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

2.1 Trend modeling

2.2 State space model

2.3 Kalman filter and smoother

2.4 Estimation of hyperparameters

The hyperparameters are defined as

$$\begin{aligned} \psi = 0.5 \log \begin{bmatrix} \sigma ^2_{\varepsilon }&\sigma ^2_{\eta } \end{bmatrix}^\mathrm{T} = 0.5 \log \begin{bmatrix} \sigma ^2_{\varepsilon }&\sigma ^2_{\zeta }&\sigma ^2_{\varsigma _1}&\sigma ^2_{\varsigma _2} \end{bmatrix}^\mathrm{T}, \end{aligned}$$

(17)

which ensures that they are non-negative, since here they represent standard deviations.

Open image in new window

2.4.1 Optimization

Maximizing \(\log L\) is equivalent to minimizing \(-\log L\). We search numerically for a set of optimal parameters that provides the minimum value for negative \(\log L\), given the process and the observed data. This optimization problem is carried out by using an interior-point (IP) algorithm as described in Byrd et al. (1999). The function \(-\log L(Y_n| \psi )\) to be minimized is called the objective function. Since the IP algorithm of Byrd et al. (1999) is a gradient-based local solver, the gradient for the objective function is computed analytically according to Durbin and Koopman (2012, chap. 7):

$$\begin{aligned} \frac{\partial \log L(Y_n|\psi )}{\partial \psi }= & {} \frac{1}{2}\sum _{t=1}^{n} {{\mathrm{tr}}}\left\{ \begin{matrix} \left( u_t u_t^{T} -D_t\right) \frac{\partial H_t}{\partial \psi } \end{matrix}\right\} \nonumber \\&+\,\frac{1}{2}\sum _{t=2}^{n} {{\mathrm{tr}}}\left\{ \begin{matrix} \left( r_{t-1} r_{t-1}^{T} -N_{t-1}\right) \frac{\partial R_t Q_t R_t^{T}}{\partial \psi } \end{matrix}\right\} \nonumber \\ \end{aligned}$$

(18)

using quantities calculated in Sect. 2.3 with \(u_t = F^{-1}_t v_t - K^{T}_t r_t\) and \(D_t = F^{-1}_t + K^{T}_t N_t K_t\).

Open image in new window

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.

2.4.3 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.

Firstly, we perform a so-called basic test to check the plausibility of the applied constraints. For this, we compute the absolute difference between the constrained and unconstrained case which should be smaller than the estimated standard deviations of the unconstrained hyperparameters:

$$\begin{aligned} | \psi _{\mathrm{con}} - \psi _{\mathrm{uncon}} | < \sigma _{ \psi _{\mathrm{uncon}}}, \end{aligned}$$

(19)

where \(\sigma _{ \psi _{\mathrm{uncon}}}\) is derived using the corresponding Hessian. This is a quick test for serious mistakes meaning that constraints are absolutely not supported by the data if the left-hand side of the equation is larger than the right-hand side. If the basic test does not reject introduced constraints (meaning the test is positive), the second, computationally more comprehensive likelihood ratio test (LR-test) is performed.

The basic idea of the LR-test is the following: if the constraint is valid, imposing it should not lead to a large reduction in the loglikelihood function (Greene 1993). Therefore, the test statistic is

$$\begin{aligned} \mathrm{LR} = 2(\log L(Y_n|\psi _{\mathrm{uncon}}) - \log L(Y_n|\psi _{\mathrm{con}})). \end{aligned}$$

(20)

LR is asymptotically \(\chi ^2\) distributed with degrees of freedom equal to the number of constraints imposed (Wilks 1938). The null hypothesis is rejected (the test is negative) if this value exceeds the appropriate critical value from the \(\chi ^2\) tables, meaning that the data do not support the constraints applied.

According to Greene (1993) the parameter spaces, and hence the likelihood functions of the two cases must be related. Moreover, the degrees of freedom of the \(\chi ^2\) statistic for the LR-test (Eq. 20) equals the reduction in the number of dimensions in the parameter space that results from imposing the constraints. Hence, the degree of freedom equals the amount of active constraints. A constraint is called active (or binding) if it is exactly satisfied, and therefore, holds as an equality constraint (Boyd and Vandenberghe 2004, p. 128). In short, the LR-test is usually applied in the equality constrained case. However, if one constraint is active it reduces the number of dimensions in the parameter space by one, since it defines one parameter on which this constraint is applied. If a constraint is active it simultaneously means that this constraint strongly influences the solution. Since we are dealing with a non-convex optimization problem having multiple local minima, it may be the case that a constraint can still strongly affect the solution without becoming active, e.g., by simply shifting the solution to the next minima. Therefore, to estimate the degree of freedom for the LR-test performed in the context of a non-convex optimization problem with inequality constraints, we have to estimate how many restrictions do indeed influence the solution. This will be achieved by using a brute force method summarized in Algorithm 1. The idea of the method is to successively apply the constraints until all restrictions are satisfied and thereby, to control the number of degrees of freedom for the LR-test. As it might be the case that applying a constraint to one parameter will already satisfy the constraints to the other parameters, we check whether newly added restrictions make previously added ones superfluous.

Open image in new window

It is important to note that the degrees of freedom of the \(\chi ^2\) statistic may differ already because of the used state space form; for details, the reader is referred to Harvey (1989, chap. 5). If both tests, i.e., the basic test and the LR-test, indicate that the data do not support the constraints, the constraints are relaxed towards unconstrained values until both tests are positive (see Algorithm 2). In this context, it is worth mentioning again that the basic test is performed for the purpose of reducing the computational complexity: if the constraints do not pass the basic test, there is no need to perform the LR test.

Open image in new window

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.

2.5.3 State space model

GPS data are often contaminated by offsets (Gazeaux et al. 2013). If undetected, they might produce an error in trend estimates (Williams 2003b). For Antarctica, the offsets are usually related to hardware changes and thus are step-like. To incorporate an offset into state space form we define a variable \(w_t\) as:

$$\begin{aligned} w_t = {\left\{ \begin{array}{ll} 0, &{} t < \tau ,\\ 1, &{} t \ge \tau . \end{array}\right. } \end{aligned}$$

(23)

Adding this to the observation Eq. (6) gives

$$\begin{aligned} y_t = \mu _t+ c_{1,t} + c_{2,t} + \delta \, w_t + \varepsilon _t, \quad t = 1,\ldots ,n, \end{aligned}$$

(24)

where \(\delta \) measures the change in the offset at a known epoch \(\tau \). For k offsets, the state vector can be written as

$$\begin{aligned} \alpha _t^{[\delta ]} = [\delta _1 \cdots \delta _k]^\mathrm{T}. \end{aligned}$$

(25)

Colored noise \({\varepsilon }_t\) can be included into the state space model as:

$$\begin{aligned} \alpha _t^{[\varepsilon ]} = \begin{bmatrix} \varepsilon _t \\ \phi _2\varepsilon _{t-1}+\cdots + \phi _l\varepsilon _{t-l+1}+ \theta _1\varkappa _t+ \cdots + \theta _{l-1}\varkappa _{t-l+2} \\ \phi _3\varepsilon _{t-1}+\cdots + \phi _l\varepsilon _{t-l+2}+ \theta _2\varkappa _t+ \cdots + \theta _{l-1}\varkappa _{t-l+3} \\ \vdots \\ \phi _l\varepsilon _{t-1} + \theta _{l-1}\varkappa _t \end{bmatrix}\nonumber \\ \end{aligned}$$

(26)

with \( \eta ^{[\varepsilon ]}= \varkappa _{t+1}\); then, the corresponding system matrices are given by

$$\begin{aligned} \begin{array}{lllll} &{} T^{[\varepsilon ]}= \begin{bmatrix} &{} \phi _1 &{} 1 &{} \quad &{} 0 \\ &{} \vdots &{} \quad &{} \ddots \\ &{} \phi _{l-1} &{} 0 &{} \quad &{} 1\\ &{} \phi _l &{} 0 &{} \cdots &{} 0 \end{bmatrix}, \quad R^{[\varepsilon ]}= \begin{bmatrix} 1&{} \theta _1 &{} \cdots &{} \theta _{l-1} \end{bmatrix} ^\mathrm{T}, \\ &{} Z^{[\varepsilon ]}= \begin{bmatrix} 1 &{} 0 &{} 0 \cdots 0\end{bmatrix}. \end{array} \end{aligned}$$

(27)

It is worth noting that for irregularly spaced observations, it is less straightforward to put an ARMA(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.

Combining the parameterization of 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

$$\begin{aligned} \alpha _t = \left( \alpha _t^{[\varepsilon ]}, \alpha _t^{[b]},\alpha _t^{[\delta ]}\right) , \end{aligned}$$

(28)

and the system matrices as

$$\begin{aligned} \begin{array}{ll} &{} Z_t = \left( Z ^{[\varepsilon ]}, Z,I_k\right) , \quad T = \mathrm{diag}\left( T^{[\varepsilon ]}, T, {I}_k\right) ,\\ &{} R = \mathrm{diag}\left( R^{[\varepsilon ]}, R, {0}_k\right) , \\ &{} Q = I\sigma ^2_{\eta } = \mathrm{diag}\left( \begin{bmatrix} \sigma ^2_{\varkappa _{t+1}} &{} \sigma ^2_{\zeta } &{} \sigma ^2_{\varsigma _1} &{} \sigma ^2_{\varsigma _1} &{}\sigma ^2_{\varsigma _2} &{}\sigma ^2_{\varsigma _2} \end{bmatrix}\right) \end{array} \end{aligned}$$

(29)

with Z, T and R being defined in Eqs. (9)–(10).

After defining this modified state space model, GPS time series can be processed in the same way as the GRACE time series. In particular, the search space for the global solver associated with the ARMA parameters does not experience any bounds.

Open image in new window

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.

3.1 Data

We use daily GPS vertical site velocities at the CAS1 station, which is located in Wilkes Land, East Antarctica. There are three reasons for selecting this GPS station: first, it is a GPS site with continuous long-term observations; second, the time series data contain all the features described in Sect. 2.5; and third, because of its geolocation. A significant accumulation anomaly event was concentrated along the Wilkes Land coast in 2009 (Luthcke et al. 2013). Due to a high signal-to-noise ratio, we expect this event to be detected by both observing systems, GPS and GRACE. Consequently, we can use this prior knowledge about geophysical processes to verify the plausibility of the proposed methodology.

Open image in new window

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.

Figure 5 demonstrates similar results as Fig. 4, but for GPS vertical site displacements. LSA results shown in Fig. 5a were generated by fitting intercept, slope, annual and semi-annual terms and two offsets to the time series without modeling colored noise. Time-correlated noise model is usually used to estimate more realistic parameter uncertainties than those resulting from white noise assumption (Thomas et al. 2011). However, to generate the KF results, we co-estimated time-correlated noise parameters as well. Following the procedure described in Sect. 2.5.2, we computed the noise PSD function of the LSA residuals, which is shown in black in Fig. 6. An AR(3) model (red in Fig. 6) was found to provide the best fit to the PSD of the approximate colored noise. Colored noise in the GPS time series was then parameterized with three autoregressive coefficients and co-estimated along with signal components. The results in Fig. 5 suggest that the KF method (Fig. 5b) outperforms the LSA method (Fig. 5a).

Open image in new window

Open image in new window

Because estimating rates/slopes as accurately as possible is the primary motivation of this study, Fig. 7 outlines the corresponding results. A constant slope as a result of a deterministic fitting along with the stochastically modeled time-varying slope are shown for GRACE (Fig. 7a) and GPS time series (Fig. 7b). To allow for a direct comparison between LSA and KF results, we compute a mean slope from the time-varying slope. If there were no changes in the rates of vertical deformation, the two constant values should be the same. In fact they differ significantly, as the proposed methodology suggests the presence of low-frequency variability in the slope component (black curves in Fig. 7) that cannot be explained by any other modeled component. For GRACE, the constant slope estimated using LSA is \(0.20 \pm 0.07\) mm, whereas the mean slope determined from the time-varying estimates is \(0.36\pm 0.12\) mm. Although these are small numbers in absolute terms, their relative difference is larger than 50 \(\%\). For GPS, the slope derived based on KF is almost 2.5 times smaller than the LSA based slope estimate being \(0.77 \pm 0.46\) and \(1.89 \pm 0.11\) for KF and LSA, respectively.

Open image in new window

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 prove the presence of low-frequency variability in the slope component estimated with the KF technique, we compute the amplitude spectrum of the GRACE and GPS time series (cf. Fig. 8a, b, respectively). The results confirm the presence of long-term variations that deviate from a linear trend in both time series. While these inter-annual variations are absorbed in the residuals when using LSA (blue graphs in Fig. 8), they are captured by the KF (red graphs in Fig. 8) and map into the time-varying slope component (Fig. 7). Considering root mean square (RMS) misfits for quantitative comparison, there is an approximate 41 % and 13 % reduction in RMS misfits for GRACE and GPS time series, respectively, as a result of using the proposed KF instead of the LSA technique. As can be seen from Fig. 8, the dominant reduction of the RMS misfit is due to the time-varying slope with a smaller part being explained by the time-varying annual signal (the amplitude of the KF residuals around the 1 cycle-per-year frequency is smaller than the amplitude of the LSA residuals). The signals in the high-frequency domain are considered as noise.

Open image in new window

Open image in new window

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.