Corresponding author: Y. Reuveni, Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Dr., MS 238-634, Pasadena, CA 91109, USA. (firstname.lastname@example.org)
 We present an improved GPS analysis strategy that reduces the noise level of GPS-based sub-daily strain measurements by a factor of ~5 or more and improves sub-daily resolution of positions and baseline estimates by a factor of ~2–5. These improvements are accomplished by reducing the key sources of error due to diurnal effects from path delays caused by reflections and refractions of the GPS signal near the receiver (multipath), and from tropospheric delays. Errors due to poorly determined tropospheric path delays are mitigated by using the tropospheric parameters estimated in static positioning runs as fixed values. The multipath effects are treated as periodic errors and are mitigated by a modified sidereal filter applied to the phase prior to processing. This combination of path error modeling results in sub-daily strain resolution on the order of ~0.1μstrain for a ~100 km baseline.
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 Studies of sub-daily strain have focused mostly on the more accurate point strain measurements that are made by borehole strain meters and long (~1 km or less) baseline laser strain meters. Both tools provide strain measurements that are typically 2–3 orders of magnitude less noisy than GPS-derived high-rate strain, depending on the period of interest. However, their cost and demanding siting and installation requirements [Roeloffs, 2004, 2005; Agnew and Wyatt, 2003] have greatly limited their numbers (fewer than 200 stations in North America) and geographic extent as compared with the much greater number and distribution of GPS stations (Figure 1). With networks of over a thousand GPS stations in the US, and a similar number in Japan, comprising numerous baselines of multiple length scales, current GPS networks have the spatial density to measure sub-daily strain, particularly in areas devoid of strainmeters.
 Here we evaluate the resolution of sub-daily strain measurements derived from GPS data while applying algorithms to reduce sub-daily tropospheric and multipath noise. We use these algorithms to establish the noise floor for sub-daily baseline strain estimates from GPS measurements.
2. Technical Approach and Methodology
 Generally, sub-daily GPS position estimates are affected by the same error sources as daily average positions. Nevertheless, with GPS orbits available with radial precisions of less than 1 cm, the most important sources of error impacting sub-daily GPS precisions are primarily from path effects, such as path delays caused by reflections of the GPS signal in the receiver's environment (multipath), and delays caused by the troposphere [Larson et al., 2010]. By mitigating these key sources of error, sub-daily resolution of positions and baseline estimates can be improved. Following is a description of the results of noise reduction GPS data analysis strategies that were tested using the Jet Propulsion Laboratory's (JPL's) GIPSY-OASIS precise point positioning software and products [Zumberge et al., 1997; Bertiger et al., 2010].
2.1. Tropospheric Correction
 In the last two decades, several tropospheric modeling strategies have been developed in order to improve positioning precision [Larson et al., 2010]. The developments in mapping functions [Niell, 1996; Boehm et al., 2006a] and tracking satellites to lower elevation angles are the main causes for these improvements. Tregoning and Herring  recognized that the use of accurate surface pressure could eliminate errors caused by poor a priori Zenith Hydrostatic Delay (ZHD) values, which corrupt the estimates of station heights and Zenith Total Delay (ZTD) values in GPS analysis. Furthermore, the introduction of azimuthal tropospheric gradient terms [Bar-Sever et al., 1998] has also enhanced positioning precision [Miyazaki et al., 2003; Gegout et al., 2011]. The GIPSY-OASIS software used in this study treats the tropospheric zenith delay and gradients as stochastic parameter to allow time-varying behavior. Stochastically time-varying parameters are assumed to be constant within each time interval, but may vary from one time interval to another. After a measurement has been processed (and the parameter estimation had been updated), a time update is performed, adding process noise to the parameter uncertainties in order to allow for unmodeled or mismodeled effects. The first step in the noise reduction strategy is to mitigate the errors, due to poorly determined tropospheric path delays by using the tropospheric parameters (tropospheric wet zenith delay and gradient values) pre-estimated in static position calculations (i.e., estimating a single station position over a period of 24 hours) as fixed values. This is done because the kinematic position parameters and the time-varying tropospheric parameters cannot be estimated simultaneously without some degradation of position precision [Tralli and Lichten, 1990]. Rather than the VMF1 mapping function [Boehm et al., 2006a], we used the GMF mapping function [Boehm et al., 2006b], an empirical model that has different static values for the mapping function coefficients for each day, since we find that it yields slightly less noisy sub-daily horizontal positions. In the static position calculations, we used a 7° minimum elevation cutoff for the satellite observations and 30-second observations. The troposphere wet delay and gradient parameters were modeled as random walk stochastic processes, whereas the tropospheric dry delay parameters remain fixed. This fixed value is a function of the station height, based on the calculation of surface pressure given by the hydrostatic equations used in the standard atmosphere. Using this fixed hydrostatic value yields the same horizontal positions results within 0.04–0.2 millimeters as using 6-hourly hydrostatic delay values derived from the VMF1 or the GPT models. The tropospheric zenith wet delay and the gradient parameters are allowed to vary within 5.0e–8 km/√sec (corresponds to about 3 mm in an hour) and 5.0e–9 km/√sec (corresponds to about 0.3 mm in an hour), respectively.
2.2. Multipath and Sidereal Filtering
 It was first demonstrated by Genrich and Bock that sidereal filtering can reduce the effect of sub-daily noise sources in GPS solutions by mitigating periodic errors associated with the repeated relative geometry between each satellite and station. These errors are mostly attributable to signal multipath from reflections off objects near the receiving antenna with the approximately sidereal period of the multipath error arising from the approximately sidereal change in the relative geometry between the transmitter, reflective surface, and the receiving antenna. In standard daily position solutions using 24 hours of data, these errors tend to average out and are absorbed in the modeling residuals [Segall and Davis, 1997]. However, for sub-daily GPS positioning, they dominate the noise of the position solutions.Bock et al.  had demonstrated that filtering the multipath noise improves high rate positioning precision and reduces the scatter by about 50% in the horizontal and vertical components.
 As was demonstrated by Choi et al. , in order to accurately remove the multipath noise it is necessary to calculate the period it takes a given satellite to repeat its position in the sky as seen from a GPS receiver on the ground. Seeber et al.  and later Agnew and Larson  pointed out that the satellite repeat period is not strictly sidereal (23 h 56 m and 4 s), but differs slightly for each satellite. This is due to the requirement set by the Department of Defense that the GPS ground tracks be fixed (instead of the orbit periods being sidereal) in order to keep the longitude of the ascending node within +/−2° of its nominal value [Chao and Schmitt, 1991]. In order to correct for the nodal drift caused by the earth's nodal precession, each satellite's orbital period is set to be ∼4 s faster than a half-sidereal day, which sets the GPS satellite above the same longitude every day [Choi et al., 2004].
Choi et al.  have improved the sidereal filter performance by using the actual orbital period in a technique commonly referred to as Modified Sidereal Filtering (MSF), which applied the sidereal filter to 1 Hz position estimates. An extension of this technique was implemented at JPL and used with GIPSY precise point positioning to remove the sidereal noise from the phase and pseudorange observations rather than the receiver positions [Owen and Webb, 2005]. Site-specific multipath effects are removed by stacking phase and pseudorange residuals offset by the orbit period of each satellite (without the restriction of completely removing first any clock, atmospheric, or orbital errors by double-differencing as suggested byZhong et al. ). The MSF corrections are determined by time shifting and then stacking the 30 second satellite-specific phase residuals at a given station over +/−n days before and after the target day by the orbit repeat period of each satellite. The time shift, or repeat period offset relative to the target day, is calculated from the JPL precise orbits for the target day and for the specific satellite. The stacked residual values are then interpolated using a 4th-order polynomial and applied as a correction to the phase and pseudorange observations of the target day. We used the phase residuals from 5 consecutive days prior to and following the target date to generate filtered input observation files for the GIPSY position analysis. This is the number of days necessary for a stable interpolation, yet is not likely to be greatly influenced by changes in ground reflection characteristics that might take place over a higher number of days. These filtered postfit residuals, along with the fixed tropospheric parameters to those estimated from the static position calculations, form the input for the sub-daily point positions and consequently baseline strain estimates.
3.1. Noise Reduction in Sub-daily Station Position
 Using the above noise reduction strategies, we observed nearly an order of magnitude reduction in the noise level of GPS sub-daily station position estimates (Figure 2). The impact of each correction is shown relative to a regular kinematic solution. Using the tropospheric parameters estimated in static position calculations as fixed values reduces the standard deviation values from 0.49 cm, 0.52 cm, and 1.3 cm, to 0.47 cm, 0.4 cm, and 0.75 cm, for the north, east, and vertical components, respectively. Using the MSF for the multipath noise reduces the standard deviation values to 0.3 cm, 0.23 cm, and 0.83 cm, for the north, east, and vertical components, respectively. The combined troposphere and multipath corrections result in a marked improvement in sub-daily station position noise as the standard deviation values are reduced to 0.22 cm, 0.17 cm, and 0.31 cm, for the north, east, and vertical components, respectively. The multipath noise that is introduced into the position solution and typically originates from higher frequencies is substantially reduced, consistent with the finding ofChoi et al. .
3.2. Baseline Solutions
 The reduction in GPS receiver position noise translates into a reduction of inaccuracy in sub-daily baseline estimates. Since long baseline laser strainmeters are the most accurate measurement of strain available we use them as the measuring rod for evaluating sub-daily GPS strain measurements.Figure 3shows the improvement in sub-daily baseline stability for a ~260 km baseline (GMRC-SIO5,Figure 1), using kinematic 30 second solutions for a period of one day (in (a) time and (b) frequency domains) in relation to both laser and borehole strainmeters. To calculate the GPS strain we first calculate the mean baseline Euclidean length of the GMRC-SIO5 vectors over the period of interest (day), which is then subtracted from the baseline measurement at each epoch. The remainder (i.e., the variation about the mean) is then divided by the baseline mean length. This relatively long baseline (~260 km) was chosen to represent an upper limit of baseline length that might observe a transient strain signal, and for its geographic bracketing of the laser and borehole strainmeters at Pinon Flats (Figure 1), which will later be used for strain noise level comparison. Analysis of shorter baselines will follow. The purpose of this comparison is to qualitatively evaluate how close the improved GPS baseline time series may get to the regional ‘true’ strain measurements from borehole and laser strainmeters. The standard deviation over a day of the estimated daily time series strain derived from our noise reduction strategy is on the order of 9.7e–9, compared to 1.7e–9 and 1.6e–9 for the borehole and laser strainmeters, respectively. This means that the time domain noise levels obtained by our noise reduction strategy are 5.5–6 times larger than the ones obtained by the borehole and laser strainmeters over a period of a day. However, in order to evaluate the noise reduction strategy relative to the borehole and laser strainmeters in shorter periods relevant to transient strain detection, a frequency domain comparison is shown Figure 3b. Implementing the above noise reduction strategies improves the baseline strain noise by a factor of ~10 depending on the period, yet it is still ~2–3 orders of magnitude noisier than the borehole and laser strainmeters. This implies that at sub-daily periods and a long (~250 km) baseline, we can only detect strain signals that are ~2–3 order of magnitude higher than the ones typically detected by the borehole and laser strainmeters depending on the period.
 Since the multipath noise is independent of baseline length, the noise level (as measured by the standard deviation of the strain time series over a day) scales inversely with the baseline length. We analyzed the dependence of baseline strain accuracy on baseline length over baselines varying in length from 50 m to 260 km. The noise level of sub-daily strain goes down from ~0.1μstrain at sub-fault-depth scales (<20 km) to ~0.01μstrain at tectonic scales (>100 km). For reference, the slow earthquake described by Linde et al.  produced a local (fault of several kilometers in size) slow earthquake signal of ~0.5 μstrain over a day. Based on the above analysis a signal of that magnitude may be just within the detection capabilities of a short (~10 km) baseline of two GPS stations.
 It should be noted that a small portion (less than 10%) of the noise associated with orbit and clock (satellite and receiver) error is introduced into the GPS point positioning solutions, and manifests itself as a common-mode error on 24-hours time scale periods. For station position estimates, this noise is easily removed as the improved noise reduction strategy enables the detection of these diurnal jumps that are typically hidden in the multipath noise. These common-mode errors are canceled out in the baseline strain estimates.
 The noise reduction strategies employed in this study can significantly reduce the sub-daily station position noise and consequently the baseline noise caused by multipath and sub-daily troposphere variations. This is accomplished by applying pre-calculated tropospheric corrections pre-estimated from static station positioning calculations, and by applying a Modified Sidereal Filter (MSF) correction to the phase data prior to performing a kinematic estimation of the station positions. Together, these improve the sub-daily noise levels of station positions by a factor of ~2–5 and the baseline strain estimates by up to a factor of ~10 depending on baseline length and the strain signal period. For a ~100 km baseline this translates to a sub-daily strain noise level on the order of ~0.1μstrain over a day. This noise reduction strategy enables a new capability for routinely measuring tectonic strain using GPS on time scales not typically examined by GPS and provides new observations of the deformation field in the underexplored period band of ~1 hour to a day. By employing these troposphere and the MSF reduction strategies, modern GPS networks can serve as exploratory tools for mapping regional variations in sub-daily strain in regions where no laser or borehole strainmeters exist and GPS stations are abundant. In such regions, although still ~2 orders of magnitude noisier than laser and borehole strainmeters, routine GPS estimates of the sub-daily strain field should be used for continuous monitoring of transient deformation. It is possible that further reduction of GPS based strain estimates may be achieved if multiple baselines are combined to measure sub-daily strain at a single location. This may be the focus of future research.
 The authors wish to thank the editor, reviewers and Duncan Agnew for their thoughtful and constructive critique of the manuscript. The research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration.
 The Editor thanks two anonymous reviewers for assisting in the evaluation of this paper.