JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 110, B08410, doi: /2005jb003642, 2005

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 110,, doi: /2005jb003642, 2005 Noise properties of continuous GPS data from concrete pillar geodetic monuments in New Zealand and comparison with data from U.S. deep drilled braced monuments John Beavan Institute of Geological and Nuclear Sciences, Lower Hutt, New Zealand Received 22 January 2005; revised 3 April 2005; accepted 26 April 2005; published 24 August [1] Reducing noise levels in geodetic data is critical for the interpretation and modeling of geophysically interesting signals. In this paper, I estimate noise properties of continuous GPS time series collected on concrete pillar monuments and compare them with similar data from deep drilled braced monuments that are considered the most stable three-dimensional monuments available. I show that the time series noise properties are very similar between the monument types, at least for time series up to 4.5 years duration. Differences between the monument types are therefore not a dominant influence on these time series, implying that the less expensive concrete pillar monuments are a valid choice for recording many signals of geophysical interest. Citation: Beavan, J. (2005), Noise properties of continuous GPS data from concrete pillar geodetic monuments in New Zealand and comparison with data from U.S. deep drilled braced monuments, J. Geophys. Res., 110,, doi: /2005jb Introduction [2] Geodetic signals of tectonic or volcanological interest include those due to short-term aseismic deformation episodes or magma inflation events, as well as linear velocity estimates and many other signals. In all cases, the detectability of the interesting signal above any noise present in the data is important for the interpretation and modeling of such signals. [3] One source of noise in geodetic signals is random motion occurring within the connection of the geodetic instrument to the ground. This ground connection noise is present in all types of geodetic measurements, whether from sensitive instruments such as borehole strainmeters and tiltmeters, or from less sensitive surface instruments such as electronic distance measurement (EDM) or GPS antennas and receivers. [4] In the case of surface instruments, the connection to the ground is through a geodetic monument, and the motion of this monument with respect to a representative volume of the Earth s near surface in its vicinity is termed monument noise. Monument noise results from processes such as soil swelling in response to rainfall, freeze-thaw cycles, and general rock and soil weathering effects [e.g., Wyatt, 1989]. [5] In order to reduce monument noise, a particular style of deep drilled braced geodetic monument that is designed to be stable in three dimensions [Wyatt et al., 1989; see also PBO Operations, monumentation/monumentation.html] has been adopted for several large continuous GPS (CGPS) networks in the western U.S. The monument uses a pyramid structure of 4 or 5 stainless steel rods (or cement-filled tubing) that are grouted into 1 vertical and 3 or 4 inclined 10 m long drill Copyright 2005 by the American Geophysical Union /05/2005JB003642$09.00 holes. The rods are grouted below about 4 m depth, but are isolated from the presumed less stable ground in the top 4 m. The rods meet at a point about 1.5 m above the ground surface, where they are welded together. The GPS antenna is mounted just above this point, which is the most stable point of the structure. This is considered to be the most stable style of 3-D monument available at the present time, but it is quite expensive to construct, and because of the drilling equipment needed during construction it is not applicable to all potential geodetic sites. I refer to this monument as deep braced for the remainder of this paper. [6] My aim is to investigate the noise levels within time series of daily CGPS positions collected from concrete pillar monuments in New Zealand (Figure 1), and to compare these noise levels with those from deep braced monuments in several U.S. CGPS networks that have been studied by Williams et al. [2004] (see section 2). I will also investigate whether monument noise is the limiting noise source in the CGPS data. 2. Monument Noise 2.1. Previous Studies [7] Monument noise is thought to follow an approximately random walk process, and the evidence for this is discussed in section 2.1. Random walk means that the expected value of the monument position relative to an initial position increases as the square-root of time. For a random walk noise level thought to be typical of a highquality geodetic monument (0.3 mm/yr 1/2 [Johnson and Agnew, 2000]), this means that the monument is expected to have moved 0.3 mm from its initial position after 1 year, 0.6 mm after 4 years, 1.2 mm after 16 years, and so on. The deviation of the monument position tends to increase with increasing time, and this means that a tectonic signal of a 1of13

2 Figure 1. Continuous GPS sites in New Zealand with more than 500 days of data from 12 February 2000 through 24 July Fifteen sites (larger dots and typeface) are used in this study (Table 1). Good quality concrete pillar monuments are shown in black; other monuments in gray with italicized site codes. Four sites with sufficient data, DUNT, GISB, LYTT, and NETT, were not used; these are shown with smaller dots and typeface. The LYTT monument is a mast mounted on a wooden wharf that is not considered stable. NETT is a concrete pillar, but its data include large seasonal signals due to snow loading and/or thermoelastic effects, as well as substantial data loss during winter due to snow cover [Beavan et al., 2004]. GISB is a high-quality concrete pillar, but this station has experienced several aseismic strain episodes since installation so the data are not suitable for a noise study [e.g., Beavan et al., 2003]. The DUNT monument is a mast mounted on a wharf; it was omitted in error from the group of other monuments analyzed, but this does not affect the statistics of the concrete pillar monuments that are the main focus of the study. The light gray lines show active faults from the September 1997 version of the GNS Active Faults Database. The inset map shows the locations of IGS stations used in reference frame stabilization. particular amplitude is easier to detect above the noise if it occurs over a short time period, than if it occurs over a longer time period. One way of thinking of random walk noise is that the position of the monument at a particular time is dependent on the history of its position at previous times. In other words, there is temporal correlation between subsequent monument positions. This is in contrast to the case of white noise, which is often used in elementary statistics and commonly used in geophysical applications, where one sample of the noise process is completely independent of previous samples; in other words, white noise is uncorrelated. [8] It is often useful to consider noise processes in the frequency domain. If a power spectrum of a time series of monument positions is plotted on a log-log scale it will generally show a line of negative slope. For random walk noise, the slope of the power spectrum is 2. By contrast, white noise has a slope of zero, and an intermediate type of noise called flicker noise has a slope of 1. An increasing negative slope in the frequency domain indicates an increasing level of temporal correlation within the time series. The value of the slope is generally referred to as the power law index, and this terminology will be used here. [9] Time series of monument positions from CGPS and high-precision EDM instruments have been studied by several authors [e.g., Zhang et al., 1997; Langbein and Johnson, 1997; Mao et al., 1999; Johnson and Agnew, 1995, 2000; Langbein, 2004; Williams et al., 2004]. After removing predictable effects, such as linear tectonic motion, seasonal cycles, and offsets due to earthquakes or changes in measuring equipment, the remaining time series may be considered as a noise process with the potential to obscure signals of interest. It has been found that CGPS time series include noise processes that are less correlated than random walk, and that are of higher amplitude than the random walk processes expected from monument noise, at least for periods up to years or even decades [e.g., Williams et al., 2004]. The limiting noise source in CGPS data, at least up to periods of many years, may therefore be from these other sources rather than from the geodetic monuments. In this case, there may be no benefit in using the expensive deep 2of13

3 braced monuments as opposed to alternatives such as wellfounded and well-constructed concrete pillars. [10] Williams et al. [2004] have addressed this problem using data from a variety of monument types in the Southern California Integrated GPS Network (SCIGN). Even though they do not detect random walk noise as the limiting error source for any of their monument types, they do find that the time series collected on deep braced monuments significantly outperform those from all other monument types, using two measures of time series quality. They conclude that there is presently no scientific basis to cease using deep braced monuments in favor of a less expensive alternative. [11] Williams et al. [2004] also find that the six monuments classified as concrete piers in their study perform among the worst of all monument types, only excepting oil platforms. This finding is of concern in New Zealand as high-quality concrete pillar monuments have been adopted for the national PositioNZ CGPS network ( and we are presently considering whether to use concrete pillars or deep braced monuments (or a mixture) for future CGPS networks aimed at tectonic problems in New Zealand. It is possible that a comparison of the New Zealand concrete pillar monuments with those of Williams et al. [2004] is unfair, as the six concrete pier monuments within the SCIGN network are almost all of particularly low quality construction or in particularly noisy environments. For example, two are at the ends of a dam wall, and one is in an actively deforming caldera. This is discussed further in section Is Monument Noise a Random Walk Process? [12] Monument noise is generally believed to follow a random walk process, though there are also proponents of a first-order Gauss-Markov (FOGM) process (see discussion by Langbein and Johnson [1997, p. 602]). FOGM looks similar to random walk over short time periods but more like white noise over long time periods, where the division between short and long time depends on the parameters of the FOGM process. The evidence for random walk monument behavior comes largely from studies by Wyatt [1982, 1989], Langbein and Johnson [1997], and Johnson and Agnew [2000]. [13] Wyatt [1982, 1989] studied time series of horizontal and vertical motion from a variety of monument types, some of which were believed to be the highest quality available at the time. He found in the frequency domain that these time series had slopes of 2, or even more negative, suggesting that the monument noise was a random walk process, or perhaps even more correlated. [14] Wyatt [1982] measured horizontal motion of massive pier monuments constructed of gabbro (commonly used for optical tables) and emplaced in holes excavated several meters into weathered but competent granodiorite. These were expected to be particularly stable monuments, but in fact significant motion was observed using two different techniques, a tiltmeter bonded to the monument, and an optical anchor that measures the monument position relative to depth by use of an equal-arm Michelson interferometer. The power spectral slopes [Wyatt, 1982, Figure 9] are slightly more negative than 2, but if interpreted as random walk noise they imply amplitudes ranging from mm/yr 1/2 (using equation 5 of Agnew [1992]). For vertical motion, Wyatt [1989] found random walk noise amplitudes of 0.2 mm/yr 1/2 for National Geodetic Survey (NGS) Class A rod marks emplaced at several meters depth, and 0.65 mm/yr 1/2 for the better surface benchmarks in his study. [15] Langbein and Johnson [1997] studied the noise properties of some two-color EDM baselines in three geographic regions, using time series as long as 15 years. Using a spectral technique they demonstrated that a power spectral slope of 2 was appropriate at the lower frequencies, consistent with a random walk noise process. They then used a maximum likelihood method to estimate the parameters of a random walk plus white noise model for the motion of the monuments at the ends of the EDM baselines. They found a range of horizontal random walk amplitudes between about 0.5 and 3.3 mm/yr 1/2. After removing five monuments known to be unstable, and allowing for a small, baseline length-dependent, noise component (which they assumed not to be due to monument noise), they found a mean and standard deviation of 0.9 ± 0.4 mm/yr 1/2 averaged over all their monuments. [16] Langbein and Johnson [1997] also reported on power spectra from short-baseline (10-m long) strain meters buried at 2 m depth. Their power spectra imply horizontal random walk monument noise of only mm/yr 1/2, but it is possible that this estimate is artificially low because of correlation between the monument noise at the two ends of the strain meters. Langbein and Johnson concluded that surface monuments installed using traditional geodetic techniques have a random walk component that is greater than about 0.5 mm/yr 1/2 but may be as large as 3 mm/yr 1/2. [17] Johnson and Agnew [2000] carefully analyzed a 6.5- year time series from two prototype deep braced monuments on a short (50 m) baseline and showed that these monuments exhibited random walk noise of 0.3 mm/yr 1/2 on each horizontal component. The power spectra in their study clearly show the change from random walk behavior at lower frequencies to white noise behavior at higher frequencies. Recent analysis of additional data from these two monuments suggest an even lower random walk noise of 0.1 mm/yr 1/2 (S. D. P. Williams, personal communication, 2004). 3. GPS Sites and Monuments [18] The 15 GPS sites whose data I use are shown in Figure 1, and their monuments are described in Table 1. I have used continuous stations with more than 500 days of data, following Williams et al. [2004], and have started my processing on 12 February 2000 even though a few sites have continuous data earlier than this. The longest series I use are of 4.45 years duration, the shortest is of 2.05 years, and the average is 4.15 years. However, the numbers of points in the analyzed time series are fewer than the total length of data, due to gaps in data collection and rejection of outliers in the processed time series (see Table 2). [19] Ten sites have what I consider high-quality reinforced concrete pillar monuments and four are on buildings or masts. One site, CHAT, has a concrete pillar that is considered of low quality because the underlying material is about 30 m of Eocene tuff which has weathered to a 3of13

4 Table 1. Continuous GPS Site Information Site Monument Information Pillars AUCK Reinforced concrete pillar, 0.35 m diameter, extending 1.3 m above ground and set 1.2 m into weathered Miocene sandstone and mudstone. No stainless steel rods connecting concrete to rock. CNCL Reinforced concrete pillar, 0.5 m high, 0.3 m diameter, set directly on schist bedrock. Pillar connected to bedrock via three 12 mm diameter stainless steel reinforcing rods set with epoxy into holes drilled about 100 mm into the rock. HOKI Reinforced concrete pillar, 0.35 m diameter, extending 2.2 m above ground and with 1 m deep concrete foundation in mid-late Pleistocene outwash gravels. Four 3 m long, 38 mm diameter stainless rods hammered to refusal at about 3 m depth to help to stabilize the pillar against tilting. KARA Reinforced concrete pillar, 0.5 m high, 0.3 m diameter, set directly on schist bedrock. Pillar connected to bedrock via three 12 mm diameter stainless steel reinforcing rods set with epoxy into holes drilled about 100 mm into the rock. MQZG Reinforced concrete pillar, 0.95 m high, 0.4 m diameter, set directly on massive outcrop of Late Cretaceous andesite or rhyolite. Pillar connected to rock via three 15 mm diameter threaded stainless steel rods set with epoxy into holes drilled about 300 mm into the rock. MTJO Reinforced concrete pillar, 2.4 m high, 0.4 m diameter, set directly on greywacke bedrock. Pillar connected to bedrock via eight 20 mm diameter stainless steel reinforcing rods set with epoxy into holes drilled mm into the rock. PAEK Reinforced concrete pillar, 0.4 m diameter, extending 1.2 m above ground with 1.2 m deep concrete foundation in weathered greywacke bedrock. Four 38 mm diameter, 3 m long stainless steel rods hammered to refusal to help stabilize the pillar against tilting. QUAR Reinforced concrete pillar, 0.3 m diameter, extending 0.8 m above ground, with concrete foundation set on good granitic outcrop at ca. 0.6 m below ground surface. Pillar connected to bedrock via three 12 mm diameter stainless steel reinforcing rods set with epoxy into holes drilled about 100 mm into the rock. TAUP Reinforced concrete pillar, 0.3 m diameter, constructed in hole augered 2.5 m into recent (1800 yr) Taupo pumice/tephra. Pillar extends 1.15 m above ground surface. No stainless rods connecting the pillar to the rock. WGTN Reinforced concrete pillar, 0.3 m diameter, extending 0.7 m above ground, with foundation constructed on weathered greywacke bedrock about 0.3 m below ground surface. No stainless steel rods connecting the pillar foundation to bedrock. CHAT GRAC OUSD TAKL WGTT Other Monuments Reinforced concrete pillar, 0.3 m square, set into a firm layer at 1 m depth and extending 0.5 m above ground surface. Underlying the firm layer is about 30 m of weathered and sometimes waterlogged tuff, above volcanic basement. No stainless steel rods connecting pillar to subsurface. A 3 m long stainless steel mast bolted to side of two-storey building. Building foundation is in well-consolidated river gravels. Concrete pillar, 1.7 m high, 0.25 m diameter, mounted above structural elevator shaft on top of four-story building. Building foundation is in a competent cobble layer within poorly sorted fluvial gravels. A 3.5 m long stainless steel mast attached to surface of concrete wharf. Wharf foundations set in early Miocene Waitemata sandstone. A 2.5 m long stainless steel mast mounted on major structural wall on top of five-story building. Building foundations are in fill and sediments that were extensively compacted prior to construction. sometimes sticky clay. This site is also well outside the network formed by the remaining sites, so is less relevant to this noise study. [20] Five of the concrete pillars are anchored directly to underlying rock using stainless steel rods epoxied into holes drilled vertically into the bedrock (Table 1). In two cases (HOKI, PAEK), where hard bedrock is not reached at shallow depth, stainless rods of 38 mm diameter have been hammered to refusal in directions inclined to the vertical. Three monuments (AUCK, TAUP, WGTN) have been connected to the underlying rock with concrete but without stainless rods linking the monument and the rock. [21] Some 30 new concrete pillar monuments have been installed in New Zealand since early None of these meet the 500-day test of data quantity so they are not included in this study. However, all these new monuments have been designed to be as good or better than any of our preexisting monuments, so I expect their monument noise to be as low or lower than that of the monuments in the present study. 4. GPS Time Series Analysis [22] The time series whose noise properties are to be analyzed must have as much as possible of the spatially correlated noise removed prior to the noise analysis, and predictable signals must be estimated simultaneously with the noise properties. Examples of predictable signals are linear tectonic motion, offsets due to coseismic displacements or equipment changes, and seasonal cycles. Spatially correlated noise is that which is common to most or all stations in a geodetic network. Such common mode noise must come from a large-scale source such as broad-scale weather patterns, atmospheric loading, or errors in satellite 4of13

5 Table 2. Statistics of Data Quantity and Outlier Removal Site Possible Number of Points, a days Actual Number of Points, a days Number of Points After Regional Filtering and Outlier Removal N E U Number of Points in MLE Analysis b Percent of Points Removed as Outliers b Percent of Possible Points in MLE Pillars AUCK CNCL HOKI KARA MQZG MTJO PAEK QUAR TAUP WGTN Other Monuments CHAT GRAC OUSD TAKL WGTT Averages Mean Median a The possible number of points is given by the start and end times of the time series, while the actual number is smaller because of data gaps. b The number of points in the MLE analysis is fewer than the number in each coordinate time series because outliers are identified separately in each coordinate direction then applied to all three directions. orbits, and is not related to the noise introduced by the geodetic monuments at each site. (For a small-size geodetic network it is possible that signals of tectonic interest could be misconstrued as common mode noise, but this is unlikely for networks of hundreds of kilometers extent like those studied here.) 4.1. GPS Data Analysis [23] I analyze each day s GPS data using Bernese version 4.2 software [Beutler et al., 2001] in a single network solution including the New Zealand continuous sites and a few regional International GPS Service (IGS) sites. I constrain station AUCK to within 2 mm (1s) of its ITRF2000 coordinates, holding final IGS00 orbits and earth orientation parameters fixed, and solving for ambiguities using the quasi ionosphere-free (QIF) technique [e.g., Beutler et al., 2001]. Ocean load tides are not incorporated, because many of the New Zealand CGPS sites are very close to the ocean and we have not yet tested whether including estimated ocean load tides improves the daily solution repeatability. Troposphere zenith delays are estimated every two hours. Tropospheric gradient estimation is not used as we have not yet tested its efficacy in New Zealand. Each day s solution is placed in the ITRF2000 reference frame by a sixparameter Helmert transformation that gives a best fit of a set of regional stations to their predicted IG00 coordinates on that day. The IGS stations used for this reference frame stabilization are TIDB, HOB2, MAC1, AUCK, CHAT and NOUM (Figure 1). [24] A velocity change that is postulated to be due to a >1 yr duration slow slip episode on the subduction interface north of Wellington has been observed since the first half of 2003 on GPS stations in the Wellington region. In order that this signal should not bias the noise estimates from these sites, a single velocity change has been modeled and subtracted from the east components of PAEK, GRAC, WGTT and WGTN, and the up component of PAEK, prior to further analysis. The modeled velocity change at PAEK was +11 mm/yr east and +11 mm/yr up starting at 23 March Smaller velocity changes were removed from the east components at GRAC, WGTT and WGTN Regional Filtering and Outlier Removal [25] The resulting time series for each component of motion (north, east, up) at each station are then filtered using a robust filtering process to detect and remove outliers and to estimate a common mode signal that is then subtracted from each original time series. This is similar to the regional filtering algorithm described by Wdowinski et al. [1997]. In detail, we do the following. [26] 1. Remove known offsets (initially estimated by eye if size not known). Offsets are generally due to equipment changes at the site. [27] 2. Calculate common mode signal by averaging a set of stations distributed throughout the network. These stations are chosen so they don t include obviously noisy stations or those with substantial numbers of data gaps. In this analysis we use stations AUCK, HOKI, MQZG, QUAR, and WGTN to estimate the common mode signal. [28] 3. Remove linear trend from common mode signal. [29] 4. Subtract common mode signal from each individual series. [30] 5. Estimate and remove linear trend and annual sinusoid from each series. [31] 6. Remove outliers in each series, by calculating the median and interquartile range (IQR) within overlapping 5of13

6 Figure 2a. North component daily time series from NZ continuous GPS sites between day 043/2000 and day 206/2004, after regional filtering, outlier rejection with a ±3 IQR criterion, and subtraction of linear velocity estimate, annual cycle estimate, and estimated offsets at designated times. Station codes on the left axis are black for good concrete pillar monuments and italicized gray for other monument types. The stations are ordered, top to bottom, as CHAT followed by mainland stations from north to south. The time series traces are plotted at different densities to assist in discriminating between traces. 60-day time windows and omitting points that exceed ±3 IQR. [32] 7. Add linear trend and sinusoid back to each series. [33] 8. Repeat steps 2 through 7 until no more outliers are removed (in practice, just two iterations). [34] 9. Add offsets back into time series. [35] The resulting time series have had a common mode signal and clear outliers removed, but retain their original linear trends, seasonal cycles and offsets. These time series are plotted in Figures 2a 2c, but for the purposes of the plot the linear trends, seasonal cycles and offsets have been subtracted from the time series so that the residuals are plotted. [36] This regional filtering and data cleaning strategy is very similar to that of Nikolaidis [2002], whose filtered data were used by Williams et al. [2004]. The only difference is that Nikolaidis [2002] used a 1-year sliding window to detect outliers whereas I use slightly overlapping 2-month windows. My outlier rejection removes 1 6% (mean 2.7%) of the points from each regional time series, similar to the 1 4% of points removed from global time series by Nikolaidis [2002]. For the noise estimation, data are rejected from all coordinate components at a particular station if an outlier is found in any component. This results in an average of 4.8% of the original points being removed during outlier rejection (Table 2). Because the original time series contain gaps due to equipment malfunctions, etc., the percentage of points analyzed in the MLE compared to the possible number of points is, on average, 91% (Table 2). [37] The time series in Figures 2a 2c generally give the impression of stochastic noise. However, there are a few cases, particularly in the data from stations KARA and CNCL, where apparently correlated signals occur. These two sites are about 10 km apart on the range front of the Southern Alps to the east of the Alpine Fault. Another station, QUAR, about 10 km from KARA but to the west of the Alpine Fault, does not show the signals. It is tempting to ascribe these signals to snow loading/unloading and associated hydrological effects, except that the two most prominent examples (KARA and CNCL horizontal components) are centered on January 2002 and June 2003, the first in the 6of13

7 Figure 2b. As for Figure 2a, but for east component. summer and the other in the winter. Further discussion of these signals is beyond the scope of this paper, except to note that KARA and CNCL show some of the larger power law noise levels in the analysis below Noise Analysis [38] The time series (from step 9 above) are analyzed for their noise properties using maximum likelihood estimation (MLE). The reasons for using MLE are thoroughly explained by Langbein [2004] and Williams et al. [2004]. Most importantly, the technique allows simultaneous estimation of the noise structure together with the parameters of a time-dependent model of the data. Also, the method doesn t require evenly sampled data, and the estimates of power law index appear to be less biased when using MLE than when using power spectral methods [see Williams et al., 2004]. [39] Quantities estimated in the MLE analysis are linear trend, offsets at designated times, annual sinusoid, power law noise index and amplitude, and white noise amplitude. In one case, CHAT, a semiannual sinusoid is also estimated. Once the offset values are estimated by MLE, the regional filtering is restarted at step 1 using the updated offset values, and the MLE analysis is rerun. I designate this power law plus white noise analysis as PL. Two independent software packages were used and found to give generally good agreement (section 5.1 and Tables S1 S4 of the auxiliary material) 1. [40] As well as estimating the annual sinusoid (plus the semiannual for CHAT), I include the estimation of a sinusoid at a fortnightly tidal frequency, which is detectable at low amplitude in the New Zealand time series. The amplitude is typically a few tenths of a mm in horizontal components and occasionally more than 1 mm in the vertical. Though the amplitude is low, the presence of this signal can cause the MLE estimates of power law amplitude to be biased high (S. D. P. Williams, personal communication, 2004). The reason this small signal is present in the time series is likely to be due to aliasing of tidal signals (principally unmodeled M2 ocean loading) with the 24-hour GPS processing window, though there is also a possible component due to the repeat period of the GPS satellite orbits being longer than the Nyquist period of diurnal and semidiurnal tides [Penna and Stewart, 2003]. [41] I also run the MLE algorithm, using the Williams et al. [2004] software, for the cases of flicker plus white noise (FL) and random walk plus white noise (RW). The reason for these analyses are so that comparisons can be made with 1 Auxiliary material is available at ftp://ftp.agu.org/apend/jb/ 2005JB of13

8 Figure 2c. As for Figure 2a, but for up component. Note the factor of two change in scale. the similar analyses undertaken by Williams et al. [2004]. These results are given in Tables S5 S7 for FL and Tables S8 S10 for RW. The values in Tables S1 S3 and S5 S10 may be converted to the functional form of a power spectrum using equations (10) (11) of Williams [2003]. [42] Langbein [2004] used a variety of other more complex noise models in his analysis of two-color EDM data. I do not investigate most of these models because the New Zealand data series are relatively short, and long series are generally required to discriminate the more complex models from the simpler ones. 5. Results 5.1. Comparison of Noise Estimates Using Independent Software Packages [43] I performed an MLE analysis of the New Zealand data using two independent software packages, cats_mle version 2.2 [Williams et al., 2004] and estnoise version 6aa [Langbein, 2004]. As shown in Tables S1 S4, the two packages give similar results for the PL case, with parameters generally agreeing within two standard errors of the estimates. One noticeable difference is that the cats_mle software often finds the best model to be a power law of low index (between 0.2 and 0.6) with a zero white noise component, whereas estnoise finds both a power law and white component in these cases. These differences do not translate into significant differences in the estimates and uncertainties of linear motion, offsets or sinusoidal components. The power law indices from both software packages are almost always between 0.2 and 1 (Tables 3 and S1 S3), and the few that are more negative than 1 are closer to flicker noise than random walk. For the remainder of this paper (with one exception) I use the cats_mle package, in order to provide the closest comparison with the Williams et al. [2004] analysis Comparison of Noise Models [44] The maximum likelihood values for the three noise models, PL, FL and RW, are compared in Table 3. The PL model invariably has an equal or higher likelihood than FL, which in turn has an equal or higher likelihood than RW. [45] A larger maximum likelihood (ML) value indicates a better model, where the ML value is the natural logarithm of the likelihood function [e.g., Langbein, 2004, equation (5)]. The difference (dml) in ML value that indicates whether one model is statistically superior to another depends on a number of factors, probably including the length of the data set and the number of points. Langbein [2004] used Monte Carlo simulations to show that for his data the lowerranking model could be rejected at the 95% confidence level if dml > 2.8. If we assume that 2 dml is distributed 8of13

9 Table 3. Maximum Likelihood Values for the PL Model and Differences (dml) With FL and RW Models a Site PL Index PL ML value NORTH EAST UP dml (PL-FL) dml (PL-RW) PL Index PL ML value dml (PL-FL) dml (PL-RW) PL Index PL ML value dml (PL-FL) Pillars AUCK CNCL HOKI KARA MQZG MTJO PAEK QUAR TAUP WGTN Other Monuments CHAT GRAC OUSD TAKL WGTT dml (PL-RW) a The values in columns headed PL ML value are the natural logarithm of the likelihood function. The ML values for the PL model are invariably greater than or equal to those for the FL model, which in turn are greater than or equal to those for the RW model. as c 2 ( [Edwards, 1992], then dml > 1.92 indicates the 95% confidence criterion between two models with a 1 degree of freedom difference (such as the PL and FL models). Using either of these criteria, a dml much greater than 3 indicates that the model with the higher ML value is superior to the other at a high confidence level. Examination of Table 3 shows that the PL model is significantly better than the FL model in nearly 70% of cases, and is better than the RW model in all cases but three Is RW Noise Detectable in Any of the New Zealand Data? [46] The indications are that RW noise is not detectable in the NZ time series, but I attempt one additional test using one of Langbein s [2004] more complex models and his software. This is a model of white noise, plus power law noise (with index estimated), plus random walk noise. The rationale is to see whether random walk noise (which might be due to monument instability) can be detected above the background of power law and white noise (which might be due to orbital, meteorological and other effects). I do not give a table of these results, as there were only four series out of 45 where a random walk component was detected with amplitude above the 2s uncertainty level. Since this is barely more than expected by chance, I infer that random walk noise cannot be reliably detected above other noise sources in the present time series. This is not to say that such noise is not present; it may simply mean that the time series are not long enough for it to be detectable Comparison With U.S. Regional Networks [47] I compare the NZ network results with those derived by Williams et al. [2004] for three U.S. regional networks, SCIGN, BARGEN and PANGA. SCIGN is concentrated in Southern California, BARGEN in the Basin and Range, and PANGA in the Pacific Northwest. The SCIGN and BARGEN networks, particularly the latter, are in relatively dry regions with moderate weather conditions. The NZ network is environmentally more similar to PANGA. Basic information on the network dimensions is given in Table 4, and network maps are given by Williams et al. [2004, Figure 3]. [48] In Table 5 I compare the results of the FL noise model between the NZ sites and several U.S. regional networks, with the U.S. results taken from Table 4 of Williams et al. [2004]. The NZ results are in general slightly noisier than those from the U.S. networks. The significantly lower noise level on the Scripps Orbit and Permanent Array Center (SOPAC) processing of the BARGEN network was noted and discussed by Williams et al. [2004]. [49] Williams et al. [2004] adopted two measures to discriminate between the noise properties of different monument types. Their first measure assumed a random walk plus white noise (RW) model and compared the median random walk amplitudes, b 2, for each monument type. The b 2 results for the NZ network are compared with results from the SCIGN network in Table 6. As acknowledged by Williams et al. [2004], this measure only provides information about monument noise if the random walk noise due to the monuments is actually detectable above other noise in the time series. For the Williams et al. [2004] regional networks this appears not to be the case, and it is even less true for the NZ data (section 5.2). However, I make the comparison for completeness. The data from the NZ pillars have b 2 values slightly lower (better) than data from the SCIGN deep braced monuments, while data from the other NZ monuments have b 2 values slightly higher than data from the next best performing SCIGN monuments. Table 4. Approximate Network Dimensions Network E-W, km N-S, km Area, Mm 2 NZ SCIGN BARGEN PANGA of13

10 Table 5. Mean and Median of the White Noise and Flicker Noise Amplitude Estimates for the North, East, and Vertical Components of the Site Time Series of the New Zealand and Selected U.S. Regional Solutions a Sites Used T avg, years T max, years Solution N E U N E U White Noise, mm Flicker Noise, mm/yr 1/4 NZ, PILLARS Mean 1.0 ± ± ± ± ± ± Median 1.1 ± ± ± ± ± ± 3.8 NZ, ALL Mean 1.1 ± ± ± ± ± ± Median 1.2 ± ± ± ± ± ± 3.7 SOPAC SCIGN b Mean 0.8 ± ± ± ± ± ± Median 0.7 ± ± ± ± ± ± 3.7 JPL SCIGN b Mean 0.9 ± ± ± ± ± ± Median 0.9 ± ± ± ± ± ± 4.3 SOPAC PANGA b Mean 0.6 ± ± ± ± ± ± Median 0.6 ± ± ± ± ± ± 2.5 SOPAC BARGEN b Mean 0.5 ± ± ± ± ± ± Median 0.5 ± ± ± ± ± ± 3.1 a The range indicated by the plus/minus symbol is the standard deviation for the mean and the interquartile range (IQR) for the median. T avg and T max give the average and maximum time series durations. b U.S. results taken from Williams et al. [2004]. I have transcribed their results for SOPAC processing of the SCIGN, PANGA and BARGEN networks plus JPL processing of the SCIGN network. [50] Williams et al. s [2004] second measure is the median time, T 1mm/yr, taken to achieve a linear velocity uncertainty of 1 mm/yr using the best fitting PL noise model. T 1mm/yr is calculated using the values in Tables S1 S3 and equations (25) (30) of Williams [2003] and is tabulated in Table 6. T 1mm/yr is again not strictly relevant to monument noise if the noise in the time series is predominantly from other sources, but it is a value that is frequently of interest for geodetic data. It also has relevance to the detection of changes in velocity like those induced by slow slip events at subduction zones [e.g., Ozawa et al., 2001, 2003; Dragert et al., 2001] or magma inflation episodes at volcanoes [e.g., Mattioli et al., 1998; Miura et al., 2000]. The data from NZ concrete pillars have values of T 1mm/yr that are almost as good as those from SCIGN deep braced monuments. The values of T 1mm/yr from other NZ monuments are very similar to values from the next best performing SCIGN monuments. [51] These comparisons suggest that time series data from good quality concrete pillar monuments in New Zealand are no worse than time series data from U.S. deep braced monuments. However, because of the small numbers in the New Zealand data set it is not possible to give high statistical confidence to this result. 6. Discussion [52] Williams et al. [2004] found that deep braced monuments outperformed all other types in their study, and concluded it was important to continue to use this monument type. However, it is fairly certain that that the major noise source they were seeing was not monument noise but some other noise source, perhaps associated with regionalscale atmospheric or hydrological mass redistributions or orbital errors. The dominant noise source in their regional time series analyses is close to flicker noise, and this argues against monument noise being a dominant process (if one accepts the arguments in section 2 that monument noise follows a random walk process). Also, the deep braced monuments in the U.S. were established as part of careful construction programs, with some assessment of site suitability and geological stability. This is not generally the case for the other monument types in their study. In other words, the improved noise behavior in the time series from deep Table 6. Median Random Walk Noise Amplitude b 2 and Median Time T 1mm/yr for the Velocity Uncertainty to Reach 1 mm/yr Compared Between New Zealand and SCIGN Monuments a b 2, mm/yr 1/2 T 1mm/yr,yr Monument Type Number of Monuments North East Up North East Up SCIGN deep braced NZ all NZ pillars NZ other SCIGN roof/chimney b SCIGN metal tripod b a SCIGN results taken from Williams et al. [2004]. b These are the next best performing SCIGN monument types after the deep braced [Williams et al., 2004]. 10 of 13

11 braced monuments may simply be due to better site selection, and high-quality concrete pillars at this set of sites might have produced time series that were just as quiet Concrete Pier Monuments Used in the U.S. Study [53] The six concrete pier monuments in the Williams et al. [2004] study were very diverse and some were in particularly poor environments. Some information on these monuments, CASA, CIC1, ESRE, ESRW, QUIN and WLSN, can be found in the Site Information section of the SOPAC web site ( CASA is on a short concrete pillar on the resurgent dome of Long Valley caldera and its data contain pronounced signals of volcanological origin that would have been interpreted as monument noise in Williams et al. s [2004] study. CIC1 is on a concrete pad about 0.2 m above ground level, but there is no information readily available about the foundation of the pad. ESRE and ESRW are on concrete pillars established close to the ends of a newly constructed earth dam (S. D. P. Williams, personal communication, 2004), which might be a noisy environment. QUIN is set at ground level on a concrete monument buried to 9 m depth, so is expected to be reasonably stable. WLSN is set on a low concrete pad on bedrock, but appears to suffer from a number of obstructions due to nearby trees. The SOPAC time series from WLSN are very noisy prior to mid-1997, and if these early data were used in Williams et al. s study they would have biased the noise estimates upward. These facts are not a criticism of Williams et al. [2004], since they were assessing mean and median properties of a very large number of time series, and therefore could not examine individual series in detail. Because the Williams et al. concrete pier designation was a default category for a range of monuments than didn t fit elsewhere, rather than a specific monument type, it is not useful to make a direct comparison between the results from the SCIGN concrete piers and the New Zealand concrete pillars. Had the New Zealand concrete pillar monument type been present in the SCIGN network it would have been given its own designation in the Williams et al. [2004] study (S. D. P. Williams, personal communication, 2005) Possible Biases in Noise Estimates Between New Zealand and U.S. Data [54] In order to use the values in Tables 5 and 6 as a basis for comparison between NZ concrete pillar and U.S. deep braced monuments, any significant biases between the data sets must be evaluated Southern Hemisphere Bias [55] Both flicker and white noise amplitudes in GPS global time series are biased high in the Southern Hemisphere [Williams et al., 2004, Figure 6]. It is possible that this bias is also present in Southern Hemisphere regional solutions. However, my working assumption is that this effect will be removed as a common mode signal in regional filtering, and will not upwardly bias the NZ regional noise estimates compared to those in the U.S Regional Filtering Strategy [56] Williams et al. [2004] note that estimated noise levels depend to some extent on the common mode signal removed during regional filtering. Eight sites out of 147 were used to form the common mode signal for the SCIGN data set analyzed by Williams et al., while preliminary analysis of SOPAC production time series, which use a larger number of sites, indicates lower noise levels [Williams et al., 2004, section 5.3]. We have used five out of 15 NZ sites to form the common mode signal (section 5.2), and our use of a larger proportion of sites may mean that our noise estimates are biased low compared to Williams et al. s Data Gaps [57] The removal of outliers and the presence of other data gaps in the time series can influence the noise estimates in two ways. A stricter removal of outliers may reduce either or both the power law and white noise amplitudes, simply due to removal of energy from the time series. The presence of data gaps will also reduce the power law amplitude because it is a function of the sample interval. For a time series with 91% data completeness (Table 2) the apparent sample interval is (1/0.91) days, implying a reduction of flicker noise amplitude to (1/0.91) 1/4, or 98%, compared to the gap-free series. I believe this is small enough to be insignificant, and the effect should be even smaller for less correlated noise signals Outlier Rejection Strategy [58] I use an almost identical outlier detection strategy to that used for Williams et al. s [2004] data, so I do not expect a bias from this source. Slightly more data were removed from the NZ time series, but the bias due to this has already been considered in section Network Size and Environmental Conditions [59] I expect networks of larger geographical extent or less uniform environmental conditions to have higher noise levels, since the common mode signal removal will be less effective in these cases. The NZ network is both larger (Table 4) and subject to more changeable weather than the SCIGN network, so I would expect the NZ noise levels to be biased high relative to the SOPAC SCIGN noise levels. The BARGEN network has the driest and most stable environmental conditions of the networks analyzed by Williams et al. [2004], and has a similar area to the NZ network. The BARGEN noise levels, as analyzed by SOPAC, are substantially lower than in any other network (Table 5). I suggest that this may be mainly due to the more stable environmental conditions, including distance from the ocean, though Williams et al. [2004] also attribute it to the fact that the network consists almost exclusively of deep braced monuments in rock. The PANGA network, as analyzed by SOPAC, also has lower white noise levels than the SCIGN or NZ networks. This is despite the fact that it has twice the area of the NZ network and is also in an area of changeable meteorological conditions. The reason for this is not clear. [60] Overall, there are some factors that may bias the NZ noise estimates upward relative to Williams et al. s [2004] U.S. estimates, and one factor that may bias them downward. While the only fully reliable comparison would require the NZ and U.S. data to be passed through an identical processing system, from raw GPS data to time series analysis, I feel that the results given in Tables 5 and 6 are sufficiently robust to draw useful conclusions Reduction of GPS Noise Sources in Future? [61] The NZ data are dominated by noise sources other than monument noise, at least for periods up to 4 5 years. 11 of 13

12 Any strategy for future monument installations therefore depends on whether one expects these noise sources to be reduced in the future. It should be noted that Williams et al. [2004, Figure 7] find a decrease in GPS noise levels through time, though the trend is much clearer before 1998 than it has been since. If a significant part of the noise is from orbital uncertainties or failures in ambiguity resolution, then I expect these noise sources will reduce, both as more Southern Hemisphere stations contribute to the GPS orbital solution, as more frequencies become operational in the GPS system, and as other global navigation satellite systems become operational. If a major contribution to the noise is atmospheric [e.g., Williams et al., 1998], then improved solutions for atmospheric parameters (e.g., better mapping functions, solving for tropospheric gradients as well as zenith delay) may help. If the noise is predominantly due to atmospheric or oceanic loading (seasonal atmospheric mass redistributions and monthly to decadal oceanic variations, for example) there may not be a straightforward solution [e.g., van Dam et al., 1994, 2001; Dong et al., 2002] Limits on Random Walk Noise at New Zealand Monuments [62] The noise in the NZ 4.5-year regionally filtered CGPS time series is not dominated by random walk noise but by noise of a lower spectral index. The spectral index is probably between 0 and 1, though it is possible that noise with a higher temporal correlation will come to dominate for longer time series lengths. If the current data are interpreted as random walk plus white noise, the median random walk amplitude is about 1.2 mm/yr 1/2 in the horizontal and 3.7 mm/yr 1/2 in the vertical (Tables 6 and S8 S10). These are, of course, upper limits on the true random walk component, which may be very much smaller. [63] If a small but significant random walk component is present in the data, it may be contributing to the uncertainties in the parameters derived from our time series even if it is not yet detectable in the time series analysis. Johnson and Agnew [2000] estimate that for a time series consisting of 1 mm white and 0.33 mm/yr 1/2 random walk components, a time series length of about 8 years would be required to reliably detect the random walk component at a 2s level, even though the random walk component would be contributing half the uncertainty to estimated parameters after only about 6 months. Equations (25) (30) of Williams [2003] may also be used to derive these results. [64] Using the same equations and assuming 1 mm white noise (Table 5) plus some level of random walk noise in our horizontal time series, I estimate the amplitude of random walk that would be detectable at the 2s level in the NZ 4.5-yr time series to be 0.65 mm/yr 1/2. For the estimated 3.4 mm white noise in our vertical time series the corresponding random walk amplitude is about 2.2 mm/yr 1/2. These are more stringent upper limits than those derived earlier in this section. [65] The power law noise level of data collected on other monument types tends to be higher than the noise level from either deep braced monuments in the U.S. or concrete pillar monuments in New Zealand (Table 6). Yet the power law indices from the other monuments do not show a noticeable tendency toward random walk behavior compared to the carefully constructed monuments (Tables S1 S3). This is either because the random walk monument noise component is present but can t be detected with the present time series length, as discussed above, or it may be because the noise introduced by the other monument types is not in fact random walk in character, but is less temporally correlated. In either case, it is clear that carefully designed monuments well connected to the ground give a significant improvement in data quality over less well designed and constructed monuments Detection of Aseismic Deformation and Long-Term Vertical Signals [66] Two of the tectonically most important reasons for continuous GPS measurements are the detection of timevarying deformation in both tectonic and volcanic environments, and the accurate measurement of vertical rates. Many of the time-varying signatures detected in recent years [e.g., Linde et al., 1996; Dragert et al., 2001; Ozawa et al., 2003] have had periods of days or weeks. For these signals my results show there is little or no benefit in using an expensive deep braced monument over a carefully constructed concrete pillar, since the noise at these shorter periods is dominated by sources other than monument noise. [67] For measurement of long-term vertical rates the situation may be different. Long-term vertical signals are typically at least an order of magnitude smaller than horizontal ones, while GPS system, atmospheric, and loading noise levels are all higher. Given these higher noise levels, studies of much longer time series will be required to determine whether deep braced monuments provide a significant improvement over concrete pillar monuments for these types of signal. 7. Conclusions [68] In this study I have examined data from a set of ten carefully constructed concrete pillar monuments, as well as a few monuments of lesser quality. Using the same criteria adopted by Williams et al. [2004], I find that the differences between the noise properties of the SCIGN time series analyzed at SOPAC and the NZ concrete pillar time series analyzed at GNS are small. This suggests that the data from the NZ concrete pillar monuments are on average nearly as good as the data from the SCIGN deep braced monuments, at least for data sets of 4 5 years duration. [69] There are several caveats to this conclusion, including the relatively short lengths of the NZ time series. Also, I do not believe that the noise we are presently seeing in the time series is dominated by monument instability, but rather by other sources that have not been explicitly identified. In other words, even though similar noise levels are seen in the SCIGN and NZ time series this does not mean that I have demonstrated that the NZ monuments are as good (i.e., low noise) as the SCIGN deep braced monuments. As lengths of time series increase, and more particularly as global navigation satellite systems and data processing techniques improve in quality, monument instability may become the limiting noise source. Long-term controlled experiments to address this issue would provide important information for future CGPS installations. 12 of 13