In this subsection, we briefly summarize the key data sets we use in this study.

GPS-IR is a technique that uses the interference between the direct and reflected GPS signals to infer ground properties such as snow depth, soil moisture, and vegetation water content (Larson et al., 2008, 2009; Small et al., 2010). Larson (2016) provides an overview of the GPS-IR technique. Here we only describe the method of using GPS-IR to measure the reflector height, which refers to the height of the GPS receiver antenna phase center above the reflecting surface.

GPS-IR uses the interference between the direct signal and the reflected signal from the ground surface. Figure 2 illustrates the interference geometry. The strength of the interference, quantified by the SNR of the received power, oscillates with the elevation angle (e). For a horizontal planar reflector, such as the flat surface surrounding SG27, the SNR oscillation is characterized by a dependency on sine of the elevation angle (Larson, 2016): SNR=Asin⁡4πHλsin⁡e+ϕ, where A is the amplitude, H is the reflector height, λ is the wavelength of the GPS signal, and ϕ is the phase offset of the oscillation. Given a measure of varying SNR with sin⁡e, we calculate its periodogram using the Lomb–Scargle spectral analysis (Press et al., 1996), determine the dominant frequency f, and eventually obtain the reflector height H as fλ/2. The reflection observed in SNR data using a geodetic antenna is most sensitive to the interface between air and the top soil layer.

We apply this method to the L1 SNR data (λ=0.19029 m) recorded by SG27 to retrieve the reflector height at daily intervals. Using the SNR data from individual satellite track with an elevation angle range of 5 to 20∘, we estimate the reflector height and repeat this for all tracks. To avoid the obstructions from buildings and other infrastructure located nearby (Fig. 1c), we only keep the H with azimuth angles of 90 to 180∘ (i.e., in the southeast quadrant). Then we average H from all usable tracks and use the average to represent the reflector height within the GPS-IR footprint. We calculate the standard error of the mean as the uncertainty of the averaged H. In fact, each track has a different reflecting point, which depends on the azimuth and elevation angles, as well as the antenna height. Using the first Fresnel zone of the reflected signals for the elevation angle of 5∘ (Larson and Nievinski, 2013), we estimate the average extent of the footprints as having a radius of 90 m from SG27. To avoid the ambiguous interpretation about reflector height changes as caused by snow depth changes or by the thawing/freezing of soil, we only consider reflector height on snow-free days between 1 July and 31 August in each summer from 2004 to 2015. We exclude the data before 2004 to avoid a significant offset due to the GPS equipment change on 1 June 2004.

By combining the daily reflector height and the vertical position of the GPS receiver, we can calculate the change of ground surface elevation at SG27 in a geocentric frame. Let V be the vertical position of the GPS receiver, then the vertical position of the ground S is simply V minus H (Fig. 2). It is worth pointing out that in a geocentric frame, the surface elevation changes (from either our GPS-IR retrieval or the GPS campaigns) include contributions from two independent processes: one is due to the dynamics of the active layer and near-surface permafrost (referred to as “frozen ground dynamics”), and the other due to the movement of solid earth. Assuming the anchor position (P) is stable as it is deeply frozen in permafrost (at ∼ 5 m depth) and the wooden pole is rigid, any change of the receiver's vertical position (V) is due to the solid earth movement. This solid earth component needs to be removed for studying frozen ground dynamics. After this correction (i.e., subtracting by V), the surface elevation change due to frozen ground dynamics, denoted as SF as a function of time t, is reduced to a simple negative relation with the reflector height: SF(t)=-H(t). Therefore, the GPS-IR framework is intrinsically convenient: we only need the reflector height H, rather than the solid earth movement V, for studying frozen ground dynamics. If not explicitly stated, all surface elevation results presented and discussed in the remainder of this paper are given with the SF(t) term in the geocentric NA12 frame. To directly compare SF values retrieved using GPS-IR and those obtained by the GPS campaigns of Streletskiy et al. (2017), we first convert their vertical position values from NAD83 to NA12, then remove V measured at SG27 from theirs. The solid earth movement is nearly the same at SG27 and the sites of Streletskiy et al. (2017), within 2 km distance in this tectonically inactive area.

We also model seasonal ground surface subsidence due to the melting of pore ice in the active layer and further assess the subsidence from the melting of segregated ice. For simplicity, the following conceptual and mathematical framework is for a given thaw season. Our model only considers one component in the seasonal subsidence that is caused by the volume decrease from ice to water in the pores of the active layer as it thaws. Another component is the subsidence caused by thawing of segregated ice. We denote these two subsidence components as dpore and dseg, respectively. The total thaw subsidence d is the sum of these two, i.e., d=dpore+dseg. We note that d is directly comparable to SF. Throughout this paper, we use capitalized and lower-case symbols for the observed and modeled variables associated with vertical movement, respectively, and symbols with a hat accent for the best fit variables (e.g., S^F in Sect. 4.5). Both dpore and dseg reach their seasonal maxima (denoted as dporemax and dsegmax, respectively) at the end of each thaw season. Because we know little about segregated ice within the active layer (when it is frozen) near SG27, let alone its temporal changes, we cannot directly quantify dseg. Instead, we only model dpore and interpret the difference between the observed or best fit seasonal subsidence and dpore as the contribution from melted segregated ice throughout a thaw season. In this flat area, surface runoff is negligible and can be ignored.

For a fully saturated active layer, dporemax can be expressed as an integral over the entire active layer soil column (Liu et al., 2014): dporemax=∫0Lϕ(z)ρw-ρiρidz, where z is the soil depth, dz is the incremental thickness of the thawed active layer soil column, ϕ is the soil porosity, ρw is the density of water, ρi is the density of pore ice, and L is the ALT, which typically varies in different years. The mean ALT within the CALM grids was 40 cm in 2016. Assuming a constant ratio between the ALTs at SG27 and CALM (i.e., 53 cm / 40 cm) throughout the past years, we extrapolate the ALT at SG27 for 2004–2015 by multiplying the CALM ALT by this ratio. We follow Liu et al. (2012) to model ϕ as a function of depth by assuming a surface organic layer with organic content decreasing exponentially with depth. We also estimate the uncertainties of dporemax by propagating the standard deviation of the ALT measured within the footprint (i.e., 6 cm) and the uncertainties in the assumed model parameters for calculating water content (see Eq. 16 of Liu et al., 2012).

Next, we model cumulative subsidence due to top-down thawing of active layer and the corresponding progressive melting of pore ice on any day t since the thaw onset (Tthaw, late May to early June) until the freeze onset (Tfreeze, late August to early September) as dporet=A(t)Amaxdporemaxif Tthaw≤t≤Tfreeze, where A is the degree day of thawing (DDT; units: ∘C days), defined as the sum of the daily surface air temperatures for all days with above 0 ∘C since the thaw onset. In Eq. (4), Amax⁡ is the maximum DDT, corresponding to the end of the thaw season. The square root relationship derives from the Stefan equation that describes the progressive downward migration of the thawing front (French, 2007; Liu et al., 2012).

Considering both the GPS-IR measurement uncertainties and that some random processes other than the gradual downward thawing may introduce random errors into our retrieved SF, we fit the time series of SF for each summer using the Stefan function in the same form as Eq. (4). The best fit time series, denoted as S^F(t), differs from SF(t) by a random error term ε(t), i.e., SFt=S^Ft+εt=A(t)AmaxS^max+εt, where S^max is the maximum accumulative subsidence within each thaw season. This S^max term is the only coefficient that we fit with the data SF using the weighted least-squares inversion. We also estimate the uncertainties of S^max using the weighted least-squares optimization.

Since the DDT records spanned thaw onset to freeze onset, we can also use Eq. (4) to extrapolate our observed SF that spanned 1 July to 31 August back to the thaw onset, around 1 June. Because surface subsidence can be rapid in early thaw season, this extrapolation is important if one needs to consider the net change during the entire thaw season.

Soil moisture greatly affects the dielectric constant of the ground and thus the multipath modulation (Nievinski and Larson, 2014a). Temporal changes of soil moisture can cause apparent changes in the retrieved reflector heights. As this apparent change is due to surface compositional properties, we follow Nievinski (2013) to refer to this as the “compositional height”. We need to assess the compositional heights due to soil moisture changes.

We first estimate the general varying pattern of the compositional height with changing soil moisture (in volumetric water contents, VWC) that increase from 0 to 100 % by an interval of 1 %. For organic-rich soils with a given VWC, we run the GPS multipath simulator of Nievinski and Larson (2014b), MPSImulator (publicly available at https://www.ngs.noaa.gov/gps-toolbox/MPsimul.htm), to simulate SNR data using the same settings as the real SNR data at SG27 (see Table 2 for a list of simulator settings). Then we apply the same GPS-IR data processing method as described earlier in Sect. 4.2 to calculate the compositional heights. Because of the changes of antenna and radome on 24 August 2010, we run the simulator using two antenna models and obtain two relationships of compositional heights versus soil moisture. Because the simulator does not include gain pattern models of the two antenna–radome combinations at SG27, we set the radome models as “SCIS” and “NONE” for the two cases, respectively. Next, we simulate a time series of compositional heights by using the soil moisture measured at U1-1.

Figure 3 shows the time series of V in the geocentric NA12 frame. The solid earth underwent regular cyclic vertical movements at the annual and semi-annual periods, due to surface mass loading from the atmosphere, ocean, and surface hydrology (van Dam et al., 1994, 2001), and a steady subsidence trend. The mean seasonal subsidence from 1 July to 31 August during 2004–2015 was 3.3 ± 0.2 cm. The best fit linear subsidence trend was 0.27 cm year-1.

Figure 4a shows the time series of surface elevation changes due to frozen ground dynamics from 2004 to 2015. We only present the values of SF on snow-free days between 1 July and 31 August. The ground surface underwent gradual seasonal subsidence. Figure 4a also shows prominent interannual variability, which is associated with summer air temperatures. Using the DDT at the end of each thaw season as an indicator of warm/cool summers (Fig. 4b), we observe a general trend that larger seasonal subsidence occurred during warm summers such as 2004 and 2007 and smaller subsidence within cool summers such as 2005, 2006, and 2014. However, the subsidence was comparatively small during a warm summer in 2012, deviating from the general correlation. At secular scales, the ground surface underwent a steady subsidence of 1.05 ± 0.03 cm year-1 from 2004 to 2010, followed by an uplift trend of 1.82 ± 0.06 cm year-1 from 2011 to 2014, and then a subsidence from 2014 to 2015. The overall linear subsidence trend between 2004 and 2015 was 0.26 ± 0.02 cm year-1.

Our estimated surface elevation changes generally agree with the GPS campaign measurements of Streletskiy et al. (2017). Since the campaign measurements were conducted in late August, we can only compare these two in the interannual sense (Fig. 4a). Both sets of elevation change results are consistent within the uncertainties in individual years except 2008, 2010, 2011, and 2012. Both show similar subsidence trends between 2004 and 2010, and similar uplift trends during 2012–2014, and the subsidence from 2014 to 2015. After removing V, which has a linear subsidence trend of 0.27 cm year-1, from the campaign measurements, we obtain an overall subsidence trend of 0.19 ± 0.14 cm year-1 between 2004 and 2015. This is consistent with our GPS-IR trend of 0.26 ± 0.02 cm year-1 within the uncertainties.

Out of the four mismatched years, the campaign measurements show strong heave relative to the previous August in three of them (i.e., heave from 2007 to 2008, from 2009 to 2010, from 2010 to 2011). Streletskiy et al. (2017) did not explicitly explain these observed heaves. The campaign measurements also show ∼ 13 cm of subsidence from August 2011 to August 2012, in contrast to the nearly zero changes between these two Augusts from our GPS-IR-based observations.

Year-by-year comparison shows that the simple square-root-of-DDT model (Eq. 5) generally fits the GPS-IR-retrieved elevation changes (Fig. 5). The R2 values of the fitting ranging from 0.24 to 0.9 and a mean of 0.6 for all the years (Table 3). According to the best fit results, the net subsidence between 1 July and 31 August in each year ranged from 1.1 to 7.4 cm, with a 12-year mean of 3.4 cm and a standard deviation of 2.1 cm (the solid magenta lines in Fig. 5 and the second column of Table 3). Extending the best fit results to 1 June, we infer that the total subsidence within each thaw season ranged from 1.8 to 12.5 cm, with a 12-year mean of 5.8 cm and a standard deviation of 3.5 cm.

Our modeled subsidence due to the melting of pore ice spanning each thaw season (i.e., dporemax) was 2.8 cm on average, with small interannual variability. Such small variability is largely because the ALT varied little during the study period (Fig. 4c), which means that the total pore water volume in the active layer did not change much over the years. In terms of multiple-year average, the modeled seasonal subsidence is smaller than the best fit by 3.0 cm. Such residual is our estimated seasonal subsidence due to the melting of segregated ice (i.e., dsegmax, listed in the last column of Table 3). The uncertainties of dsegmax are obtained by error propagation. In 8 out of 12 years, our inferred dsegmax are larger than their corresponding uncertainties, which will be used in the following analysis.

Since the subsidence caused by melting of segregated ice is controlled by the total amount of segregated ice in the active layer before thawing, we hypothesize that more segregated ice may develop after a wet thaw season, therefore resulting in larger subsidence during the following thaw season. Due to lack of soil moisture data throughout the study period, we use the cumulative precipitation in August as a proxy for excess soil water, which we refer to as the extra water that is more than a fully saturated active layer can hold before it starts to freeze. Figure 6 shows a scatter plot between dsegmax and the precipitation in the previous August. We observe two distinct groups: for dsegmax that are larger than 5 cm, they increase nearly linearly with the precipitation, which confirms our hypothesis; yet for small dsegmax (around 2 cm), they are independent of the precipitation.

Figure 7 shows the possible range of compositional height changes due to soil moisture changes from 0 to 100 %. The two curves correspond to the two antenna models, before and after the equipment change on 24 August 2010. Both curves show that reflector height increases monotonically with soil moisture. In the post-2010 case, the reflector height shows a higher sensitivity to soil moisture than the pre-2010 case.

Figure 8a shows the time series of VWC at 5 cm depth, measured at U1-1. Five centimeters is the resolved depth range for soil moisture retrievals using GPS-IR (Larson et al., 2008). Between July and August in each year, the change range of VWC was up to 15 %. The only exception was 2009 when the range was the largest, ∼ 25 %. In many summers, the soil moisture first decreased from June to July, then increased in August. Rainfall events sharply increased the soil moisture (e.g., in 2010 and 2011).

Figure 8b shows the time series of the simulated composition height changes. Because the soil moisture records are not from within the GPS-IR footprint and their period does not fully overlap with our GPS-IR records, we cannot use the simulated compositional heights to “correct” the GPS-IR reflector height results. Instead, we interpret the simulated results to assess the possible effects of soil moisture, in the following aspects. First, in our case using the settings listed in Table 2, the compositional heights are always positive. However, because we are only interested in temporal changes, any systematic bias due to soil moisture changes is irrelevant. Second, the changes of compositional height within each summer were within in 0.5 cm, much smaller than the reflector height changes at seasonal scales. Third, the largest interannual change in compositional height was between 2010 and 2011, due to the antenna change. For instance, the compositional height increased by ∼ 0.8 cm between 1 July 2010 and 1 July 2011. This is still smaller than the ∼ 3.2 cm subsidence based on the GPS-IR reflector height records. Fourth, because the near-surface soil moisture in Barrow did not undergo any significant decadal changes, the secular trend of compositional height was negligible (e.g., 1.4×10-5 cm year-1 for 1996–2011). For the above reasons, we conclude that our retrieved changes of ground surface elevation are not significantly affected by these soil moisture effects.

We postulate that both large seasonal subsidence and the decadal subsidence trend are due to thawing of the transition layer in warm summers. Figure 4a shows that the largest seasonal surface subsidence occurred in 2004 and 2007, which were the warmest summers during the 12 years (Fig. 4b). The thaw indices also increased from 2005 to 2013 with a trend of 20.3 ∘C days year-1, which may cause the gradual thawing of the transition layer and thus the linear surface subsidence trend during the same period. In years when excess water remains in the active layer before freezing, significant accretions of segregated ice can develop within the transition layer and cause surface heave during winter (Fig. 6).

The transition layer is widely thought to act as a buffer between the thawing of the active layer and ice-rich permafrost in that it protects the permafrost beneath from thawing (Hinkel and Nelson, 2003; Shur et al., 2005). The progressive thawing of the transition layer causes a gradual surface subsidence that is barely observable without accurate measurements over decades. In addition to the GPS campaigns conducted by Shiklomanov et al. (2013) and Streletskiy et al. (2017), as well as the InSAR study of Liu et al. (2010) over Prudhoe Bay, our GPS-IR results provide another set of observations of such subtle decadal changes on the North Slope of Alaska. The two independent estimates of linear trends at Barrow agree well (i.e., 0.26 ± 0.02 cm year-1 from this work and 0.19 ± 0.14 cm year-1 from the GPS campaigns of Shiklomanov et al., 2013). The InSAR measurements of Liu et al. (2010) revealed linear subsidence trends of 0.1 to 0.4 cm year-1 between 1992 and 2002 over Prudhoe Bay, consistent with the two Barrow studies within the same order of magnitude.

The surface elevation changes retrieved from our GPS-IR measurements are daily and long-lasting, which are unique and valuable for quantifying subtle surface changes over permafrost areas. In cases of no major instrumental changes or when any vertical shift in GPS antenna phase center due to instrument change is known (e.g., 2 mm in 2010 for SG27), the GPS-IR-based measurements are consistent, sustained, and progressively increasing. This is important for studying seasonal, interannual, and long-term dynamics of the active layer and permafrost. In situ ALT or GPS campaign measurements were typically conducted annually, but not always on the same day of the year due to logistical constraints. Our GPS-IR results show that the seasonal changes are more significant than the interannual and long-term changes. It is possible that the interannual and long-term changes estimated from a poorly sampled record of elevation changes (e.g., annual measurements) may be aliased by the seasonal changes (Liu et al., 2015). Our daily sampled and long-lasting records from GPS-IR can avoid such aliasing problem and give robust estimates on the interannual and long-term variations.

Since the GPS-IR-estimated reflector height directly reflects the frozen ground dynamics, it is unnecessary to process geodetic-level GPS positioning data or correcting for the solid earth movement. Knowing the GPS receiver position, nonetheless, we can obtain the “absolute” surface ground elevation changes in a geocentric frame (i.e., the S term in Fig. 2). These types of geocentric records can be directly compared with altimetry observations and be used to tie locally reference measurements, such as InSAR. For instance, the surface elevation changes we have obtained at SG27 would serve as a good reference point to tie InSAR measurements to the geocentric Earth frame. The daily records would also complement InSAR measurements by filling their temporal gaps.

However, GPS-IR suffers from a few limitations. First, the day-to-day variations in our retrieved reflector height are unreliable, due to relatively large uncertainties (centimeter level in the case of SG27) and the soil moisture effects. Therefore, we choose not to interpret the daily changes in our time series as associated with frozen ground dynamics. Second, GPS-IR signals on snow-covered days are dominated by snow depth changes, limiting the use of this type of data for studying ground surface changes only on snow-free days. Nonetheless, as we have demonstrated in this study, noisy daily records continuously spanning 60 days in each summer and 12 years can provide robust estimates of seasonal, interannual, and decadal changes. Third, similar to typical in situ observations, GPS-IR only offers site-specific measurements. We note that the GPS-IR method takes spatial averages within the reflection footprint (∼ 90 m radius in the case of SG27). This averaging helps to mitigate the spatial heterogeneities due to changes in soil and vegetation, as well as active layer and ground ice conditions. Lastly, reliable GPS-IR retrieval requires a smooth surface within the footprint. Therefore, this method is not applicable for studying thermokarst landforms.