For further improvements of gravity field models based on Gravity Recovery and Climate Experiment (GRACE) observations, it is necessary to identify the error sources within the recovery process. Observation residuals obtained during the gravity field recovery contain most of the measurement and modeling errors and thus can be considered a realization of actual errors.

In this work, we investigate the ability of wavelets to help in identifying specific error sources in GRACE range-rate residuals. The multiresolution analysis (MRA) using discrete wavelet transform (DWT) is applied to decompose the residual signal into different scales with corresponding frequency bands. Temporal, spatial, and orbit-related features of each scale are then extracted for further investigations.

The wavelet analysis has proven to be a practical tool to find the main error contributors. Besides the previously known sources such as K-band ranging (KBR) system noise and systematic attitude variations, this method clearly shows effects which the classic spectral analysis is hardly able or unable to represent. These effects include long-term signatures due to satellite eclipse crossings and dominant ocean tide errors.

For more than 15 years, the Gravity Recovery and Climate Experiment (GRACE) satellite mission measured the time variation of Earth's gravity field with high temporal and spatial resolutions

Based on these observations, various time-variable gravity models with monthly resolution were published by different analysis centers

In recent years, significant research efforts have been made to identify and parametrize the systematic errors such as uncertainties in star camera alignment

If the calibration parameters are correctly adjusted and the stochastic model fully describes the observation noise, it is expected that all of the mentioned errors are completely contained within the residuals. In reality, however, these errors might affect the gravity parameters due to imperfections in modeling. Therefore, residual analysis becomes a research topic as it is not only a way to study measurement and physical modeling errors, but also helps to evaluate and improve the gravity field solutions.

The studies in this field have been conducted mainly on the theoretical residuals, which are the difference between the actual GRACE ranging observation and simulated observation computed through force models.

The main challenge in the spectral analysis of the residuals is that several noisy signals and disturbances are known to be superimposed at each frequency. Furthermore, the analysis is based on the assumption of the stationary behavior of these signals. However, in reality, most of these signals have nonstationary behavior, meaning that they have dynamic frequency components over time. Classical spectral analysis using Fourier transforms only represents the frequency content of such signals (Fig.

In an attempt to consider time variations in the sought-after signals, time–frequency methods can be applied to identify and localize the content of the nonstationary signals in the time and frequency domains simultaneously. The simplest method is the short-time Fourier transform (STFT), which is implemented by sliding a window throughout a signal and applying a Fourier transform to each windowed data segment. The squared magnitudes of the STFT coefficients form a spectrogram, representing the variation of the signal's spectrum over time (Fig.

To overcome STFT drawbacks, the wavelet analysis was introduced as a more effective technique for representation, decomposition, and reconstruction of nonstationary signals

The aim of this paper is to exploit the advantages of the wavelet transform to investigate the major contributors to GRACE range-rate residuals and ideally detect nonstationary noise sources in sensors and background models which cannot be observed with traditional spectral analysis. The results of this study will further improve gravity field modeling based on GRACE data. In addition, they will be beneficial for the preparation of GRACE Follow-On data processing infrastructure. To reach this goal, we decompose the residual signal into three groups of scale and compare the characteristics of each group with known or supposed sources.

In the upcoming Sect. 2, we explain how the residual signal is obtained in the frame of computing the ITSG-Grace2016 model and review the performed data processing steps in order to introduce potential error sources. Section 3 discusses the methodology of the multiresolution analysis and the wavelet transform. In Sect. 4, results of the employed method on the residuals are described. Finally, Sect. 5 presents the interpretation of results and a discussion.

In this study, we use GRACE range-rate residuals obtained in the course of computing the ITSG-Grace2016

In the ITSG-Grace2016 gravity field processing, high-precision kinematic orbits

Summary of ITSG-Grace2016 force models.

In the course of the adjustment process, nongravity parameters are also co-estimated for each day. These parameters include the initial orbit states of both satellites, accelerometer scale factor matrices, accelerometer biases modeled by cubic splines with 6 h nodes, and daily gravity field variations up to degree and order 40.

It is worth mentioning that unlike in the standard GRACE monthly solutions, in ITSG-Grace2016 the correlations between observations within a data block of 3 h are taken into account. For each observation type, a stochastic model of the observation noise is built under the assumption of stationarity. This model is estimated once per month directly from the observation residuals.

The weights for the different frequency components of the observations are determined through the residual power spectral density (PSD). This PSD is iteratively computed directly from the residuals through variance component estimation (VCE)

The wavelet transform

For the wavelet transform

The approximation of the signal at the scale

The original signal can be reconstructed by adding all layers of details up to decomposition scale

Mathematically, the convolution of the filter response with the discrete signal is expressed as follows:

The scaling function, defined by the filter coefficients

A fast inverse DWT reconstructs the initial signal

As described before, the DWT decomposes the original signal into an approximation subsignal and detailed subsignals. The MRA algorithm suggested by

Three-level MRA decomposition tree, consisting of a high-pass filter

We applied MRA using a discrete Daubechies wavelet transform with 20 vanishing moments

Daubechies-20

As shown in Fig.

short timescale details, containing the details at levels 1 to 3, corresponding to the frequency band above 12.5

medium timescale details, containing the details at levels 4 to 5, corresponding to the frequency range from 3.125 up to 12.5

long timescale details, containing the details at levels 6 to 8, corresponding to the frequency range from 0.391 up to 3.125

As mentioned in the first section, a spectrogram shows the variation of a signal's energy as a function of time and frequency. Another tool which can be used directly on the wavelet coefficients is the scalogram, in which the amplitude of the coefficients are plotted as a function of the scale and transition parameters. In our analyses, we used spectrograms because the interpretation of a signal in terms of frequency is more accessible than in terms of scale (Fig.

Plotting each time series with respect to the satellite ground track is useful to identify any features of geophysical origin in the data (Fig.

Plotting each time series as a function of satellite position and time reveals features related to the orbit geometry or instrument errors caused by orbital conditions. As the GRACE orbits are near-circular, the position of each satellite can be specified without loss of accuracy by the argument of latitude, ranging from

The proposed MRA scheme, implemented according to the characteristics of the residual signal.

The proposed MRA bandwidth division of the residuals with frequency sampling

Time–frequency analysis of

Spatial distribution of

Orbital analysis of

These analyses are carried out on the whole ITSG-Grace2016 time span (April 2002–June 2017). However, due to low data quality before 2004 and several data gaps and degraded quality of the measurements after 2016, these time periods are excluded from the illustrations. Highlights of this analysis are presented in the next section.

To prove whether or not our applied method using the DWT is applicable to detect the error sources, we initially focused on the investigation of known issues. For instance, it is known that the K-band system noise is dominant in the frequency range above 12.5

According to

Spectrograms of

These first investigations already show that our applied method is well suited to identify error sources. However, compared to the spectral analysis, the advantage of the implemented method of DWT is a better separation of superimposed signals in frequencies lower than 12.5

Analysis of the medium timescale details throughout the GRACE time span reveals long-term systematic signatures (Fig.

Each satellite passes through partial or full eclipse phases when it enters Earth's shadow. Occasionally the Moon also casts a shadow on the satellites. The eclipse factor is defined as the fraction of the Sun's light that reaches the satellite. It has a minimum value of zero if the satellite is in the umbra of the occulting body and a maximum value of one if the satellite is in direct sunlight. For a detailed calculation, the reader is referred to

The difference between GRACE-B and GRACE-A eclipse factors indicates if the mission, i.e., one of the satellites, is in a transit mode. Difference values not equal to zero are interpreted as transit events, in which one of the satellites is passing through a partial eclipse phase. Figure

The GRACE formation mission started with GRACE-A as leading and GRACE-B as trailing satellite. After 3 years in orbit, the satellites had to exchange their positions to limit the damage on the K-band horn caused by atomic oxygen. This swap maneuver happened at the end of 2005. Before this time, eclipse crossing signature occurs when the pair entered sunlight. After the orbit swap maneuver in December 2005, when GRACE-B became the leading satellite, the signatures are visible when the pair enters the shadow area.

However, after the year 2011, these rules cannot explain the eclipse crossing signatures in the residuals as they appear in both entering and leaving shadow conditions with different intensities. The unstable thermal condition due to the disabled thermal controls might be a possible reason.

We compared the temperature measurements obtained from Level-1A High-Resolution Temperature data (HRT1A) for November 2008 and October 2011 with these signatures. It becomes obvious that there is a high correlation between the GRACE-B K-band antenna horn temperature variation and the disturbances during eclipse crossing events (Fig.

Medium timescale details of the residuals compared to the GRACE-B K-band antenna horn temperature for

Errors in the background force models of temporal gravity field variations can be found in the long timescale details. Due to the spatial nature of these errors and the periodicity of satellite passes over their source regions, different model errors are superimposed at the same

The two main potential error sources at this scale are (a) inaccuracies in the employed ocean tide model EOT11a

Dynamic orbits are computed based on the background models mentioned in Table

Error-free observations for position, velocity, nongravitational accelerations, and the K-Band instrument are synthesized from these ideal orbits.

Realistic models of instrument noise are used to degrade synthesized observations. White Gaussian noise with a standard deviation of 3

The final step is to recover a monthly gravity field using the simulated degraded observations. To this end, the dynamic orbits are reintegrated using the artificially degraded accelerometer observations and the separate models under study, each in a dedicated scenario. The respective obtained residuals are then analyzed and compared.

In the first scenario, the same background models as mentioned in the first step of the simulation process are used to compute the reintegrated dynamic orbits. Therefore the results only show the effects of instrument noise. As expected, the propagated noise is 1 order of magnitude smaller than the real residuals in frequency range from 0.391 up to 3.125

The second scenario studies the propagated errors due to inaccuracies of the nontidal mass variation model. In order to recover a gravity field in this scenario, the AOD1B RL05 model and the van Dam–Ray atmospheric tide model

Spatial analysis of long timescale propagated errors from

In the third scenario, we study the contribution of the ocean tide model. To recover a gravity field in this scenario, the EOT11a ocean tide model is substituted for the FES2014 model. After decomposition of the simulated residual signal, its long timescale components are compared to the real data. These errors have comparable magnitude and spatial pattern (Fig.

These results showcase the capability of wavelet analysis in studying the signals due to geophysical processes in GRACE range-rate residuals. The implemented method efficiently finds structures in the signal which are not explicitly apparent in the PSD of the residuals. The wavelet analysis proves to be an efficient tool in decomposing the background model errors and finding the most prominent sources.

The results presented in this paper show the advantages of using a DWT in analyzing the range-rate residuals from the ITSG-Grace2016 gravity field model. Several improvements in ITSG-Grace2016 resulted in a cumulative noise reduction of 20 %–40 % compared to its predecessor ITSG-Grace2014. The proposed analysis framework confirms known and reveals previously unknown systematics in the residuals that allow for a specifically tailored parametrization in the gravity field retrieval.

We showed that the short timescale details of the residuals, equivalent to frequencies above 12.5

Besides the previously known instrument error sources, long-term signatures due to eclipse transits of the satellites were identified. They appear as a bias term in the K-band range-rate observations. As this is a clearly deterministic effect, its influence can be reduced by co-estimation of additional calibration parameters in the gravity field recovery process.

Analysis of the results from the implemented discrete wavelet transform brings new insights and a new understanding of the signals at the long timescale level. At this level, spectral analysis is unable to differentiate between the individual contributing sources, due to the nonstationary nature of the errors. Knowing that this scale level contains valuable information about the time-variable gravity field signal, we introduced nontidal mass variation and ocean tide models as the potential dominant sources. Comparing simulation results with the real data scenario, the EOT11a ocean tide errors are identified as the dominant error source within this scale. This means that using a more accurate ocean tide model can lower the residuals in this frequency band.

It has been shown that the wavelet-based MRA approach can properly represent the major error sources in GRACE processing data. These error sources have the largest impact on the accuracy of gravity field solutions derived from observations by GRACE. Even if the purpose of this study is to find the degrading factors in monthly gravity field models, which are mainly affecting the observations in the millihertz (mHz) frequency band, the investigation will be further continued by looking for physical interpretations for features at the lower frequencies of the residuals. This can be achieved by using a wavelet base with higher vanishing moments and thus higher decomposition level.

Besides the range-rate observations, the presented framework is also beneficial for the data processing of the other sensors aboard GRACE or similar satellite missions. The results can potentially detect inconsistent time periods in each set of measurements and provide an initial interpretation of their possible origin.

The GRACE Level-1B data (

SB and TMG developed and carried out the analysis. JF, BK, and SG provided reviews and suggestions on the GRACE instrument characteristics. SB was the lead author and all co-authors contributed to drafting and editing of the paper.

The authors declare that they have no conflict of interest.

We would like to thank Srinivas Bettadpur from the Center for Space Research, The University of Texas at Austin for providing us the GRACE Level-1A test datasets. We would also like to thank two anonymous reviewers for their suggestions and comments that contributed to improving the article.

This research has been supported by the German Research Foundation (DFG) (Relativistic Geodesy and Gravimetry with Quantum Sensors (geo-Q), grant no. SFB 1128).

This paper was edited by Lev Eppelbaum and reviewed by two anonymous referees.