# Ranging Sensor Fusion in LISA Data Processing: Treatment of Ambiguities,
Noise, and On-Board Delays in LISA Ranging Observables

Jan Niklas Reinhardt<EMAIL_ADDRESS>Max-Planck-Institut für
Gravitationsphysik (Albert-Einstein-Institut),
Callinstraße 38, 30167 Hannover, Germany  Leibniz Universität Hannover,
Welfengarten 1, 30167 Hannover, Germany  Martin Staab  Max-Planck-Institut für
Gravitationsphysik (Albert-Einstein-Institut),
Callinstraße 38, 30167 Hannover, Germany  Leibniz Universität Hannover,
Welfengarten 1, 30167 Hannover, Germany  Kohei Yamamoto  Max-Planck-Institut
für Gravitationsphysik (Albert-Einstein-Institut),
Callinstraße 38, 30167 Hannover, Germany  Leibniz Universität Hannover,
Welfengarten 1, 30167 Hannover, Germany  Jean-Baptiste Bayle  University of
Glasgow, Glasgow G12 8QQ, United Kingdom  Aurélien Hees  SYRTE, Observatoire
de Paris, Université PSL, CNRS, Sorbonne Université,
LNE, 61 avenue de l’observatoire 75014 Paris, France  Olaf Hartwig  SYRTE,
Observatoire de Paris, Université PSL, CNRS, Sorbonne Université,
LNE, 61 avenue de l’observatoire 75014 Paris, France  Karsten Wiesner  Max-
Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut),
Callinstraße 38, 30167 Hannover, Germany  Leibniz Universität Hannover,
Welfengarten 1, 30167 Hannover, Germany  Gerhard Heinzel  Max-Planck-Institut
für Gravitationsphysik (Albert-Einstein-Institut),
Callinstraße 38, 30167 Hannover, Germany  Leibniz Universität Hannover,
Welfengarten 1, 30167 Hannover, Germany

###### Abstract

Interspacecraft ranging is crucial for the suppression of laser frequency
noise via time-delay interferometry (TDI). So far, the effect of on-board
delays and ambiguities in the LISA ranging observables was neglected in LISA
modelling and data processing investigations. In reality, on-board delays
cause offsets and timestamping delays in the LISA measurements, and PRN
ranging is ambiguous, as it only determines the range up to an integer
multiple of the pseudo-random noise (PRN) code length. In this article, we
identify the four LISA ranging observables: PRN ranging, the sideband
beatnotes at the interspacecraft interferometer, TDI ranging, and ground-based
observations. We derive their observation equations in the presence of on-
board delays, noise, and ambiguities. We then propose a three-stage ranging
sensor fusion to combine these observables in order to gain optimal ranging
estimates. We propose to calibrate the on-board delays on ground and to
compensate the associated offsets and timestamping delays in an initial data
treatment (stage 1). We identify the ranging-related routines, which need to
run continuously during operation (stage 2), and implement them numerically.
Essentially, this involves the reduction of ranging noise, for which we
develop a Kalman filter combining the PRN ranging and the sideband beatnotes.
We further implement crosschecks for the PRN ranging ambiguities and offsets
(stage 3). We show that both ground-based observations and TDI ranging can be
used to resolve the PRN ranging ambiguities. Moreover, we apply TDI ranging to
estimate the PRN ranging offsets.

##  I Introduction

The Laser Interferometer Space Antenna (LISA), due for launch in 2034, is an
ESA-led mission for space-based gravitational-wave detection in the frequency
band between $0.1\text{\,}\mathrm{mHz}$ and $1\text{\,}\mathrm{Hz}$ [1]. LISA
consists of three satellites forming an approximate equilateral triangle with
an armlength of $2.5\text{\,}\mathrm{Gm}$, in a heliocentric orbit that trails
or leads Earth by about $20$ degrees. Six infrared laser links with a nominal
wavelength of $1064\text{\,}\mathrm{nm}$ connect the three spacecraft (SC),
whose relative motion necessitates the usage of heterodyne interferometry.
Phasemeters are used to extract the phases of the corresponding beatnotes [2],
in which gravitational-waves manifest in form of microcycle deviations
equivalent to picometer variations in the interspacecraft ranges.

The phasemeter output, however, is obscured by various instrumental noise
sources. They must be suppressed to fit in the LISA noise budget of
$10\text{\,}\mathrm{pm}\text{\,}{\mathrm{Hz}}^{-0.5}$ (single link) [3],
otherwise they would bury the gravitational-wave signals. Dedicated data
processing algorithms are being developed for each of these instrumental noise
sources, their subsequent execution is referred to as initial noise reduction
pipeline (INReP). The dominating noise source in LISA is by far the laser
frequency noise due to the armlength differences in the order of $1\%$
($25\,000\text{\,}\mathrm{km}$). It must be reduced by more than 8 orders of
magnitude. This is achieved by time-delay interferometry (TDI), which combines
the various beatnotes with the correct delays to virtually form equal-optical-
path-length interferometers, in which laser frequency noise naturally cancels
[4, 5]. The exact definition of these delays depends on the location of TDI
within the INReP (see fig. 1) [6], but wherever we place it, some kind of
information about the absolute interspacecraft ranges is required.

Yet, absolute ranges are not a natural signal in a continuous-wave heterodyne
laser interferometer such as LISA. Therefore, a ranging scheme based on
pseudo-random noise (PRN) codes is implemented [7, 8, 9]. Each SC houses a
free-running ultra-stable oscillator (USO) as timing reference. It defines the
spacecraft elapsed time (SCET). PRN codes generated according to the
respective SCETs are imprinted onto the laser beams by phase-modulating the
carrier. The comparison of a PRN code received from a distant SC, hence
generated according to the distant SCET, with a local copy enables a
measurement of the pseudorange: the pseudorange is commonly defined as the
difference between the SCET of the receiving SC at the event of reception and
the SCET of the emitting SC at the event of emission [10]. It represents a
combination of the true geometrical range (light travel time) with the offset
between the two involved SCETs (see eq. 70).

In the baseline TDI topology (upper row in fig. 1), TDI is performed after
SCET synchronization to the barycentric coordinate time (TCB), the light
travel times are used as delays. The pseudoranges comprise information about
both the light travel times and the SCET offsets required for synchronizing
the clocks (see appendix A). A Kalman filter can be used to disentangle the
pseudoranges in order to retrieve light travel times and SCET offsets [11]. In
the alternative TDI topology (lower row in fig. 1), the pseudoranges are
directly used as delays. In that topology, TDI is executed on the
unsynchronized beatnotes sampled according to the respective SCETs [6].

Figure 1: In the baseline TDI topology (upper part) we perform TDI after clock
synchronization to TCB, the delays are given by the light travel times. In the
alternative TDI topology (lower part) we execute TDI without clock
synchronization and apply the pseudoranges as delays [6]. Both topologies rely
on a ranging sensor fusion.

However, PRN ranging (PRNR) does not directly provide the pseudoranges but
requires three treatments. First, due to the finite PRN code length (we assume
$400\text{\,}\mathrm{km}$), PRNR measures the pseudoranges modulo an ambiguity
[7]. Secondly, PRNR is limited by white ranging noise with an RMS amplitude of
about $1\text{\,}\mathrm{m}$ when sampled at $4\text{\,}\mathrm{Hz}$ [9].
Thirdly, on-board delays due to signal propagation and processing cause
offsets and timestamping delays in the PRNR. There are three additional
pseudorange observables to resolve these difficulties: ground-based
observations provide inaccurate but unambiguous pseudorange estimates; time-
delay interferometric ranging (TDIR) turns TDI upside-down seeking a model for
the delays that minimizes the laser frequency noise in the TDI combinations
[12]; the sideband beatnotes include information about the time derivatives of
the pseudoranges [6]. The combination of these four pseudorange observables in
order to form optimal pseudorange estimates is referred to as ranging sensor
fusion in the course of this article. It is common to both TDI topologies (see
fig. 1) and consequently a crucial stage of the INReP.

In section II, we first specify the pseudorange definition. We then derive the
observation equations of the four pseudorange observables carefully
considering the effects of the on-board delays. In section III, we introduce a
three-stage ranging sensor fusion consisting of an initial data treatment, a
core ranging processing, and crosschecks. In the initial data treatment, we
propose to compensate for the offsets and timestamping delays caused by the
on-board delays. We identify PRNR unwrapping and noise reduction as the
ranging processing steps that need to run continuously during operation. In
parallel to this core ranging processing, we propose crosschecks of the PRNR
ambiguities and offsets. We implement the core ranging processing and the
crosschecks numerically. In section IV we discuss the performance of this
implementation, and conclude in section V.

##  II Ranging measurements

Each SC houses an ultra-stable oscillator (USO) generating an
$80\text{\,}\mathrm{MHz}$ clock signal, the phasemeter clock (PMC). The PMC
can be considered as the timing reference on board the SC (see fig. 3), its
associated counter is referred to as spacecraft elapsed time (SCET):

$\displaystyle\text{SCET}(n)=\sum_{1}^{n}\>\frac{1}{$80\text{\,}\mathrm{MHz}$}.$
(1)

The SCET, denoted by $\hat{\tau}_{i}$, differs from the barycentric coordinate
time (TCB), denoted by $t$, due to instrumental clock drifts and jitters, and
due to relativistic effects. Following the notation of [13], we use
superscripts to indicate a quantity to be expressed as function of a certain
time scale, e.g., $\hat{\tau}_{1}^{t}$ denotes the SCET of SC 1 as function of
TCB. Note that

$\displaystyle\hat{\tau}_{i}^{\hat{\tau}_{i}}(\tau)=\tau.$ (2)

Figure 2: LISA labeling conventions (from [14]). The SC are labeled clockwise.
The MOSAs are labeled by 2 indices: the first one indicates the SC they are
located at, the second one the SC they are oriented to. The measurements and
related quantities (optical links, pseudoranges, etc.) share the indices of
the MOSAs they are measured at. Below, we distinguish between left-handed
MOSAs (12, 23, 31) and right-handed MOSAs (13, 32, 21). Figure 3: We trace a
local laser (red arrows) and a distant laser (yellow arrows) to the ISI BSs on
both SC, where they interfere and form beatnotes (orange arrows). Before, the
carriers are phase-modulated with the $\mathrm{GHz}$ clock and the PRN signals
(follow the arrows from the PMC and the PRN to the EOM). We show the USO
frequency distribution (follow the blue arrows after the USOs) and illustrate
the on board signal processing (follow the arrows after the QPRs).
Constituents of the pseudorange are marked purple. These are the light travel
time between the PBSs (at the telescopes) and the transformation between the
two SCETs (considered at the PBS of the receiving SC). In light blue, we mark
the PRN ranging offset from the pseudorange. We identify the common carrier,
sideband, and PRN timestamping delays in green, dark yellow, and pink,
respectively.

Each SC contains two movable optical sub-assemblies (MOSAs) connected by an
optical fibre (see fig. 2 for the labeling conventions). Each MOSA has an
associated laser and houses a telescope, a free-falling test mass marking the
end of the corresponding optical link, and an optical bench with three
interferometers: the interspacecraft interferometer (ISI), in which the
gravitational-wave signals eventually appear, the reference interferometer
(RFI) to compare local and adjacent lasers, and the test-mass interferometer
(TMI) to sense the optical bench motion with respect to the free-falling test
mass in direction of the optical link. The $\mathrm{MHz}$ beatnotes in these
interferometers are detected with quadrant-photo-receivers (QPRs). They are
digitized in analog-to-digital converters (ADCs) driven by the PMCs.
Phasemeters extract the beatnote phases111In the current design, the
phasemeters deliver the beatnote frequencies with occasional phase anchor
points. using digital phase-locked loops (DPLLs), which are then downsampled
to $4\text{\,}\mathrm{Hz}$ in a multi-stage decimation procedure (DEC) and
telemetered to Earth.

###  II.1 The pseudorange and on-board delays

The pseudorange, denoted by $R_{ij}^{\hat{\tau}_{i}}$, is commonly defined as
the difference between the SCET of the receiving SC at the event of reception
and the SCET of the emitting SC at the event of emission [10]. It represents a
combination of the light travel time between the emission at SC $j$ and the
reception at SC $i$, and the differential SCET offset (see eq. 70). However,
considering the complexity of the LISA metrology system, this definition
appears to be rather vague: to what exactly do we relate the events of
emission and reception? Two specifications are required here: we need to
locate emission and reception, and we need to define the actual events. It is
convenient to consider emission and reception at the respective polarizing
beam splitters (PBSs) in front of the telescopes (denoted PBS1 in [15]), and
to treat the on-board signal propagation and processing on both SC as on-board
delays. Thus, we clearly separate the pseudorange from on-board delays. This
definition is not unique, the events of emission and reception could be
located elsewhere, assuming that the on-board delays are defined accordingly.
The LISA optical links do not involve delta-pulse-like events. In order to
define the actual events of emission and reception we, instead, use the
instants when the light phase changes at the beginning of the first PRN code
chip. At first glance, the PRN code might seem unfavorable for the pseudorange
definition, as PRN and carrier phase are oppositely affected by the solar
wind: the PRN phase is delayed by the group-delay, while the carrier phase is
advanced by the phase delay. However, these effects are at the order of
$10\text{\,}\mathrm{pm}$ (see appendix C), whereas our best pseudorange
estimates are at $0.1\text{\,}\mathrm{mm}$ accuracy. Consequently, the solar
wind dispersion can be neglected in the pseudorange definition.

When expressing the interferometric measurements according to this specified
pseudorange definition, we need to consider the excluded on-board signal
propagation and processing. For that purpose, we introduce two kinds of delay
operators by their action on a function $f^{\hat{\tau}_{j}}$. The on-board
delay operator describes delays due to on-board signal propagation and
processing and is defined on the same SCET as the function it is acting on:

$\displaystyle\textbf{D}_{x}^{\hat{\tau}_{j}}\>f^{\hat{\tau}_{j}}(\tau)$
$\displaystyle=f^{\hat{\tau}_{j}}\left(\tau-
d_{x}^{\hat{\tau}_{j}}(\tau)\right).$ (3)

$x$ is a place holder for any on-board delay, e.g., $\textbf{D}_{\text{pbs
$\leftarrow$ l}}$ denotes the optical path length from the laser to the PBS
and $\textbf{D}_{\text{dec}}$ the decimation filter group delay. The
interspacecraft delay operator is defined on a different SCET than the
function it is acting on and applies the pseudorange as delay:

$\displaystyle\textbf{D}_{ij}^{\hat{\tau}_{i}}\>f^{\hat{\tau}_{j}}(\tau)$
$\displaystyle=f^{\hat{\tau}_{j}}\left(\tau-
R_{ij}^{\hat{\tau}_{i}}(\tau)\right).$ (4)

For on-board delays that differ between carrier, PRN, and sideband signals, we
add the superscripts _car_ , _prn_ , and _sb_ , respectively. To trace the
full path of a signal from the distant SC, we need to combine the
interspacecraft delay operator for the interspacecraft signal propagation and
the SCET conversion (considered at the PBS of the receiving SC) with on-board
delay operators on both SC. The application of a delay operator to another
time-dependent delay operator results in nested delays:

$\displaystyle\textbf{D}_{x}^{\hat{\tau}_{i}}\>\textbf{D}_{ij}^{\hat{\tau}_{i}}f^{\hat{\tau}_{j}}(\tau)=f^{\hat{\tau}_{j}}\left(\tau-
d_{x}^{\hat{\tau}_{i}}(\tau)-R_{ij}^{\hat{\tau}_{i}}\left(\tau-
d_{x}^{\hat{\tau}_{i}}(\tau)\right)\right).$ (5)

For a constant delay operator $\textbf{D}_{x}$ we can define the associated
advancement operator $\textbf{A}_{x}$ acting as its inverse:

$\displaystyle\textbf{A}^{\hat{\tau}_{j}}_{x}\>f^{\hat{\tau}_{j}}(\tau)$
$\displaystyle=f^{\hat{\tau}_{j}}\left(\tau+d_{x}^{\hat{\tau}_{j}}\right),$
(6) $\displaystyle\textbf{A}_{x}\>\textbf{D}_{x}\>f^{\hat{\tau}_{j}}(\tau)$
$\displaystyle=f^{\hat{\tau}_{j}}\left(\tau-
d_{x}^{\hat{\tau}_{j}}+d_{x}^{\hat{\tau}_{j}}\right)=f^{\hat{\tau}_{j}}(\tau).$
(7)

For advancement operators associated to propagation delays, e.g., the optical
path length from the laser to the PBS, we write

$\displaystyle\textbf{D}_{\text{pbs $\leftarrow$ l}}^{-1}=\textbf{A}_{\text{l
$\leftarrow$ pbs}},$ (8)

the subscript underlines that the advancement operator undoes the signal
propagation. Below, we consider on-board delays as constant or slowly time
varying so that their associated advancement operators are well-defined.

What does the specified pseudorange definition imply for TDI in the context of
on-board delays? In [6] the pseudoranges are said to be the delays that are to
be applied in TDI in the alternative topology. To find out whether this
statement holds, we write down the ISI carrier beatnotes in the presence of
on-board delays using the above defined delay operators:

$\displaystyle\text{ISI}_{ij}^{\hat{\tau}_{i}}(\tau)=\textbf{D}_{\text{dec
$\leftarrow$ bs}}^{\text{car},\>\hat{\tau}_{i}}\Big{(}$
$\displaystyle\textbf{D}_{\text{bs $\leftarrow$
pbs}}^{\hat{\tau}_{i}}\>\textbf{D}_{ij}^{\hat{\tau}_{i}}\>\textbf{D}_{\text{pbs
$\leftarrow$ l}}^{\hat{\tau}_{j}}\Phi_{ji}^{\hat{\tau}_{j}}(\tau)$
$\displaystyle-$ $\displaystyle\textbf{D}_{\text{bs $\leftarrow$
l}}^{\hat{\tau}_{i}}\>\Phi_{ij}^{\hat{\tau}_{i}}(\tau)\Big{)}.$ (9)

$\textbf{D}_{\text{pbs $\leftarrow$ l}}$ denotes the optical path length from
the laser to the PBS (before transmission), $\textbf{D}_{\text{bs $\leftarrow$
pbs}}$ is the optical path length from the PBS to the recombining beam
splitter of the interspacecraft interferometer (ISI BS) (after reception), and
$\textbf{D}_{\text{bs $\leftarrow$ l}}$ denotes the optical path length from
the local laser to the ISI BS. These optical path lengths are in the order of
$10\text{\,}\mathrm{cm}$ to $1\text{\,}\mathrm{m}$ [15].
$\textbf{D}_{\text{dec $\leftarrow$ bs}}^{\text{car}}$ denotes the delay from
the ISI BS to the decimation filters, it differs for sideband and PRN signals.
The dominating part of $\textbf{D}_{\text{dec $\leftarrow$ bs}}^{\text{car}}$
is the group delay of the decimation filters in the order of
$1\text{\,}\mathrm{s}$. To identify the delay we need to apply in TDI, it is
convenient to split the delays in section II.1 into a common and an uncommon
delay by inserting $\textbf{D}_{\text{bs $\leftarrow$
l}}^{\hat{\tau}_{i}}\>\textbf{A}_{\text{l $\leftarrow$
bs}}^{\hat{\tau}_{i}}=\textbf{1}$ in front of the bracket:

$\displaystyle\text{ISI}_{ij}^{\hat{\tau}_{i}}(\tau)$
$\displaystyle=\textbf{C}_{i}^{\text{car},\>\hat{\tau}_{i}}\left(\textbf{U}_{ij}^{\hat{\tau}_{i}}\>\Phi_{ji}^{\hat{\tau}_{j}}(\tau)-\Phi_{ij}^{\hat{\tau}_{i}}(\tau)\right),$
(10) $\displaystyle\textbf{C}_{i}^{\text{car},\>\hat{\tau}_{i}}$
$\displaystyle=\textbf{D}_{\text{dec $\leftarrow$
bs}}^{\text{car},\>\hat{\tau}_{i}}\>\textbf{D}_{\text{bs $\leftarrow$
l}}^{\hat{\tau}_{i}},$ (11) $\displaystyle\textbf{U}_{ij}^{\hat{\tau}_{i}}$
$\displaystyle=\textbf{A}_{\text{l $\leftarrow$
bs}}^{\hat{\tau}_{i}}\>\textbf{D}_{\text{bs $\leftarrow$
pbs}}^{\hat{\tau}_{i}}\>\textbf{D}_{ij}^{\hat{\tau}_{i}}\>\textbf{D}_{\text{pbs
$\leftarrow$ l}}^{\hat{\tau}_{j}}.$ (12)

$\textbf{C}_{i}^{\text{car}}$ denotes the common delay of the local and the
distant carrier phase. $\textbf{U}_{ij}$ is the uncommon delay that only
applies to the distant carrier phase. We refer to
$\textbf{C}_{i}^{\text{car}}$ and $\textbf{U}_{ij}$ as common and uncommon
carrier delay, respectively. To see how these delays affect the carrier
beatnotes, we expand eq. 10:

$\displaystyle\text{ISI}_{ij}^{\hat{\tau}_{i}}(\tau)=$
$\displaystyle\>\Phi_{ji}^{\hat{\tau}_{j}}\left(\tau-
c_{i}^{\hat{\tau}_{i}}-u_{ij}^{\hat{\tau}_{i}}\big{(}\tau-
c_{i}^{\hat{\tau}_{i}}\big{)}\right)$ $\displaystyle-$
$\displaystyle\>\Phi_{ij}^{\hat{\tau}_{i}}\left(\tau-
c_{i}^{\hat{\tau}_{i}}\right).$ (13)

The common carrier delay causes a timestamping delay in both the laser phases
and the uncommon carrier delay (essentially the pseudorange). It can be
compensated by application of its associated advancement operator:

$\displaystyle\left(\textbf{C}_{i}^{\text{car},\>\hat{\tau}_{i}}\right)^{-1}\text{ISI}_{ij}^{\hat{\tau}_{i}}(\tau)$
$\displaystyle=\textbf{U}_{ij}^{\hat{\tau}_{i}}\>\Phi_{ji}^{\hat{\tau}_{j}}(\tau)-\Phi_{ij}^{\hat{\tau}_{i}}(\tau),$
(14)
$\displaystyle\left(\textbf{C}_{i}^{\text{car},\>\hat{\tau}_{i}}\right)^{-1}$
$\displaystyle=\left(\textbf{D}_{\text{bs $\leftarrow$
l}}^{\hat{\tau}_{i}}\right)^{-1}\left(\textbf{D}_{\text{dec $\leftarrow$
bs}}^{\text{car},\>\hat{\tau}_{i}}\right)^{-1}$
$\displaystyle=\vcentcolon\textbf{A}_{\text{l $\leftarrow$
dec}}^{\text{car},\>\hat{\tau}_{i}}.$ (15)

TDI is blind to the common carrier delay, as it equally delays the laser
phases and the pseudorange. Hence, from the perspective of TDI section II.1
and eq. 14 are equivalent. Nevertheless, the compensation of the common
carrier delay is important for the synchronization of the measurements to TCB.
We propose to calibrate $\textbf{C}_{i}^{\text{car}}$ on ground, so that
during operation it can be compensated in an initial data treatment by
application of its associated advancement operator (see eq. 14). After this
initial data treatment, the uncommon carrier delay constitutes the delay that
is to be applied in TDI in the alternative topology. It is composed of the
optical path length delay from the distant laser source to the local ISI BS
and the optical path length advancement from the ISI BS to the local laser
source. Hence, it can be thought of as the differential optical path length
from both lasers to the ISI BS. To construct the uncommon carrier delay, we
need to measure the optical path lengths laser to PBS, PBS to ISI BS, and
laser to ISI BS on ground, and we need to measure the pseudorange during
operation. The sections II.2 to II.5 cover the four pseudorange observables.
Before, we close this section with a few comments on the common carrier delay.

Parts of the common carrier delay are slowly time varying. To analyze the
origin of this time dependence we decompose $\textbf{C}_{i}^{\text{car}}$ into

$\displaystyle\textbf{C}_{i}^{\text{car}}=$
$\displaystyle\>\textbf{D}^{\text{car}}_{\text{dec}}\>\textbf{D}^{\text{car}}_{\text{dpll}}\>\textbf{D}_{\text{dpll
$\leftarrow$ abee}}\>\textbf{D}^{\text{car}}_{\text{abee}}$
$\displaystyle\>\textbf{D}_{\text{abee $\leftarrow$
qpr}}\>\textbf{D}^{\text{car}}_{\text{qpr}}\>\textbf{D}_{\text{qpr
$\leftarrow$ bs}}\>\textbf{D}_{\text{bs $\leftarrow$ l}},$ (16)

these constituents are marked green in fig. 3. The dominating contribution is
by far the decimation filter group delay
$\textbf{D}^{\text{car}}_{\text{dec}}$ in the order of $1\text{\,}\mathrm{s}$.
It is constant and predetermined by the design of the decimation filters. The
group delays of the quadrant-photo-receiver
$\textbf{D}^{\text{car}}_{\text{qpr}}$ and the analog backend
electronics222The analog backend electronics comprise analog signal
amplifiers, analog low-pass filters, and the ADC.
$\textbf{D}^{\text{car}}_{\text{abee}}$ depend amongst others on the beatnote
frequency [16]. Hence, they change over time and differ between carrier,
sideband, and PRN signals. Together with the cable delays
$\textbf{D}_{\text{abee$\leftarrow$ qpr}}$ and
$\textbf{D}_{\text{dpll$\leftarrow$ abee}}$ they can amount to
$10\text{\,}\mathrm{m}$. The DPLL delay
$\textbf{D}^{\text{car}}_{\text{dpll}}$ depends on the time-dependent beatnote
amplitude. The higher this amplitude the smaller
$\textbf{D}^{\text{car}}_{\text{dpll}}$ [2, 17]. $\textbf{D}_{\text{qpr
$\leftarrow$ bs}}$ and $\textbf{D}_{\text{bs $\leftarrow$ l}}$, for
completeness, denote the optical path lengths from the local laser to the QPR
in the order of $10\text{\,}\mathrm{cm}$ to $1\text{\,}\mathrm{m}$ [15]. We
propose to individually calibrate all constituents of
$\textbf{C}_{i}^{\text{car}}$ on ground. The time-dependent ones should be
calibrated for all combinations of the time-dependent parameters. Hence,
during operation they can be constructed with the help of the SC monitors,
which provide the corresponding parameter values, e.g., beatnote frequency and
amplitude.

###  II.2 PRN ranging (PRNR)

A set of 6 pseudo-random noise (PRN) sequences has been computed such that the
cross-correlations and the auto-correlations for nonzero delays are minimized.
These PRN codes are associated to the 6 optical links in the LISA
constellation. The PRN codes are generated according to the respective PMCs
and imprinted onto the laser beams by phase-modulating the carriers in
electro-optical modulators (EOMs). In each phasemeter, DPLLs are applied to
extract the beatnote phases. The PRN codes show up in the DPLL error signals
since the DPLL bandwidth of $10\text{\,}\mathrm{kHz}$ to
$100\text{\,}\mathrm{kHz}$ is lower than the PRN chipping rate of about
$1\text{\,}\mathrm{MHz}$. In a delay-locked loop (DLL), these error signals
are correlated with PRN codes generated according to the local SCET. This
correlation yields a pseudorange measurement, we refer to it as PRN ranging
(PRNR) [7, 8].

We now derive the PRNR observation equation carefully taking into account on-
board delays. We model the path of the PRN code from the distant SC to the
local DLL by applying delay operators to the distant SCET:

$\displaystyle\textbf{D}_{\text{dll $\leftarrow$
pbs}}^{\text{prn},\>\hat{\tau}_{i}}\>\textbf{D}_{ij}^{\hat{\tau}_{i}}\>\textbf{D}_{\text{pbs
$\leftarrow$
pmc}}^{\text{prn},\>\hat{\tau}_{j}}\>\hat{\tau}_{j}^{\hat{\tau}_{j}}(\tau).$
(17)

The two on-board delays can be decomposed into

$\displaystyle\textbf{D}_{\text{pbs $\leftarrow$ pmc}}^{\text{prn}}=\>$
$\displaystyle\textbf{D}_{\text{pbs $\leftarrow$ eom}}\>\textbf{D}_{\text{eom
$\leftarrow$ prn}}$
$\displaystyle\textbf{D}_{\text{prn}}\>\textbf{D}_{\text{prn $\leftarrow$
pmc}},$ (18) $\displaystyle\textbf{D}_{\text{dll $\leftarrow$
pbs}}^{\text{prn}}=\>$
$\displaystyle\textbf{D}_{\text{dll}}\>\textbf{D}_{\text{dpll}}^{\text{prn}}\>\textbf{D}_{\text{dpll
$\leftarrow$
abee}}\>\textbf{D}_{\text{abee}}^{\text{prn}}\>\textbf{D}_{\text{abee
$\leftarrow$ qpr}}$
$\displaystyle\textbf{D}_{\text{qpr}}^{\text{prn}}\>\textbf{D}_{\text{qpr
$\leftarrow$ bs}}\>\textbf{D}_{\text{bs $\leftarrow$ pbs}}.$ (19)

$\textbf{D}_{\text{pbs $\leftarrow$ pmc}}^{\text{prn}}$ consists of the cable
delays from the PMC to the EOM, the processing delay due to the PRN code
generation, and the optical path length from the EOM to the PBS. All these
delays are constant at the sensitive scale of PRNR, so that we do not have to
consider delay nesting in $\textbf{D}_{\text{pbs $\leftarrow$
pmc}}^{\text{prn}}$. We added the superscript _prn_ because this path is
different for the sideband signal. $\textbf{D}_{\text{dll $\leftarrow$
pbs}}^{\text{prn}}$ is explained in the next paragraph as part of the PRN
timestamping delay. At the DLL, the received PRN codes are correlated with
identical codes generated according to the local SCET. We model this
correlation as the difference between the local SCET and the delayed distant
SCET (eq. 17), and we apply $\textbf{D}^{\text{prn}}_{\text{dec}}$ to model
the group delay of the decimation filters applicable to PRN ranging:

$\displaystyle\textbf{D}^{\text{prn}}_{\text{dec}}\left(\hat{\tau}_{i}^{\hat{\tau}_{i}}(\tau)-\textbf{D}_{\text{dll
$\leftarrow$
pbs}}^{\text{prn},\>\hat{\tau}_{i}}\>\textbf{D}_{ij}^{\hat{\tau}_{i}}\>\textbf{D}_{\text{pbs
$\leftarrow$
pmc}}^{\text{prn},\>\hat{\tau}_{j}}\>\hat{\tau}_{j}^{\hat{\tau}_{j}}(\tau)\right).$
(20)

To see how the on-board delays affect the PRNR we expand eq. 20 applying eq.
2:

$\displaystyle\>\textbf{D}^{\text{prn}}_{\text{dec}}\Big{(}\hat{\tau}_{i}^{\hat{\tau}_{i}}(\tau)-\hat{\tau}_{j}^{\hat{\tau}_{j}}\big{(}\tau-d^{\hat{\tau}_{i}}_{\text{dll
$\leftarrow$ pbs}}$ $\displaystyle\hskip 91.6179pt-
R_{ij}^{\hat{\tau}_{i}}\big{(}\tau-d^{\hat{\tau}_{i}}_{\text{dll $\leftarrow$
pbs}}\big{)}$ $\displaystyle\hskip 91.6179pt-d^{\hat{\tau}_{j}}_{\text{pbs
$\leftarrow$ pmc}}\big{)}\Big{)}$ $\displaystyle=$
$\displaystyle\>\textbf{D}_{\text{dec $\leftarrow$
pbs}}^{\text{prn},\>\hat{\tau}_{i}}\>R_{ij}^{\hat{\tau}_{i}}\left(\tau\right)+O^{\text{prn}}_{ij}.$
(21)

The on-board delays cause a timestamping delay $\textbf{D}_{\text{dec
$\leftarrow$ pbs}}^{\text{prn}}$, the PRN timestamping delay, and an offset
$O^{\text{prn}}_{ij}$, the PRNR offset:

$\displaystyle\textbf{D}_{\text{dec $\leftarrow$ pbs}}^{\text{prn}}=$
$\displaystyle\>\textbf{D}^{\text{prn}}_{\text{dec}}\>\textbf{D}_{\text{dll}}\>\textbf{D}_{\text{dpll}}^{\text{prn}}\>\textbf{D}_{\text{dpll
$\leftarrow$ abee}}$
$\displaystyle\>\textbf{D}_{\text{abee}}^{\text{prn}}\>\textbf{D}_{\text{abee
$\leftarrow$ qpr}}\>\textbf{D}_{\text{qpr}}^{\text{prn}}$
$\displaystyle\>\textbf{D}_{\text{qpr $\leftarrow$ bs}}\>\textbf{D}_{\text{bs
$\leftarrow$ pbs}},$ (22) $\displaystyle O_{ij}^{\text{prn}}=$
$\displaystyle\>d^{\hat{\tau}_{i}}_{\text{dll $\leftarrow$
pbs}}+d^{\hat{\tau}_{j}}_{\text{pbs $\leftarrow$ pmc}}.$ (23)

The PRN timestamping delay has similar constituents as the common carrier
delay, they are marked pink in fig. 3. However, most of them are frequency or
amplitude dependent. Therefore, they differ between carrier and PRN signals.
As for the common carrier delay, we propose to individually calibrate all
constituents of the PRN timestamping delay on ground before mission start.
Hence, during operation $\textbf{D}_{\text{dec $\leftarrow$
pbs}}^{\text{prn}}$ can be compensated in an initial data treatment by
application of its associated advancement operator $\textbf{A}_{\text{pbs
$\leftarrow$ dec}}^{\text{prn}}$. After that, the PRNR observation equation
including ranging noise and PRN ambiguity can be written as:

$\displaystyle\textbf{A}_{\text{pbs $\leftarrow$
dec}}^{\text{prn},\>\hat{\tau}_{i}}\>\text{PRNR}_{ij}^{\hat{\tau}_{i}}(\tau)=$
$\displaystyle\>R_{ij}^{\hat{\tau}_{i}}(\tau)+O^{\text{prn}}_{ij}+N^{\text{prn}}_{ij}(\tau)$
$\displaystyle-$ $\displaystyle\>a^{\text{prn}}_{ij}(\tau)\cdot l.$ (24)

$l$ denotes the finite PRN code length. We use $400\text{\,}\mathrm{km}$ as a
placeholder, the final value has not been decided. The finite PRN code length
leads to an ambiguity, $a^{\text{prn}}_{ij}$ denote the associated ambiguity
integers [8]. $N^{\text{prn}}_{ij}$ is the white ranging noise with an RMS
amplitude of about $\mathrm{1}\mathrm{m}$ at $4\text{\,}\mathrm{Hz}$. This
ranging noise is mainly due to shot noise and PRN code interference [9]. The
PRNR offset $O^{\text{prn}}_{ij}$ involves contributions on the emitter and on
the receiver side (see eq. 23), they are marked light blue in fig. 3. It can
amount to $10\text{\,}\mathrm{m}$ and more [17, 18]. Similar to the common
carrier and the PRN timestamping delay, we propose to calibrate the PRNR
offset on ground, so that it can be subtracted in an initial data treatment.

###  II.3 Sideband ranging (SBR)

For the purpose of in-band clock noise reduction in the INReP, a clock noise
transfer between the SC is implemented [9]: the $80\text{\,}\mathrm{MHz}$ PMC
signals are up-converted to $\nu^{\text{m}}_{l}=$2.400\text{\,}\mathrm{GHz}$$
and $\nu^{\text{m}}_{r}=$2.401\text{\,}\mathrm{GHz}$$ for left and right-
handed MOSAs, respectively (see fig. 2 for the definition of left and right-
handed MOSAs). The EOMs phase-modulate the carriers with the up-converted PMC
signals, thereby creating clock sidebands.333We focus on the first order upper
clock sidebands, because the lower sidebands contain almost the same
information. We show below that the beatnotes between these clock sidebands
constitute a pseudorange observable.

Considering on-board delays, the difference between carrier and sideband
beatnotes can be written as

ISI
${}_{ij}^{\hat{\tau}_{i}}(\tau)-\text{ISI}_{\text{sb},\>ij}^{\hat{\tau}_{i}}(\tau)=-\>\textbf{D}^{\text{sb},\>\hat{\tau}_{i}}_{\text{dec
$\leftarrow$ bs}}$ $\displaystyle\Big{\\{}$
$\displaystyle\textbf{D}^{\hat{\tau}_{i}}_{\text{bs $\leftarrow$
pbs}}\>\textbf{D}^{\hat{\tau}_{i}}_{ij}\left(\textbf{D}^{\text{sb},\>\hat{\tau}_{j}}_{\text{pbs
$\leftarrow$
pmc}}\>\nu^{\text{m}}_{ji}\>\hat{\tau}_{j}^{\hat{\tau}_{j}}(\tau)+\nu^{\text{m}}_{ji}\>M^{\hat{\tau}_{j}}_{ji}(\tau)\right)$
$\displaystyle-\Big{(}$
$\displaystyle\textbf{D}^{\text{sb},\>\hat{\tau}_{i}}_{\text{bs $\leftarrow$
pmc}}\>\nu^{\text{m}}_{ij}\>\hat{\tau}_{i}^{\hat{\tau}_{i}}(\tau)+\nu^{\text{m}}_{ij}\>M^{\hat{\tau}_{i}}_{ij}(\tau)\Big{)}\Big{\\}}.$
(25)

$\textbf{D}^{\text{sb}}_{\text{pbs $\leftarrow$ pmc}}$ and
$\textbf{D}^{\text{sb}}_{\text{bs $\leftarrow$ pmc}}$ are the delay operators
associated to the paths from the PMC to the PBS and to the ISI BS,
respectively. They can be decomposed into:

$\displaystyle\textbf{D}^{\text{sb}}_{\text{(p)bs $\leftarrow$ pmc}}=$
$\displaystyle\>\textbf{D}_{\text{(p)bs $\leftarrow$
eom}}\>\textbf{D}_{\text{eom $\leftarrow$ pmc}}\>\textbf{D}_{\text{up}},$ (26)

$\textbf{D}_{\text{up}}$ is the up-conversion delay due to phase-locking a
$2.40(1)\text{\,}\mathrm{GHz}$ oscillator to the the $80\text{\,}\mathrm{MHz}$
PMC signal, $\textbf{D}_{\text{eom $\leftarrow$ pmc}}$ is the cable delay from
the PMC to the EOM. $\nu^{\text{m}}_{ij}$ is the up-converted USO frequency
associated to $\text{MOSA}_{ij}$. Since section II.3 is expressed in the SCET,
all clock imperfections are included in
$\hat{\tau}_{i}^{\hat{\tau}_{i}}(\tau)$. The modulation noise
$M^{\hat{\tau}_{i}}_{ij}$ contains any additional jitter collected on the path
$\textbf{D}^{\text{sb}}_{\text{(p)bs $\leftarrow$ pmc}}$, e.g., due to the
electrical frequency up-converters. The amplitude spectral densities (ASDs) of
the modulation noise for left and right-handed MOSAs are specified to be below
[6, 19]

$\displaystyle\sqrt{S_{M_{l}}(f)}$ $\displaystyle=2.5\times
10^{-6}$\mathrm{m}\text{\,}{\mathrm{Hz}}^{-0.5}$\left(\frac{f}{$\mathrm{H}\mathrm{z}$}\right)^{-2/3},$
(27) $\displaystyle\sqrt{S_{M_{r}}(f)}$ $\displaystyle=2.5\times
10^{-5}$\mathrm{m}\text{\,}{\mathrm{Hz}}^{-0.5}$\left(\frac{f}{$\mathrm{H}\mathrm{z}$}\right)^{-2/3}.$
(28)

The modulation noise on left-handed MOSAs is one order of magnitude lower,
because the pilot tone used for the ADC jitter correction, hence being the
ultimate phase reference, is derived from the $2.400\text{\,}\mathrm{GHz}$
clock signal.

To derive a pseudorange observation equation from the sideband beatnote we
expand section II.3 using eq. 2. We apply the advancement operator
$\textbf{A}^{\text{sb}}_{\text{pbs $\leftarrow$ dec}}$ to avoid nested delays
in the pseudorange:

$\displaystyle\hskip
17.07182pt\textbf{A}^{\text{sb},\>\hat{\tau}_{i}}_{\text{pbs $\leftarrow$
dec}}\left(\text{ISI}_{ij}^{\hat{\tau}_{i}}(\tau)-\text{ISI}_{\text{sb},\>ij}^{\hat{\tau}_{i}}(\tau)\right)$
$\displaystyle=$
$\displaystyle\>\nu^{\text{m}}_{ij}\>\textbf{A}^{\hat{\tau}_{i}}_{\text{pbs
$\leftarrow$ bs}}\left(\textbf{D}^{\text{sb},\>\hat{\tau}_{i}}_{\text{bs
$\leftarrow$
pmc}}\>\hat{\tau}_{i}^{\hat{\tau}_{i}}(\tau)+M^{\hat{\tau}_{i}}_{ij}\right)$
$\displaystyle-$
$\displaystyle\>\nu^{\text{m}}_{ji}\>\textbf{D}^{\hat{\tau}_{i}}_{ij}\left(\textbf{D}^{\text{sb},\>\hat{\tau}_{j}}_{\text{pbs
$\leftarrow$
pmc}}\>\hat{\tau}_{j}^{\hat{\tau}_{j}}(\tau)+M^{\hat{\tau}_{j}}_{ji}(\tau)\right)$
$\displaystyle=$
$\displaystyle\>\left(\nu^{\text{m}}_{ij}-\nu^{\text{m}}_{ji}\right)\tau+\nu^{\text{m}}_{ji}\>R_{ij}^{\hat{\tau}_{i}}(\tau)$
$\displaystyle+$ $\displaystyle\>\nu^{\text{m}}_{ji}\cdot
d^{\hat{\tau}_{j}}_{\text{pbs $\leftarrow$
pmc}}-\nu^{\text{m}}_{ij}\cdot\left(d^{\hat{\tau}_{i}}_{\text{bs $\leftarrow$
pmc}}-d^{\hat{\tau}_{i}}_{\text{pbs $\leftarrow$ bs}}\right)$ $\displaystyle+$
$\displaystyle\>\nu^{\text{m}}_{ij}\>\textbf{A}^{\hat{\tau}_{i}}_{\text{pbs
$\leftarrow$
bs}}\>M^{\hat{\tau}_{i}}_{ij}(\tau)-\nu^{\text{m}}_{ji}\>\textbf{D}^{\hat{\tau}_{i}}_{ij}M^{\hat{\tau}_{j}}_{ji}(\tau).$
(29)

We subtract the $1\text{\,}\mathrm{MHz}$ ramp and then refer to section II.3
as sideband ranging (SBR). Taking into account that the SBR phase is defined
up to a cycle, the SBR can be written as

$\displaystyle\text{SBR}_{ij}^{\hat{\tau}_{i}}(\tau)$
$\displaystyle=\textbf{A}^{\text{sb},\>\hat{\tau}_{i}}_{\text{pbs $\leftarrow$
dec}}\left(\text{ISI}_{ij}^{\hat{\tau}_{i}}(\tau)-\text{ISI}_{\text{sb},\>ij}^{\hat{\tau}_{i}}(\tau)\right)\pm$1\text{\,}\mathrm{MHz}$\>\tau$
$\displaystyle=\nu^{\text{m}}_{ji}\>R_{ij}^{\hat{\tau}_{i}}(\tau)+O^{\text{sb}}_{ij}+N^{\text{sb}}_{ij}(\tau)-a^{\text{sb}}_{ij}(\tau).$
(30)

$a^{\text{sb}}_{ij}$ denote the SBR ambiguity integers. Expressed as length,
the SBR ambiguity is $12.5\text{\,}\mathrm{cm}$ corresponding to the
wavelength of the $\mathrm{GHz}$ sidebands. The SBR offset

$\displaystyle O^{\text{sb}}_{ij}=$ $\displaystyle\>\nu^{\text{m}}_{ji}\cdot
d^{\hat{\tau}_{j}}_{\text{pbs $\leftarrow$ pmc}}$ $\displaystyle-$
$\displaystyle\>\nu^{\text{m}}_{ij}\cdot\left(d^{\hat{\tau}_{i}}_{\text{bs
$\leftarrow$ pmc}}-d^{\hat{\tau}_{i}}_{\text{pbs $\leftarrow$ bs}}\right)$
(31)

can be thought of as the differential phase accumulation of local and distant
PMC signals on their paths to the respective PBSs. Similar to the PRNR offset
and the various delays, the SBR offset should be measured on ground.
$N^{\text{sb}}_{ij}$ denotes the appearance of the modulation noise in the
SBR:

$\displaystyle N^{\text{sb}}_{ij}(\tau)=$
$\displaystyle\>\nu^{\text{m}}_{ij}\>\textbf{A}^{\hat{\tau}_{i}}_{\text{pbs
$\leftarrow$
bs}}\>M^{\hat{\tau}_{i}}_{ij}(\tau)-\nu^{\text{m}}_{ji}\>\textbf{D}^{\hat{\tau}_{i}}_{ij}M^{\hat{\tau}_{j}}_{ji}(\tau).$
(32)

This is a combination of left and right-handed modulation noise, their RMS
amplitudes are $2.9\times 10^{-5}\>$\mathrm{m}$$ and 2.9 $\times
10^{-4}\>$\mathrm{m}$$, respectively. As shown in [6], it is possible to
combine carrier and sideband beatnotes from the RFI to form measurements of
the dominating right-handed modulation noise, which can, thus, be subtracted
from the SBRs (see appendix B).

The advancement operator $\textbf{A}_{\text{pbs $\leftarrow$
dec}}^{\text{sb}}$ (see section II.3) is associated to the delay operator
$\textbf{D}_{\text{dec $\leftarrow$ pbs}}^{\text{sb}}$, to which we refer as
sideband timestamping delay. The sideband timestamping delay can be decomposed
into:

$\displaystyle\textbf{D}_{\text{dec $\leftarrow$ pbs}}^{\text{sb}}=$
$\displaystyle\>\textbf{D}^{\text{sb}}_{\text{dec}}\>\textbf{D}^{\text{sb}}_{\text{dpll}}\>\textbf{D}_{\text{dpll
$\leftarrow$ abee}}\>\textbf{D}^{\text{sb}}_{\text{abee}}$
$\displaystyle\>\textbf{D}_{\text{abee $\leftarrow$
qpr}}\>\textbf{D}^{\text{sb}}_{\text{qpr}}\>\textbf{D}_{\text{qpr $\leftarrow$
bs}},$ (33)

these constituents are marked dark yellow in fig. 3. As for the common carrier
and the PRN timestamping delay, we propose to individually calibrate all its
constituents on ground. The sideband timestamping delay can then be
compensated in an initial data treatment by application of its associated
advancement operator (see section II.3).

In reality, the beatnotes are expected to be delivered not in phase, but in
frequency with occasional phase anchor points. Therefore, we consider the
derivative of section II.3, we refer to it as sideband range rate
($\dot{\text{SBR}}$):

$\displaystyle\dot{\text{SBR}}_{ij}^{\hat{\tau}_{i}}(\tau)$
$\displaystyle=\nu^{\text{m}}_{ji}\>\dot{R}_{ij}^{\hat{\tau}_{i}}(\tau)+\dot{N}^{\text{sb}}_{ij}(\tau).$
(34)

The sideband range rates are an offset-free and unambiguous measurement of the
pseudorange time derivatives. Phase anchor points enable their integration, so
that we recover section II.3.

###  II.4 Time-delay interferometric ranging (TDIR)

TDI builds combinations of delayed ISI and RFI carrier beatnotes to virtually
form equal-arm interferometers, in which laser frequency noise is suppressed.
In the alternative TDI topology, the corresponding delays are given by the
pseudoranges in combination with the small optical path lengths between laser,
PBS, and ISI BS (see the uncommon carrier delay eq. 12). Time delay
interferometric ranging (TDIR) turns this approach upside-down: it minimizes
the power integral of the laser frequency noise in the TDI combinations by
varying the delays that are applied to the beatnotes [12]. When doing this
before clock synchronization to TCB, i.e., with the beatnotes sampled
according to the respective SCETs, the uncommon delays show up at the very
minimum of that integral. Thus, TDIR constitutes a pseudorange observable.

Below, we consider TDI in frequency [14]. We introduce the Doppler-delay
operator, which can be considered as the time derivative of the
interspacecraft delay operator (see eq. 4):

$\displaystyle\dot{\textbf{D}}^{\hat{\tau}_{i}}_{ij}\>f^{\hat{\tau}_{j}}(\tau)=\left(1-\dot{R}_{ij}^{\hat{\tau}_{i}}(\tau)\right)\cdot
f^{\hat{\tau}_{j}}\left(\tau-R_{ij}^{\hat{\tau}_{i}}(\tau)\right).$ (35)

We use the shorthand notation

$\displaystyle\dot{\textbf{D}}^{\hat{\tau}_{i}}_{ijk}=\dot{\textbf{D}}^{\hat{\tau}_{i}}_{ij}\>\dot{\textbf{D}}^{\hat{\tau}_{j}}_{jk}$
(36)

to indicate chained interspacecraft Doppler-delay operators. In this paper we
neglect on-board delays in the RFI beatnotes. We start our consideration of
TDIR from the intermediary TDI variables $\eta_{ij}$. These are combinations
of the ISI and RFI carrier beatnotes to eliminate the laser frequency noise
contributions of right-handed lasers. In terms of the $\eta_{ij}$ the second-
generation TDI Michelson variables can be expressed as [20]

$\displaystyle
X_{2}^{\hat{\tau}_{1}}=\left(1-\dot{\textbf{D}}^{\hat{\tau}_{1}}_{121}-\dot{\textbf{D}}^{\hat{\tau}_{1}}_{12131}+\dot{\textbf{D}}^{\hat{\tau}_{1}}_{1312121}\right)\left(\eta^{\hat{\tau}_{1}}_{13}-\dot{\textbf{D}}^{\hat{\tau}_{1}}_{13}\eta^{\hat{\tau}_{3}}_{31}\right)$
$\displaystyle-\left(1-\dot{\textbf{D}}^{\hat{\tau}_{1}}_{131}-\dot{\textbf{D}}^{\hat{\tau}_{1}}_{13121}+\dot{\textbf{D}}^{\hat{\tau}_{1}}_{1213131}\right)\left(\eta^{\hat{\tau}_{1}}_{12}-\dot{\textbf{D}}^{\hat{\tau}_{1}}_{12}\eta^{\hat{\tau}_{2}}_{21}\right)$
(37)

$Y_{2}^{\hat{\tau}_{2}}(\tau)$ and $Z_{2}^{\hat{\tau}_{3}}(\tau)$ are obtained
by cyclic permutation of the indices. For later reference, we also state the
first generation TDI Michelson variables:

$\displaystyle X_{1}^{\hat{\tau}_{1}}=$
$\displaystyle\>(1-\dot{\textbf{D}}^{\hat{\tau}_{1}}_{121})\left(\eta^{\hat{\tau}_{1}}_{13}-\dot{\textbf{D}}^{\hat{\tau}_{1}}_{13}\eta^{\hat{\tau}_{3}}_{31}\right)$
$\displaystyle-$
$\displaystyle\>(1-\dot{\textbf{D}}^{\hat{\tau}_{1}}_{131})\left(\eta^{\hat{\tau}_{1}}_{12}-\dot{\textbf{D}}^{\hat{\tau}_{1}}_{12}\eta^{\hat{\tau}_{2}}_{21}\right).$
(38)

In the framework of TDIR, the delays applied in TDI are parameterized by a
model, e.g., by a polynomial model. We minimize the power integral of the TDI
combinations by varying the model parameters. TDIR attempts to minimize the
in-band laser frequency noise residual. Therefore, we apply a band-pass filter
to first remove other contributions appearing out-of-band, i.e., slow drifts
and contributions above 1Hz that are dominated by aliasing and interpolation
errors. The TDIR pseudorange observables for the second generation TDI
Michelson variables can then be expressed as

$\displaystyle\text{TDIR}_{ij}^{\hat{\tau}_{i}}$
$\displaystyle=\min_{\Theta}\frac{1}{T}\int_{\frac{1}{T}}^{T}\left[\tilde{X}_{2}^{\hat{\tau}_{1}}\right]^{2}+\left[\tilde{Y}_{2}^{\hat{\tau}_{2}}\right]^{2}+\left[\tilde{Z}_{2}^{\hat{\tau}_{3}}\right]^{2}\>\text{d}t,$
(39)

$\Theta$ denotes the parameters of the delay model, the tilde indicates the
filtered TDI combinations.

The TDIR accuracy, we denote it by $\sigma^{\text{tdir}}$, increases with the
integration time $T$ (length of telemetry dataset). It is in the order of
[12]:

$\displaystyle\sigma^{\text{tdir}}(T)\propto$10\text{\,}\mathrm{cm}$\>\sqrt{\frac{$\mathrm{d}$}{T}},$
(40)

where $\mathrm{d}$ stands for day.

###  II.5 Ground-observation based ranging (GOR)

The mission operation center (MOC) provides orbit determinations (ODs) via the
ESA tracking stations and MOC time correlations (MOC-TCs). When combined
properly, these two on-ground measurements form a pseudorange observable
referred to as ground-observation based ranging (GOR). It has an uncertainty
of about $50\text{\,}\mathrm{km}$ due to uncertainties in both the OD and the
MOC-TC. Yet, it yields valuable information. It is unambiguous, hence it
allows to resolve the PRNR ambiguities.

The OD yields information about the absolute positions and velocities of the
three SC. New orbit determinations are published every few days. For the
position and velocity measurements in the line of sight, radial (with respect
to the sun) and cross-track direction conservative estimations by ESA state
the uncertainties as $2\text{\,}\mathrm{km}$ and
$4\text{\,}\mathrm{mm}\text{\,}{\mathrm{s}}^{-1}$, $10\text{\,}\mathrm{km}$
and $4\text{\,}\mathrm{mm}\text{\,}{\mathrm{s}}^{-1}$,
$50\text{\,}\mathrm{km}$ and
$5\text{\,}\mathrm{cm}\text{\,}{\mathrm{s}}^{-1}$, respectively [21]. The MOC-
TC is a measurement of the SCET desynchronization from TCB. It is determined
during the telemetry contacts via a comparison of the SCET associated to the
emission of a telemetry packet and the TCB of its reception on Earth taking
into account the down link delay. We expect the accuracy of the MOC-TC to be
better than $0.1\text{\,}\mathrm{ms}$ (corresponds to
$30\text{\,}\mathrm{km}$). This uncertainty is due to unexact knowledge of the
SC-to-ground-station separation, as well as inaccuracies in the time tagging
process on board and on ground.

As shown in appendix A, the pseudoranges can be expressed in TCB as functions
of the reception time:

$\displaystyle R_{ij}^{t}(t)$
$\displaystyle=(1+\delta\dot{\hat{\tau}}_{j}^{t}(t))\cdot
d_{ij}^{t}(t)+\delta\hat{\tau}_{ij}^{t}(t).$ (41)

$d_{ij}^{t}$ denotes the light travel time from SC $j$ to SC $i$,
$\delta\hat{\tau}_{ij}^{t}$ the offset between the involved SCETs, and
$\delta\dot{\hat{\tau}}_{j}^{t}$ the SCET drift of the emitting SC with
respect to TCB. The light travel times can be expressed in terms of the ODs
[22]:

$\displaystyle d^{t}_{\text{od},\>ij}(t)$
$\displaystyle=\frac{1}{c}L^{t}_{ij}(t)+\frac{1}{c^{2}}\>\vec{L}^{t}_{ij}(t)\cdot\vec{v}_{j}^{t}(t)+O(c^{-3}),$
(42) $\displaystyle\vec{L}_{ij}$
$\displaystyle=\vec{r}_{i}-\vec{r}_{j},\>L_{ij}=|\vec{L}_{ij}|,$ (43)

$\vec{r}_{i}$ denoting the position of the receiving SC, $\vec{r}_{j}$ and
$\vec{v}_{j}$ the position and the velocity of the emitting one, respectively.
The terms of order $O(c^{-3})$ contribute to the light travel time at the
order of $10\text{\,}\mathrm{m}$ and are therefore negligible compared to the
large uncertainties of the orbit determination. Combining the light travel
times obtained this way with the MOC-TC allows to write the GOR as

$\displaystyle\text{GOR}_{ij}^{t}(t)$
$\displaystyle=d^{t}_{\text{od},\>ij}(t)+\delta\hat{\tau}^{t}_{\text{tc},\>i}(t)-\delta\hat{\tau}^{t}_{\text{tc},\>j}(t)+N^{\text{gor}}_{ij}(t).$
(44)

$\delta\hat{\tau}^{t}_{\text{tc},\>i}$ denotes the MOC-TC of SC $i$ and
$N_{\text{gor}}^{t}\sim$50\text{\,}\mathrm{km}$$ the GOR uncertainty. Note
that OD and MOC-TC, and hence also the GOR, are given in TCB, while all other
pseudorange observables are sampled in the respective SCETs. This
desynchronization is negligible: the desynchronization can amount up to
$10\text{\,}\mathrm{s}$ after the ten year mission time, the pseudoranges
drift with $\mathrm{1}\mathrm{0}$ to
$100\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$ (see central plot in fig.
5). Hence, neglecting the desynchronization leads to an error in the order of
$\mathrm{1}\mathrm{0}\mathrm{0}$ to $1000\text{\,}\mathrm{m}$, which is
negligible compared to the large GOR uncertainty.

##  III Ranging sensor fusion

Figure 4: We illustrate the three-stage ranging sensor fusion. Processing
elements are drawn with a black frame. In the upper part we show the initial
data treatment. Products of the on-ground calibration (the various delays and
offsets) are drawn green. Raw datasets are drawn yellow, after the initial
data treatment we add a green frame. In the central part we show the core
ranging processing. Its output, the pseudoranges, are drawn with a blue frame.
In the right box we show how the pseudoranges are combined with the small
optical path length to form the uncommon delays (the delays for TDI). In the
lower part we show simultaneous crosschecks of PRNR ambiguity, PRNR offset,
and SBR ambiguity. Products of these crosschecks are drawn with a red frame.
We do not consider the on-board delays of the RFI and TMI beatnotes.

To combine the four pseudorange observables, we propose a three-stage ranging
sensor fusion consisting of an initial data treatment, a ranging processing,
and crosschecks. The ranging processing (central part of fig. 4) refers to the
ranging-related routines, which need to run continuously during operation.
These are the PRNR unwrapping, and the reduction of ranging and right-handed
modulation noise. Simultaneously, the PRNR ambiguities and offsets are
steadily crosschecked using TDIR and GOR (lower part of fig. 4). Both ranging
processing and crosschecks rely on a preceding initial data treatment (upper
part of fig. 4), in which the various delays and offsets are compensated for.
Ranging processing and crosschecks can be categorized into four parts
demonstrated below: PRNR ambiguity, noise, PRNR offset, and SBR ambiguity.

###  III.1 PRNR ambiguity

As part of the ranging processing, the PRNR needs to be steadily unwrapped:
due to the finite PRN code length, the PRNR jumps back to
$0\text{\,}\mathrm{km}$ when crossing $400\text{\,}\mathrm{km}$ and vice versa
(see upper plot in fig. 5). These jumps are unphysical but easy to track and
to remove. Apart from that, the PRNR ambiguities need to be crosschecked
regularly. For that purpose we propose two independent methods below.

The combination of PRNR and GOR enables an identification of the PRNR
ambiguity integers $a^{\text{prn}}_{ij}$:

$\displaystyle\text{GOR}_{ij}^{t}(t)-\text{PRNR}_{ij}^{\hat{\tau}_{i}}(\tau)=N^{\text{gor}}_{ij}+a^{\text{prn}}_{ij}(\tau)\cdot$400\text{\,}\mathrm{km}$$
$\displaystyle\hskip
56.9055pt+\underbrace{R_{ij}^{t}(t)-R_{ij}^{\hat{\tau}_{i}}(\tau)-O^{\text{prn}}_{ij}-N^{\text{prn}}_{ij}(\tau)}_{\text{negligible}},$
(45) $\displaystyle
a^{\text{prn}}_{ij}(\tau)=\text{round}\left[\frac{\text{GOR}_{ij}^{t}(t)-\text{PRNR}_{ij}^{\hat{\tau}_{i}}(\tau)}{$400\text{\,}\mathrm{km}$}\right],$
(46)

$400\text{\,}\mathrm{km}$ is the value we assumed for the PRN code length.
However, this procedure only succeeds if $|N^{\text{gor}}_{ij}|$ does not
exceed the PRN code’s half length, i.e., $200\text{\,}\mathrm{km}$. Otherwise,
a wrong value for the associated PRN ambiguity integer is selected resulting
in an estimation error of $400\text{\,}\mathrm{km}$ in the corresponding link.
Note that $\text{GOR}_{ij}^{t}(t)$ and
$\text{PRNR}_{ij}^{\hat{\tau}_{i}}(\tau)$ are sampled according to different
time frames, but this desynchronization is negligible considering the low
accuracy that needs to be reached here (see section II.5).

TDIR constitutes an unambiguous pseudorange observable too. It can be applied
as an independent crosscheck of the PRNR ambiguities. We linearly detrend the
ISI, RFI, and TMI beatnotes. We then form the first-generation TDI Michelson
variables (see section II.4) assuming constant delays. It is not necessary to
apply second-generation TDI, the first-generation already accomplishes the
task (see fig. 8). The pseudoranges are actually drifting by
$\mathrm{1}\mathrm{0}$ to $100\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$
mainly due to differential USO frequency offsets (see central plot in fig. 5).
Therefore, we choose a short integration time (we use
$150\text{\,}\mathrm{s}$), otherwise the constant delay model is not
sufficient. We use the GOR for the initial delay values of the TDIR estimator.
The TDIR pseudorange estimates can then be used to crosscheck the PRNR
ambiguity integers:

$\displaystyle
a^{\text{prn}}_{ij}(\tau)=\text{round}\left[\frac{\text{TDIR}_{ij}^{\hat{\tau}_{i}}(\tau)-\text{PRNR}_{ij}^{\hat{\tau}_{i}}(\tau)}{$400\text{\,}\mathrm{km}$}\right].$
(47)

###  III.2 Noise reduction

For the ranging noise reduction in the ranging processing, we propose to
combine PRNR and sideband range rates in a linear Kalman filter (KF). The
conventional KF requires all measurements to be sampled according to one
overall time grid. However, in LISA each SC involves its own SCET. We
circumvent this difficulty by splitting up the system and build one KF per SC.
Each KF only processes the measurements taken on its associated SC, so that
the individual SCETs serve as time-grids.

The state vector of the KF belonging to SC 1 and its associated linear system
model can be expressed as

$\displaystyle x^{\hat{\tau}_{1}}$
$\displaystyle=(R^{\hat{\tau}_{1}}_{12},\>R^{\hat{\tau}_{1}}_{13},\>\dot{R}^{\hat{\tau}_{1}}_{12},\>\dot{R}^{\hat{\tau}_{1}}_{13},\>\ddot{R}^{\hat{\tau}_{1}}_{12},\>\ddot{R}^{\hat{\tau}_{1}}_{13})^{\intercal},$
(48) $\displaystyle x^{\hat{\tau}_{1}}_{k+1}$
$\displaystyle=\begin{pmatrix}1&0&\Delta t&0&\frac{\Delta t^{2}}{2}&0\\\
0&1&0&\Delta t&0&\frac{\Delta t^{2}}{2}\\\ 0&0&1&0&\Delta t&0\\\
0&0&0&1&0&\Delta t\\\ 0&0&0&0&1&0\\\ 0&0&0&0&0&1\end{pmatrix}\cdot
x^{\hat{\tau}_{1}}_{k}+w^{\hat{\tau}_{1}}_{k},$ (49)

$k$ being a discrete time index. Eq. 49 describes the time evolution of the
state vector from $k$ to $k+1$. $w^{\hat{\tau}_{1}}_{k}$ denotes the process
noise vector, its covariance matrix is given by

$\displaystyle\text{E}\left[w_{k}\cdot w_{l}^{\text{T}}\right]$
$\displaystyle=\delta_{k,\>l}\>W,$ (50) $\displaystyle W$
$\displaystyle=\text{diag}\Big{(}$ $\displaystyle 0,\>0,\>0,\>0,$
$\displaystyle
10^{-15}${\mathrm{s}}^{-1}$,\>10^{-15}${\mathrm{s}}^{-1}$\Big{)}^{2}.$ (51)

$\delta_{k,\>l}$ denotes the Kronecker delta. Hence, eq. 50 indicates that
each component of $w^{\hat{\tau}_{1}}_{k}$ is a white random process. The
process noise covariance matrix we used in our implementation is given in eq.
51. The measurement vector and the associated observation model are given by

$\displaystyle y^{\hat{\tau}_{1}}$
$\displaystyle=(\text{PRNR}^{\hat{\tau}_{1}}_{12},\>\text{PRNR}^{\hat{\tau}_{1}}_{13},\>\dot{\text{SBR}}^{\hat{\tau}_{1}}_{12},\>\dot{\text{SBR}}^{\hat{\tau}_{1}}_{13})^{\intercal},$
(52) $\displaystyle y^{\hat{\tau}_{1}}_{k}$
$\displaystyle=\begin{pmatrix}1&0&0&0&0&0\\\ 0&1&0&0&0&0\\\
0&0&$2.401\text{\,}\mathrm{GHz}$&0&0&0\\\
0&0&0&$2.400\text{\,}\mathrm{GHz}$&0&0\end{pmatrix}\cdot
x^{\hat{\tau}_{1}}_{k}+v^{\hat{\tau}_{1}}_{k}.$ (53)

Eq. 53 relates the measurement vector to the state vector.
$v^{\hat{\tau}_{1}}_{k}$ denotes the measurement noise vector, its covariance
matrix is given by

$\displaystyle\text{E}\left[v_{k}\cdot v_{l}^{\text{T}}\right]$
$\displaystyle=\delta_{k,\>l}\>V,$ (54) $\displaystyle V$
$\displaystyle=\text{diag}\Big{(}$ $\displaystyle 3\cdot
10^{-9}$\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$,\>3\cdot
10^{-9}$\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$,$ $\displaystyle 5.2\cdot
10^{-13},\>\>5.2\cdot 10^{-13}\Big{)}^{2}.$ (55)

The measurement noise covariance matrix we used in our implementation is given
in eq. 55. The diagonal entries denote the variances of the respective
measurements. We assume the measurements to be uncorrelated, so that the off-
diagonal terms are zero. The KFs for SC 2 and SC 3 are defined accordingly. In
this manner, we remove the ranging noise and obtain estimates for the six
pseudoranges and their time derivatives.

These pseudorange estimates are dominated by the right-handed modulation
noise, which is one order of magnitude higher than the left-handed one. As
pointed out in [6], the right-handed modulation noise can be subtracted (see
appendix B): we combine the RFI measurements to form the $\Delta M_{i}$, which
are measurements of the right-handed modulation noise on SC $i$ (see eq. 75).
For right-handed MOSAs, the local right-handed modulation noise enters the
sideband range rates and we just need to subtract the local $\Delta M_{i}$
(see eq. 76b). For left-handed MOSAs the Doppler-delayed right-handed
modulation noise from the distant SC appears in the sideband range rates. Here
we need to apply the Kalman filter estimates for the pseudoranges and their
time derivatives to form the Doppler-delayed distant $\Delta M_{i}$, which
then can be subtracted (see eq. 76a). We then process the three KFs again,
this time with the corrected sideband range rates. Nowe they are limited by
left-handed modulation noise, so that the respective noise levels are lower.
Therefore, we need to adjust the measurement noise covariance matrix for the
second run of the KFs:

$\displaystyle V_{\text{ cor}}$ $\displaystyle=\text{diag}\Big{(}$
$\displaystyle 3\cdot 10^{-9}$\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$,\>3\cdot
10^{-9}$\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$,$ $\displaystyle 7.4\cdot
10^{-14},\>\>7.4\cdot 10^{-14}\Big{)}^{2}.$ (56)

In this way we obtain estimates for the pseudoranges and their time
derivatives, which are limited by the left-handed modulation noise.

###  III.3 PRNR offset

The PRNR offset is calibrated on ground before mission start. During
operation, it is constructed with the help of SC monitors and subtracted in
the initial data treatment.

TDIR can be used as a crosscheck for residual PRNR offsets, as it is sensitive
to offsets in the delays. To obtain optimal performance we choose the second-
generation TDI Michelson variables to be ultimately limited by secondary
noises. In-band clock noise is sufficiently suppressed, since we operate on
beatnotes in total frequency and make use of the in-band ranging information
provided by the preceding noise reduction step. Accordingly, the offset delay
model is parameterized by

$\displaystyle
d_{ij}^{\hat{\tau}_{i}}(\tau)=\hat{R}_{ij}^{\hat{\tau}_{i}}(\tau)-O_{ij},$
(57)

$\hat{R}_{ij}^{\hat{\tau}_{i}}$ denote the pseudorange estimates after noise
reduction, $O_{ij}$ are the 6 offset parameters. As discussed in section II.4,
computing TDI in total frequency units generally results in a variable with
residual trends. Those trends need to be removed prior to calculation of the
TDIR integral to be sensitive to residual laser noise in band. This is
achieved by an appropriate band-pass filter with a pass-band from
$0.1\text{\,}\mathrm{Hz}1\text{\,}\mathrm{Hz}$. The TDIR integral then reads

$\displaystyle\hat{O}_{ij}=\operatorname*{arg\,min}_{O_{ij}}\int_{0}^{T}\\!\tilde{X}^{2}(t)+\tilde{Y}^{2}(t)+\tilde{Z}^{2}(t)\,\mathrm{d}t$
(58)

where tilde indicates the filtered quantity.

###  III.4 SBR ambiguity

Phase anchor points, together with the pseudorange estimates after noise
reduction, enable the resolution of the SBR ambiguity (see section II.3):

$\displaystyle
a^{\text{sb}}_{ij}(\tau)=\text{round}\left[\nu^{\text{m}}_{ji}\>\hat{R}_{ij}^{\hat{\tau}_{i}}(\tau)-\text{SBR}_{ij}^{\hat{\tau}_{i}}(\tau)\right].$
(59)

$\text{SBR}_{ij}^{\hat{\tau}_{i}}$ are the phase anchor points,
$\hat{R}_{ij}^{\hat{\tau}_{i}}$ the pseudorange estimates after noise
reduction. Thus, we obtain estimates of the SBR ambiguity integers
$a^{\text{sb}}_{ij}$. The resolution is successful if the pseudorange
estimates are more accurate than $6.25\text{\,}\mathrm{cm}$ (half the
ambiguity). From the perspective of noise reduction, this is feasible (see
section IV). Having resolved the SBR ambiguity, the pseudorange estimates
associated to the phase anchor points serve as initial values for the
integration of the sideband range rates. The resolution of the SBR ambiguity
is worthwhile: SBR constitutes a very accurate pseudorange observable, as both
its stability and accuracy are limited by the modulation noise.

##  IV Results

Figure 5: Upper plot: raw PRNR. The ambiguity jumps at $0\text{\,}\mathrm{km}$
and $400\text{\,}\mathrm{km}$ can be seen. Central plot: ambiguous PRNR, the
jumps have been removed but the PRNR ambiguities have not been resolved yet.
The large slopes are mainly due to USO frequency offsets. Lower plot:
unambiguous PRNR. The large differences between the links are caused by
differential SCET offsets.

In this section, we demonstrate the performance of our implementation of the
core ranging processing and the crosschecks as proposed in section III
(central and lower part of fig. 4). We did not implement the initial data
treatment. Instead we assume that the common carrier, PRN, and sideband
timestamping delays are compensated beforehand. We further consider offset-
free PRNR and apply TDIR as a crosscheck for residual offsets.

Figure 6: ASDs of the residual pseudorange estimates for link 12 (upper plot)
and link 21 (lower plot). In blue, residual PRNR. In orange, residual
pseudorange estimates after ranging noise reduction. In green, residual
pseudorange estimates after subtraction of right-handed modulation noise. In
red, residual SBR. Dashed black lines, right-handed modulation noise model.
Dash-dotted black lines, left-handed modulation noise model.

We use telemetry data simulated by LISA Instrument [23] and LISANode [24]
based on orbits provided by ESA [21, 25]. We simulate phase anchor points for
the SBR (see section II.3). The SCET deviations from the respective proper
times are modeled by

$\displaystyle\delta\hat{\tau}_{i}(\tau)=\delta\hat{\tau}_{i,\>0}+y_{i}\>\tau+\frac{\dot{y}_{i}}{2}\>\tau^{2}+\frac{\ddot{y}_{i}}{3}\>\tau^{3}+\int_{\tau_{0}}^{\tau}\text{d}\tilde{\tau}\>y_{i}^{\epsilon}(\tilde{\tau}),$
(60)

the $\delta\hat{\tau}_{i,\>0}$ denote the initial SCET deviations set to
$1\text{\,}\mathrm{s}$, $-1.2\text{\,}\mathrm{s}$, and
$0.6\text{\,}\mathrm{s}$ for SC 1, 2, and 3, respectively. The $y_{i}$ model
the PMC frequency offsets corresponding to linear clock drifts. They are set
to $10^{-7}$, $-2\times 10^{-7}$, and $0.6\times 10^{-7}$ for SC 1, 2, and 3,
respectively. $\dot{y}_{i}\sim 10^{-14}\>${\mathrm{s}}^{-1}$$ and
$\ddot{y}_{i}\sim 10^{-23}\>${\mathrm{s}}^{-2}$$ are constants modeling the
linear and quadratic PMC frequency drifts. The $y_{i}^{\epsilon}$ denote the
stochastic clock noise in fractional frequency deviations, the associated ASD
is given by

$\displaystyle\sqrt{S_{y^{\epsilon}}(f)}=6.32\times
10^{-14}${\mathrm{Hz}}^{-0.5}$\left(\frac{f}{$\mathrm{Hz}$}\right)^{-0.5}.$
(61)

We simulate laser frequency noise with an ASD of

$\displaystyle\sqrt{S_{\dot{N}^{p}}(f)}=$30\text{\,}\mathrm{Hz}\text{\,}{\mathrm{Hz}}^{-0.5}$,$
(62)

and ranging and modulation noise as specified in the sections II.2 and II.3.
Furthermore, we consider test-mass acceleration noise

$\displaystyle\sqrt{S_{N^{\delta}}(f)}$ $\displaystyle=4.8\times
10^{-15}$\mathrm{m}\text{\,}{\mathrm{s}}^{-2}\text{\,}{\mathrm{Hz}}^{-0.5}$\sqrt{1+\left(\frac{$0.4\text{\,}\mathrm{mHz}$}{f}\right)^{2}}$
(63)

and readout noise

$\displaystyle\sqrt{S_{N^{\text{ro}}}(f)}=A\>\sqrt{1+\left(\frac{$2\text{\,}\mathrm{mHz}$}{f}\right)^{4}},$
(64)

where $A=6.35\times 10^{-12}$\mathrm{m}\text{\,}{\mathrm{Hz}}^{-0.5}$$ for the
ISI carrier and $A=1.25\times
10^{-11}$\mathrm{m}\text{\,}{\mathrm{Hz}}^{-0.5}$$ for the ISI sideband
beatnotes. For the readout noise we set a saturation frequency of
$f_{\text{sat}}=$0.1\text{\,}\mathrm{mHz}$$, below which we whiten. The orbit
determinations are simulated by LISA Ground Tracking with the noise levels
specified in section II.

###  IV.1 Ranging processing

Here we demonstrate the performance of our implementation of the core ranging
processing for one day of telemetry data simulated by LISA Instrument [23].
The first ranging processing step covers the PRNR unwrapping (see fig. 5). The
upper plot shows the raw PRNR, which jumps back to $0\text{\,}\mathrm{km}$
when crossing $400\text{\,}\mathrm{km}$ and vice versa. These jumps are easy
to track and to remove. In our implementation we remove all PRNR jumps bigger
than $200\text{\,}\mathrm{km}$. The central plot shows the unwrapped but yet
ambiguous PRNR. Here you can see PRNR drifts of the order of
$\mathrm{1}\mathrm{0}$ to $100\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$,
which are mainly due to differential USO frequency offsets. Inserting the PRNR
ambiguity integers obtained from GOR and TDIR yields the unambiguous PRNR
shown in the lower plot.

In the second step, we use the Kalman filter presented in section III to
reduce the ranging noise. Subsequently, we subtract the right-handed
modulation noise applying the $\Delta M$ measurements constructed from the RFI
beatnotes (see appendix B). After noise reduction, we resolve the SBR
ambiguities combining the estimated pseudoranges with the simulated SBR phase
anchor points (see eq. 59). We then integrate the sideband range rates, to
obtain unambiguous SBR.

In fig. 6, we plot the ASDs of the residual pseudorange estimates (deviations
of the estimates from the true pseudorange values in the simulation) for link
12 (upper plot) and link 21 (lower plot). Blue lines show the ASDs of the
residual PRNR, which are essentially the ASDs of the white ranging noises. The
residual pseudorange estimates after ranging noise reduction are plotted in
orange. They are obtained by combining the PRNR with the sideband range rates.
Therefore, they are limited by right-handed modulation noise (dashed black
line). In green, we plot the residual pseudorange estimates after subtraction
of right-handed modulation noise with the RFI beatnotes. Now the estimates are
limited by left-handed modulation noise (dash-dotted black line). The residual
SBR are drawn red, they are limited by left-handed modulation noise as well,
but involve a smaller offset, since the SBR phase anchor points are more
accurate than PRNR after ranging noise reduction (see fig. 7). In the case of
left-handed MOSAs (see link 12) the RFI beatnotes need to be time shifted to
form the delayed $\Delta M$ measurements. We apply the time shifting method of
PyTDI [26], which consists in a Lagrange interpolation (we use order 5). The
interpolation introduces noise in the high frequency band (see the bump at
$2\text{\,}\mathrm{Hz}$ in the upper plot) but this is out of band.

Figure 7: Upper plot: residual pseudorange estimates after ranging noise
reduction. Second plot: residual pseudorange estimates after subtraction of
right-handed modulation noise. Third plot: residual SBR estimates.

Fig. 7 shows the different residual pseudorange estimates as time series. The
upper plot shows the 6 residual pseudorange estimates after ranging noise
reduction, the second plot after subtraction of right-handed modulation noise.
The third plot shows the SBR residuals. The subtraction of right-handed
modulation noise reduces the noise floor, but it does not increase the
accuracy of the pseudorange estimates. The accuracy can be increased by one
order of magnitude through the resolution of the SBR ambiguities. After
ambiguity resolution, SBR constitutes pseudorange estimates with
sub-$\mathrm{mm}$ accuracy.

Figure 8: PRNR ambiguity resolution via GOR (upper plots) and TDIR (lower
plots). The histogram plots show the residual GOR and TDIR pseudorange
estimates for the different links.

###  IV.2 Crosschecks

Here we demonstrate the performance of our implementation of the crosschecks
for PRNR ambiguity and PRNR offset.

The PRNR ambiguities can be resolved using either GOR (see eq. 46) or TDIR
(see eq. 47). To evaluate the performance of both methods, we simulate 1000
short ($150\text{\,}\mathrm{s}$) telemetry datasets with LISA Instrument [23],
and one set of ODs and MOC-TCs for each of them. We compute the GOR and TDIR
pseudorange estimates for each of the 1000 datasets. Fig. 8 shows the GOR
residuals (first row) and the TDIR residuals (second row) in $\mathrm{km}$ as
histogram plots. We see that the GOR accuracy depends on the arm, because we
obtain more accurate ODs for arms oriented in line of sight direction than for
those oriented cross-track. The PRNR ambiguity resolution via GOR is
successful for GOR deviations smaller than $200\text{\,}\mathrm{km}$.

Figure 9: We simulate one year of telemetry data with PRNR offsets in the
order of $100\text{\,}\mathrm{m}$. We divide this dataset into 1 day chunks
(left plots), 2 day chunks (central plots), and 3 day chunks (right plots). We
then we apply TDIR to each of these chunks in order to estimate the PRNR
offsets. Upper plots: Residual offset estimates in $\mathrm{m}$ for the
different chunk sizes. Lower plots: residual offset estimates after cumulative
averaging in $\mathrm{m}$ for the different chunk sizes. Dashed-black lines:
half the SBR ambiguity.

In the case of the links 23, 31, 13, and 32 all PRNR ambiguity resolutions via
GOR are successful. For each of the links 12 and 21, 2 out of the 1000 PRNR
ambiguity resolutions fail. The GOR estimates are passed as initial values to
TDIR, which then reduces the uncertainty by almost one order of magnitude
(lower plot of fig. 8), such that eventually all PRNR ambiguity resolutions
are successful.

TDIR can also be applied to estimate the PRNR offsets. Hence, it constitutes a
cross-check of the on-ground PRNR offset calibration. We simulate one year of
telemetry data using LISANode [24]. We set the PRNR offsets to
$160.3\text{\,}\mathrm{m}$, $-210.2\text{\,}\mathrm{m}$,
$137.3\text{\,}\mathrm{m}$, $-250.3\text{\,}\mathrm{m}$,
$-188.8\text{\,}\mathrm{m}$, and $105.1\text{\,}\mathrm{m}$ for the links 12,
23, 31, 13, 32, and 21, respectively. We divide the dataset into 1 day chunks
(left plots in fig. 9), 2 day chunks (central plots in fig. 9), and 3 day
chunks (right plots in fig. 9). In each partition we apply the TDIR estimator
presented in section III.3 to each chunk in order to estimate the PRNR
offsets. This computation was parallelized and executed on the ATLAS cluster
at the AEI Hannover. In the upper part of fig. 9 we show the offset estimation
residuals for the three chunk sizes. The offset estimation accuracy increases
with the chunk size in agreement with the order of magnitude estimate through
eq. 40. In the lower part of fig. 9 we plot the residual cumulative averages
of the PRNR offset estimates for the different chunk sizes. Here, it can be
seen that the TDIR estimator performs similarly for the different chunk sizes.
With the 3 day chunk size we can estimate all PRNR offsets with an accuracy of
better than $20\text{\,}\mathrm{cm}$ after 10 days. The dashed-black lines
indicate $6.25\text{\,}\mathrm{cm}$ (half the SBR ambiguity). This is the
required PRNR offset estimation accuracy for a successful SBR ambiguity
resolution. With the 3 day chunk size all offset estimation residuals are
below these $6.25\text{\,}\mathrm{cm}$ after 179 days.

##  V Conclusion

The reduction of laser frequency noise in TDI crucially depends on information
about the pseudoranges. There are four pseudorange observables each having
advantages and disadvantages. In this article, we first derived their
observation equations carefully taking into account ambiguities, noise, and
on-board delays, which cause offsets and timestamping delays. We then proposed
a three-stage ranging sensor fusion (initial data treatment, ranging
processing, crosschecks, compare fig. 4) to combine the four pseudorange
observables, such that we obtain optimal pseudorange estimates.

We pointed out that the common carrier, PRN, and sideband timestamping delays
(see eqs. II.1, 22, and II.3), as well as the PRNR and SBR offsets (see eqs.
23 and 26) need to be calibrated on ground, so that they can be compensated in
the initial data treatment. We further derived that the small optical path
lengths between laser and PBS, PBS and ISI BS, and laser and ISI BS show up in
the uncommon delays (see eq. 12), which are to be applied in TDI. We proposed
to measure these optical path lengths on ground, so that during operation they
can be combined with the pseudorange estimates to form the uncommon delays.

We identified the processing steps, which need to be performed continuously
during operation. These are the PRNR unwrapping, and the reduction of ranging
and right-handed modulation noise, we referred to them as ranging processing.
We implemented the ranging processing numerically: we showed that the white
ranging noise can be reduced by combining the PRNR with the sideband range
rates in a KF. We split up the system and implemented one KF per SC, such that
the individual SCETs served as KF time-grids. We further applied the RFI
beatnotes to subtract the right-handed modulation noise. The pseudorange
estimates we obtained this way were at sub-$\mathrm{cm}$ accuracy. We showed
that in combination with phase anchor points they allow for the resolution of
the SBR ambiguity resulting in pseudorange estimates at sub-$\mathrm{mm}$
accuracy.

We implemented crosschecks for the PRNR ambiguities and offsets. We showed
that both GOR and TDIR allow for the resolution of the PRNR ambiguity. We
applied TDIR as a crosscheck for the PRNR offset calibration and demonstrated
its performance for one year of telemetry data: after about 180 days all PRNR
offset estimates reached an accuracy of better than $6.25\text{\,}\mathrm{cm}$
allowing for the resolution of the SBR ambiguity.

In reality, the PRNR offsets are slowly time-varying. The investigation of the
PRNR offset estimation via TDIR could be extended for linearly time varying
PRNR offsets. The delay model for the TDIR estimator would then become
(compare eq. 57):

$\displaystyle
d_{ij}^{\hat{\tau}_{i}}(\tau)=\hat{R}_{ij}^{\hat{\tau}_{i}}(\tau)-(O^{0}_{ij}+O^{1}_{ij}\cdot\tau),$
(65)

The TDIR estimator would now have to fit the 12 parameters $O^{0}_{ij}$ and
$O^{1}_{ij}$. Apart from that, tone-assisted TDIR [27] could be applied for
the PRNR offset estimation in order to reach faster convergence. As a further
follow-up investigation, time-varying on-board delays and the associated SC
monitors could be included into the simulation, which would enable an
inspection of the feasibility of the initial data treatment as proposed in
section III. Furthermore, the ranging sensor fusion could be included into the
different INReP topologies. Apart from that, the algorithms could be applied
to real data as, e.g., produced by the hexagon experiment [28], [29].

## Acknowledgements

J. N. R. acknowledges the funding by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) under Germany’s Excellence Strategy within the
Cluster of Excellence PhoenixD (EXC 2122, Project ID 390833453). Furthermore,
he acknowledges the support by the IMPRS on Gravitational Wave-Astronomy at
the Max Planck Institute for Gravitational Physics in Hannover, Germany. This
work is also supported by the Max-Planck-Society within the LEGACY (“Low-
Frequency Gravitational-Wave Astronomy in Space”) collaboration
(M.IF.A.QOP18098). O. H. and A. H. acknowledge support from the Programme
National GRAM of CNRS/INSU with INP and IN2P3 co-funded by CNES and from the
Centre National d’Études Spatiales (CNES). The authors thank Miles Clark,
Pascal Grafe, Waldemar Martens, and Peter Wolf for useful discussions. The
study on PRNR offset estimation via TDIR was performed on the ATLAS cluster at
AEI Hannover. The authors thank Carsten Aulbert and Henning Fehrmann for their
support.

##  Appendix A Pseudoranges in TCB

The pseudorange can be expressed in TCB by writing the SCETs of receiving and
emitting SC as functions of TCB evaluated at the events of reception and
emission, respectively:

$\displaystyle
R_{ij}^{t}(t_{\text{rec}})=\hat{\tau}_{i}^{t}(t_{\text{rec}})-\hat{\tau}_{j}^{t}(t_{\text{emit}}),$
(66)

$\hat{\tau}_{i}^{t}$ denotes the SCET of SC $i$ expressed as a function of
TCB. The TCB of emission can be expressed as the difference between the TCB of
reception and the light travel time from SC $j$ to SC $i$, denoted by
$d_{ij}^{t}$:

$\displaystyle
R_{ij}^{t}(t_{\text{rec}})=\hat{\tau}_{i}^{t}(t_{\text{rec}})-\hat{\tau}_{j}^{t}\left(t_{\text{rec}}-d_{ij}^{t}(t_{\text{rec}})\right),$
(67)

in the following we drop the subscript, hence $t$ refers to the TCB of
reception. The SCET can be expressed in terms of the SCET deviation from TCB

$\displaystyle\hat{\tau}_{i}^{t}(t)$
$\displaystyle=t+\delta\hat{\tau}_{i}^{t}(t),$ (68)

which allows us to write eq. 67 as

$\displaystyle R_{ij}^{t}(t)$
$\displaystyle=\delta\hat{\tau}_{i}^{t}(t)+d_{ij}^{t}(t)+\delta\hat{\tau}_{j}^{t}(t-d_{ij}^{t}(t).$
(69)

Expanding the emitting SC SCET deviation from TCB around the reception TCB
yields:

$\displaystyle R_{ij}^{t}(t)$
$\displaystyle=\delta\hat{\tau}_{ij}^{t}(t)+\left(1+\delta\dot{\hat{\tau}}_{j}^{t}(t)\right)\cdot
d_{ij}^{t}(t),$ (70) $\displaystyle\delta\hat{\tau}_{ij}^{t}(t):$
$\displaystyle=\delta\hat{\tau}_{i}^{t}(t)-\delta\hat{\tau}_{j}^{t}(t).$ (71)

Hence, in a global time frame like TCB, the pseudorange can be expressed in
terms of the light travel time $d_{ij}^{t}$ and the differential SCET offset
$\delta\hat{\tau}_{ij}^{t}$.

##  Appendix B Subtraction of right-handed modulation noise

Following the notation in [6], we express the RFI beatnotes in frequency:

$\displaystyle\text{RFI}_{ij}^{\hat{\tau}_{i}}(\tau)=$
$\displaystyle\>\nu^{\hat{\tau}_{i}}_{ik}(\tau)-\nu^{\hat{\tau}_{i}}_{ij}(\tau),$
(72) $\displaystyle\text{RFI}_{\text{sb},\>ij}^{\hat{\tau}_{i}}(\tau)=$
$\displaystyle\>\nu^{\hat{\tau}_{i}}_{\text{sb},\>ik}(\tau)-\nu^{\hat{\tau}_{i}}_{\text{sb},\>ij}(\tau),$
(73) $\displaystyle\nu^{\hat{\tau}_{i}}_{\text{sb},\>ij}(\tau)=$
$\displaystyle\>\nu^{\hat{\tau}_{i}}_{ij}(\tau)+\nu_{ij}^{\text{m}}\cdot(1+M_{ij}^{\hat{\tau}_{i}}).$
(74)

In this article we do not consider on-board delays in the RFI beatnotes. We
combine the RFI carrier and sideband beatnotes to form measurements of the
right-handed modulation noise:

$\displaystyle\Delta M_{i}^{\hat{\tau}_{i}}\vcentcolon=$
$\displaystyle\>\frac{\text{RFI}_{ij}^{\hat{\tau}_{i}}-\text{RFI}_{\text{sb},\>ij}^{\hat{\tau}_{i}}+$1\text{\,}\mathrm{MHz}$}{2}$
$\displaystyle-$
$\displaystyle\>\frac{\text{RFI}_{ik}^{\hat{\tau}_{i}}-\text{RFI}_{\text{sb},\>ik}^{\hat{\tau}_{i}}-$1\text{\,}\mathrm{MHz}$}{2},$
$\displaystyle=$ $\displaystyle\>\nu^{\text{m}}_{ij}\cdot
M^{\hat{\tau}_{i}}_{ij}-\nu^{\text{m}}_{ik}\cdot M^{\hat{\tau}_{i}}_{ik},$
(75)

$i,\>j,$ and $k$ being a cyclic permutation of 1, 2, and 3. We can now
subtract the $\Delta M_{i}^{\hat{\tau}_{i}}$ measurements from the sideband
range rates (eq. 34). Thus, we reduce the right-handed modulation noise, so
that we are limited by the one order of magnitude lower left-handed modulation
noise:

$\displaystyle\dot{\text{SBR}}_{\text{cor},\>ij}^{\hat{\tau}_{i}}=\dot{\text{SBR}}_{ij}^{\hat{\tau}_{i}}-\dot{\textbf{D}}^{\hat{\tau_{i}}}_{ij}\cdot\Delta
M_{j}^{\hat{\tau}_{j}}(\tau),$ $\displaystyle=$
$\displaystyle\nu^{\text{m}}_{ji}\cdot\dot{R}_{ij}^{\hat{\tau}_{i}}+\nu^{\text{m}}_{ij}\left(M^{\hat{\tau}_{i}}_{ij}-\dot{\textbf{D}}^{\hat{\tau_{i}}}_{ij}\cdot
M^{\hat{\tau}_{j}}_{jk}(\tau)\right),$ (76a)
$\displaystyle\dot{\text{SBR}}_{\text{cor},\>ik}^{\hat{\tau}_{i}}=\dot{\text{SBR}}_{ik}^{\hat{\tau}_{i}}(\tau)+\Delta
M_{i}^{\hat{\tau}_{i}}(\tau),$ $\displaystyle=$
$\displaystyle\nu^{\text{m}}_{ki}\cdot\dot{R}_{ik}^{\hat{\tau}_{i}}+\nu^{\text{m}}_{ki}\left(M^{\hat{\tau}_{i}}_{ij}+\dot{\textbf{D}}^{\hat{\tau_{i}}}_{ik}\>M^{\hat{\tau}_{k}}_{ki}(\tau)\right),$
(76b)

$i,\>j,$ and $k$ being a cyclic permutation of 1, 2, and 3.

##  Appendix C Solar wind dispersion

The average solar wind particle density at the LISA orbit is about
$10\text{\,}{\mathrm{cm}}^{-3}$. Hence, at the scales of optical wavelengths
the solar wind plasma can be treated as a free electron gas with the plasma
frequency [30]

$\displaystyle\nu_{p}^{2}=\frac{n_{e}\>e^{2}}{4\pi^{2}\>\epsilon_{0}\>m_{e}}\approx
8\times 10^{8}\>${\mathrm{s}}^{-2}$,$ (77)

$n_{e}$ denotes the electron density, $e$ the elementary charge, $m_{e}$ the
electron mass, and $\epsilon_{0}$ the the vacuum permittivity. Contributions
from protons and ions can be neglected as the plasma frequency is inversely
proportional to the mass. We describe the refractive index of the solar wind
plasma by the Appleton equation. Neglecting collisions and magnetic fields it
denotes

$\displaystyle n(\nu)=\sqrt{1-\left(\frac{\nu_{p}}{\nu}\right)^{2}}.$ (78)

In a dispersive medium we need to distinguish between phase and group
velocity. The phase velocity is given by

$\displaystyle v_{\text{p}}(\nu)$
$\displaystyle=\frac{c}{n(\nu)}=\frac{c}{\sqrt{1-\left(\frac{\nu_{p}}{\nu}\right)^{2}}}\approx
c\cdot\left(1+\frac{1}{2}\frac{\nu_{p}^{2}}{\nu^{2}}\right),$ (79)

where we applied the expansion for $\nu\gg\nu_{p}$, as we consider optical
frequencies. The product of group and phase velocity yields $c^{2}$.
Consequently, the group velocity is

$\displaystyle v_{\text{g}}(\nu)$ $\displaystyle=c\cdot
n(\nu)=c\cdot\sqrt{1-\left(\frac{\nu_{p}}{\nu}\right)^{2}}\approx
c\cdot\left(1-\frac{1}{2}\frac{\nu_{p}^{2}}{\nu^{2}}\right).$ (80)

Group and phase delay can now be written as

$\displaystyle\Delta\tau_{\text{g}}(\nu)$
$\displaystyle=L\left(\frac{1}{c\cdot\sqrt{1-\left(\frac{\nu_{p}}{\nu}\right)^{2}}}-\frac{1}{c}\right)\approx\frac{L\>\nu_{p}^{2}}{2\>c}\cdot\frac{1}{\nu^{2}},$
(81) $\displaystyle\Delta\tau_{\text{p}}(\nu)$
$\displaystyle=L\left(\frac{\sqrt{1-\left(\frac{\nu_{p}}{\nu}\right)^{2}}}{c}-\frac{1}{c}\right)\approx-\frac{L\>\nu_{p}^{2}}{2\>c}\cdot\frac{1}{\nu^{2}},$
(82)

where $L=$2.5\text{\,}\mathrm{Gm}$$ denotes the LISA armlength. PRN and
sideband signals propagate at the group velocity, hence they are delayed by
the group delay:

$\displaystyle\Delta\tau_{\text{g}}^{\text{prn}}$
$\displaystyle=\Delta\tau_{\text{g}}($281\text{\,}\mathrm{THz}$\pm$1\text{\,}\mathrm{MHz}$)\approx$12.7\text{\,}\mathrm{pm}$,$
(83) $\displaystyle\Delta\tau_{\text{g}}^{\text{sb}}$
$\displaystyle=\Delta\tau_{\text{g}}($281\text{\,}\mathrm{THz}$\pm$2.4\text{\,}\mathrm{GHz}$)\approx$12.7\text{\,}\mathrm{pm}$.$
(84)

The phase delay is negative, because the phase velocity is bigger than $c$.
Therefore, the laser phase is advanced with respect to a wave propagating in
vacuum. For the LISA carrier this phase advancement corresponds to

$\displaystyle\Delta\tau_{\text{p}}($281\text{\,}\mathrm{THz}$)\approx-$12.7\text{\,}\mathrm{pm}$.$
(85)

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