequencies (blue dashed segment). The light
blue line denotes a $1\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ tone
amplitude (computed via the ENBW). The ranging measurement is dominated by
laser frequency noise at highest frequencies (above
$30\text{\,}\mathrm{mHz}\text{/}$), and by the differential gravitational and
non-gravitational forces below.

Figure 2 illustrates the significance of the static scale factor error
$\varepsilon_{\mathrm{SCF}}=10^{-6}$ (green) and timeshift
$\zeta=$50\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$$ (red), which
represent the common orders of magnitude in current LRI data processing. The
effects of these exceed or are close to the LRI noise requirement for
frequencies between $0.63\text{\,}\mathrm{m}\mathrm{H}\mathrm{z}$, indicating
that the scale factor and timeshift need to be known to better precision, e.
g., at the level of ${10}^{-7}{10}^{-8}$ for the scale and at a level of a few
microseconds or better for the timeshift. Fourier frequencies below
$0.6\text{\,}\mathrm{m}\mathrm{H}\mathrm{z}$ are dominated by sinusoidal
errors at integer multiples of the orbital frequency
$f_{\mathrm{orb}}\approx$0.18\text{\,}\mathrm{mHz}\text{/}$$. These peaks need
to be compared to tone error requirements with the unit of a meter (rms or
peak) instead of spectral densities with the unit of
$\mathrm{m}\mathrm{/}\sqrt{\mathrm{Hz}}$. An indicative level for tone errors
is $1\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ (peak), which is a
requirement for the MWI at 2/rev frequency in GRACE-FO Kornfeld et al. (2019),
is shown in light blue. The displayed errors exceed the one micron tone level
by approximately two orders of magnitude.

By combining the effects due to a scale factor error (eq. 13) and an
uncompensated timeshift (eq. 14), we obtain the error of the LRI measured
range w.r.t. the truth as

$\rho_{\mathrm{LRI}}^{\mathrm{inst}}(t)-\rho^{\mathrm{inst}}_{\mathrm{true}}(t)\approx\varepsilon_{\mathrm{SCF}}(t)\cdot
L(t)+\zeta\cdot\dot{\rho}(t)\ .$ (15)

##  4 Scale Factor Determination

The scale factor $\varepsilon_{\mathrm{SCF}}$ implicitly defined in eq. 1
provides an estimate $\nu_{\mathrm{R,est}}$ for the actual laser frequency
$\nu_{\mathrm{R}}$ of the LRI reference unit, which in turn is needed for
accurately converting the phase measurement into a range in meter. Here we
present three approaches to either directly calculate the absolute laser
frequency $\nu_{\mathrm{R}}$ or through the scale factor
$\varepsilon_{\mathrm{SCF}}$ that is related to absolute laser frequency
through eq. 8.

Since GRACE-FO hosts the KBR and LRI, which are designed to measure the same
quantity in parallel, the obvious way to obtain the LRI scale
$\varepsilon_{\mathrm{SCF}}$ (or frequency $\nu_{\mathrm{R}}$) is to compare
the ranging data of the two instruments. We define the instantaneous KBR range
as

$\rho_{\mathrm{KBR}}^{\mathrm{inst}}(t)=\rho_{\mathrm{KBR}}^{\mathrm{raw}}(t)+\rho_{\mathrm{KBR}}^{\mathrm{LTC}}(t)+\rho_{\mathrm{KBR}}^{\mathrm{AOC}}(t)\
.$ (16)

The three quantities on the right-hand side are the ionosphere-free K/Ka-band
range $\rho_{\mathrm{KBR}}^{\mathrm{raw}}$, the KBR Light Time Correction
(LTC) and the antenna offset correction (AOC), which are regarded as error-
free here. They are given in the KBR1B data product Wen et al. (2019).
Ultimately, daily arcs of LRI phase measurements can be calibrated against the
KBR ranging data, i. e., by minimizing the KBR-LRI residuals as

$\left\|\rho_{\mathrm{KBR}}^{\mathrm{inst}}(t)-\lambda_{\mathrm{est}}^{\mathrm{SDS}}\cdot\varphi_{\mathrm{LRI}}(t+\zeta)-\rho^{\mathrm{LTC}}_{\mathrm{LRI}}\right\|\rightarrow
0$ (17)

using a daily constant laser wavelength
$\lambda_{\mathrm{est}}^{\mathrm{SDS}}$ and timeshift $\zeta$ as fit
parameters. The KBR scale factor error can be regarded as negligible since the
relevant Ultra-Stable Oscillator (USO) frequency is determined during precise
orbit determination by referencing it to global positioning system (GPS). The
USO fractional frequency varies by about ${10}^{-11}$, mainly at 1/rev (Müller
et al., , Fig. 1), and we assume the knowledge error to be even smaller. The
processing of daily chunks of data essentially decomposes the scale factor
$\varepsilon_{\mathrm{SCF}}$ into a static and time-variable part as

$\varepsilon_{\mathrm{SCF}}(t)=\langle\varepsilon_{\mathrm{SCF}}\rangle+\delta\varepsilon_{\mathrm{SCF}}(t)\
,$ (18)

of which only the static part $\langle\varepsilon_{\mathrm{SCF}}\rangle$ is
determined separately on every day with discontinuities at the day boundary.

This cross-calibration scheme is the official processing strategy employed by
the Science Data System (SDS) for the LRI1B data product in version 04, where
a conversion factor from phase to range and a timeshift $\zeta$ is estimated
once per day. The scale
$\langle\varepsilon_{\mathrm{SCF}}^{\mathrm{SDS}}\rangle$ is reported in the
ionospheric correction (iono_corr) column of the LRI1B files, whereas the
timeshift $\zeta$ is applied through LLK1B. The scale value relates to the
laser frequency and wavelength estimates of the reference laser through

$\displaystyle\lambda_{\mathrm{est}}^{\mathrm{SDS}}(t_{\mathrm{daily}})=(1+\langle\varepsilon_{\mathrm{SCF}}^{\mathrm{SDS}}\rangle(t_{\mathrm{daily}}))\cdot\lambda_{0}$
(19) and
$\displaystyle\nu_{\mathrm{est}}^{\mathrm{SDS}}(t_{\mathrm{daily}})=\frac{\nu_{0}}{\left(1+\langle\varepsilon_{\mathrm{SCF}}^{\mathrm{SDS}}\rangle(t_{\mathrm{daily}})\right)}\
,$ (20)

where $\nu_{0}$ is a nominal frequency for the LRI lasers, given in the
documentation as $\nu_{0}=$281\,616\,393\text{\,}\mathrm{MHz}\text{/}$$ and
$\nu_{0}=$281\,615\,684\text{\,}\mathrm{MHz}\text{/}$$ for GF-1 and GF-2,
respectively Wen et al. (2019). The two nominal values were determined pre-
flight, but do not represent the best knowledge of the actual frequency. The
corresponding nominal wavelength is $\lambda_{0}=c_{0}/\nu_{0}$. Equation 20
is in principle a reformulation of eqs. 8 and 18, however, the time-varying
part $\delta\varepsilon_{\mathrm{SCF}}(t)$ is neglected by the processing of
daily segments. Therefore, eq. 4 simplifies to

$\rho_{\mathrm{LRI}}^{\mathrm{SDS}}(t)=\frac{c_{0}}{2}\frac{(1+\langle\varepsilon_{\mathrm{SCF}}^{\mathrm{SDS}}\rangle)}{\nu_{0}}\cdot\varphi_{\mathrm{LRI}}(t)\
.$ (21)

Figure 3: Comparison of LRI scale factor
$\langle\varepsilon_{\mathrm{SCF}}\rangle$ using the conventional cross-
calibration method. _Blue:_ using the AEI ranging phase
$\varphi_{\mathrm{LRI}}$, cf. eq. 4. _Green:_ the SDS LRI1B-v04 result.
_Orange dashed:_ Exponential model for the cavity resonance frequency.

The minimization result for
$\langle\varepsilon_{\mathrm{SCF}}^{\mathrm{SDS}}\rangle$, as given in the
LRI1B-v04 data product, is shown in green in fig. 3. Our recomputation with an
in-house Level 1A to 1B processing is denoted as
$\langle\varepsilon_{\mathrm{SCF}}^{\mathrm{AEI}}\rangle$ (blue), which will
be used later on as one possible frequency model. The plot covers the timespan
from 2018-Dec-13 until 2022-Jan-01, where GF-1 acts as the LRI reference unit.
Due to spacecraft-related outages, the LRI was not in science mode from
2019-Feb-06 to 2019-Mar-17. Smaller gaps in the data originate from phase
breaks, e. g., due to spacecraft maneuvers or diagnostic data recording. The
frequent data gaps starting in mid-2021 are due to nadir-pointing of the
spacecraft, occurring roughly two days per week. In these periods, the
pointing angles between the spacecraft-fixed coordinate system and the line-
of-sight exceed the LRI pointing capabilities.

Even though the blue and green traces roughly match, the SDS implementation
seems less robust as it shows more outliers, which may be related to imperfect
phase jump removal Abich et al. (2019). The number of outliers reduced after
2020-Jun-27, when the deglitching algorithm was adjusted by the SDS Wen . Both
traces show a slow drift that seems to converge and peaks and dips occurring
roughly every three months, indicating an apparent change of the laser
frequency with a magnitude of $\pm{10}^{-7}$ or $\pm
20\text{\,}\mathrm{MHz}\text{/}$. It is noteworthy that we can not distinguish
which instrument contributes to those periodic variations, as we always use
the difference between LRI and KBR. However, sections 8 and 9 will address
these variations in more detail. The slow drift in fig. 3 was fitted as
exponential decay of the form

$\varepsilon^{\mathrm{Cav}}(t)=\varepsilon^{\mathrm{Cav}}_{\infty}-\varepsilon^{\mathrm{Cav}}_{0}\exp(-\lambda^{\mathrm{Cav}}\cdot
t)$ (22)

and is shown in orange. The decay rate is
$\lambda^{\mathrm{Cav}}=$4.006\text{\times}{10}^{-8}\text{\,}\text{/}\mathrm{s}$$,
with $\varepsilon_{\infty}^{\mathrm{Cav}}=$2.221\text{\times}{10}^{-6}$$ and
$\varepsilon_{0}^{\mathrm{Cav}}=$-1.190\text{\times}{10}^{-7}$$, the time $t$
is GPS seconds past 2018-May-22, 00:00:00 UTC. Exponential shrinkage (and thus
increasing frequency) has already been observed in similar cavities made from
ultra-low expansion (ULE) materials, and the suspected cause is aging of the
spacer material Alnis et al. . Equation 22 can of course be converted into an
equivalent frequency model $\nu_{\mathrm{R}}^{\mathrm{Cav}}$ via eq. 20. This
exponential model is the second model for the laser frequency, resulting in
similar values as the SDS scheme, but without the periodic features. Until
now, we only derived these exponential model coefficients for GF-1. The
derivation for GF-2 is beyond the scope of this manuscript as GF-2 was only
for short times in reference mode.

The scheme of cross-calibration is only possible due to the unique situation
of having two independent ranging measurements by KBR and LRI. However, it can
not resolve intraday frequency variations of the LRI and introduces small
discontinuities at day bounds. Furthermore, it depends on the KBR, which will
likely not be present in future missions. Therefore we present an on-ground
calibration that has been performed for the two laser flight models as a third
method to determine the laser frequency and derive a calibrated frequency
model only using telemetry data of the LRI. It is based on the fact that the
laser frequency $\nu_{\mathrm{R}}$ can be deduced from the setpoints of the
laser’s frequency-lock control loop and thermal state. With this model, we can
continuously evaluate the optical frequency in orbit with moderate accuracy.
The laser frequency may change due to varying environmental conditions, e. g.,
temperatures of the optical reference cavity. This model will be derived in
sections 5, 6 and 7, and all models are analyzed and compared to each other in
section 8.

##  5 LRI Laser and Telemetry Description

The LRI Reference Laser Units (RLUs) were built by Tesat Spacecom and are
comparable to the laser onboard LISA Pathfinder and to the seed laser of one
possible LISA implementation Armano et al. (2017). They are based on an Nd:YAG
non-planar ring oscillator (NPRO) crystal and are fiber-connected to an
optical reference cavity built by Ball Aerospace Pierce et al. (2012);
Thompson et al. (2011) and to the Optical Bench Assembly (OBA). The laser’s
output power is in the order of $25\text{\,}\mathrm{mW}\text{/}$ in the near-
infrared regime ($\lambda\approx$1064\text{\,}\mathrm{nm}\text{/}$$) Nicklaus
et al. (2017). The laser frequency is actively controlled by feedback control
loops either from the reference cavity using the PDH scheme Drever et al.
(1983) (in reference mode) or to the incoming beam using a frequency-offset
lock (transponder mode). The tuning is achieved through a thermal element for
slow variations and a Piezo-Electric Transducer (PZT) actuator for fast
variations. The actuator signals are downlinked in the laser telemetry and
published within the LHK1A/B data products at a rate of
$1\text{\,}\mathrm{Hz}\text{/}$, if the LRI is in science mode, i. e., when
the laser link is established. The data type is unsigned integer of 32 bits
depth. The corresponding normalized signed data streams are computed via the
two’s complement and scaling by the bit depth as

$\displaystyle\texttt{u2i}(x,N)=$
$\displaystyle\left\\{\begin{matrix}x/2^{N}-1,&\mathrm{if~{}}x\geq 2^{N-1}\\\
x/2^{N},&\mathrm{if~{}}x<2^{N-1}\end{matrix}\right.$ (23)

with $N=32$ and $x$ denoting the unsigned value from the telemetry. The value
range is $-1/2<\texttt{u2i}(x,\cdot)\leq 1/2$. In the following, these
normalized data streams are denoted as pztIL, pztOOL, thermIL and thermOOL.
The temperature of the laser can be retrieved from so-called “OFFRED” data,
which is recorded by the OBC. The measurement is taken at the Thermal
Reference Point (TRP) of the RLU, which is located at the laser’s housing. By
the time of writing, the laser TRP temperature is not publicly available.
Still, it will be shown later that the influence of the TRP coupling is small
during nominal operations.

The notations in-loop (IL) and out-of-loop (OOL) are not referring to
different sensors as in conventional feedback control circuits, but two parts
that are added to form the final setpoint. The OOL channel is used for manual
control with some logic (e. g., to drive a frequency ramp for locking to the
cavity or during acquisition). In contrast, the IL value represents the
evolution of the actuator value in closed-loop operation. The actuator range
of the PZT and thermal actuators is $\pm 1\text{\,}\mathrm{V}\text{/}$ and
$\pm 9\text{\,}\mathrm{V}\text{/}$, respectively, with nominal frequency
coupling coefficients of $5\text{\,}\mathrm{MHz}\text{/}\mathrm{V}$ and
$500\text{\,}\mathrm{MHz}\text{/}\mathrm{V}$, respectively, also shown in
table 1.

##  6 RLU On-Ground Calibration

The laboratory setup used to calibrate the LRI lasers is shown in fig. 4. It
consisted of the LRI flight laser, the Laser Ranging Processor (LRP)
(including the phasemeter), a frequency-controlled reference laser and a
wavemeter. The measurements were performed by parts of the LRI teams at
JPL/NASA and AEI Hannover.

Figure 4: Laboratory setup for the LRI flight laser frequency calibration
measurement. Blue lines denote optical fibers; red lines are laser beams in
free space. Black arrows denote electric signals.

During these activities, the frequency of the reference laser was tuned using
a computer, and its frequency was recorded using a wavemeter. The LRI unit in
transponder mode locks its laser frequency to the incoming beam and adds a
$10\text{\,}\mathrm{MHz}\text{/}$ offset and is thus known as well. During the
activities, the RLU temperature, as well as the PZT and thermal telemetry, is
recorded. The frequency of the LRI laser was not measured directly, since it
was more convenient to use the second output port of the reference laser (one
fiber to the wavemeter, one to the optical bench), while the LRI laser light
is free-beam on the optical bench. We use a linear model to estimate the laser
frequency based on the telemetry (TM):

$\nu^{\mathrm{TM}}(t)=\begin{pmatrix}c_{\texttt{pztIL}}\\\
c_{\texttt{pztOOL}}\\\ c_{\texttt{thermIL}}\\\ c_{\texttt{thermOOL}}\\\
c_{\texttt{lasTRP}}\end{pmatrix}\cdot\begin{pmatrix}\texttt{pztIL}(t)\\\
\texttt{pztOOL}(t)\\\ \texttt{thermIL}(t)\\\ \texttt{thermOOL}(t)\\\
\texttt{lasTRP}(t-\tau)-$26\text{\,}\mathrm{\SIUnitSymbolCelsius}\text{/}$\end{pmatrix}+\nu_{0,\mathrm{air}}+\Delta\nu_{\mathrm{AirToVac}}\
,$ (24)

which depends on the actuator states, i. e., the telemetry data streams pztIL,
pztOOL, thermIL, thermOOL as well as the surrounding temperature, which is
measured at the TRP of the laser. Since the TRP is located outside the thermal
shielding, a time delay of $\tau=$520\text{\,}\mathrm{s}\text{/}$$ is applied
to the temperature measurements, which represents the propagation time of
outer temperature changes to the NPRO crystal. The manufacturer provided this
numerical value. Furthermore, only deviations from the nominal temperature of
$26\text{\,}\mathrm{\SIUnitSymbolCelsius}\text{/}$ are considered.

The nominal values for the coupling factors given by the laser manufacturer
are shown in table 1. However, we refine the individual laser units’ thermal
coupling coefficients with our measurements. The PZT and TRP coupling are not
refined since they were not modulated strongly enough during the calibration
measurements to derive reliable coupling factors. We expect the TRP coupling
to be non-critical since the lasers’ TRP temperature varies only in the sub-
Kelvin domain in flight as shown in fig. 5. The blue trace depicts the daily
averaged laser TRP recording of GF-1 and its respective daily minimum (green)
and maximum (red) values.

Figure 5: Daily mean, min and max of the GF-1 lasTRP data.

The temperature of the GF-1 laser is stable when averaged daily. It shows sub-
daily variations of $\pm$0.2\text{\,}\mathrm{K}\text{/}$$, which translates
into $\Delta\nu=\pm$2.4\text{\,}\mathrm{MHz}\text{/}$$ in frequency, or
$\varepsilon_{\mathrm{SCF}}\approx\Delta\nu/\nu\approx$\pm
8.5\text{\times}{10}^{-9}$$ using a coupling of
$-12\text{\,}\mathrm{MHz}\text{/}\mathrm{K}$ (cf. table 1).

Table 1: Coupling factors for the two LRI laser flight units. Shown are the design values provided by the laser manufacturer and fit results from on-ground measurements. PZT and TRP coupling factors were not refined, since the measurements were not suitable to derive these couplings. The static value $\Delta\nu_{\mathrm{AirToVac}}$ was provided by the manufacturer and represents the frequency change from air to vacuum. The shown uncertainties are the formal errors of the least squares estimation. Coupling | Unit | Design value |  GF-1 (fit) |  GF-2 (fit)
---|---|---|---|---
$c_{\texttt{pztIL}}$ | [MHz] | $$2\text{\,}\mathrm{V}\text{/}$\cdot$5\text{\,}\mathrm{MHz}\text{/}\mathrm{V}$=$10\text{\,}\mathrm{MHz}\text{/}$$ |  - |  -
$c_{\texttt{pztOOL}}$ | [MHz] | $$18\text{\,}\mathrm{V}\text{/}$\cdot$5\text{\,}\mathrm{MHz}\text{/}\mathrm{V}$=$90\text{\,}\mathrm{MHz}\text{/}$$ |  - |  -
$c_{\texttt{thermIL}}$ | [MHz] | $$2\text{\,}\mathrm{V}\text{/}$\cdot$500\text{\,}\mathrm{MHz}\text{/}\mathrm{V}$=$1000\text{\,}\mathrm{MHz}\text{/}$$ |  $$1097$\pm$0.383$$ |  $$1094$\pm$1.383$$
$c_{\texttt{thermOOL}}$ | [MHz] | $$18\text{\,}\mathrm{V}\text{/}$\cdot$500\text{\,}\mathrm{MHz}\text{/}\mathrm{V}$=$9000\text{\,}\mathrm{MHz}\text{/}$$ |  $$9155$\pm$7.396$$ |  $$8857$\pm$27.565$$
$c_{\texttt{lasTRP}}$ | [MHz/K] | $-12$ |  - |  -
$\nu_{0,\mathrm{air}}$ | [MHz] |  $281\,614\,803$ (GF-1) $281\,614\,780$ (GF-2) |  $281\,614\,682.081$ $\pm$0.378$$ |  $281\,614\,631.999$ $\pm$1.094$$
$\Delta\nu_{\mathrm{AirToVac}}$ | [MHz] |  $37$ (GF-1) $27$ (GF-2) |  - |  -

(a) PZT control loop signals (b) Thermal control loop signals (c) Temperature
at the laser TRP. (d) Measured frequency $\nu^{\mathrm{ref}}$ and model
$\nu^{\mathrm{TM}}$.

(e) Frequency residuals $\nu^{\mathrm{ref}}-\nu^{\mathrm{TM}}$

Figure 6: Regression results for the GF-2 laser. The numbering on top of the
individual panels of each subfigure corresponds to the measurement campaigns,
as explained in the text. Note the offset in the residuals in (e) when the
less precise wavemeter WS6-600 was used in campaign (3). The average bias here
is $51.494\pm 0.24\text{\,}\mathrm{MHz}\text{/}$. All x-axes are in arbitrary
time units.

Several calibration measurements were performed on both RLUs between July 2017
and January 2018. For the laser integrated into GF-2, four measurements were
taken. In the following, we label these four measurements (1)…(4). They all
differ a little in their procedure. In (1) the reference laser’s frequency was
commanded in discrete steps, which caused the LRP to lose lock and forced
reacquisition and thus a temporary data loss. Afterward, the reference laser
was put into a free-running cool-down mode without active stabilization.
Reacquisition was avoided in (2) by sweeping continuously over the same
frequency range. (3) was a diagnostic test for the Differential Wavefront
Sensing (DWS), and the absolute frequency measurement was a secondary result.
Test (4) consists of very few sample points only since the used wavemeter had
no digital output port but only a display to retrieve the data. Thus, this
analysis does not use the data of (4). The measurements (1) and (2) were
performed in July 2017 using a HighFinesse WS7-60 wavemeter with an absolute
accuracy of $60\text{\,}\mathrm{MHz}\text{/}$. Test (3) in November 2017 used
a HighFinesse WS6-600 ($600\text{\,}\mathrm{MHz}\text{/}$ accuracy) and in
(4), a Burleigh WA1500 ($60\text{\,}\mathrm{MHz}\text{/}$ accuracy) was used.
The GF-1 laser was tested twice: once with a WS6-600 in November 2017 and a
Burleigh WA1500 in January 2018, and again, the latter one is not used in this
analysis.

We use a least squares approach to estimate the linear coupling factors and
constants of eq. 24. Additionally, we weigh the WS7-60 measurements higher by
a factor of 5 compared to the WS6-600, which has lower accuracy. We
furthermore estimate a relative offset of the WS6-600 wavemeter, which we can
deduce by analyzing the residuals. This offset of the WS6-600 is also apparent
when measuring an absolute frequency reference like an iodine cell; see
appendices A and B for more information.

Figure 6 shows the regression result using the measurements for GF-2. The
individual measurement campaigns are labeled (1)…(3). The subfigures (a) and
(b) show the normalized telemetry of the laser control loops, and the
temperature of the laser’s TRP is shown in subfigure (c). Panel (d) contains
the absolute frequency of the reference laser $\nu^{\mathrm{ref}}$ (shifted by
$10\text{\,}\mathrm{MHz}\text{/}$ to compensate for the offset-frequency lock
of the LRP) and the resulting laser frequency model $\nu^{\mathrm{TM}}$ of the
LRI laser. The trace in (e) shows the residuals
$\nu^{\mathrm{ref}}-\nu^{\mathrm{TM}}$, which clearly exhibits an offset of
approximately $50\text{\,}\mathrm{MHz}\text{/}$ beginning at (3), where the
WS6-600 was used. The high-frequency variations are higher in (3) due to the
lower precision of the WS6-600. The resulting coupling factors from the linear
least squares minimization are shown in table 1. Generally, the resulting
values match the manufacturer’s design values with only slight deviations.

##  7 Empirical Refinement of Telemetry-Based Laser Frequency Model

Figure 7: Purely ground calibration-based models $\nu^{\mathrm{TM}}_{1/2}$ and
empirically corrected TM models $\nu^{\mathrm{TME}}_{1/2}$ for GF-1 and GF-2
laser frequencies alongside the SDS frequency
$\nu^{\mathrm{SDS}}_{\mathrm{est}}$ from KBR-LRI cross-calibration. Outliers
in SDS curve removed. The right axis shows approximate equivalent laser
frequency variations, cf. eq. 20.

The calibrated telemetry models $\nu^{\mathrm{TM}}_{1/2}$ are now compared to
the frequency $\nu_{\mathrm{est}}^{\mathrm{SDS}}$ (cf. eq. 20) from the KBR-
LRI cross-correlation, where the flight data spanning from 2018-Dec-13 until
2022-Jan-01 is used. Figure 7 shows the frequency estimates from the TM models
for both spacecraft (blue and yellow) alongside the SDS frequency (green). The
latter is already shown in fig. 3, but outliers were removed this time. The
GF-2 curves are shifted down by $10\text{\,}\mathrm{MHz}\text{/}$ to remove
the intended transponder frequency offset (cf. section 2). The subscript $1/2$
refers to GF-1 or GF-2, respectively.

The models (blue and yellow) differ by $20\text{\,}\mathrm{MHz}\text{/}$ at
maximum, which is within the accuracy of the better wavemeter WS7-60, defining
the model accuracy. However, a drift of the models w. r. t. the KBR cross-
calibration method (green) is visible. The current hypothesis to explain this
drift is an aging effect of the NPRO crystal or the electronics within the
LRP. However, there is little literature on aging-induced frequency changes of
NPRO lasers, and this theory might need verification in a laboratory
experiment. The drift appears only in the laser setpoint telemetry but not in
the frequency, which is tightly locked to the cavity resonance.

The curves show some data gaps starting in mid-2021, caused by regular nadir-
pointing periods, in which the LRI was not creating science data. The steep
slopes and the dip in February and March 2020 in $\nu^{\mathrm{TM}}_{\,1/2}$
are due to spacecraft-related non-science phases of the LRI, after which the
units had to heat up to reach the nominal temperatures. This heating process
is visible at the laser TRP (cf. fig. 5) and thus affects not only the TM
model but also the green SDS curve with comparable magnitude, which confirms
the temperature coupling estimate in the TM model. However, we found that the
link acquisition happened before the lasers reached their thermal equilibrium,
which led to an apparent small step in the $\nu_{1/2}^{\mathrm{TM}}$ frequency
model (see appendix C). Imperfect coupling factors could possibly cause this.
To account for these steps $\nu^{\mathrm{step}}_{i}$ in our telemetry-based
laser frequency model, as well as for the drifts $a$ and offsets $\Delta\nu$
from the NPRO aging, we define an empirical correction and estimate its
parameters by least squares minimization using
$\nu_{\mathrm{est}}^{\mathrm{SDS}}$ as the reference. The empirical model
reads

$\nu^{\mathrm{emp}}(t)=a\cdot(t-t_{0})+\Delta\nu\ +\,\nu^{\mathrm{step}}(t)\
,$ (25)

where the reference epoch $t_{0}=1210982400$ GPS is 2018-May-22 at midnight.
The steps are defined as

$\displaystyle\nu^{\mathrm{step}}(t)=\nu_{i}^{\mathrm{step}}\mathrm{~{}~{}if~{}~{}}t^{\mathrm{step}}_{i}\leq
t<t^{\mathrm{step}}_{i+1}\ .$ (26)

The estimated parameters $a$ and $\Delta\nu$ are shown in table 2a, while the
steps $\nu^{\mathrm{step}}_{i}$ and the corresponding time-tags
$t^{\mathrm{step}}_{i}$ are shown in table 2b. Unfortunately, this empirical
model makes the telemetry-based frequency model still dependent on the KBR. In
principle, one could overcome the needs of an empirical model by better
calibrating the laser before launch.

Table 2: Parameters for the empirical part of the laser frequency model for
GF-1 and GF-2 of eq. 25.

Coupling |  Unit | Value
---|---|---
|  |  GF-1 |  GF-2
$a$ |  [$\mathrm{Hz}\text{/}\mathrm{s}$] |  $1.419$ |  $1.110$
$\Delta\nu$ |  [$\mathrm{MHz}\text{/}$] |  $-5.881$ |  $-1.089$
(a) Drift Parameters

$i$ |  $t^{\mathrm{step}}_{i}$ |  $\nu^{\mathrm{step}}_{i}$ [MHz]
---|---|---
|  |  GF-1 |  GF-2
1 |  2018-05-22 |  0 |  $0$
2 |  2020-01-10 |  0 |  $-1.770$
3 |  2020-02-10 |  $-11.593$ |  $2.881$
4 |  2020-05-11 |  $-10.446$ |  $2.881$
5 |  2022-01-01 |  undefined |  undefined
(b) Steps in the telemetry-based laser frequency model. Time tags refer to
midnight.

This empirical model is subtracted from the telemetry model to form the final
telemetry-based and empirically corrected TME frequency estimate

$\nu_{1/2}^{\mathrm{TME}}(t)=\nu^{\mathrm{TM}}_{1/2}(t)-\nu^{\mathrm{emp}}_{1/2}(t)\
.$ (27)

After applying the empirical model, the numerical values of the total
frequency models $\nu^{\mathrm{TME}}_{1/2}$ for GF-1 and GF-2 (orange and
purple traces in fig. 7) are in the range of
$\nu^{\mathrm{SDS}}_{\mathrm{est}}$. Besides the cross-calibration method, the
telemetry-based model does not show seasonal or periodic features. Note that
the exponential drift of the cavity is contained in
$\nu^{\mathrm{SDS}}_{\mathrm{est}}$ and $\nu_{1/2}^{\mathrm{TME}}$, even
though it is hard to see in the fig. 7. However, the empirical model in eq. 25
does not absorb the effect of exponentially increasing frequency (cf. figs. 3
and 22), since that cavity drift is present in both, $\nu^{\mathrm{TME}}_{1}$
and the reference $\nu^{\mathrm{SDS}}_{\mathrm{est}}$, thus it is not apparent
in the metric of the least squares adjustment.

##  8 Comparison of the LRI1B-Equivalent Data Sets

We define the pre-fit range error based on the instantaneous range difference
between the LRI and KBR,

$\rho_{\mathrm{err,v5X}}^{\mathrm{pre}}(t)=\rho_{\mathrm{LRI,v5X}}^{\mathrm{inst}}(t)-\rho_{\mathrm{KBR}}^{\mathrm{inst}}(t)-\rho_{\mathrm{KBR}}^{\mathrm{FV}}(t)\
.$ (28)

The subscript v5X denotes three different versions of the LRI1B data product
derived at the Albert-Einstein Institute (AEI) Müller (2021); Müller et al. .
They differ by the models for the laser frequency
$\nu_{\mathrm{R}}^{\mathrm{G}}$. At first, the data product using the
telemetry-based model described in the previous section (cf. eq. 27) is called
LRI1B-v51. The exponential cavity model (cf. eq. 22) forms LRI1B-v52. The last
data product, LRI1B-v53, uses the predetermined, constant value $\nu_{0}$
only, which makes it, in principle, a pre-release of LRI1B-v04 without the
daily scale $\langle\varepsilon_{\mathrm{SCF}}^{\mathrm{SDS}}\rangle$ and
timeshift $\zeta$ applied. The other differences between all three versions
and the official v04 data are the improved deglitching algorithm Müller (2021)
and the LTC implementation according to Yan et al. Yan et al. . The LRI
ranging data for these three versions at a $10\text{\,}\mathrm{Hz}\text{/}$
data rate is derived for the time spanning from 2018-Dec-13 until 2022-Jan-01.
We further used a correction $\rho_{\mathrm{KBR}}^{\mathrm{FV}}$ for the
intra-day carrier frequency variations of the KBR. This correction mainly
contains signal at 1/rev and 2/rev frequency Müller et al. and improves the
consistency between SDS-derived KBR and AEI-derived LRI data products, since
the AEI-derived LRI products inlude such a correction arising from the
difference between proper time and coordinate GPS time (cf.
$\nu_{\mathrm{R}}(t)$ vs. $\nu_{\mathrm{R}}^{\mathrm{G}}(t)$ in section 3).
However, the magnitude of this effect is small, and the results barely change
when the correction is omitted.

In general, the range error exhibits long-term drifts in the order of a few
$10\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}\mathrm{/}\mathrm{d}\mathrm{a}\mathrm{y}$,
which we remove through a high-pass Finite Impulse Response (FIR) filter with
a cutoff frequency of $$0.08\text{\,}\mathrm{mHz}\text{/}$\approx
f_{\mathrm{orb}}/2$. Future studies may address the reason for these long-term
drifts, but this is beyond the scope of this article at the current stage. As
mentioned in section 4, the LRI scale factor is sensitive to variations or
errors at 1/rev and 2/rev frequencies, which are unaffected by the filter. Due
to the filtering, half a day of data is cropped at the start and end of each
continuous segment, i. e., at every loss of the interferometric link of either
KBR or LRI, to remove the initialization of the FIR filter. Hence, all gaps
appear longer than they actually are. In the following, filtered quantities
are denoted with a tilde, e. g.,

$\tilde{\rho}_{\mathrm{err,v5X}}^{\mathrm{pre}}(t)=\mathrm{HPF}(\rho_{\mathrm{err,v5X}}^{\mathrm{pre}},$0.08\text{\,}\mathrm{mHz}\text{/}$)\
.$ (29)

The filtered pre-fit range errors for the three different LRI data products
(v51, v52, v53) are shown as blue traces in fig. 8. For saving computational
costs, the range error is decimated to a sampling rate of
$3.3\text{\,}\mathrm{mHz}\text{/}$.

(a) LRI1B-v51: Telemetry-based laser frequency model $\nu_{1/2}^{\mathrm{TM}}$
(b) LRI1B-v52: Exponential cavity frequency decay model
$\nu_{\mathrm{R}}^{\mathrm{Cav}}$ (c) LRI1B-v53: Pre-calibrated fixed
frequency value $\nu_{0}$. The prefit difference KBR-LRI is large, since the
constant frequency $\nu_{0}$ is a few $\mathrm{MHz}\text{/}$ off from the
truth.

Figure 8: Prefit range error (eq. 29, blue) and postfit range error before
(eq. 30, orange) and after (eq. 32, yellow) TC fitting for all three frequency
models. Initially, the prefit KBR-LRI range error shows rms variations of
$25\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$,
$25\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$,
$664\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ for v51, v52 and v53,
respectively. For the orange traces, the effect of a global scale
$\langle\varepsilon_{\mathrm{SCF}}^{\mathrm{glo}}\rangle$ and timeshift
$\zeta^{\mathrm{glo}}$ is subtracted, which already removes large parts of the
residual signal ($5.7\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$,
$5.3\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$,
$6.7\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$). After removal of the
full TC including five thermistors (yellow), the postfit range error is
further reduced to a rms level of $1.6\text{\,}\mathrm{\SIUnitSymbolMicro
m}\text{/}$, $1\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ and
$2.3\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$.

The signal in the pre-fit range error mainly oscillates at 1/rev and 2/rev
frequencies, with varying amplitude over the months. The rms values for the
traces are approximately $25\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$
for v51 and v52, and $664\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ for
v53. By estimating a global static scale
$\varepsilon_{\mathrm{SCF}}^{\mathrm{glo}}$ and time-shift
$\zeta^{\mathrm{glo}}$, we can obtain post-fit residuals of the range error

$\tilde{\rho}^{\mathrm{post}}_{\mathrm{err,v5X}}(t)=\left(1+\langle\varepsilon_{\mathrm{SCF}}^{\mathrm{glo}}\rangle\right)\cdot\tilde{\rho}_{\mathrm{LRI,v5X}}^{\mathrm{inst}}(t+\zeta^{\mathrm{glo}})-\tilde{\rho}_{\mathrm{KBR}}^{\mathrm{inst}}(t)-\tilde{\rho}_{\mathrm{KBR}}^{\mathrm{FV}}(t)\
,$ (30)

which are significantly lower at the level of approx.
$6\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}~{}\mathrm{r}\mathrm{m}\mathrm{s}$
(cf. orange traces in fig. 8).

Table 3: Global scale factor $\langle\varepsilon_{\mathrm{SCF}}^{\mathrm{glo}}\rangle$ and timeshift $\zeta^{\mathrm{glo}}$ for the data span from 2018-Dec-13 to 2022-Jan-01. Here, prefit denotes the first step of the algorithm, before temperature sensors are added. Postfit denotes the parameters when all five temperature sensors were added. LRI1B | $\langle\varepsilon_{\mathrm{SCF}}^{\mathrm{glo}}\rangle$ |  $\zeta^{\mathrm{glo}}$ | $\langle\varepsilon_{\mathrm{SCF}}^{\mathrm{glo}}\rangle$ |  $\zeta^{\mathrm{glo}}$
---|---|---|---|---
version | postfit |  postfit | postfit+TC |  postfit+TC
v51 | $-1.992\text{\times}{10}^{-9}$ |  $71.15\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$ | $-1.626\text{\times}{10}^{-8}$ |  $67.81\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$
v52 | $-1.430\text{\times}{10}^{-9}$ |  $71.13\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$ | $-3.810\text{\times}{10}^{-9}$ |  $67.95\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$
v53 | $2.235\text{\times}{10}^{-6}$ |  $71.11\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$ | $2.387\text{\times}{10}^{-6}$ |  $68.40\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$

The estimated global parameters are given in columns 2 and 3 of table 3
(without TC). They indicate that the high magnitude of the pre-fit range error
was mainly caused by a static timeshift
$\zeta^{\mathrm{glo}}\approx$71\text{\,}\mathrm{\SIUnitSymbolMicro
s}\text{/}$$ between LRI and KBR in the case of v51 and v52, and by the scale
($2.2\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$) and timeshift in case of v53.
These results were expected, e. g., the
$2.2\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$ scale offset was already apparent
from fig. 3.

We expect the KBR noise level to limit the postfit range error. When assuming
a
$10\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}\mathrm{/}\sqrt{\mathrm{Hz}}$
white noise in the KBR at low Fourier frequencies and with a
$3.3\text{\,}\mathrm{mHz}\text{/}$ sampling rate, we obtain

$$10\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}\mathrm{/}\sqrt{\mathrm{Hz}}$\cdot\sqrt{$3.3\text{\,}\mathrm{mHz}\text{/}$/2}\approx$0.4\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}~{}\mathrm{r}\mathrm{m}\mathrm{s}$$
(31)

as the KBR noise limit, however the postfit range error is still above this
level.

In addition to estimating and correcting a global mean scale and timeshift, we
also estimated the scale and timeshift on a daily basis, which are shown as
blue traces in fig. 9.

(a) Scale factor $\langle\varepsilon_{\mathrm{SCF}}\rangle$ for v51. (b)
Timeshift $\zeta$ for v51. Blue line shifted by
$75\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$. (c) Scale factor
$\langle\varepsilon_{\mathrm{SCF}}\rangle$ for v52. (d) Timeshift $\zeta$ for
v52. Blue line shifted by $75\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$.

(e) Scale factor $\langle\varepsilon_{\mathrm{SCF}}\rangle$ for v53. Blue line
shifted by $-2.23\text{\times}{10}^{-6}$.

(f) Timeshift $\zeta$ for v53. Blue line shifted by
$75\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$.

Figure 9: Comparison of scale and timeshift for LRI1B-v51, v52 and v53. Each
subplot shows the results from the raw data product (blue) and including the
Thermal Coupling (TC) in red. Left column: Scale factor
$\langle\varepsilon_{\mathrm{SCF}}\rangle$ for different frequency models v51,
v52, v53. Right column: corresponding timeshift $\zeta$. From top to bottom:
v51, v52, v53. Outliers removed for computing the rms values.

Here, the scale factor shows some seasonal patterns with approx. 3 month
period and with an amplitude of $\pm
0.05\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$ in all products; in v51 and v52
around zero, and in v53 around a $2.2\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$
offset. The blue trace in the lower left panel for v53 also includes the
exponential decay shown in fig. 3. The timeshift in the right panels exhibits
a $75\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$ offset and seasonal
variations with $\pm 25\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$
amplitude for all three products.

We note that the variations in the red traces of fig. 8 with approx.
$6\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}~{}\mathrm{r}\mathrm{m}\mathrm{s}$
could be explained to a large extent with a daily varying scale and timeshift
shown in fig. 9. However, if these peaks and dips forming the seasonal
variations with $\pm 0.05\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$ or
equivalently $\pm 14\text{\,}\mathrm{MHz}\text{/}$ amplitude are physical
variations in the laser and the cavity resonance frequency, we would expect to
see such variations in the laser telemetry and thus the telemetry-based laser
frequency $\nu^{\mathrm{TME}}(t)$. We also lack an explanation for variations
in the timeshift between KBR and LRI. A static timeshift could be produced by
delays and uncertainties in the timing chain, though the exact contributor is
not yet found (see appendix D for a brief discussion of the LRI time frame).
Therefore, in the next section, we investigate if the post-fit range error as
defined by eq. 30 can be further reduced when temperature coupling
coefficients are co-estimated with the global scale and timeshift.

##  9 Thermal Coupling in KBR-LRI Residuals

Changes in the thermal environment at many spacecraft units predominantly
appear at 1/rev and 2/rev frequencies and may alter the measured range. We
identified two possible coupling mechanisms. First, the coupling could be in
the laser frequency regime, like temperature changes of the cavity or USO
acting as an additional scaling term. Second, errors could occur in the phase
(pathlength) regime, e. g., due to temperature-dependent alignment of
components or temperature-driven effects in the electronics. In this section,
we estimate linear coupling factors for different temperature sensors, with
units of $\mathrm{1}\mathrm{/}\mathrm{K}$ for the (fractional) frequency
regime and $\mathrm{m}\text{/}\mathrm{K}$ for the phase regime, such that the
residuals between LRI and KBR are further minimized. We call the sum of these
two corrections the Thermal Coupling (TC). The TC coefficients and the global
scale and time shift are estimated simultaneously so that the postfit
residuals

$\tilde{\rho}_{\mathrm{err,v5X}}^{\mathrm{post-
TC}}(t)=\left(1+\langle\varepsilon_{\mathrm{SCF}}^{\mathrm{glo}}\rangle\right)\cdot\tilde{\rho}_{\mathrm{LRI,v5X}}^{\mathrm{inst}}(t+\zeta^{\mathrm{glo}})-\tilde{\rho}_{\mathrm{KBR}}^{\mathrm{inst}}(t)-\tilde{\rho}_{\mathrm{KBR}}^{\mathrm{FV}}(t)-\tilde{\rho}_{\mathrm{TC}}(t)$
(32)

are minimized. We define the TC as

$\tilde{\rho}_{\mathrm{TC}}(t)=\sum_{i}\delta\tilde{\rho}_{\mathrm{KBR-
LRI},\,i}^{\mathrm{freq}}(t)+\sum_{i}\delta\tilde{\rho}_{\mathrm{KBR-
LRI},\,i}^{\mathrm{phase}}(t)\ ,$ (33)

where we account for the two different error coupling mechanisms and $i$
denotes contributions from different temperature sensors $T_{i}(t)$. In case
of the frequency-domain coupling, we define

$\delta\tilde{\rho}_{\mathrm{KBR-
LRI},\,i}^{\mathrm{freq}}(t)=\mathrm{HPF}\left(L(t)\cdot\left(c_{1,i}\cdot
T_{i}(t)+c_{2,i}\cdot\dot{T}_{i}(t)\right),\
$0.08\text{\,}\mathrm{mHz}\text{/}$\right)\ .$ (34)

Importantly, we use a coupling where the scale error ($c_{1,i}\cdot
T_{i}\approx\varepsilon_{\mathrm{SCF}}$) is multiplied with the absolute
distance $L\approx$220\text{\,}\mathrm{km}\text{/}$$, cf. eq. 13. The
$\mathrm{cm}\text{/}$-precision of the GPS-based absolute range $L$ obtained
from GNI1B-v04 is sufficient here because the coupling coefficients are below
${10}^{-5}$, which yields a precision of
$0.1\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ or better. The
coefficients $c_{1,i}$ and $c_{2,i}$ have units of
$\mathrm{1}\mathrm{/}\mathrm{K}$ and $\mathrm{s}\text{/}\mathrm{K}$,
respectively. They can be converted to approximate equivalent laser frequency
couplings in units of $\mathrm{Hz}\text{/}\mathrm{K}$ by multiplying with
$\nu_{0}\approx$281\text{\,}\mathrm{THz}\text{/}$$. The second term
$c_{2,i}\cdot\dot{T}_{i}$ originates from a potential timeshift due to
propagation time from temperature changes to the measurement and is linearized
to first order. This timeshift can be computed by
$\zeta_{T_{i}}=c_{2,i}/c_{1,i}$. It should be noted that a positive sign of
$\zeta_{T,i}$ is not violating causality since the timeshift can always be
regarded as a modulus w. r. t. the orbital frequency.

The phase-domain TC contributors read

$\delta\tilde{\rho}_{\mathrm{KBR-
LRI},\,i}^{\mathrm{phase}}(t)=\mathrm{HPF}\left(c_{1,i}\cdot
T_{i}(t)+c_{2,i}\cdot\dot{T}_{i}(t),\
$0.08\text{\,}\mathrm{mHz}\text{/}$\right)\ ,$ (35)

where the coefficients $c_{1}$ and $c_{2}$ have the units
$\mathrm{m}\text{/}\mathrm{K}$ and
$\mathrm{s}\cdot\mathrm{m}\mathrm{/}\mathrm{K}$, respectively. The same high-
pass filter used for the error range (cf. eq. 29) removes frequencies below 1
CPR, i. e., long-term drifts, but maintains 1 CPR, which has high relevance
for the scale factor.

We decompose the temperature $T_{i}$ into
$T_{i}=T_{\mathrm{AC},i}+T_{\mathrm{DC},i}$ by high- and lowpass filtering,
again using the same cutoff frequency of $0.08\text{\,}\mathrm{mHz}\text{/}$.
There are 161 thermistors $T_{i}$ per spacecraft (SC), and the data is
retrieved from so-called OFFRED data and downsampled to
$3.3\text{\,}\mathrm{mHz}\text{/}$ as well. We expect that the DC parts are
more likely to cause variations in the frequency regime, while the AC parts
cause $\mathrm{m}\text{/}\mathrm{K}$-couplings. This is because the DC part
contains a large static offset with only slight variations, which would imply
a constant and hence irrelevant offset in $\tilde{\rho}_{\mathrm{err}}$, if
the phase-domain coupling would apply, but prominent 1/rev tones in case of
the frequency coupling due to the multiplication with $L$ in eq. 34.

An optimization algorithm iteratively picks a single temperature sensor that
minimizes the postfit range error
$\tilde{\rho}_{\mathrm{err,v5X}}^{\mathrm{post-TC}}$ the most. To do so, the
parameters $c_{1,i}$ and $c_{2,i}$ from eqs. 35 and 34 are determined for both
components $T_{\mathrm{AC},i}$ and $T_{\mathrm{DC},i}$ of each temperature
sensor $T_{i}$ in every iteration and the minimization gain, i. e., the
residual rms of KBR-LRI _with_ this particular correction term divided by the
residual rms _without_ , is computed. For every iteration, the parameters of
all the previously added sensors as well as
$\langle\varepsilon_{\mathrm{SCF}}^{\mathrm{glo}}\rangle$ and
$\zeta^{\mathrm{glo}}$ are always co-estimated alongside the newly added
sensor. Hence we extend the design matrix for the least squares minimization
by two columns per iteration.

The algorithm stops after adding five sensors, giving 12 coefficients in
total: two global scale
$\langle\varepsilon_{\mathrm{SCF}}^{\mathrm{glo}}\rangle$ and timeshift
$\zeta^{\mathrm{glo}}$ biases for the whole period and two coefficients for
each selected temperature sensor according to eqs. 33, 34 and 35. The
estimated constants for scale and timeshift are shown in the last two columns
of table 3. The corresponding thermistor coefficients are shown in tables 4c,
4a and 4b for the ranging products v51, v52, and v53, respectively. The
resulting residuals $\tilde{\rho}_{\mathrm{err,v5X}}^{\mathrm{post-TC}}$ are
also shown in fig. 8 (yellow). The subtraction of the full TC model reduces
the KBR-LRI rms residuals to a level of
$1.6\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ (v51),
$1.0\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ (v52) and
$2.3\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ (v53). Especially in the
case of v52, this is close to the expected KBR noise limit of
$0.4\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}~{}\mathrm{r}\mathrm{m}\mathrm{s}$
(cf. eq. 31).

Table 4: Thermal Coupling parameters. The index $i$ denotes the order of
importance, i. e., the gain in reducing the rms residuals. The
$\delta\tilde{\rho}$ type denotes the coupling in phase or frequency regime.
Thus, the unit of $c_{1}$ is $\mathrm{m}\text{/}\mathrm{K}$, if the AC-
component was used and $\mathrm{1}\mathrm{/}\mathrm{K}$, if the DC-component
was used. The coefficient $c_{2}$ has units
$\mathrm{s}\text{\,}\mathrm{m}\text{/}\mathrm{K}$ (AC) or
$\mathrm{s}\text{/}\mathrm{K}$ (DC). The last column $\zeta_{T}=c_{2}/c_{1}$
describes the timeshift of the temperature data in seconds.

$i$ | SC |  Sensor ID |  Ascii Name |  $T_{\mathrm{AC}}/T_{\mathrm{DC}}$ |  $\delta\tilde{\rho}$ type |  $c_{1}$ |  $c_{2}$ |  $\zeta_{T}$ [$\mathrm{s}\text{/}$]
---|---|---|---|---|---|---|---|---
1 | GF-1 |  THT10013 |  SaMzPx |  AC |  phase |  $-1.127\text{\times}{10}^{-07}$ |  $-6.240\text{\times}{10}^{-06}$ |  $55.4$
2 | GF-2 |  THT10133 |  BatTrp |  AC |  phase |  $-1.173\text{\times}{10}^{-05}$ |  $-1.379\text{\times}{10}^{-03}$ |  $117.6$
3 | GF-1 |  THT10144 |  Pr21 |  DC |  freq |  $-1.642\text{\times}{10}^{-09}$ |  $-1.501\text{\times}{10}^{-06}$ |  $914.0$
4 | GF-1 |  THT10022 |  LriLpcMy |  DC |  freq |  $2.640\text{\times}{10}^{-09}$ |  $1.182\text{\times}{10}^{-06}$ |  $447.9$
5 | GF-1 |  THT10143 |  Oct11 |  AC |  phase |  $4.153\text{\times}{10}^{-06}$ |  $-9.814\text{\times}{10}^{-04}$ |  $-236.3$
(a) v51: Telemetry-based laser frequency model $\nu_{1/2}^{\mathrm{TM}}$

$i$ | SC |  Sensor ID |  Ascii Name |  $T_{\mathrm{AC}}/T_{\mathrm{DC}}$ |  $\delta\tilde{\rho}$ type |  $c_{1}$ |  $c_{2}$ |  $\zeta_{T}$ [$\mathrm{s}\text{/}$]
---|---|---|---|---|---|---|---|---
1 | GF-1 |  THT10013 |  SaMzPx |  AC |  phase |  $-1.302\text{\times}{10}^{-07}$ |  $6.194\text{\times}{10}^{-06}$ |  $-47.6$
2 | GF-2 |  THT10138 |  MepFrontPy |  AC |  phase |  $-2.238\text{\times}{10}^{-07}$ |  $5.403\text{\times}{10}^{-05}$ |  $-241.4$
3 | GF-1 |  THT10007 |  GpsOccAnt |  DC |  freq |  $1.513\text{\times}{10}^{-11}$ |  $-2.382\text{\times}{10}^{-08}$ |  $-1573.9$
4 | GF-2 |  THT10089 |  LriOba |  AC |  phase |  $-1.080\text{\times}{10}^{-05}$ |  $1.091\text{\times}{10}^{-03}$ |  $-101.0$
5 | GF-2 |  THT10113 |  LriLas |  DC |  freq |  $1.049\text{\times}{10}^{-10}$ |  $-1.121\text{\times}{10}^{-06}$ |  $-10\,689.7$
(b) v52: Exponential cavity frequency decay model
$\nu_{\mathrm{R}}^{\mathrm{Cav}}$

$i$ | SC |  Sensor ID |  Ascii Name |  $T_{\mathrm{AC}}/T_{\mathrm{DC}}$ |  $\delta\tilde{\rho}$ type |  $c_{1}$ |  $c_{2}$ |  $\zeta_{T}$ [$\mathrm{s}\text{/}$]
---|---|---|---|---|---|---|---|---
1 | GF-2 |  THT10032 |  SaMzMx |  AC |  phase |  $-1.022\text{\times}{10}^{-07}$ |  $1.923\text{\times}{10}^{-05}$ |  $-188.2$
2 | GF-2 |  THT10052 |  AccPanel |  DC |  freq |  $-9.162\text{\times}{10}^{-09}$ |  $-4.827\text{\times}{10}^{-06}$ |  $526.8$
3 | GF-2 |  THT10138 |  MepFrontPy |  AC |  phase |  $-4.996\text{\times}{10}^{-06}$ |  $8.787\text{\times}{10}^{-04}$ |  $-175.9$
4 | GF-2 |  THT10157 |  Oct22 |  DC |  freq |  $5.288\text{\times}{10}^{-10}$ |  $3.349\text{\times}{10}^{-07}$ |  $633.4$
5 | GF-2 |  THT10052 |  AccPanel |  AC |  phase |  $8.433\text{\times}{10}^{-05}$ |  $-3.014\text{\times}{10}^{-03}$ |  $-35.7$
(c) v53: Pre-calibrated fixed frequency value $\nu_{0}$

We observe that in the first iteration, sensors called “SaMz**” were chosen in
all three cases, which are attached to the zenith-pointing solar panels (solar
array minus $\underline{z}$). We expect that the underlying satellite
interior’s thermal environment is highly correlated to these sensors since the
solar arrays are directly heated by the sun and thus exhibit large temperature
variations. Figure 10 shows the dominant 1/rev and 2/rev amplitudes of these
particular sensors. A 1/rev amplitude of $80\text{\,}\mathrm{K}\text{/}$
results in a tone error of roughly $8\text{\,}\mathrm{\SIUnitSymbolMicro
m}\text{/}$ at 1/rev, as apparent from the
$c_{1}\approx$0.1\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}\mathrm{K}$$
coupling factors in the first rows of tables 4c, 4a and 4b.

Figure 10: Peak amplitudes at 1/rev and 2/rev frequencies for SaMzPx of GF-1
(blue) and SaMzMx of GF-2 (orange), which are the most dominant TC
contributors. The yellow curve shows the absolute value of the $\beta$-angle
between the orbital plane and the sun. The sinusoidal 1/rev and 2/rev
temperature amplitudes are much higher when the $\beta$-angle is close to
zero, i. e., when the sun is in the orbital plane.

We highlight that most temperature sensors inside the spacecraft are highly
correlated. Thus, there might be other sets of five sensors that could produce
very similar results.

For verifying the TC, the daily scale and timeshift of KBR and LRI are
computed again using the LRI1B-v5X ranging products, but now we subtract the
TC model $\tilde{\rho}_{\mathrm{TC}}(t)$ beforehand. The new scale and
timeshift are shown in orange also in fig. 9. They clearly show less seasonal
variations than the blue curves without TC. The best performance, by means of
reducing the variations in the daily scale factor estimate
$\langle\varepsilon_{\mathrm{SCF}}\rangle$ in the KBR-LRI differences, is
achieved when using v52, which is based on the exponential cavity frequency
model in combination with the TC. Here, the rms variations of the scale factor
$\langle\varepsilon_{\mathrm{SCF}}\rangle(\rho_{\mathrm{v52}})$ are reduced
from $1.551\text{\times}{10}^{-8}$ to $2.653\text{\times}{10}^{-9}$. Also, the
corresponding timeshift
$\zeta(\rho_{\mathrm{v52}}+\mathrm{TC}_{\mathrm{v52}})$ exhibits low
variations of about $1\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$ when
including the TC correction $\tilde{\rho}_{\mathrm{TC}}$.

We emphasize that the TC parameters shown in tables 3 and 4c can also be used
to correct the LRI1B-v04 dataset by SDS. However, one has to revert the
effects of the daily $\langle\varepsilon_{\mathrm{SCF}}^{\mathrm{SDS}}\rangle$
and $\zeta$ beforehand, which are already applied in LRI1B-v04. The timeshift
$\zeta$ can be extracted by forming the difference of the time offsets
(eps_time) provided in CLK1B and LLK1B.

If we assume that $\tilde{\rho}_{\mathrm{TC}}$ is mainly caused by the KBR
instrument, e. g., due to the stable cavity resonance frequency and thermally
induced KBR antenna phase center variations, the most precise inter-satellite
ranging dataset is given by LRI1B-v52 with scale
$\langle\varepsilon_{\mathrm{SCF}}^{\mathrm{glo}}\rangle=$-3.810\text{\times}{10}^{-9}$\approx$-1\text{\,}\mathrm{MHz}\text{/}$$,
timeshift $\zeta^{\mathrm{glo}}=$67.95\text{\,}\mathrm{\SIUnitSymbolMicro
s}\text{/}$$ and without subtracting $\tilde{\rho}_{\mathrm{TC}}$.

Since the OFFRED thermistor data is not publicly available, we provide the
$0.5\text{\,}\mathrm{Hz}\text{/}$ ranging data product LRI1B-v52 and the
corresponding TC ranging correction called RTC1B-v52 (Range Thermal Coupling),
see Data Availability Statement below.

##  10 Discussion of Results and Alternative Approaches for Future Missions

The frequency of the LRI laser is needed to convert phase to range. Any error
or uncertainty can be considered a scale error in the range measurement.
Currently, the LRI scale or absolute frequency is estimated daily by
correlating KBR range with LRI range. One goal of the methods presented in
this paper is to derive an independent and reliable model for the absolute
laser frequency $\nu_{\mathrm{R}}$ of the LRI. Such models would be needed if
KBR data is missing, e. g., if the second IPU on the GF-2 satellite would
become unavailable. Moreover, since the LRI processing becomes less dependent
on the KBR, measurement errors in the KBR would not affect the LRI data
anymore. Furthermore, in a future mission, there will likely be only a single
LRI-like ranging instrument which requires a new processing scheme. There are
several options for determining the absolute laser frequency for these
missions, which will be discussed briefly in the following.

The telemetry-based models $\nu^{\mathrm{TME}}_{1/2}$, which includes the
empirical correction term $\nu^{\mathrm{emp}}_{1/2}$ from in-flight
measurements, reached an accuracy of approximately
$$60\text{\,}\mathrm{MHz}\text{/}$\approx$200\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$$
(see fig. 7). Since the laser is thermally coupled to the satellite platform,
temperature variations of the surrounding units couple into the setpoint-based
model but not into the true frequency determined by the cavity. We emphasize
that the LRI is a technology demonstrator and the calibration of the laser
frequency had only a moderate priority. However, the authors assume that such
accuracy could also be achievable in a future mission from on-ground
calibrations only if the laser is characterized more thoroughly, in
particular, if the drift of $40\text{\,}\mathrm{MHz}\text{/}\mathrm{yr}$ of
the laser setpoint in $\nu^{\mathrm{emp}}_{1/2}$ is calibrated. Additionally,
one might attempt to characterize the cavity frequency exponential decay on-
ground and derive an estimate for the flight phase.

An alternative to determining the frequency from in-flight telemetry is to co-
estimate it during gravity field recovery, as it is usually done for the
accelerometer scales and biases Helleputte et al. ; Klinger and Mayer-Gürr .
However, any LRI scale uncertainty mainly manifests at 1/rev and 2/rev
frequencies, where the gravity field recovery processing strategies often
introduce empirical parameters that partly absorb the scale factor Klinger and
Mayer-Gürr . Furthermore, the estimated scale is highly correlated with the
$C_{2,0}$ spherical harmonics coefficient, which is mainly measurable at 1/rev
and 2/rev frequencies and not estimated reliably in gravity field recovery
Klinger and Mayer-Gürr . We would expect that LRI errors from a scale factor
uncertainty would be at the level of the post-fit residuals of the gravity
field recovery, which are much higher than the LRI requirements of ${10}^{-7}$
or ${10}^{-8}$ (cf. section 3).

A well-known and broadly used approach to obtain a well-defined absolute laser
frequency relies on iodine spectroscopy, where the hyperfine transition line
of an iodine molecule is used as an absolute reference for a laser lock Arie
et al. (1992). This technology has also been used for calibrating the LRI RLUs
(cf. appendices A and B), and there are activities ongoing to qualify such
setups for the space environment, see e. g. Döringshoff et al. . However,
saturated doppler-free spectroscopy is likely incompatible with the available
optical power from Tesat RLUs used in the LRI so far. Hence, one would need to
add optical amplifiers, which significantly increase the complexity and
electric power consumption. Laboratory experiments showed absolute frequency
repeatability levels below
$$1\text{\,}\mathrm{kHz}\text{/}$\approx$3\text{\times}{10}^{-3}\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$$
Döringshoff et al. . Thus, this method is probably the most accurate mean to
fix or determine the scale factor. Optionally, there is the possibility for a
hybrid lock by using both a conventional PDH lock to an optical cavity and a
spectroscopic locking to an iodine reference. This hybrid lock combines the
stability of the cavity at high Fourier frequencies, and the absolute laser
frequency knowledge through the molecular reference Sanjuan et al. (2021).

Another approach for measuring the cavity resonance frequency is based on an
extension of the PDH lock DeVoe and Brewer . Adding an additional tone (scale
factor tone) at a few $\mathrm{MHz}\text{/}$ frequency together with upper and
lower sidebands with the approximate free spectral range (FSR) separation
enables the readout of the actual cavity FSR w. r. t. the applied sidebands.
The frequency of these sidebands is derived coherently from the USO as the two
tones and LRP time. After determination of the USO frequency during precise
orbit determination, this technique can provide an estimate for the FSR
frequency of the cavity. The cavity resonance frequency and FSR are linearly
related to each other through
$\nu=n_{\mathrm{mode}}\cdot\mathrm{FSR}+\mathrm{offset}$, where the mode
number $n_{\mathrm{mode}}$ is an integer mode number. The offset must be
calibrated on ground Rees et al. (a). The principle has been demonstrated in
laboratory experiments with an accuracy of roughly
$$3\text{\,}\mathrm{MHz}\text{/}$\approx$10\text{\,}\mathrm{p}\mathrm{p}\mathrm{b}$$.
It is noteworthy that the scale factor tone and its sidebands have little
influence on the conventional PDH readout Rees et al. (b, a). The advantage of
this technique is that only minor changes to existing flight hardware are
needed, e. g., the use of $\mathrm{GHz}\text{/}$ electro-optical modulators
instead of $\mathrm{MHz}\text{/}$. However, additional RF electronics and an
additional processing unit for the readout are required. The FSR-readout is
currently the most probable solution for upcoming gravity missions.

##  11 Conclusion

In this paper, the methodology as presented by Müller et al. to derive a
precise range from raw interferometric phase measurements was applied to in-
flight data of the GRACE-FO LRI instrument. Based on that work, we derived the
two dominant error terms, namely a time-variable scaling of the laser
frequency, expressed through a scale factor $\varepsilon_{\mathrm{SCF}}$, and
a timeshift $\zeta$ of the LRI measurement w. r. t. the reference measurement
is given by the KBR. Importantly, variations in $\varepsilon_{\mathrm{SCF}}$
couple into the range proportionally to the absolute distance $L$ between the
two satellites. The scale $\varepsilon_{\mathrm{SCF}}$ and timeshift $\zeta$
parameters can easily be compromised if errors in the range measurement at
1/rev and 2/rev frequencies are present.

In the second part, three different models to calculate the in-flight laser
frequency were shown, which are largely independent of KBR measurements, once
the model parameters are determined. Based on these models, we derive three
versions of an LRI1B-equivalent data product, namely v51, v52, and v53. The
first method (v53) uses a constant, nominal laser frequency $\nu_{0}$, which
was chosen early before launch, and without sophisticated analysis since it
was clear that the numerical values serve as a start value for the subsequent
and accurate refinement utilizing cross-calibration. Thus, v53 is, in
principle, a pre-release for the official LRI1B-v04 dataset. The latter is
further refined by daily cross-calibrating LRI and KBR ranging data, i. e. ,
estimating scale $\langle\varepsilon_{\mathrm{SCF}}\rangle$ and time-shift
$\zeta$ on a daily basis. The daily scale factor or frequency
$\nu^{\mathrm{SDS}}$ determined from the cross-calibration revealed seasonal
variations and a settling effect, which we attribute to the optical reference
cavity and which might be related to aging of the ULE cavity spacer material
as reported in Alnis et al. . We use an exponential decay function (cf. eq.
22) to describe this settling effect and to form v52. The v51 dataset uses a
laser frequency model derived from LRI laser and temperature telemetry. One
can relate the lasers’ control loop setpoints and temperatures to the output
frequency using linear coupling factors, which were calibrated on ground
before launch. The setup of the pre-flight calibration measurements was
explained, and the calibration factors were provided. The initial
$\nu^{\mathrm{TM}}$-model from on-ground calibrations was then compared to
four years of in-flight data of the GRACE-FO mission. It was found that the
$\nu^{\mathrm{TM}}$ model frequency or, more precisely, the setpoints of the
laser control loops drift over time by roughly
$40\text{\,}\mathrm{MHz}\text{/}\mathrm{yr}$. Furthermore, two steps were
observed when the lasers were operated in non-nominal conditions. The physical
reason for the drift could be a consequence of aging effects of the NPRO
crystal or electronics. However, while the exact reason remains unknown, we
compensate for the drift and steps with an empirical model here. We observed
that this telemetry-based model
$\nu^{\mathrm{TME}}_{1/2}=\nu^{\mathrm{TM}}_{1/2}-\nu^{\mathrm{emp}}_{1/2}$ of
the laser frequency does not show seasonal variations, indicating that the
seasonal variations of the frequency $\nu^{\mathrm{SDS}}$ are not actual
changes in the laser frequency. The v51 dataset uses this empirically
corrected model $\nu^{\mathrm{TME}}_{1/2}$.

Afterward, we focused on analyzing residuals of the difference of KBR-LRI,
which we call the range error. At first, the direct difference yields large
prefit range errors $\tilde{\rho}_{\mathrm{err,v5X}}^{\mathrm{pre}}$ of more
than
$25\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}~{}\mathrm{r}\mathrm{m}\mathrm{s}$
for all three v5X data sets. In the second step, the effect of a global scale
factor $\langle\varepsilon_{\mathrm{SCF}}^{\mathrm{glo}}\rangle$ and a global
timeshift $\zeta^{\mathrm{glo}}$ are subtracted, reducing the postfit
residuals $\tilde{\rho}_{\mathrm{err,v5X}}^{\mathrm{post}}$ to approximately
$6\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}~{}\mathrm{r}\mathrm{m}\mathrm{s}$
in all three cases.

These postfit residuals could be explained by seasonal variations in the scale
and timeshift as determined from daily cross-calibration of LRI w.r.t. KBR.
However, since the telemetry-based frequency model $\nu^{\mathrm{TME}}$ does
not show these seasonal variations, we described them with a Thermal Coupling
(TC). We accounted for two TC mechanisms, one in the phase domain and one in
the frequency domain. An algorithm to determine coupling coefficients for all
temperature sensors on both spacecraft was explained, and equations to compute
the TC were given. For each of the three datasets, a TC composed of 12
coefficients, which includes five temperature sensors, was derived. For each
temperature sensor, we derived a linear coupling $c_{1}$ with units of
$\mathrm{m}\text{/}\mathrm{K}$ (phase domain) or
$\mathrm{1}\mathrm{/}\mathrm{K}$ (frequency domain) and a possible time delay
$\zeta_{T}$. Furthermore, the two global parameters for the scale
$\langle\varepsilon_{\mathrm{SCF}}^{\mathrm{glo}}\rangle$ and timeshift
$\zeta^{\mathrm{glo}}$, which are constant over the whole mission span, were
refined. We showed that the differences between LRI and KBR can be reduced
from approx.
$25\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}~{}\mathrm{r}\mathrm{m}\mathrm{s}$
to
$1\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}~{}\mathrm{r}\mathrm{m}\mathrm{s}$
when using LRI1B-v52 including the TC. In all three cases, the dominant
thermal coupling originates from thermistors attached to the zenith-facing
solar array. An analysis of these thermistor timeseries’ revealed that their
1/rev and 2/rev amplitudes are highest when the angle $\beta$ between the
orbital plane and the sun is close to zero, which occurs roughly every six
months. The tone error magnitude of these sensors in the TC is in the order of
$\pm 8\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$.

We analyzed only the thermal effects that are not common in LRI and KBR, i.
e., which appear in the KBR-LRI difference. Thus, the TC does not address
potential common effects.

This paper introduced a new laser frequency model for the LRI, representing
the current best knowledge of the cavity resonance frequency on GF-1. We
showed that this resonance frequency is relatively stable after an initial
exponential convergence. As apparent from daily KBR-LRI cross-calibration,
seasonal variations can be explained with tone errors from a Thermal Coupling.

Conceptualization: Malte Misfeldt and Vitali Müller; Funding acquisition:
Gerhard Heinzel; Investigation: Malte Misfeldt, Vitali Müller, Laura Müller
and Henry Wegener; Project administration: Gerhard Heinzel; Writing – original
draft: Malte Misfeldt; Writing – review & editing: Vitali Müller, Laura
Müller, Henry Wegener and Gerhard Heinzel.

This work has been supported by: The Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation, Project-ID 434617780, SFB 1464); Clusters of
Excellence “QuantumFrontiers: Light and Matter at the Quantum Frontier:
Foundations and Applications in Metrology” (EXC-2123, project number:
390837967); the European Space Agency in the framework of Next Generation
Geodesy Mission development and ESA’s third-party mission support for GRACE-
FO; the Chinese Academy of Sciences (CAS) and the Max Planck Society (MPG) in
the framework of the LEGACY cooperation on low-frequency gravitational-wave
astronomy (M.IF.A.QOP18098).

GRACE-Follow On Level-1 instrument data is distributed by the GRACE/GRACE-FO
project via NASA PODAAC (https://podaac-
tools.jpl.nasa.gov/drive/files/allData/gracefo). The data product LRI1B-v52 as
derived in this manuscript, together with the Range Thermal Coupling product
RTC1B-v52 is available at the LUH data repository Malte Misfeldt . Thermistor
data from OFFRED telemetry are not publicly available by the time of writing.

###### Acknowledgements.

The authors would like to thank the JPL LRI team for helpful regular
discussions and insights. The authors declare no conflict of interest.
Abbreviations The following abbreviations are used in this manuscript:

## Acronyms

AEI

    Albert-Einstein Institute
ASD

    amplitude spectral density
DWS

    Differential Wavefront Sensing
ENBW

    equivalent noise bandwidth
FIR

    Finite Impulse Response
FSR

    free spectral range
GCRF

    geocentric celestial reference frame
GRACE-FO

    GRACE Follow-On
GPS

    global positioning system
GRACE

    Gravity Recovery And Climate Experiment
IL

    in-loop
IPU

    MWI Instrument Processing Unit
KBR

    K-band Ranging
LRI

    Laser Ranging Interferometer
LRP

    Laser Ranging Processor
LTC

    Light Time Correction
MCM

    Mass Change Mission
MTS

    modulation transfer spectroscopy
MWI

    Microwave Instrument
NGGM

    Next Generation Gravity Mission
NPRO

    non-planar ring oscillator
OBA

    Optical Bench Assembly
OBC

    Onboard Computer
OGSE

    Optical Ground Support Equipment
OOL

    out-of-loop
PDH

    Pound-Drever-Hall
PZT

    Piezo-Electric Transducer
RLU

    Reference Laser Unit
SC

    spacecraft
SDS

    Science Data System
TC

    Thermal Coupling
TM

    telemetry
TMA

    Triple Mirror Assembly
TRP

    Thermal Reference Point
ULE

    ultra-low expansion
USO

    Ultra-Stable Oscillator

yes

##  Appendix A Calibration of WS6-600 using an Iodine Cell

Various measurement campaigns for determining the telemetry (TM) frequency
models for the two laser flight models of the Laser Ranging Interferometer
(LRI) have been performed between July 2017 and January 2018. In these
campaigns, three different wavelength meters (or wavemeters) have been used: A
WS6-600 and a WS7-60 by HighFinesse / Angstrom and a WA1500 by Burleigh. The
first one has an absolute accuracy of $600\text{\,}\mathrm{MHz}\text{/}$,
while the latter two are more accurate by one order of magnitude. The two
HighFinesse devices were used mainly during the testing campaigns. They have a
built-in calibration via a neon lamp. The reference frequency of a well-known
iodine hyperfine transition was used to verify this internal calibration.
Iodine is a commonly used molecule for stabilizing lasers to an absolute
frequency reference, see e. g. Arie et al. (1992); Schuldt (2003).

At first, the accuracy of the WS6-600 was measured in a setup utilizing a
Prometheus laser by Coherent, Inc. Coherent, Inc. (2022) providing
$500\text{\,}\mathrm{mW}\text{/}$ output power at a wavelength of
$1064\text{\,}\mathrm{nm}\text{/}$. The secondary output provides
$20\text{\,}\mathrm{mW}\text{/}$ of frequency-doubled light at
$532\text{\,}\mathrm{nm}\text{/}$. The frequency of this reference laser can
be tuned over a range of roughly $60\text{\,}\mathrm{GHz}\text{/}$ via thermal
elements, and piezoelectric transducers Coherent, Inc. (2022).

The laser’s frequency was locked via Doppler-free modulation transfer
spectroscopy (MTS) to the R(56)32-0 iodine line, of which we used the $a_{1}$
and $a_{10}$ components. The iodine cell was manufactured by InnoLight as
well. The nominal frequency of the $a_{10}$ hyperfine component is Riehle et
al. (2018)

$\nu_{a_{10}}=$563\,260\,223.513\text{\,}\mathrm{MHz}\text{/}$\ .$ (36)

The $a_{10}$ component’s frequency is elevated by
$\delta\nu_{\text{vis}}=$572.1\text{\,}\mathrm{MHz}\text{/}$$ w. r. t. the
$a_{1}$ component Arie et al. (1992). Hence,

$\nu_{a_{1}}=\nu_{a_{10}}-\delta\nu_{\text{vis}}=$563\,259\,651.413\text{\,}\mathrm{MHz}\text{/}$\
.$ (37)

The corresponding difference frequency at $1064\text{\,}\mathrm{nm}\text{/}$
is

$\delta\nu_{\text{IR}}=$286.05\text{\,}\mathrm{MHz}\text{/}$\ ,$ (38)

which would be the accurate result.

Figure 11: Absolute frequency measurements of the reference laser locked to
different hyperfine lines of an iodine cell. The apparent quantization of
approximately $3\text{\,}\mathrm{MHz}\text{/}$ arises from the finite
resolution of ${10}^{-5}\text{\,}\mathrm{nm}\text{/}$ of the wavemeter
WS6-600. Outliers removed. The small oscillations are observed repeatedly for
this wavemeter. The color indicates the wavemeter’s exposure time.

Figure 12: OGSE laser frequency, measured with two wavemeters.
$110\text{\,}\mathrm{min}\text{/}$ waiting for thermal equilibrium, afterwards
active control using the laser’s thermal elements. Outliers removed.
$\nu_{0}=$281\,616\,307\text{\,}\mathrm{MHz}\text{/}$$.

Our first calibration measurement, shown in fig. 12, took
$5000\text{\,}\mathrm{s}\text{/}$, of which the laser was locked to the
$a_{1}$ component in the beginning and the end for approximately
$1000\text{\,}\mathrm{s}\text{/}$ and the $a_{10}$ component in between. The
optical power in the fiber going to the WS6-600 was relatively low at about
$850\text{\,}\mathrm{nW}\text{/}$. The average frequency measured for
$\nu_{a_{1}}$ at $1064\text{\,}\mathrm{nm}\text{/}$ is about
$60\text{\,}\mathrm{MHz}\text{/}$ below the nominal value, which is within the
$600\text{\,}\mathrm{MHz}\text{/}$ accuracy of the device. The measured
frequency difference between the two hyperfine components $a_{1}$ and $a_{10}$
is $\delta\nu_{\mathrm{meas.}}=$288.785\text{\,}\mathrm{MHz}\text{/}$$, being
$2.735\text{\,}\mathrm{MHz}\text{/}$ higher than $\delta\nu_{\mathrm{IR}}$.
Other measurements confirmed a bias of this device by approximately 20 to
$60\text{\,}\mathrm{MHz}\text{/}$, while the relative measurements are more
precise. The small oscillations with a magnitude of up to
$15\text{\,}\mathrm{MHz}\text{/}$ and a period of about
$300\text{\,}\mathrm{s}\text{/}$ have been observed repeatedly for the
WS6-600. The measurement is often noisier at short exposure times of about
$600\text{\,}\mathrm{ms}\text{/}$ (cf. color encoding in fig. 12).

##  Appendix B Calibration of OGSE laser and WS7-60

After calibrating the WS6-600 wavemeter, the laser of the Optical Ground
Support Equipment (OGSE) was characterized w. r. t. its thermal coupling and
drifts. The OGSE laser was used in single-spacecraft functional tests to
simulate the received light for the LRI units. The flight laser units were
operated in transponder mode and were locked to the incoming OGSE light. At
the integration facility, a second wavemeter, the WS7-60 with an absolute
accuracy of $60\text{\,}\mathrm{MHz}\text{/}$, was available alongside the
WS6-600. Hence, this allowed us to calibrate the WS7-60 against the WS6-600.

A comparison of the two wavemeters, shown in fig. 12, revealed that the
WS6-600 measures frequencies which are lowered by approximately $40$ to
$80\text{\,}\mathrm{MHz}\text{/}$ compared to the WS7-60. This is consistent
with the calibration with iodine lines (see appendices A and 12). It was found
that the OGSE laser needs at least 60 to $80\text{\,}\mathrm{min}\text{/}$ to
reach thermal equilibrium. The frequency drift after this warm-up phase is
below $20\text{\,}\mathrm{kHz}\text{/}\mathrm{s}$. Since the two measurements
agree well and the WS6-600 offset was observed before, it was concluded that
the WS7-60 is accurate within the needs.

##  Appendix C Thermal actuator signals of GF-1

Figure 13 illustrates the in-flight thermal actuator signals of the two LRI
units. The black vertical lines highlight the regions, where the LRI acquired
the link before the thermal equilibrium of the lasers was reached. This caused
the in-loop signal to reach higher values than usual. At these instances, the
telemetry-based laser frequency model shows nonphysical steps, which are then
corrected by empirical parameters in eq. 25.

(a) GF-1 actuator signals.  (b) GF-2 actuator signals.

Figure 13: Control loop actuator signals multiplied with their calibrated
coefficient (see table 1) for the thermal actuator of the RLUs from 2018-Dec
until 2022-Jan. _Left axes:_ IL signal (blue). _Right axes:_ OOL signal
(orange) and sum of both (dashed orange). The black vertical lines indicate
the times, where a step was applied in the empirical model. _(a)_ A step is
visible at the first dashed vertical line, where the LRI locked before
reaching thermal equilibrium. At the second vertical line, no frequency step
is visible, but the actuator signals jumped back to the usual range. _(b)_
Similar but smaller steps as in the left subplot are present between the two
dashed vertical lines.

##  Appendix D LRI Time Frames

The LRI time frame is initialized at the startup of the Laser Ranging
Processor (LRP), however, this initialization introduces an unknown offset of
$1.5\text{\,}\mathrm{s}\text{/}$ at maximum between the Onboard Computer (OBC)
time and the LRI time. After initialization, the LRP clock is counting eight-
fold Ultra-Stable Oscillator (USO) ticks, which implies a clock rate of
$38.656\,000\text{\,}\mathrm{MHz}\text{/}$ for GF-1 and
$38.656\,792\text{\,}\mathrm{MHz}\text{/}$ for GF-2. The offset between the
time frames of the LRI and the OBC is regularly determined using so-called
datation reports. The reported datation bias usually remains constant between
reboots of either the LRP or the MWI Instrument Processing Unit (IPU).
Furthermore, a filter is used inside the LRP to reduce the phase telemetry
sampling rate to roughly $10\text{\,}\mathrm{Hz}\text{/}$. This filter
introduces a delay of
$28\,802\,038\text{\,}\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{k}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{k}\mathrm{s}$
(approximately $0.75\text{\,}\mathrm{s}\text{/}$, slightly different for the
two spacecraft) that has to be accounted for Ware (2018); Wen et al. (2019).
In actual flight-data processing, when K-band Ranging (KBR) and LRI range data
are cross-calibrated, an estimated additional offset of
$\zeta\approx$70\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$$ is observed,
whose origin is unknown and that is not measured with LRI datation reports.
During the analysis presented in this paper, this time shift $\zeta$ is
numerically estimated in every comparison between LRI and KBR ranging data.

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