Instrument Data Simulations for GRACE Follow-on: Observation and Noise Models

The Gravity Recovery and Climate Experiment (GRACE) mission has yielded data on the Earth’s gravity field to monitor temporal changes for more than fifteen years now. The GRACE twin satellites use microwave ranging with micrometer precision to measure distance variations between two satellites caused by the Earth’s global gravitational field. GRACE Follow-on (GRACE-FO) will be the first satellite mission to use inter-satellite laser interferometry in space. The laser ranging instrument (LRI) will provide two additional measurements compared to the GRACE mission: interferometric inter-satellite 5 ranging with nanometer precision and inter-satellite pointing information. We have designed a set of simulated GRACE-FO data, which include LRI measurements, apart from all other GRACE instrument data needed for the Earth’s gravity field recovery. The simulated data files are publicly available via https://doi.org/10.22027/AMDC2 and can be used to derive gravity field solutions like from GRACE data. This paper describes the scientific basis and technical approaches used to simulate the GRACE-FO instrument data. 10 Copyright statement. TEXT


Introduction
The space gravimetry mission GRACE (Tapley et al., 2004) observes the Earth's gravity field changes with time.GRACE is the first low-low satellite-to-satellite tracking mission: the principal measurement is the distance variability between low-orbit GRACE twin satellites, which translates into the monthly gravity models (Wahr et al., 1998).Kim (2000) published the first GRACE satellite simulation study before the launch of the GRACE satellites (in 2002).Now, 17 years later, GRACE satellites are at the end of their lifetime and GRACE-FO data will be available soon.Although the GRACE-FO mission and its instrument data streams will be very similar to GRACE, the necessity for a GRACE-FO instrument data simulation emerges from the additional interferometric inter-satellite ranging.Flechtner et al. (2016) have performed a full-scale simulation over the nominal GRACE-FO mission lifetime of 5 years and showed notable improvements with the LRI on a global scale of the order of 23 %.Also, the GRACE-FO science data system team at Jet Propulsion Laboratory (JPL) has planned to release a GRACE-FO "Grand Simulation" data set before the real GRACE-FO data are available (Watkins et al., 2016).
Most importantly, the operation of the LRI in addition to the primary K-band ranging (KBR) instrument yields extra information not only in the ranging measurement, but also in the attitude determination.Therefore GRACE-FO LRI data processing will contain precise measurements of the satellites' pitch and yaw angles.In this paper, simulated LRI pitch and yaw angles are provided for the first time.Exploitation of the new GRACE-FO measurements has great potential to improve the spatial and temporal resolution of the Earth's gravity field solutions.
N. Darbeheshti et al.: Instrument data simulations for GRACE Follow-on form medium to test and improve different gravity field recovery techniques.
We have generated a set of simulated GRACE-FO data for a period of 1 month with a 5 s sampling rate.The data set is available for download via https://doi.org/10.22027/AMDC2.The recovered gravity field solutions using this data set can be submitted via the same link.The goals of generating this set of simulated data are to improve different gravity field recovery techniques by comparing the input gravity field for the simulation and the recovered gravity fields; and to use new LRI data, such as LRI ranging and LRI attitude information, in different gravity field recovery techniques.
The analysis of seasonal or sub-seasonal geophysical features is not the focus of this simulated data set, as the duration of the simulated data is short.
The main purpose of this paper is to describe the chain of instrument data simulation procedures.The first section presents the preliminaries for the data simulation, including the coordinate systems and symbols, followed by each section describing each instrument data simulation with details of the instrument noise models.

Preliminaries
The following coordinate systems are used to define the various simulated data.
-International Celestial Reference Frame (ICRF) -inertial frame origin: centre of mass (CoM) of the Earth axes: according to IERS 2010 conventions (Petit and Luzum, 2010) -International Terrestrial Reference Frame (ITRF) -Earth-fixed (co-rotating) frame origin: CoM of the Earth axes: according to IERS 2010 conventions (Petit and Luzum, 2010) -Line-of-sight frame (LOSF), one per satellite for GRACE A origin: satellite CoM , where r is the satellites' position vector in the ICRF (i.e.line-of-sight vector and roll axis)

yaw axis;
for GRACE B, the A and B indices should be exchanged) -Satellite frame (SF), one per satellite according to Case et al. (2002) origin: satellite CoM x SF = from the origin to a target location of the phase centre of the K-or Ka-band horn y SF = forms a right-handed triad with x SF and z SF z SF = normal to x SF and to the plane of the main equipment platform and positive towards the satellite radiator on the bottom of the GRACE-FO The LOSF and SF are shown in Fig. 1.Since we did not model variations in the satellites' CoM (and the CoM coinciding with the on-board accelerometer's proof masses) for data simulation, the SF coincides with the science reference frame defined in Case et al. (2002).
All simulated data are published in GRACE Level-1B data format: daily files with a 5 s sampling rate (Case et al., 2002).They can be considered pre-processed like GRACE Level-1B data.Time tags are given in GRACE GPS seconds, i.e. seconds since epoch 1 January 2000, 12:00:00 (no leap seconds applied).Five instrument data types were simulated; the following sections in this paper describe each simulated instrument's observations and errors, respectively.
-GPS navigation data (GNV1B) Simulated GPS positions and velocities are the output of the orbit integrator, which are rotated from ICRF to ITRF, and a GPS error is added to each.The error-free positions can be considered a kinematic orbit.
-K-band ranging system (KBR1B) Simulated KBR ranging data are derived from the errorfree GPS positions and velocities with added KBR errors.
-Star camera (SCA1B) Simulated star camera quaternions are derived from the simulated roll, pitch, and yaw angles with added errors.
-Accelerometer (ACC1B) Simulated linear accelerations are calculated from the non-gravitational accelerations acting on the satellites.The error-free simulated star camera quaternions are used to transform the linear accelerations from ICRF to SF.Then accelerometer noise, scale, and bias are added.The angular accelerations are calculated from the errorfree simulated star camera quaternions.-Laser ranging instrument (LRI1B) Simulated LRI ranging data are derived from error-free GPS positions and velocities with added LRI errors.
Figure 2 shows a flow chart of the procedure used for the simulations.For each instrument, first the error-free observation was generated, and then the errors including instrument noise, bias, and scale were applied to each instrument observation.
In this paper, the symbols δ and are used for timevarying and constant errors, respectively.The symbol δ denotes amplitude spectral densities (ASDs).For data simulations, a five-point numerical differentiation method was used for the numerical differentiations.The LISA Technology Package Data Analysis (LTPDA) toolbox (https://www.elisascience.org/ltpda/)for MATLAB was used for the generation of time series based on instrument noise models given in terms of ASD.LTPDA uses Franklin's random noise generator method (Franklin, 1965) to generate arbitrarily long time series with a prescribed spectral density.

Simulating GNV1B data
An orbit integrator is used to calculate the trajectories of the GRACE-FO satellites (GRACE-FO A and GRACE-FO B) through the numerical integration of Newton's second law of motion based on knowledge of the forces acting on the satellite.Table 1 summarises the orbit integrator parameters.
The IERS2010 conventions are used for rotation between the ITRF and the International Celestial Reference Frame (ICRF).Two types of force models were used for orbit integration.
-Gravitational forces: -A static gravity field of a certain degree and order -The ocean tide model EOT11a (Rieser et al., 2012) up to degree and order 80 -Direct tides of the Moon and Sun using NASA JPL DE405 ephemeris (Standish, 1998) -Anelastic solid Earth tides according to IERS2010 -Non-gravitational forces: -Atmospheric drag model -Solar radiation pressure model The static gravity model and its exact degree and order are the unknowns for the gravity field recovery.The degree and order that were used as input are between 75 and 95.The atmospheric drag and solar radiation pressure models are described in Appendix A. Other gravitational forces, such as atmosphere and ocean short-term mass variations, are not used as this simulation data set focuses on the impact of instrument data errors.
The input to the orbit integrator is the initial time and state (position and velocity vectors) of GRACE-FO A and GRACE-FO B at time 00:00:00 on 1 May 2005.It calculates the two trajectories separately in addition to the time series of accelerations along the trajectory from the gravitational White noise with a level of a few cm Hz −1 was generated along the x, y, and z axes independently and added to each satellite position: (1) Then the noise was differentiated numerically and added to the velocities along the x, y, and z axes separately for each satellite: 4 Simulating SCA1B data The satellite attitude with respect to the ICRF is determined from the star cameras on-board the satellites.The measured attitude is expressed in terms of quaternions q: q = q 0 q 1 q 2 q 3 . (3) Here, q 0 denotes the real component and q 1 , q 2 , and q 3 are the imaginary components of the quaternion.The time series of quaternions is provided in the SCA1B product.An attitude and orbit control system keeps the satellite orientation near its nominal attitude within a certain boundary for each of the three pointing angles.These boundaries have been lowered for GRACE-FO compared to GRACE for two reasons.The first is due to the coupling of pointing angle errors into the ranging data; experience has shown that improved pointing would enhance the quality of gravity field solutions (Horwath et al., 2011).Secondly, the LRI requires better satellite pointing in order to guarantee its functionality; otherwise there is a risk that the laser beam will start to hit obstacles.Hence, the combined effect of pointing jitter on one hand and frame misalignments on the other hand cannot exceed a certain value (a few milliradians in terms of pitch and yaw angles for GRACE-FO).This yields strict requirements for the construction and mounting of the LRI components and also the necessity for an improved pointing control.
The pointing jitter angles describe how the "true" satellite orientation (as it actually is) deviates from the "nominal" orientation (as it should be ideally in the absence of pointing angles).The nominal orientation is satellites' attitude reference.We assumed that the satellites' attitude reference is the alignment of SF and LOSF for the simulations.
Accordingly, satellite pointing angles can be computed from star camera quaternions and orbital positions (described in Appendix B).For simulating star camera quaternions, one has to go the opposite way.The pointing angles from GRACE-FO attitude and orbital control system performance predictions were provided to us by JPL and AIRBUS Defense and Space.A model based on the spectrum of these predicted angles was used to simulate the pointing angles.The common approach for generating time series with a known spectrum is to use a random noise generator.Figure 3 shows the ASD of the simulated roll (θ x ), pitch (θ y ), and yaw (θ z ) angles.One can see that all three angles have peaks mostly in the frequency band between 10 −4 and 2 × 10 −3 .These peaks disturb the functionality of the random noise generator, and thus they were modelled individually.The result is a time series of error-free inter-satellite pointing angles.
To simulate star camera measurements, white noise (δθ SCA1B ) and a bias ( θ SCA1B ) were added to each errorfree angle separately: Here, θ x , θ y , and θ z are the error-free simulated roll, pitch, and yaw angles; θ x,SCA , θ y,SCA , and θ z,SCA are simulated star camera roll, pitch, and yaw angles.
The GRACE-FO satellites are equipped with improved star cameras compared to GRACE, and the number of star camera heads will increase from two to three per satellite (Gath, 2016).Bandikova et al. (2012) suggested that a proper combination of the different star camera heads reduces the high-frequency noise of the pointing angles.Accordingly, it is expected that a better estimation of pointing angles from GRACE-FO star camera data will be available.Therefore, white noise with a level of a few tens of µrad Hz −1/2 was chosen, which is lower than the current noise level in roll, pitch, and yaw angles estimated from the GRACE star camera data.The GRACE star cameras are strong in the roll axis and weak in the pitch and yaw axes due to the orientation in which they were mounted (Harvey, 2016).GRACE data (Fig. 4) confirm the 150-300 µrad Hz −1/2 accuracy for pitch and yaw and 25-35 µrad Hz −1/2 for roll, which meet the mission requirements (Stanton et al., 1998).
The value of bias for each angle was chosen in the range of a few milliradians.This level of bias has been investigated by Horwath et al. (2011) based on GRACE Level-1B data.Figure 5 shows simulated star camera roll, pitch, and yaw angles, which are similar to the GRACE inter-satellite pointing variations plot in Bandikova et al. (2012).
From the contaminated simulated pointing angles of Eq. ( 4), the rotation matrix R LOSF SF was built, and with  the error-free simulated orbit positions, the rotation matrix R ICRF LOSF was built.With these two matrices, the matrix containing the simulated star camera quaternions (Fig. 6).Finally, the simulated quaternions can be recovered from the rotation matrix R ICRF SF by using the equation series (Wu et al., 2006): where R ij represents the elements of R ICRF SF .Note that the series in Eq. ( 6) is only numerically stable as long as the trace of R is non-negative (i.e.not close to −1).A numerically stable pseudocode that was used is shown in Appendix C.
Two other sets of quaternions were generated: error-free quaternions from error-free pointing angles in Eqs.(4) and noisy quaternions that come from white-noise-contaminated pointing angles without the bias.We will refer to these two set of quaternions in the following sections.

Simulating ACC1B data
Figure 2 shows that the non-gravitational accelerations were computed along the orbit in ICRF.

Linear accelerations
The non-gravitational accelerations are the sum of atmospheric drag and solar radiation pressure accelerations (Appendix A) along the orbit in inertial frame (ICRF).The nongravitational accelerations rICRF were transformed into the satellite frame rSF using the rotation matrix R SF ICRF from error-free simulated quaternions: After being transformed into the SF, the linear accelerations were multiplied by the scale factors s x , s x , and s z , and then the accelerometer noise time series (δ rACC1B ) and the biases ( rACC1B ) were added along the x, y, and z axes independently: The ASD noise model of Kim (2000) was used to generate accelerometer noise (δ rACC1B ): The y axis in SF (y SF in Fig. 1) is considered the leastsensitive axis for accelerometer measurements (Kim, 2000).
The noise ASD of the sensitive axes and the less-sensitive axis are shown in Fig. 7.A 1-month time series of the accelerometer noise was generated separately for the x, y, and z axes from the ASD models and added to the accelerations (Eq.8).Values close to the GRACE accelerometer scale and bias along each axis were chosen and kept constant for 1 month of the simulated data.Therefore, in total for both satellites, six accelerometer scale parameters and six accelerometer bias parameters should be estimated during the gravity field recovery using 1 month of the simulated data.The scale and bias parameters will be available via https://doi.org/10.22027/AMDC2for comparison with the estimated ones.

Angular accelerations
The error-free simulated quaternions were used to generate angular accelerations based on the relations between the quaternions and angular accelerations ( ωx , ωy , ωz ) (Müller, 2010): where qm are the numerically differentiated simulated quaternions.

Simulating KBR1B data
The position, velocity, and acceleration differences of GRACE-FO A and GRACE-FO B are calculated as follows: in order to calculate simulated error-free range, range rate, and range acceleration according to where • is the vector dot product.The GRACE-FO KBR instrument (as in GRACE) will measure the biased range between the twin satellites; a bias ( ρ) of a few centimetres was added to the error-free range (ρ).The KBR instrument noise is dominated by system and oscillator noise (δρ SO ).It was added to the error-free ranging products, as was a geometric error, which is a pointing jitter coupling effect caused by an offset of the KBR antenna phase centre for each satellite A and B (δρ APC ): In the following, these two error sources are described.

System and oscillator noise
The following ASD model was used to generate KBR noise: This ASD model is in agreement with the system and oscillator KBR noise for the satellite pair separation of 238 km in Kim (2000).Figure 8 illustrates the ASD model.Based on this model, a 1-month time series of the range noise was generated.Then numerical differentiation was used to generate range rate noise and range acceleration noise from the range noise time series (Fig. 9).

Antenna phase centre pointing jitter coupling
The KBR instrument measures the distance between the antenna phase centres, which are placed nominally on the SF www.earth-syst-sci-data.net/9/833/2017/ Earth Syst.Sci.Data, 9, 833-848, 2017 x axis almost 1.5 m away from the satellites' CoM.However, due to manufacturing imperfections and due to the large acceleration of the system during launch, the actual positions differ from the nominal ones.Consequently, any pointing jitter (deviations of the satellites' attitudes from their nominal attitudes) causes a geometric error in the ranging measurement.In the absence of such misplacements and in the absence of pointing jitter, this effect would be zero (rather, constant) and hence not effect the measured (biased) range.
Given the antenna phase centre (APC) offset vector (p SF A ) in SF and the matrix rotating from SF to ICRF, this error is computed as: i.e. it is the APC offset (w.r.t.CoM) projected onto the line of sight.For the simulation, the R ICRF SF was calculated from Eq. (B1) using the error-free simulated quaternions, and the line-of-sight vector (e ICRF AB ) was calculated from the errorfree satellite positions in ICRF: For GRACE-FO B, the indices A and B should be swapped in Eqs. ( 16) and ( 17). Figure 10 shows time series of the APC offset pointing jitter coupling for 1 month of GRACE-FO A.
In GRACE, there have been calibration manoeuvres in order to try and estimate the APC offset vectors (p SF A , p SF B ).The estimates have been published by JPL in the VKB1B files (Case et al., 2002).For the simulations, values of similar magnitude were chosen.These values are not directly given to the user; however, the simulated KBR1B files include a column of simulated estimated correction terms.This means that it is computed from the imperfect attitude information that is provided via simulated SCA1B files.Real GRACE KBR1B data also contain this column, which is called the antenna offset correction (AOC) term (Case et al., 2002).It has to be added to the KBR ranging measurement, so it describes the negative of the error term: For the simulations, the correction term of AOC ρ was computed according to Eq. ( 16), with the difference that the matrix R ICRF SF was derived from the simulated noisy quaternions without the bias.
A second and third column are also provided that are computed by using numerical differentiation and describe the correction for range rate and range acceleration: 7 Simulating LRI1B data The structure of the LRI1B data file is similar to the KBR1B file, but it contains two additional observations of pitch and yaw angles.Tables D1 and D2 in Appendix D show the format of the data records for simulated KBR1B and LRI1B files.The simulated error-free range, range rate, and range accelerations are calculated from Eqs. ( 11), ( 12), and ( 13).
Apart from a bias of a few centimetres, various other errors were added: In Eq. ( 20), α = 1 + 10 −6 is a scale factor which is due to the limited accuracy of the absolute laser frequency value for the phase to length conversion.The three main LRI noise sources in Eq. ( 20) are laser frequency (LF) noise (δρ LF ), the coupling of the pointing jitter into the length measurement via triple mirror assembly (TMA; Fig. 11) for each satellite A and B (δρ TMA ), and the additional linear and quadratic pointing jitter coupling (δρ ALQ ).This is a selection of relatively well-known LRI error sources in which LF and TMA errors are expected to be the dominating ones.For the range rate and range acceleration noise, the errors were numerically differentiated.In the following, the LRI error sources are described in detail.

Laser frequency noise
Based on LRI cavity performance tests carried out by JPL, the current best estimate of the ASD of the laser frequency Earth Syst.Sci.Data, 9, 833-848, 2017 www.earth-syst-sci-data.net/9/833/2017/  noise (i.e. the ranging noise which is induced by the frequency jitter of the LRI master laser) for a satellite separation of 238 km is Figure 12 illustrates this noise.Note that this specific ASD corresponds to a constant satellite separation (of 238 km), which is a sufficient simplification for the purpose of generating noise time series.A 1-month time series of the range noise δρ LF was generated from the ASD model (Fig. 12).Then numerical differentiation was used to generate range rate noise and range acceleration noise from the noise range time series (Fig. 13).

Triple mirror assembly pointing jitter coupling
With a good approximation, the LRI measures the biased distance between the TMA vertices of the twin satellites (Fig. 11).Both the pointing jitter and frame misalignments couple into the LRI ranging measurement.This effect is in principle the same as the geometric error effect due to the APC position in the KBR measurement.The only difference is that the nominal positions of the TMA vertices are in the CoM, whereas the nominal positions of the APC are almost 1.5 m in the SF x direction away from this point.
An offset of the TMA vertex from the satellites'CoM leads to the coupling of satellite pointing jitter into the round-trip length variations measured by the LRI (Fig. 14).The magnitudes of TMA vertex offset vectors (v SF ) along the x, y, and z axes were chosen in the order of a few hundred micrometres.The real values after the GRACE-FO launch are unknown and will have to be calibrated.To calculate δρ TMA , the TMA vertex offset vector (v SF ) is rotated from the SF into the ICRF and then projected onto the line of sight: where e ICRF AB is the line-of-sight vector in ICRF.Again, for GRACE-FO B, indices A and B should be swapped.Similar to the KBR1B files, LRI1B files contain correction terms -vertex point correction (VPC) terms -for range, range rate, and range acceleration, which were calculated using the simulated noisy quaternions without the bias:

Additional linear and quadratic pointing jitter coupling
There is additional linear and quadratic coupling (ALQ) of the pointing jitter angles (θ x , θ y , and θ z ) into the length measurements, which can be described as For GRACE-FO B, the indices A should be exchanged into B in Eq. ( 24).Linear coefficients of c x , c y , and c z are estimated to be in the order of a few µm rad −1 and quadratic coefficients of c xy and c xz are in the order of a few µm rad −2 .Error-free time series of θ x , θ y , θ z were used to simulate δρ ALQ .

Differential wavefront sensing: pitch and yaw measurements
Differential wavefront sensing (DWS) is a well-known technique for measuring the relative wavefront misalignment between two laser beams with high sensitivity (Sheard et al., 2012).Figure 17 illustrates the basic principle of DWS.DWS provides two extra measurements of the satellite attitude: yaw and pitch pointing angles with respect to the line of sight.DWS angle measurements on-board GRACE-FO are obtained from the steering mirror on the LRI optical bench (Sheard et al., 2012).The steering mirror orientation is controlled using the DWS error signals constantly driving the error signals back to zero.The steering mirror orientation is recorded as pitch and yaw angles.However, the steering mirror can only turn in discrete units of 4.5 µrad around the pitch axis and 6 µrad around the yaw axis.Therefore, the angle determination is limited to integer multiples of these units.
where "round" means rounding towards the nearest integer; θ y,DWS and θ z,DWS are the simulated DWS pitch and yaw angles, and θ y and θ z are the error-free pitch and yaw angles.
The biases ( θ y,DWS , θ y,DWS ) stem mainly from a misalignment of the LRI frame with respect to the SF, which is expected to be within the range of a few milliradians.Figure 18 shows simulated DWS pitch and yaw angles.

Conclusions
We have described the simulation of observation and noise models for the GRACE-FO multi-sensor system consisting of inter-satellite ranging with microwave and laser ranging instruments, GPS orbit tracking, accelerometry, and attitude sensing.For the first time, simulated LRI data that include DWS attitude information were generated.The simulated LRI ranging and attitude data may be used in different data analysis scenarios for GRACE-FO, such as the combination of KBR and LRI data and the calibration or estimation of geometric corrections for both KBR and LRI ranging.
On the other hand, different Earth's gravity field solutions derived from actual satellite data can only be compared against each other because the real Earth's gravity field is not known.This is a major problem in the evaluation of the performance of gravity field recovery approaches.A closedloop simulation starting with a known gravity field provides the opportunity to overcome this problem by comparing the www.earth-syst-sci-data.net/9/833/2017/ Earth Syst.Sci.Data, 9, 833-848, 2017

Figure 1 .
Figure 1.Illustration of SF and LOSF for GRACE satellites.Small positive yaw (left) and pitch (right) angles indicate the direction of rotation defining the sign of the pointing angles.

Figure 2 .
Figure 2. Flow chart of the simulation steps for GRACE-FO instrument data; please refer to Fig. 6 for a detailed description of SCA simulated data.

Figure 5 .
Figure 5. Simulated star camera roll, pitch, and yaw angles during two orbital revolutions for GRACE-FO A.

Figure 6 .
Figure 6.Flow chart of the simulation steps for SCA1B data.

Figure 8 .
Figure 8. ASD of KBR system and oscillator noise for range (a), range rate (b), and range acceleration (c).

Figure 9 .
Figure 9.Time series of KBR oscillator and system noise for range (a), range rate (b), and range acceleration (c).

Figure 10 .
Figure 10.Time series of APC offset pointing jitter coupling with subtracted mean value during two orbital revolutions for GRACE-FO A.

Figure 11 .
Figure 11.GRACE-FO laser ranging instrument optical layout (from Sheard et al., 2012).BS is beam splitter, CP is compensation plate, and TMA is triple mirror assembly.

Figure 12 .
Figure 12.ASD of laser frequency noise for range (a), range rate (b), and range acceleration (c).

Figure 13 .
Figure 13.Time series of laser frequency noise for range (a), range rate (b), and range acceleration (c).

Figure 14 .
Figure 14.Triple mirror assembly vertex offset from the satellites' centre of mass in two dimensions.
shows time series of TMA pointing jitter coupling for 1 month of GRACE-FO A.

Figure 15 .
Figure 15.Time series of TMA pointing jitter coupling with subtracted mean value during two orbital revolutions for GRACE-FO A.

Figure 16 .
Figure 16.Time series of ALQ pointing jitter coupling with subtracted mean value during two orbital revolutions for GRACE-FO A.
shows time series of ALQ pointing jitter coupling for 1 month of GRACE-FO A.

Figure 17 .
Figure 17.Differential wavefront sensing principle.Two beams of radius r with a relative wavefront tilt of α are detected by a quadrant photodetector.The two beams also have a slight frequency difference (from Sheard et al., 2012).

Figure 18 .
Figure 18.Simulated DWS pitch and yaw angles during two orbital revolutions for GRACE-FO A.