# An approach for estimating time-variable rates from geodetic time series

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## Abstract

There has been considerable research in the literature focused on computing and forecasting sea-level changes in terms of constant trends or rates. The Antarctic ice sheet is one of the main contributors to sea-level change with highly uncertain rates of glacial thinning and accumulation. Geodetic observing systems such as the Gravity Recovery and Climate Experiment (GRACE) and the Global Positioning System (GPS) are routinely used to estimate these trends. In an effort to improve the accuracy and reliability of these trends, this study investigates a technique that allows the estimated rates, along with co-estimated seasonal components, to vary in time. For this, state space models are defined and then solved by a Kalman filter (KF). The reliable estimation of noise parameters is one of the main problems encountered when using a KF approach, which is solved by numerically optimizing likelihood. Since the optimization problem is non-convex, it is challenging to find an optimal solution. To address this issue, we limited the parameter search space using classical least-squares adjustment (LSA). In this context, we also tested the usage of inequality constraints by directly verifying whether they are supported by the data. The suggested technique for time-series analysis is expanded to classify and handle time-correlated observational noise within the state space framework. The performance of the method is demonstrated using GRACE and GPS data at the CAS1 station located in East Antarctica and compared to commonly used LSA. The results suggest that the outlined technique allows for more reliable trend estimates, as well as for more physically valuable interpretations, while validating independent observing systems.

## Keywords

Time-variable trend Kalman filter Non-convex optimization problem Colored noise## 1 Introduction

Antarctica is one of the largest contributors to global sea-level rise (IPCC 2014), making the knowledge of its ice-mass change of considerable societal and ecological importance. The GRACE satellite gravimetry mission has provided an extremely useful tool for observing the global integral effects of mass changes. Nevertheless, Antarctic ice-mass change estimates derived from GRACE remain challenging due to, among others, the effect of glacial isostatic adjustment (GIA, Velicogna and Wahr 2006). GPS vertical site displacements are gaining importance in constraining GIA-induced rates in Antarctica (Whitehouse et al. 2012; Ivins et al. 2013; Sasgen et al. 2013; van der Wal et al. 2015).

Normally, all the above-mentioned processes related to sea-level involving ice mass and GIA rates are estimated as constant trends along with deterministically modeled seasonal components (e.g., Velicogna et al. 2014; Gunter et al. 2014; Shepherd et al. 2012) without allowing for inter-annual and seasonal variability, which might not have captured the true trend estimates (Davis et al. 2012). Accurately modeling known sources of temporal variation is crucial for interpreting geodetic data properly, especially because of large inter-annual variations in the Antarctic climate (Ligtenberg et al. 2012). Moreover, very few geophysical processes are exactly periodic; instead there are signal constituents which fluctuate around a reference value, e.g., around a 1-year period with slightly varying amplitudes. Therefore, modeling seasonal processes using traditional deterministic fitting methods may not provide very accurate results. In this study, we model them stochastically within a KF framework allowing for physically natural variations of signal constituents in time. This idea was brought to the geodetic community by Davis et al. (2012) while being a well-established technique in econometrics since the 1980s (Harvey 1989). However, Davis et al. (2012) assumed the statistical noise parameters to be known. Moreover, the econometric literature lacks methods for a robust estimation of the noise parameters as the optimization problem to be solved for those parameters turns out to be non-convex (i.e., there can be multiple local minima).

Therefore, the goal of this study is to provide a robust tool for estimating time-variable trends from geodetic time series. For this purpose, detailed descriptions are provided on how different components, such as trend and known periodicities, can be modeled stochastically and put into KF form (Sect. 2). Special attention is paid towards carefully estimating the noise parameters, which is an essential step in the KF. The presented statistical framework is appropriate to any time series, but is demonstrated in this study on GRACE and GPS time series that have been widely used in the context of trend estimation (Sect. 3). A spectral analysis of the results shows that the developed tool yields more reliable estimates compared to those derived from commonly used LSA. Moreover, the technique presented allows different geodetic time series to be analyzed for validation purposes.

## 2 Methodology

The theory described below is largely based on Durbin and Koopman (2012) and Harvey (1989).

As we demonstrate the methodology on GRACE and GPS data, Sects. 2.1–2.4 are relevant for both types of datasets, whereas Sect. 2.5 is devoted to the analysis of features typical of GPS time series. Section 2.6 summarizes the major steps of the time-series analysis by the suggested method.

### 2.1 Trend modeling

*integrated random walk*. The larger the variance \(\sigma ^2_{\zeta }\), the greater the stochastic movements in the trend. In other words, \(\sigma ^2_{\zeta }\) defines how much the slope \(\beta \) in Eq. (3) is allowed to change from one time step to another.

*c*and

*s*and hence the corresponding amplitude and phase to evolve over time. Note that \(c_t\) in Eq. (5) is the current value of the harmonic signal and \(s_{t-1}\) appears by construction to form \(c_t\).

### 2.2 State space model

*Z*links \(y_t\) to \(\alpha _t\), is called the observation equation and the second is called the state equation. Any model that includes an observation process and a state process is called a

*state space model*. The observation equation has the structure of a linear regression model where the vector \(\alpha _t\) varies over time. The second equation represents a first-order vector autoregressive model. The transition matrix

*T*describes how the state changes from

*t*to \(t+1\), and \(\eta _t\) is the process noise with \(Q=I\sigma ^2_{\eta }\). The initial state \(\alpha _1\) is \(N(a_1,P_1)\) where \(a_1\) and \(P_1\) are assumed to be known.

*Z*,

*T*,

*R*,

*H*, and

*Q*are independent of time. Therefore, the corresponding index

*t*is dropped out hereinafter. Another reason for not including any time reference is that we use equally spaced data. It is worth pointing out that a state space model can also be defined for time series containing data gaps or for unevenly spaced time series. While dealing with missing observations is particularly simple as shown in Durbin and Koopman (2012, chap. 4.10), some modifications might be required for unevenly spaced time series depending on the complexity of the desired state space model (Harvey 1989, chap. 9).

### 2.3 Kalman filter and smoother

*t*

### 2.4 Estimation of hyperparameters

#### 2.4.1 Optimization

The IP algorithm is used because it accounts for a potential non-convexity, and the problem we are dealing with is non-convex. If an optimization problem is non-convex, there can be multiple local minimum points with objective function values different from the global minimum (Horst et al. 2000). Finding a globally optimal solution of a multivariate objective function that has many local minima is very challenging. One of the main difficulties is the choice of the initial guess for the starting point \({\psi _{0}}\) (initial solution) that is required for the optimization. If the initial guess is sufficiently close to a local minimum, the optimization algorithm terminates at this local minimum (Fig. 1). Visualizing the objective function is helpful to choose a suitable initial guess, but the problem we are describing is at least four-dimensional. Dimensionality may further increase, for instance, if other periodic constituents are considered (e.g., the S2 tidal alias in GRACE data analysis); another example of a higher dimension is discussed in Sect. 2.5.3. Therefore, our approach is to compute the objective function for a number of starting points and use the solution in further computations that provides the smallest objective function value and thus is more likely to be a global minimum (Anderssen and Bloomfield 1975). The question, however, is how to define suitable starting points that allow all or as many as possible local minima to be identified, which in turn will increase the probability of finding the global minimum. For this, a set of uniformly distributed starting points is randomly generated within a finite search space. As a result, the same optimal solution is obtained after each run despite the fact that the method is heuristic, ensuring the existence of an optimal solution within the predefined bounds.

#### 2.4.2 Limiting the parameter space

In the following, we limit the parameter search space in the context of a non-convex optimization problem to improve the chance of finding a global optimum. First, all lower bounds are set equal to zero. The upper bounds are chosen from LSAs to the given data as follows. We fit the model described by Eq. (1) to the data, and use the variance of the postfit residuals as an upper bound for \(\sigma ^2_{\varepsilon }\) in Eq. (9). This choice is justified, since LSA-residuals contain the unmodeled signal, measurement noise and possible fluctuations in the modeled terms (in our case in trend, annual and semi-annual components), whereas \(\sigma ^2_{\varepsilon }\) in Eq. (9) does not include possible fluctuations in the modeled terms, because we model them stochastically as described in Sect. 2.1. Similarly, the upper bounds for annual and semi-annual terms are found. After subtracting a deterministic trend from the time series, annual and semi-annual signals are simultaneously estimated using LSA within a sliding window that has a minimum timespan of 2 years. The maximum size of the sliding window corresponds to the length of the time series used. Done this way, a sufficient amount of annual and semi-annual amplitudes are estimated and the corresponding variances are used as upper bounds for \(\sigma ^2_{\varsigma _1}\) and \(\sigma ^2_{\varsigma _2}\), respectively. The choice of the upper bounds is justified by the fact that the standard deviation of the signal computed for different time intervals is never smaller than the process noise of this signal, since here standard deviations indicate possible signal variations within the considered time span, whereas process noise represents the signal variations from one time step to the next only. Moreover, these upper bounds still include possible variations within the trend component supporting the idea of being the upper limits for the process noise associated with estimated harmonics. Regarding the process noise associated with the trend component \(\sigma ^2_{\zeta }\), no upper bound is set.

By bounding the search space for \(\psi \) in the manner described above and by setting the amount of start points to 200 (chosen by trial and error), we obtain after each run numerically the same optimal solution. To substantiate the reliability of the estimated hyperparameters, we additionally analyze the amplitude distribution of the estimated signal constituents (Eq. 8) as a function of frequency. Investigating whether the amplitude spectrum shows a peak around the expected frequency allows us to draw conclusions on the reasonableness of the estimated noise parameters, since they determine the estimation of the signal constituents.

To illustrate the idea of the analysis in the spectral domain, an example based on GPS time series, which will be described later, is presented in Fig. 2. To produce this figure, we first estimated noise parameters stored in \(\psi \) (Eq. 17) with and without limiting the parameter space for \(\sigma ^2_{\varepsilon }, \sigma ^2_{\varsigma _1}\) and \(\sigma ^2_{\varsigma _2}\). For these two cases, we then estimated the state vector \(\alpha _t\) and computed the amplitude spectrum for the rate \(\beta _t\), annual \(c_{1,t}\) and semi-annual \(c_{2,t}\) estimates. Figure 2a provides an indication of reasonably estimated hyperparameters, since the amplitude spectrums of the corresponding signal estimates show significant peaks over the expected frequencies and there are no significant peaks elsewhere. For comparison, Fig. 2b provides an example generated without limiting the parameter space, where the hyperparameter associated with the annual signal is overestimated including variations of the rate/slope component while the amplitude of the slope has an unrealistically small magnitude of zero mm. This example also emphasizes the importance of limiting the parameter search space within a non-convex optimization problem.

The solution we obtain for the hyperparameters \(\psi \) is referred to as an unconstrained solution hereinafter, since only the search space for the global solver has been limited, but no restrictions are applied yet to the parameters themselves.

#### 2.4.3 Constrained optimization

Introducing constraints on some of the noise parameters may improve the chance of finding a global minimum within a non-convex optimization. Sometimes, we have prior knowledge about some noise parameters, e.g., we know that \(\sigma ^2_{\varepsilon }\) must be larger than some threshold. This inequality constraint can be easily applied within the numerical optimization (Nocedal and Wright 2006). However, if the introduced constraints are not supported by the data, applying them may significantly change the estimated noise parameters and, in turn, the estimate of the state vector \(\alpha _t\), yielding erroneous geophysical interpretations. As we are dealing with a non-convex problem, the testing procedure proposed in Roese-Koerner et al. (2012) cannot be applied. Therefore, we outline a method to verify whether the data support the applied constraints, paying particular attention to non-convexity.

By doing so, we avoid using constraints that are too strong and not supported by the data, but still try to find a compromise between a statistically based and a physically meaningful estimate.

### 2.5 GPS

The analysis of GPS time series often differs substantially from that of GRACE data. GRACE time series have a sampling period of typically one month, data gaps are sparse, and noise correlations between the monthly data (if there are any) are negligible. GPS data are known to contain colored (temporally correlated) observational noise that cannot be neglected (Williams 2003a). Moreover, GPS time series are frequently unevenly spaced in time and may contain large data gaps as well as outliers. In the following sections, we describe how we handle these different features present in the GPS data.

#### 2.5.1 Pre-processing

A KF can easily deal with unevenly distributed observations. However, equally spaced data will be beneficial when we later define the state space model for temporally correlated noise. Therefore, we generate equally spaced data by filling short gaps with interpolated values and long gaps with NaN values. We define a gap to be long if more than seven consecutive measurements are missing, i.e., more than 1 week of daily GPS data.

Since the KF is not robust to outliers, they should be removed beforehand. Outliers are detected here by a Hampel filter according to Pearson (2011). The measurements are removed from the time series where horizontal or vertical site displacements of a GPS station were identified as outliers.

#### 2.5.2 Colored noise

*p*,

*q*). The ARMA process is defined as

*p*; if \(p = 0\), the process is a moving-average (MA) process of order

*q*.

The postfit residuals obtained after fitting a deterministic model to the data represent colored noise. It is important to understand that it is only an approximation of the observational noise, since the residuals contain a potentially unmodeled time-dependent portion of the signal. To parameterize this approximate colored noise using an ARMA(*p*, *q*) model, we need to determine how *p* and *q* should be chosen. For this, we follow the idea of Klees et al. (2003) and use the ARMA(*p*, *q*) model that best fits the noise power spectral density (PSD) function. Thus, using the PSD function of the approximate colored noise we estimate the pure recursive part of the filter (MA) and non-recursive part of the filter (AR) by applying the standard Levinson–Durbin algorithm (Farhang-Boroujeny 1998). The parameters of the MA and AR models are computed using a defined *p* and *q*, which are then used to compute the PSD function of the combined ARMA(*p*, *q*) solution. To control the dimension of the state vector \(\alpha _t\) we limit the maximum order of the ARMA process to 5, which means we compute the PSD for ARMA(*p*, *q*) generated for \(p,\,q \in \{0, \ldots , 5\}\) (including two special cases AR(*p*) and MA(*q*)). Then, we use GIC (Generalized Information Criterion) order selection criterion to select the PSD of the ARMA model that best fits the PSD of the approximate colored noise. The *p* and *q* of this ARMA model define the number of \(\phi \) and \(\theta \) coefficients used to parameterize colored noise \(\varepsilon _t\). More details about the use of ARMA models in the context of GPS time series can be found in accompanying Supplement.

#### 2.5.3 State space model

*k*offsets, the state vector can be written as

*p*,

*q*) process for models of order \(p > 2\) into state space form. Therefore, the data were pre-processed as outlined in Sect. 2.5.1.

*k*offsets (Eq. 25) and of the “shaping filter” (Eq. 26) with the basic model defined in Eq. (8) (hereafter \(\alpha _t\) used with the index

*b*for

*basic*), we take the state vector as

*Z*,

*T*and

*R*being defined in Eqs. (9)–(10).

### 2.6 Summary of the developed framework

The flow diagram in Fig. 3 outlines the major steps of the time-series analysis by the suggested method. The method can be applied to any equally spaced data; it can cope with missing observations and different stochastic properties of the data. Once the components of interest are defined in the state vector, the corresponding state space model with all required matrices can be formulated. If present, time-correlated observational noise can be modeled using a general ARMA model that subsumes two special cases (AR and MA) as described in Sect. 2.5.3 or in more detail in the accompanying Supplement. Another representation of the colored observational noise within the state space formalism can be found in, e.g., Dmitrieva et al. (2015), in which a linear combination of independent first-order Gauss–Markov (FOGM) processes is used to approximate the noise.

Once in the state space form, the parameters governing the stochastic movements of the state components are estimated by numerically optimizing likelihood. The likelihood function is computed using the by-products of the Kalman filter (Eq. 16). Finding an optimal solution as demonstrated in Sect. 2.4 is the key of the proposed methodology, since it ensures optimal estimates for the hyperparameters, which in turn determine the estimates of the signal constituents. Limiting the parameter search space (Sect. 2.4.2), as well as imposing constraints (Sect. 2.4.3) that are supported by the data, both increase the likelihood of getting the optimal solution. Once the hyperparameters are estimated, the Kalman filter and smoother can be used (Sect. 2.3) for obtaining the best estimate of the state at any point within the analyzed time span. This can be important for investigating the way in which a component such as trend has evolved in the past.

## 3 Application to real data

### 3.1 Data

The GPS data at CAS1 are processed similar to Thomas et al. (2011). The GPS time series contains two step-like offsets (in Oct. 2004 and Dec. 2008) within the chosen estimation period, which is Feb. 2003 to Dec. 2010. For the same period, GRACE monthly time series are computed at Delft University of Technology (Farahani 2013) complete to spherical harmonic degree/order 120 and optimally filtered using a Wiener filter (Klees et al. 2008). Stokes coefficients representing the monthly gravity fields were converted into vertical deformations following Kusche and Schrama (2005) making GRACE data comparable with GPS observations. This conversion is done for potential validation which, as will be shown later, leaves room for physical interpretations if the proposed methodology is applied.

### 3.2 Results

Results derived by modeling signal constituents stochastically within the KF framework are called hereinafter KF results for brevity. We show plots in the time and frequency domain for GPS and GRACE time series at the same geolocation. Both time series represent vertical deformations due to GIA and the elastic response of the solid Earth to the surface load. Before discussing the results it is worth noting that what is called *trend* (in mm) thereafter is the integrated random walk part of the signal (\(\mu _{t}\) in Eq. 8) with deterministically modeled intercept and time-varying *slope* (or rate) in mm/year introduced as \(\beta _{t}\) in Eq. (8).

For GRACE time series, we estimate the slope, and annual, semi-annual and tidal S2 periodic terms deterministically using LSA and stochastically using the KF framework. In both cases, the intercept is co-estimated deterministically. Figure 4 shows vertical deformation derived based on GRACE data, the LSA fit and the KF fit, as well as estimated trends using different techniques. Error bars represent one-sigma uncertainties. Figure 4a, b serves as a visual inspection and indicate that the model which allows signal components to vary in time represents the data considerably better than the model that assumes a linear trend and exactly periodic processes with constant amplitudes.

*I*, which has a dimension of the estimated state vector. To compute the uncertainty estimates from LSA, formal errors were rescaled by the a posteriori variance. This is a commonly used approach (e.g., Baur 2012) which yields over-optimistic uncertainties (e.g., Williams 2003a).

In the context of slope estimation, we find it worth noting that, especially for Antarctic GPS site velocities that are used to constrain GIA rates, each erroneously estimated millimeter of vertical deformation corresponds to significantly erroneous ice-mass change estimates (Gunter et al. 2014) highlighting the need to estimate these rates as accurately as possible.

To validate the results based on the proposed methodology from the geophysical point of view, we plot the estimated time-varying rates derived from GPS and GRACE time series, respectively, in Fig. 9. The known accumulation anomaly event from 2009 is clearly evident. In this year, GPS and GRACE observe maximum subsidence of the solid Earth as an immediate response to the high levels of accumulation within the analyzed time period. Although the two observing systems do not agree perfectly, they do observe similar processes starting from 2005. In fact, there are a number of different factors to be considered when comparing GPS and GRACE time series, such as the spatial resolutions of the data sets (GPS-derived deformations are discrete point measurements, while GRACE results represent a spatial average), the effects of geocenter motion should be considered when converting GRACE coefficients into vertical deformation, etc. Though the validation of different geodetic observing techniques is beyond the scope of this study, we feel the proposed methodology provides better interpretation opportunities (Fig. 9) than the traditional LSA approach. It should also be noted that once GRACE and GPS time series are corrected such that they represent the same signal, it is straightforward to combine them within the described approach. However, the GRACE and GPS time series are used in this study only to validate the proposed methodology, and their data combination is also beyond the scope of this paper. It is also worth mentioning here that we have chosen this GPS station because of the existing prior knowledge about the geophysical process (accumulation anomaly) that took place there in 2009. Two different observing systems, GPS and GRACE, detected this geophysical process because of its high magnitude. While estimating time-variable rates, the time series from these two different observing systems were treated in two different ways with respect to the observational noise model used: white noise for GRACE and colored noise for GPS time series. Nonetheless, the time-varying trends derived from the GRACE and GPS time series show the same behavior. We therefore interpret this behavior as a signal and not as potentially mismodeled observational noise.

The target of this study is to provide a robust tool for reliable trend estimation. The robustness of the proposed methodology is determined by finding an optimal minimum that is necessary for estimating the noise parameters (Sect. 2.4) which, in turn, are the key for reliable rate estimates. To demonstrate the role of the noise parameters on the estimated signal components, we use the example shown in Fig. 2. Based on the GPS time series, we estimate the noise parameters by limiting the parameter search space for finding an optimal solution (as it is done through this section) and without limiting the parameter search space. Using these differently estimated noise parameters, we estimate modeled signal constituents. In Fig. 10, we illustrate the results for the slope and the annual component in the time domain (there is no evident difference in the semi-annual component, as can be seen in Fig. 2). By limiting the parameter search space, the process noise for the slope and annual component is estimated to be 0.37 mm/year and 0.06 mm, respectively. The corresponding estimates are shown in Fig. 10a suggesting a correlation between both the changes in the rates of vertical deformation and their annual variability. This is physically reasonable, as both are responses to the changing climate.

If the parameter search space is not limited, the process noise for slope and annual signal is \(5.75\times 10^{-8}\) mm/year and 0.34 mm, respectively. Figure 10b shows the corresponding plots. Because the slope is not allowed to vary much, it is comparable with the LSA estimate shown in Fig. 7b. However, the variance of the annual component is much higher than the one used in Fig. 10a, which is why the corresponding annual amplitude in Fig. 10b shows an erratic behavior.

We could also assume the noise parameters to be known, e.g., by modeling the slope deterministically and using a fixed standard deviation for the annual signal. The higher we set this standard deviation the more we force the annual signal to absorb long-term variations and possible variations originating from other sources, yielding wrong interpretations. Therefore, we recommend to limit the parameter search space as described in Sect. 2.4.2 and to verify potentially existing prior knowledge about noise parameters according to Sect. 2.4.3 to ensure the reliability of the estimated signal constituents. Moreover, we suggest modeling all signal components stochastically to ensure a reliable noise parameter estimation, unless there are good reasons not to do so.

## 4 Conclusions

We developed a robust method for estimating time-variable trends from geodetic time series. This method is more sophisticated compared to commonly used LSA, as it allows the rate and seasonal signals to change in time. The advantages are twofold: more reliable trend estimation, because (i) there is no contamination by seasonal variability and (ii) it accounts for any long-term evolution in the time series, which would appear as noise when modeled as a time-invariant slope.

The right choice of the noise parameters is at the heart of the proposed methodology. We suggested a method which allows a robust estimation of the noise parameters. We verified the reliability of the estimates using spectral analysis. The plausibility of the estimated time-varying rates was additionally confirmed by existing geophysical knowledge. Furthermore, the results estimated using the KF framework were visually compared with those derived using LSA in the time and frequency domains. Visual inspections and RMS misfits suggested that the KF outperforms LSA. The proposed methodology is not limited to GPS and GRACE time series, but can be used for any other time series.

Our results suggest that potential changes in rates may yield significantly different trends when post-processed compared to the deterministic linear trend. Indeed, the longer the time series, the more deviations can be expected from the deterministic linear trend assumption as well as from the constant seasonal amplitudes and phases. Moreover, any change in the trend term reflects an acceleration, making the stochastic approach much more flexible than the deterministic one. It can therefore be reasonable to consider signal as a stochastic process in particular when analyzing climatological data.

## Notes

### Acknowledgments

The authors would like to thank Matt King for providing GPS data and comments on a draft version of this manuscript, and also H. Hashemi Farahani and P. Ditmar for DMT2 optimally filtered monthly GRACE solutions. The authors would also like to thank two anonymous reviewers and the editor S. Williams for their insightful comments.The freely available software provided by Peng and Aston (2011) was used as an initial version for the state space models. MATLAB’s Global Optimization Toolbox along with the Optimization Toolbox was used to solve the described optimization problem. This research was financially supported by the Netherlands Organization for Scientific Research (NWO) as part of the New Netherlands Polar Programme.

## Supplementary material

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