Keywords

1 Introduction

Each topocentriccomponent of station coordinates (by means of North, East and Up) is considered to follow the sum of:

$$ \begin{array}{ll} x(t)&={x}_0+{\textit v}_x\cdot t+{\displaystyle \sum_{i=1}^n\left[{A}_i\cdot \sin \left({\omega}_i\cdot t+{\varphi}_i\right)\right]}\\ &\quad +{O}_x+{\displaystyle \sum_{j=1}^m{p}_j\cdot {x}_j^{{\it off}}}+{\varepsilon}_x(t) \end{array} $$
(1)

where x 0 is the initialvalue of the coordinate component, v is the velocity, A, B, ω, ϕ are the amplitudes, angular velocity and phase shift of the i-th periodic component of a time series, O x stands for any known outliers, x off for offsets, p is the Heaviside step function that is equal to 0 or 1 depending on the position of the offset, ε x is the noise component. The noise component ε x is, in most cases, a combination of white noise and coloured noise with amplitudes of a and b κ , respectively (Zhang et al. 1997):

$$ {\varepsilon}_x(t)=a\cdot \alpha (t)+{b}_{\kappa}\cdot \beta (t) $$
(2)

As stated previously, Agnew (1992), noises in a geophysical time series are correlated in time and are well described by power-law process with a power spectrum equal to:

$$ {P}_x(f)={P}_0{\left(\frac{f}{f_0}\right)}^{\kappa } $$
(3)

where f is the spatial or temporal frequency, P 0 and f 0 are normalising constants and κ is the spectral index (Mandelbrot and Van Ness 1968). Mandelbrot (1983) and Feder (1988) discussed processes with different spectral indexes and attempted to attribute causes to. Agnew (1992) proved that the power spectra of most geophysical phenomena can be described by the power-law process with spectral indexes often falling in the range of −3 up to −1. The integer values of indexes indicate special types of noises. Processes with “κ = 0” correspond to white noise (WH) with a flat power spectrum, “κ = −1” stands for flicker noise (FL) (Mandelbrot 1983) which is commonly recognized in most GPS coordinate time series and can be present in data time series due to GNSS signal propagation errors (Wielgosz et al. 2012; Hadas et al. 2013), finally, “κ = −2” is described as random-walk (RW) noise and is considered to be related to instability of monuments that GPS antennae are attached to (Johnson and Agnew 1995; Williams et al. 2004; Beavan 2005; Hill et al. 2009). In order to improve the detection of a random-walk influence from a geodetic monument is to repeat the high-precision measurements in a tectonically stable region. In addition, its appearance in a time series can be reliably detected only in time series where the appropriate length of data, sampling frequency and favourable (low) amplitudes of other noises are present in the data. The issue of noise analysis is of great importance in the determination of the reliability of the velocity field on local and regional scales (Bogusz et al. 2012, 2013). In addition, most of the permanent stations belong to active geodetic networks supporting precise positioning, hence their stability influences the user position (Grejner-Brzezinska et al. 2009).

There are a few different techniques which can be used to easily detect noise in a time series. The first one, often considered to be the mosteffective, accurate and precise (Beran 1994; Williams et al. 2004), is the technique of Maximum Likelihood Estimation (MLE) (Langbein and Johnson 1997). MLE has already been used in many papers that describe noise evaluation, e.g. Beavan (2005), Bergstrand et al. (2007), Teferle et al. (2008), Bos et al. (2008). It is calculated (e.g. Williams et al. 2004) using the following:

$$ \begin{array}{ll} {\it lik}\left(\hat{{\textit v}},C\right)&=\frac{1}{{\left(2\cdot \pi \right)}^{N/2}\cdot {\left( \det C\right)}^{1/2}}\cdot \\ &\quad \times \exp \left(-0.5\cdot {\hat{{\textit v}}}^T\cdot {C}^{-1}\cdot \hat{{\textit v}}\right) \end{array} $$
(4)

where lik is the likelihood function, \( \hat{{\textit v}} \) stands for postfit residuals from linear or nonlinear models applied to data, N is the number of epochs, C is the data covariance matrix. The second method of evaluating noise, spectral analysis, is based on the evaluation of the power spectrum of the data (as in Zhang et al. 1997; Mao et al. 1999). Both King and Watson (2010) and Bogusz and Kontny (2011) used it in their analysis but Langbein and Johnson (1997) and Pilgrim and Kaplan (1998) have both stated that it is less precise than MLE. The classical definition of the periodogram is:

$$ {P}_x\left(\omega \right)=\frac{1}{N_0}{\big|F{T}_x\left(\omega \right)\big|}^2 $$
(5)

where \( F{T}_x\left(\omega \right)={\displaystyle \sum_{j=1}^{N_0}x\left({t}_j\right)\cdot \exp \left(-i\cdot \omega \cdot {t}_j\right)} \) for \( j=1,2,\dots {N}_0 \)

2 Data Analysis

Data used in this research was processed according to the EPN (Bruyninx et al. 2002) guidelines using Bernese 5.0 software (with absolute models consistent with IGS08 (Rebischung et al. 2012)) by the Centre of Applied Geomatics that cooperates at Military University of Technology as one of 18 EPN independent Local Analysis Centres. The processing strategy was performed in the Bernese 5.0 software and included parameters mentioned in Table 1. As the result, the coordinates in ITRF2008 Reference Frame were obtained (Altamimi et al. 2011). We have used 18 permanent Polish EPN stations with daily changes of topocentric coordinates (North, East, Up) (Fig. 1). One of the most important issues in studies of GPS noise is the monumentation used for instance whether the antenna is mounted on buildings or specialist concrete pillars. The main goal of this analysis is to investigate if the amplitudes of random-walk noise change with the different types of station monumentation. This investigation is a continuation of the author’s research concerning reliability in a GNSS time series (Bogusz et al. 2011).

Table 1 Parameters used during the processing strategy in Bernese 5.0 software
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Fig. 1
figure 1

Permanent Polish EPN stations used for the research. Grey dots stand for antennae placed on concrete pillars, white ones for antennae placed on buildings. The map was drawn in GMT software (Wessel and Smith 1998)

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The preliminary analysis of the data involved the removal of outlying values that exceeded the criterion of three times the standard deviation (3σ), the removal of seasonal components with annual and semi-annual periods as well as a linear trend with least squares. The outliers constituted about1-3%of each time series. Any missing data was interpolated using linear interpolation before further analysis (to obtain regularly sampled data). For each topocentric component, the FFT was performed and stacked power spectral densities were made. The scale for them was changed from linear into log–log one. The main advantage of using PSD is that the slope of the graph in log-log space corresponds to the spectral index of the dominant noise existing in the time series. Long-period components are thought to follow flicker or random-walk noise, whereas high-frequency ones follow a white noise model. For each graph, the theoretical values of “κ = −2”, “κ = −1” and “κ = 0” were added so as to show how noise type influence a time series at low and high frequencies (Fig. 2). In spite of the fact that the stacked PSDs are almost the same, PSDs made for individual permanent stations show a few differences (Fig. 3). Slopes of PSDs for sites where the antennae is placed on buildings are in few cases higher than for ones mounted on concrete pillars. It indicates that monument instability is probably higher for buildings and may be caused by the settling of the building or perhaps thermal changes.

Fig. 2
figure 2

Stacked Power Spectral Densities for North (left), East (middle) and Up (right) coordinate components. Stacked PSDs for concrete pillars (red) and buildings (green) are placed on one plot to compare both monument types. Theoretical values of spectral indexes for white noise (κ = 0), flicker noise (κ = −1) and random-walk (κ = −2) were also added

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Fig. 3
figure 3

Exemplary PSDs for two types of antennae monumentation. Left – station BPDL with antennae placed on buildings. Right – station LAMA with antennae situated on concrete pillar. PSDs were made for each station for N, E and U coordinate components. Theoretical values of integer spectral indexes were added to the plots

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For the MLE analysis we assume that the noise present in the coordinate time series follow a combination of white noise, flicker noise and random-walk. However due to the fact that monument instability is thought to follow a random-walk noise model and it’s amplitude is such (King and Williams 2009) that it may be masked by the FL and WH noise components we have analysed our time series with MLE using two different choices of noise combination: the first – WH, FL and RW and the second one – only WH and RW. All analyses were performed using the CATS software (Williams 2008).

The amplitudes of the noises in the first test show that FL dominates over WH and RW (Fig. 4). Although the typical range is between 2 and 4 mm year−0.25 for the horizontal components, it is much higher for the vertical (6–12 mm year−0.25). The similarity in flicker noise amplitude over the whole region are most likely explained by the fact that FL is thought to be regionally coherent and for small areas (such as in Poland) it can reach similar values. The amplitudes of the WH component are also smaller for North and East than for Up. Unfortunately, for the first combination of noises, random-walk is close to zero for the majority of stations. Some of estimates allow for RW but its maximum amplitude reach the 1 mm year−0.5 in the most extreme case (Up component, KRAW station). Such small values of RW indicate the relative stability of Polish EPN stations monuments or prove that the data is clearly not enough to detect it. The small RW noise amplitudes are consistent with the results presented in King and Williams (2009) which showed that if random-walk is considered as monument noise it is smaller than previously thought. They have analysed 10 short-baselines (two of them were created between stations analysed in the following research – BOGI-BOGO and JOZE-JOZ2) with the assumption of power-law or first-order Gauss Markov noise model. It was found that the amplitudes of noises for baselines are in general an order of magnitude smaller than in case of single station. They stated that random-walk is probably no higher than 0.5 mm year−0.5 for well monumented stations. On the other hand, we should be aware of the fact that RW amplitudes may be quite small (or estimated to be zero) because of the present domination of FL and the limited length of the data (only 5 years). On the basis of Williams et al. (2004), to detect RW with an amplitude of 0.4 mm year−0.5 a period of at least 30 years worth of data is needed to detect it easily. But of course, the longer the data period, the more reliable the estimations of noises which can be obtained. Scatter plots made for the first noise combination (Fig. 5) show some dependencies between the amplitudes of WH and FL. For the Up component the amplitudes are higher and much more spread out than for the horizontal components. Unfortunately, due to the small values of RW, no dependencies between RW and WH or FL were noticed.

Fig. 4
figure 4

Amplitudes of white noise (grey), flicker noise (pink) and random walk (green) for North (top), East (middle) and the Up (bottom) components obtained with the MLE method. For each of these amplitudes a 1-sigma error bar was added. Different types of monumentation (building or concrete pillar) were marked with different colours representing station abbreviations (black for buildings, grey for concrete pillars)

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The noise amplitudes obtained for the combinations of WH and RW with the MLE method for horizontal components (North, East) are quite varied for random-walk noise while they remain similar for white noise at around 1 mm (Fig. 6). The highest amplitude for both North and East components were found at the station, BPDL. White noise amplitudes for the Up component are greater than for horizontal ones and are at the level of 3 mm. All of the random-walk amplitudes are higher than 6 mm year−0.5 while some of them even exceed the value of 20 mm year−0.5 (20 times greater than for the first combination). They also have greater error bars which can be the result of incorrect fit and the inappropriate assumption of WH plus RW only. This rather unrealistic result is likely due to the domination of flicker noise in the series which, in the absence of a flicker noise component in the MLE, is misinterpreted as random-walk. Furthermore the length ofdata means that the MLE could not give us significant results. The scatter plot for the WH and RW combination (Fig. 7) clearly presents dependencies between noises. It can be noted that noise amplitudes for horizontal components are placed adjacent to another while they are more scattered and not centred around one specific value for the vertical. Although the second assumption of noise combination has some drawbacks as described above, amplitudes of RW obtained with this combination appear to show that concrete pillars are better as monuments for antennae than buildings. Unfortunately, due to the fact that this assumption is not correct, it cannot be stated for sure.

Fig. 5
figure 5

Scatter plots for different noise types: flicker noise vs. white noise (left), random-walk vs. white noise (middle), random-walk vs. flicker noise (right). Amplitudes for North, East and Up components are represented with different colours – red, yellow and blue, respectively

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Fig. 6
figure 6

Amplitudes of white (grey) and random-walk (green) noise for Polish EPN stations obtained with the MLE method with the assumption of WH and RW noise only. The plots are presented for North (top), East (middle) and Up (bottom) components. The 1-sigma error bars were added to each of amplitude

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Fig. 7
figure 7

Scatter plot for amplitudes of white and random-walk noise for Polish EPN stations obtained with MLE for the assumption of WH and RW noise only. The coordinate components are represented with red (North), yellow (East) and blue (Up) colours

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3 Discussion

Application of the MLE algorithm to the determination of the amplitudes of white, flicker and random-walk noise showed how they influence the accuracy and reliability of the parameters that are estimated from GPS time series (e.g. velocities of permanent stations). We compared two stochastic models that were combinations of white noise, flicker noise and random-walk and alternatively white noise plus random-walk noise. We showed that the amplitudes of the assumed models influence the coordinate time series measured by Polish EPN stations. The second combination of noise unexpectedly gave values of RW up to 20 times larger than in the first combination. However if flicker noise is present in the time series and its existence is ignored, it will influence the estimated amplitudes of the random-walk component. The flicker noise was found to be at a similar amplitude over small areas, such as the size of Poland, and therefore variations in the estimated random-walk amplitudes may still be considered to be due to the influence of monument instability. The questionable point is whether the current time series (5 years) are really long enough to detect changes related to random-walk. As showed by Williams et al. (2004) to detect easily random-walk with a magnitude of 0.4 mm year−0.5 the 30 years data will be needed. Of course, the longer the time series, the more reliable the estimation of random-walk.

We also applied FFT to the topocentric components to create their Power Spectral Density estimates. Their slopes in log-log space also indicate the characteristics of the noise that appears in GPS time series. We averaged or stacked the power spectra as a function of monument type. However, when plotted together we could not visibly discern any obvious differences. Individually, the PSDs made for each station did show small variations in their slopes. Taking into consideration the power spectral densities (some of the estimated spectral slopes, or indices, for buildings are closer to −2 than for concrete pillars) and the second noise combination (WH plus RW) in the MLE results although somewhat lacking (time series do not simply reflect such a characteristic so it cannot be stated for sure) the results hint that antennae placed on concrete pillars are apparently more stable than those mounted on buildings.