Weather‐Induced Satellite Orbit Perturbations

Satellites in Earth's orbit are exposed to Earth radiation, consisting of reflected solar and emitted thermal radiation, thereby exerting a non‐conservative force that causes acceleration and affects the orbits. Gravity Recovery and Climate Experiment Follow‐On (GRACE‐FO) mission aiming to retrieve the Earth's gravity potential is critically dependent on accounting for this force and all other non‐gravitational forces. There are both diurnal and seasonal variations in the Earth's radiation pressure, of which the seasonal variability can be represented by climatology. Nevertheless, the daily variations in the Earth's radiation pressure, due to the transient changes in the weather; for example, clouds and their properties, are not accounted for in the orbit perturbations studies. We show here that the top‐of‐atmosphere radiation fluxes computed with a numerical weather prediction (NWP) model explain most of the measured short‐term variations in the radial acceleration of the GRACE‐FO satellite. Our physics‐based modeling corrects a hitherto unexplained lack of power spectral density in the measured accelerations. For example, we can accurately model the accelerations associated with a tropical storm in the Indian Ocean in December 2020, which would not be possible when using climatological data. Our results demonstrate that using a global numerical weather prediction model significantly improves the simulation of non‐gravitational effects in the satellites' orbits. In the 7‐day data set, OpenIFS‐simulated acceleration exhibited higher accuracy than climatological‐data‐simulated acceleration (2.5 compared to 2.6 nms−2) and an improved precision (2.6 compared to 3.0 nms−2). This advancement contributes to a more precise orbit determination across various applications in Earth sciences.


Introduction
In early December 2020, a storm was developing in the middle of the South Indian Ocean.Météo-France La Réunion issued a warning bulletin about a Tropical disturbance on the 6 December and named it the moderate Tropical storm "Bongoyo" on the seventh.The storm gained the status of a severe tropical storm on the eighth, but since it was far off the Cocos Islands, no threat to inhabited lands was foreseen.Météo-France and the Joint Typhoon Warning Centre in Pearl Harbor finally saw it off as a remnant low on 11 December.Bongoyo was an inconsequential tropical storm with few witnesses outside the weather monitoring community (Joint Typhoon Warning Center, 2020;Météo France, 2020).
Nearly 500 km above these atmospheric events, the GRACE-FO satellite soared right across Bongoyo at 3 p.m. local time on 6 December 2020.The underside of the vehicle was illuminated by solar radiation reflected from the tall convective clouds below, exerting a radiation pressure force on the vehicle.The satellite surged upwards, which the onboard accelerometers recorded as a radial acceleration anomaly reaching 10 nms 2 .The overpass across Bongoyo lasted about 2 min, during which the satellite orbit was gradually lifted by about 20 μm, assuming the absence of other effects.The otherwise inconsequential Bongoyo had perturbed the GRACE-FO orbital trajectory (Figure 1).GRACE-FO stands for Gravity Recovery and Climate Experiment Follow-On mission, launched in 2018 (Landerer et al., 2020).For accurate inference of the Earth's gravity potential and derived quantities, such as variations in terrestrial water storage, all non-gravitational effects on the satellite orbit need to be well understood and separated by physics-based modeling.A novel approach is taken here to model the influence of Earth radiation on the satellite orbit by using a global numerical weather prediction model of very high forecast skill, the Open Integrated Forecasting System (OpenIFS) model of the European Centre for Medium-Range Weather Forecasts (ECMWF), and their numerical analyses of global weather (Ollinaho et al., 2021).In weather models, the up-welling radiation flux, that is, reflected solar radiation and thermal emission, is modeled with reasonable precision on average.Usually, the up-welling radiation flux is relatively unbiased, but in the case of optically thick ice clouds, weather models tend to exhibit positive up-welling shortwave radiation flux bias of tens of W/m 2 (see e.g., Ham et al., 2022;Ren et al., 2023).In relative terms, this bias equals roughly 10% of the flux.
The radiation flux fields are available as global maps on an hourly basis (ECMWF, 2017).Since weather models only provide up/down fluxes, we assume here that each model grid cell is a Lambertian surface, providing isotropic angular distribution of reflected solar radiation and thermal emission.This allows computation of incident radiation on the space vehicle from every weather model grid point in the satellite's field-of-view below.Another novelty here is that the weather model is closely coupled with a precise orbit solver that is specifically designed for GRACE-FO satellite orbit determination, the Gravity Recovery Object-Oriented Programming System (GROOPS) of the Graz University of Technology (Mayer-Gürr et al., 2021).Finally, the simulated Earth radiation pressure-induced acceleration of the GRACE-FO satellite in radial, cross-track, and along-track directions can be compared with measurements from the hyper-sensitive accelerometers onboard the GRACE-FO satellite.This physics-based simulation of the GRACE-FO satellite acceleration measurements is well beyond the state-of-the-art, where the standard approach is to apply climatological Earth radiation data collected in the Clouds and the Earth's Radiant Energy System (CERES) project.The CERES data set lacks day-to-day variability of acceleration, whereas weather models explicitly simulate this variability.Consequently, we now have two data sets for testing: the CERES-based default data set and the weather model-based data set.

Material and Methods
The non-gravitational forces affecting the satellite are measured by the accelerometer mounted on GRACE-FO satellites.Positioned at the centre of mass of the satellite, the accelerometer measures acceleration along three axes: nadir, cross-track, and along-track.The precision of the GRACE-FO accelerometer is 0.1 nms 2 in the radial and along-track axes and 1 nms 2 in th cross-track axis.The acceleration the accelerometer measures primarily originates from Earth Radiation Pressure (ERP), Solar Radiation Pressure (SRP), and Atmospheric Drag, the latter resulting from the momentum exchange between the satellite's surface and the molecules and ions in the atmosphere.The simulation of non-gravitational acceleration requires a detailed model of satellite surface properties and their orientation in interaction with non-gravitational forces.This model is contained in the GROOPS toolkit (Mayer-Gürr et al., 2021) in the form of GRACE-FO macro model (Wen et al., 2019).In the macro model, the area of each satellite surface, its geometry, and its reflectance or absorption coefficient can be found, as well as different models used for simulating the forces; such as Knocke et al. (1988) for ERP, Lemoine et al. (2013) for and Sentman (1961) and Moe and Moe (2005) for Atmospheric Drag.In this study, we focus on ERP, which is small compared to direct solar radiation but highly variable in space and time.The neutral atmosphere does not reach the height of the GRACE-FO orbit; therefore, OpenIFS cannot be used to model atmospheric drag.Since the same model for the simulation of atmospheric drag is used for both OpenIFS-simulated acceleration and CERESsimulated acceleration, a comparison between the two simulations is possible.Now we will explain how the ERP modeling is done inside GROOPS.

ERP Modeling
In the model suggested by Knocke et al. (1988), the top-of-atmosphere (TOA) visible instantaneously from the satellite is divided into 19 elements, and the data related to the albedo and emission of each TOA element are required.The properties of each TOA element in the ERP model are referred to as "reflectivity" and "emissivity," which are coefficients a and e in Equation 1.
where dA is the area of each TOA element, θ denotes the angle of incident irradiance on the TOA element, α represents the angle to the satellite from the TOA element, A c indicates the cross-sectional area of the satellite, r represents the distance from the satellite to the element dA, and E s denotes the total solar irradiance (Wm 2 ).For each TOA element, cos α and cos θ must be positive for the satellite to be able to sense the radiation emanating from the element.

Reflectivity
For each TOA element, reflectivity is defined as the ratio of total shortwave (SW) outgoing flux (measured in W) to the total solar flux received by that TOA element.
where ϕ SW,out is the integral of reflected radiation of one element into space over a hemisphere and ϕ SW,in is equal to E s cos θdA.Equation 2 then will be The assumption here is that the Earth-atmosphere system acts as a perfectly diffuse reflector, exhibiting Lambertian characteristics.Consequently, to an observer, the Lambertian surface reflectance of the TOA is constant regardless of the angle of observation.

Emissivity
The ratio of longwave (LW) emittance of the element dA to the LW emittance of a Black Body is called emissivity.Equation 3reads where M LW is ϕ LWe,out dA and M BB , the black body emittance of the Earth-atmosphere system, is equal to E s 4 .Equation 4hence reads e = 4ϕ LW,out E s dA (5)

Radiation Fields
To determine emissivity and reflectivity coefficients required for the ERP model, it is common to rely on climatological Earth radiation data, often sourced from radiation measurement archives such as those provided by the CERES mission.CERES operates a series of space-based sensors equipped with scanning radiometers, which measure both the SW solar energy reflected by the planet (albedo) and the LW thermal energy emitted by it (Wielicki et al., 1996).Utilizing data from CERES enables the calculation of monthly mean Earth radiation estimates, which are derived from processed radiation measurements accumulated over extended periods, typically spanning years or decades.CERES offers valuable climatological insights; however, by construction, the CERES data set lacks variations related to daily weather events.
The novel solution we propose involves leveraging OpenIFS, a top-tier physics-based numerical weather prediction model.This weather model simulates Earth radiation, which can be interpreted as ERP and further be used to calculate the acceleration of the GRACE-FO satellites.OpenIFS is the atmospheric forecasting component of the Integrated Forecasting System (IFS) of ECMWF, and its source code is available upon request from ECMWF.
OpenIFS shares the fundamental components with IFS, including the dynamical core, physical parametrizations, land surface, and wave model, ensuring comparable forecasting capabilities to the corresponding IFS version.However, OpenIFS does not have the data assimilation capability.Therefore, we initialize OpenIFS using the initial states provided by ECMWF.
In this study, we utilize OpenIFS version 43r3v1, corresponding to IFS version 43r3, which was operational at ECMWF between July 2017 and June 2018.We conduct simulations with a model resolution of T L 639, which equals to approximately 32 km grid spacing at the equator, and 91 vertical levels.Radiative transfer is computed hourly on a coarser grid (T L 255) which corresponds to around 80 km grid spacing at the equator.Consequently, radiation quantities are computed approximately every sixth grid point and are interpolated back to the full model resolution (ECMWF, 2017).We utilize the following TOA radiation fields retrieved from OpenIFS.
• TOA incident solar radiation (TISR) (equivalent to E s cos θ) • Top net solar radiation (TSR) (equivalent to E s cos θ ϕ SW,out dA ) • Top net thermal radiation (TTR) (equivalent to ϕ LW,out dA ) Comparing these fields with the definition of emissivity and reflectivity (Equations 3 and 5) and utilizing the specifications in Hogan (2015), the coefficients a and e will be rewritten as: The fields TISR, TSR, and TTR are accumulated over an hour (J m 2 ).To obtain the non-accumulated data, these fields are divided by 3,600 s (resulting in units of Wm 2 ).The advantage of using these fields is that they are short weather forecasts affected by meteorological Earth observations.As a result, they capture the true evolution of weather.

Data
The simulation of non-gravitational accelerations was conducted using the radiation data and the satellite data from the first week of December 2020.Specifically, Level-1B products from the GRACE-C satellite (the leading satellite in GRACE-FO mission) were employed, including GNV1B, SCA1B, and THR1B data sets covering the same period.GNV1B, short for GPS (Global Positioning System) Navigation solution data, contains the initial orbit of the satellite.SCA1B provides details about the three Star Cameras used for attitude correction, while the Journal of Geophysical Research: Atmospheres 10.1029/2023JD040009 THR1B data contains the epochs corresponding to the firing of the thrusters.The thruster firings generate spikes in acceleration, representing residual linear accelerations resulting from imperfections in the attitude control thrusters.Each satellite is equipped with 12 10 mN cold gas attitude control thrusters.The data are sampled at a 5s interval and are available online.
The radiation fields for the corresponding dates were generated using OpenIFS at a spatial resolution of T L 639, providing global coverage through 2-dimensional fields.The temporal resolution of the data is 1 hour, with the mid-point of each time interval considered the best approximation.For comparison, climatological data from the CERES Level 3 ES-4 data set was utilized.

Results and Discussion
Figure 2 shows a simulation of the Bongoyo overpass on 6 December 2020, at 9 UTC (3 p.m. local time).The influences of SRP and atmospheric drag on radial acceleration simulated and subsequently their effect was removed from all the time series depicted in Figure 2. The methodology outlined by Lemoine et al. (2013) was followed for simulating SRP, while the simulation of atmospheric drag followed the approaches described in Sentman (1961) and Moe and Moe (2005).In the right panel, a map of the reflected solar radiation (represented by blue tones) is displayed, with the ground track of GRACE-FO overlaid.Measured accelerations are color-coded along the ground track, with pink tones indicating significant radial (upward) acceleration.Bongoyo, located approximately at (14S, 80E), lies just underneath the ground track, exerting very high reflected solar radiation flux (about 800 Wm 2 ) due to tall atmospheric convection with high cloud water content.
The middle panel presents a map of the thermal radiation (depicted by orange tones), where the weak thermal emission from the cold cloud-tops of Bongoyo (about 100 Wm 2 ) is distinct from its moist and warm surroundings, exhibiting much higher emission (about 250 Wm 2 ).On the left panel are the measured acceleration (represented by the black solid line) and simulations based on OpenIFS (orange) and CERES (blue) radiation fluxes.The measured acceleration of the GRACE-FO satellite exhibits short-lived "spikes," for instance, when crossing the equator.As mentioned in the Data section, these spikes are attributed to the thrusters, which can cause linear acceleration.The use of THR1B data enables the removal of these spikes from the time series.
The measured acceleration reaches a peak of approximately 28 nms 2 , with an anomaly of about 8 nms 2 coinciding with the GRACE-FO overpass of the Bongoyo storm.Simulation using the OpenIFS model remarkably reproduces the peak both in timing and amplitude, providing a convincing explanation for the acceleration anomaly observed in the time series of GRACE-FO measurement.It is worth stressing that the radiation maps (Figure 2) and the GRACE-FO measurements are entirely independent of each other.The default simulation using CERES radiative input exhibits the correct overall level of acceleration, but it is unaware of the Bongoyo storm, thus failing to provide any explanation for the observed anomaly.Furthermore, the radiation maps at full hours represent short-term forecasts initiated with atmospheric state estimates twice a day (00 and 12 UTC).In contrast, measurements, are continuous in time, and the orbital arches are plotted on the radiation maps at ±30 min around each full hour.
There are additional features captured by OpenIFS-simulated accelerations.After the Bongoyo overpass, as the GRACE-FO satellite orbits further south, it encounters a frontal cloud system associated with a mid-latitude cyclone at latitude 40S (Figure 2).These clouds signify the intrusion of moist air from the subtropics to the Antarctic, commonly referred to as an "atmospheric river," which transports large quantities of water vapor to the relatively dry Polar regions (Guan & Waliser, 2015).There is a sharp peak of approximately 25 nms 2 in the measured acceleration, with an anomaly of more than 5 nms 2 coinciding with the overpass of the frontal cloud bank.This event is almost perfectly captured by the OpenIFS simulation, while the default simulation with CERES data fails to detect it.
Reflected Solar radiation primarily drives weather-induced variations in the measured acceleration of the GRACE-FO satellite during the Bongoyo overpass (Figure 2).To illustrate the influence of thermal emission on the measured acceleration, it is instructive to examine the night side of the Earth.Figure 3 portrays the Bongoyo overpass occurring 12 hr earlier on 5 December 2020 at 21 UTC (at 3 a.m.local time).On the night side, the GRACE-FO satellite ascends towards the north crossing the terminator at approximately 80S.The radial acceleration rapidly diminishes as the satellite transitions from the day side to the night side.The Bongoyo overpass is characterized by a local minimum of 1-2 nms 2 in the measured acceleration (depicted by the black solid line in Figure 3).The OpenIFS-simulated acceleration effectively reproduces this depression attributed to the cold cloud tops of Bongoyo and the associated weak thermal emission.Generally, the acceleration variations on the night side due to thermal emission alone are subdued, typically on the order of only a few nms 2 , and exhibit a broader horizontal scale compared to those on the day side.Nevertheless, these acceleration variations provide additional detail compared to the default simulation using CERES radiance.In conclusion, thermal emission contributes to the acceleration variation on the night side of the Earth and modulates the sharper structures resulting from solar reflection on the day side.

Seven Orbits Comparison
Next, we will study the simulation performance across multiple orbits.Figure 4 shows the seven orbital revolutions of GRACE-FO observed on 1 December 2020.An orbit is defined as the period during which the argument of latitude completes a full circle from 180°to 180°.The plot represents the accelerometer data, the accelerometer data with thruster spikes removed, the CERES-simulated accelerations, and the OpenIFS-simulated accelerations, all with SRP and atmospheric drag removed.Notably, both the climatological data and the weather model data closely match the measured accelerations.However, while the climatological data provides a smoother representation of accelerations, the simulation using OpenIFS captures the high-frequency variations attributed to weather effects.Additionally, there exists a noticeable discrepancy or residual between the measured and simulated accelerations, which will be further explored and discussed in subsequent analyses.

Residuals Analysis
The differences between accelerometer data and OpenIFS-simulated acceleration (ACC-OIFS), as well as the differences between accelerometer and CERES-simulated acceleration (ACC-CERES) over half of the week (1-3 December 2020), are depicted in Figure 5. Considering the entire week (1-7 December 2020), the accuracy of ACC-OIFS, assessed with Mean Absolute Error (MAE), slightly surpasses that of ACC-CERES (2.5 nms 2 compared to 2.6 nms 2 ), while the precision or reproducibility of ACC-IFS, evaluated by standard deviation, is superior to that of ACC-CERES (2.6 nms 2 compared to 3.0 nms 2 ).
The ACC-CERES plot reveals unexplained large values acceleration residual pulses, reaching ±10 nms 2 , whereas in ACC-OIFS only periodic once-per-revolution variations in the acceleration residuals are observed, which rarely exceeds ±6 nms 2 .This discrepancy or residual arises from unmodelled accelerations that are still sensed by the accelerometer.The once-per-revolution residuals appear to be associated with the altitude of the satellite, as supported by Figure 6.These residuals may be attributed to modeling errors of the other forces; namely, atmospheric drag and solar radiation.Atmospheric drag increases when the satellite's altitude decreases, as it correlates with atmospheric density, which is higher at lower altitudes and can cause a decay in orbital height of 500 km by 150 m (Beutler, 2004).Additionally, other effects, including thermal re-radiation of the space vehicle and the associated acceleration, are not accounted for.2019) showed that for GRACE, the predecessor of the GRACE-FO mission, thruster spikes continue to impact the acceleration signal even after the application of a 35 MHz low-pass filter.The thrusters' effect can linger more than 70 s.
To visualize the distribution of the residuals ACC-OIFS and ACC-CERES, a box plot of the two data sets is presented in Figure 7.This plot illustrates a higher variability in the ACC-CERES residual signal compared to that of ACC-OIFS.The unexplained accelerations in the CERES simulation result in a greater number of outliers in ACC-CERES.In contrast, ACC-OIFS displays a narrower range of variability in residuals, with only a few outliers at the lower boundary.Despite the slightly higher median of the residuals for ACC-OIFS compared to ACC-CERES, the narrower spread range and lower number of outliers favor the OpenIFS simulation.

Spectral Analysis
Based on these examples, it is evident that the weather-induced acceleration variations are more accurately captured by the OpenIFS model than by the CERES radiance.This qualitative observation is further quantified in Figure 8, which illustrates power spectral densities of the measured (black solid line) and the simulated acceleration time series using OpenIFS (orange solid line) and CERES radiance (blue solid line).In this analysis, the spectral window length (Welch, 1967) corresponds to the GRACE-FO satellite's orbital period of 94.5 min, and the time series covers 1 week (1-8 December 2020), totaling 106 orbits.
Weather-induced acceleration variations primarily occur within periods ranging from a couple of minutes to about 15 min, corresponding to the time it takes for GRACE-FO to pass over individual weather systems.Evidently, the OpenIFS simulation spectral density is an order of magnitude larger than that of CERES in the 2-15 min range.Comparing the power spectral density of both to that of the measured shows that highly variable accelerations are indeed induced by weather.It can be concluded that the OpenIFS simulation more accurately represents the measured spectral density within this period  range and exhibits a more realistic level of spectral density compared to the CERES simulation.In the measured time series, variability in sub-minute periods includes effects such as steering maneuvers and unknown sources.Notably, the CERES spectrum displays a spike at a period of approximately 40 s (and at the associated sub-harmonics), the cause of which remains unknown.

The Acceleration Peaks
So far, it is clear that the peaks in the acceleration time series are related to weather events and the associated variations in surface temperature and cloud properties.Hence, we study these peaks and the related statistics.Peaks are defined as the local maxima in measured accelerometer time series, with the amplitude of at least 0.5 nms 2 .
For 1-8 December 2020 a total of 474 peaks are found, and their strength, location, time of occurrence, and spatial extent are extracted.The distribution of these peaks can be seen in Figures 9 and 10.The peaks are binned based on their spatial extent (in km) and separated based on their location; that is, whether they happen in the tropical or extratropical regions, and their strength (or amplitude); that is, whether they are strong (above 10 nms 2 ), moderate (between 5 and 10 nms 2 ), or weak (lower than 5 nms 2 ).287 peaks occurred on the day side, whereas 187 peaks were seen on the night side.The higher number of peaks on the day side indicates the bigger effect of SW radiation on the orbital perturbations of the GRACE-FO satellite.
Typically, the stronger peaks and moderate peaks have happened on the day side, while on the night side, there are only two moderate peaks and no strong peaks.Some of the daytime peaks with a large horizontal extent might be associated with stratocumulus clouds, which strongly reflect the SW radiation.We refer to Chen et al. (2000) regarding how different cloud types affect the radiation in overcast and in realistic cases at different latitudes.The annual, zonal mean effect of stratocumulus clouds on the net top-of-atmosphere SW radiation varies substantially in the meridional direction.However, as shown by Wood (2012), locally, this variation can be even bigger.For example, the variation is particularly strong in the meridional direction over the eastern parts of the low-latitudeto-subtropical oceans and nearby land regions (Figure 4 in Wood ( 2012)), which can cause moderate peaks.Nonetheless, most of the peaks (400 out of 474) are weak.The night side peaks, caused by LW radiation, originate both from surface thermal emission and weather events, that is, variations in atmospheric temperature, humidity, and clouds.The simulated acceleration is a modulation of these two radiation sources.Besides, based on Figures 9 and 10 the peaks with a spatial extent of about 2,000 km are the most frequent; hence, we can identify this value as the typical length of the satellite trajectory affected by ERP and link it to the weather events in the atmosphere.Considering the ground velocity of the satellite, which is 7.5 km/s, the typical time length of the orbit perturbation caused by weather events is about 5 min.
However, the spatial extent of the peaks (equivalent to their timescale) does not exactly correspond to the spatial scale of weather patterns, since the satellite receives radiation also before and after the overpass.For example, the overpass of the Bongoyo storm at 9 UTC on 6 December 2020, is about 1,000 km wide (about 10 degrees of latitude), while the measured peak is about 2,000 km wide.In other words, the time it takes for GRACE to pass over this storm is about 2 min, but it senses the acceleration caused by the storm for about 5 min.Also, thermal emission and reflection from underlying surfaces in cloud-free areas, such as deserts of high albedo and warm ocean areas, modulate the response.

Conclusion
Numerical weather prediction models provide accurate information about the top-of-atmosphere radiation fluxes, which can be harnessed to simulate Earth radiation pressure acting on Earth-orbiting satellites.Here, the GRACE-FO satellite non-gravitational acceleration measurements are simulated using a global weather prediction model, OpenIFS, and global weather analyses by ECMWF.From the 1-week data set (1-8 December 2020) we compared the time series of measured and simulated acceleration.Compared to CERES data, OpenIFS turns out to be very powerful in explaining the measured acceleration anomalies.
The strongest acceleration peaks anomalies (above 5 nms 2 ) occur in the Tropical regions during the daytime, due to strong incident solar radiation and tall convective clouds with high water content.For example, the high SW radiation originated from the Bongoyo storm on 6 December 2020 is clearly visible from measured and simulated accelerations time series.
In particular, the missing variability in measured acceleration in the timescale of 2-15 min (in the satellite's frame-of-reference) is now quantitatively explained as compared to earlier simulations based on the CERES radiance information.Since the recovery of the Earth's gravity potential-the chief goal of the GRACE-FO mission-is sensitive to the simulation of non-gravitational effects in the satellite orbit, we foresee physicsbased weather modeling to provide a fresh perspective to improve the retrievals of such science missions, including GRACE-FO.

Figure 1 .
Figure 1.A cartoon about the influence of tropical storm "Bongoyo" on the GRACE-FO satellite orbit via the reflection of solar radiation and thermal emission (Hurricane Felix from Iss, 2007).

Figure 2 .
Figure 2. The simulation of the Bongoyo overpass on 6 December 2020 at 9 UTC (3 p.m. local time by the storm).Displayed are the reflected solar radiation (map with blue tones; unit Wm 2 ) and the thermal radiation (map with orange tones; unit: Wm 2 ), both based on the OpenIFS weather model and the acceleration time series of measured (black solid line; unit: nms 2 ) and simulated accelerations based on OpenIFS (orange solid line) and CERES radiation data (blue solid line).In the maps, the measured acceleration is color-coded to the ground track, such that pink tones are indicative of large radial acceleration upwards.The Bongoyo storm is at the approximate location of (14S, 80E).

Figure 3 .
Figure 3.As Figure 2 but depicted 12 hr earlier during the previous night over the Indian Ocean on 5 December 2020 at 21 UTC (3 a.m. by the Bongoyo storm at the approximate location of 14S, 80E).

Figure 4 .
Figure 4. Time series of measured acceleration, measured acceleration with thruster spikes removed, CERES simulation, and OpenIFS simulation during seven orbital revolutions of GRACE from 1 December 2020 1:00 to 1 December 2020 12:00.The direct solar effect and atmospheric drag are removed from all the time series.

Figure 7 .
Figure 7.The box plots of residuals of ACC-OIFS (left) and ACC-CERES (right) from 1 December 2020 1:00 to 7 December 2020 23:59.The boxes show the InterQuartile Range (IQR), the distance between the first and third quartile of data distribution (Q3 Q1), the horizontal lines are the medians, the whiskers extend from the minimum to the first quartile, and from the third quartile to the maximum.The minimum and maximum are defined as Q1 1.5 * IQR and Q3 + 1.5 * IQR, respectively.The data points outside this range are known as outliers.

Figure 8 .
Figure 8. Power spectral densities of the measured (black solid line) and the simulated acceleration time series with OpenIFS (orange solid line) and CERES radiance (blue solid line).The window length corresponds to the GRACE-FO satellite's orbital period of 94.5 min.The time series is from 1 to 8 December 2020, totaling 106 orbits.The time series are cleaned from the influences of direct Solar radiation and atmospheric drag.

Figure 9 .
Figure 9.The distribution of maxima in the acceleration time series (on the day side).

Figure 10 .
Figure 10.The distribution of maxima in the acceleration time series (on the night side).