Ground-based measurements of active layer thickness provide useful data for validating/calibrating remote sensing and modeling results. However, these in situ measurements are usually site-specific with limited spatial coverage. Here we apply interferometric synthetic aperture radar (InSAR) to measure surface deformation over permafrost on the North Slope of Alaska during the 1992–2000 thawing seasons. We find significantly systematic differences in surface deformation between floodplain areas and the tundra-covered areas away from the rivers. Using floodplain areas as the reference for InSAR's relative deformation measurements, we find seasonally varying vertical displacements of 1–4 cm with subsidence occurring during the thawing season and a secular subsidence of 1–4 cm/decade. We hypothesize that the seasonal subsidence is caused by thaw settlement of the active layer and that the secular subsidence is probably due to thawing of ice-rich permafrost near the permafrost table. These mechanisms could explain why in situ measurements on Alaskan North Slope reveal negligible trends in active layer thickness during the 1990s, despite the fact that atmospheric and permafrost temperatures in this region increased during that time. This study demonstrates that surface deformation measurements from InSAR are complementary to more traditional in situ measurements of active layer thickness, and can provide new insights into the dynamics of permafrost systems and changes in permafrost conditions.
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 The active layer, defined as “the top layer of ground subject to annual thawing and freezing in areas underlain by permafrost” [Harris et al., 1988, p. 13], plays a key role in land surface processes in cold regions. One of the important physical properties is active layer thickness (ALT), which varies across a broad spectrum of spatial and temporal scales [e.g., Brown et al., 2000] and is thus difficult to monitor.
 Several methods have been used to determine seasonal and long-term changes in ALT: mechanical probing [Brown et al., 2000], frost or thaw tubes [Mackay, 1973; Rickard and Brown, 1972; Nixon and Taylor, 1998], and inference from temperature measurements [Zhang et al., 1997; Brown et al., 2000]. All these methods can provide good quality data and information at a single site, or across a small grid with a scale that typically varies between 100 × 100 m and 1000 × 1000 m. However, the spatial coverage of these in situ measurements is extremely limited.
 Of these ground-based methods, mechanical probing is usually the most practical and has the lowest costs. Therefore, it is widely used, including in the Circumpolar Active Layer Monitoring (CALM) program established by the International Permafrost Association [Brown et al., 2000]. The CALM program currently maintains more than 125 sites in both hemispheres, including many in Alaska (see http://www.udel.edu/Geography/calm/).
 The Beaufort Coastal Plain on the North Slope of Alaska (Figure 1) is underlain by continuous permafrost that contains ground ice (pore and excess ice) up to 70% by volume [Brown and Sellmann, 1973; Reimnitz et al., 1988]. The overlying active layer ranges from 30 to 70 cm in thickness, which increases from the Arctic coast toward the inland [Zhang et al., 1997; Brown et al., 2000; Hinkel and Nelson, 2003]. Climate conditions on the Alaskan North Slope have a sharp contrast from the Arctic coast to the Arctic inland. Within about 20 km from the Arctic coastline, climate is highly marine-influenced, characterized by cool summers, relatively warm and long (8-month) winters, and low precipitation [Zhang et al., 1996]. Further south in the Arctic inland of the Alaskan North Slope, climate becomes more continental with relatively warmer summers and colder winters.
 A large portion of the study area (shown as the brown color in the Figure 1b Landsat “true-color” image) is a flat and low-elevation terrain underlain by alluvial and marine deposits with high organic and silt content, as well as a large amount of ground ice. Surface vegetation is dominated by tussock sedges. Shallow thermokarst ponds, mostly wind-oriented, are scattered across the area. In contrast to the dark features mentioned above, white linear features in Figure 1b are floodplain deposits associated with active river channels (black lines in Figure 1a). Floodplain deposits consist mainly of gravel and sand with relatively low ice content [Pullman et al., 2007]. In these areas, the surface is barren or only partially vegetated due to frequent sedimentation and scouring. The lower-central upland (elevation higher than 60 meters) is part of the Brooks Foothills. Its surficial soil, like the lowland soil, contains thick organics and silts, and is underlain by ice-rich permafrost. Also notable in Figure 1b near Prudhoe Bay are man-made structures related to oil-field activity.
 Although permafrost temperatures on the North Slope have increased by 2 to 3°C since the mid 1980s [Osterkamp, 2007], mechanical probing measurements show no significant trend in ALT [Brown et al., 2000]. This finding contradicts the general expectation that the active layer should thicken in response to climatic warming, an expectation supported by large-scale climate-driven analytic models [e.g., Anisimov et al., 1997; Oelke et al., 2004], and ALT variations in Siberia as inferred from soil temperature measurements [Frauenfeld et al., 2004; Zhang et al., 2005]. The contradictory evidence from the North Slope of Alaska is possibly due to the fact that some in situ measurements may provide incomplete information on the active layer-permafrost system, since they cannot detect changes in the underlying permafrost. For instance, thawing of ice-rich permafrost near the permafrost table would cause little or no change in ALT, because not much soil material from the permafrost would add to the overlying active layer. This change could not be measured with mechanical probing, but could be revealed using techniques such as thaw penetration measurements with thaw tubes [Nixon and Taylor, 1998].
 Surface deformation measurements provide another good method of detecting changes in an active layer-permafrost system. For instance, melting of ground ice and the subsequent outflow of meltwater can cause long-term surface subsidence across a broad spectrum of spatial scales corresponding to variations in the spatial distribution of ice-rich permafrost and ground ice. Additionally, the seasonal freeze-thaw cycle of the active layer leads to seasonal changes equal to ∼9% of the ice volume within the active layer, due to the density difference between water and ice. These variations in volume can cause a seasonal cycle of surface uplift (frost heave) and subsidence (thaw settlement). Accordingly, observations of surface deformation can serve as a useful complement to ALT measurements for understanding changes in permafrost conditions.
 Although there are theoretical models of permafrost-related deformation such as frost heave and thaw settlement [e.g., Romanovsky and Osterkamp, 1997; Pullman et al., 2007], few measurements of surface deformation have been conducted over remote areas such as the North Slope of Alaska. Global Positioning System (GPS) campaign measurements near Prudhoe Bay [Little et al., 2003] show (i) 1 cm of uplift between July 2001 and June 2002 (i.e., between late summer and early summer in the subsequent year); and (ii) 4 cm of subsidence between June 2002 and August 2002. Little et al.  interpreted their observed surface deformations as caused by seasonal thaw of the active layer. However, field measurements such as GPS campaign surveys can achieve only limited spatial and temporal resolution.
 Here we apply a remote sensing technique, interferometric synthetic aperture radar (InSAR), to monitor both seasonal and secular surface deformation near Prudhoe Bay on Alaskan North Slope. The objectives of this study are to demonstrate the capability of InSAR for monitoring permafrost-related surface deformation over areas larger than those that can be easily accessed with field measurements, to provide insight into the causes of surface deformation, and to infer changes in active layer/permafrost conditions in the study area.
 InSAR has been applied to monitor lateral surface displacement related to permafrost in short timescales of a few months, such as a permafrost landslide in northern Canada [Singhroy et al., 2007] and rock glaciers in Swiss Alps [Strozzi et al., 2004]. Previous InSAR studies over permafrost areas in Alaska focused only on the seasonal cycle in subsidence [Rykhus and Lu, 2008] and uplift [Wang and Li, 1999] related to thaw settlement and frost heave, respectively, in several permafrost areas near the Brooks Range foothills. In this paper, we study a long time series of permafrost-related surface subsidence. We show that permafrost areas in the Arctic coastal plain on the North Slope of Alaska underwent both seasonal (1–4 cm) and, more importantly, secular subsidence (1–4 cm/decade) during 1992–2000.
2. InSAR Processing
 The InSAR technique has been widely used for ground surface deformation measurements because of its ability to remotely sense mm–cm surface deformation over a typically 100-kilometer-wide swath with a spatial resolution of 10s of meters or better [Bürgmann et al., 2000]. By measuring phase differences between two synthetic aperture radar (SAR) images taken at different times by a satellite or an aircraft, InSAR constructs an interferogram that shows ground surface displacement along the line-of-sight (LOS) direction during the time interval of the SAR images.
 When applying InSAR to measure surface deformation over permafrost regions, we need to address three issues related to limitations of the InSAR technique. First, InSAR cannot measure absolute ground displacements, but can only determine relative displacements across an individual interferogram, due to satellite orbit errors and the 2π ambiguity problem in radar phase measurements [Goldstein et al., 1988]. Phase unwrapping (i.e., reconstructing the absolute phase difference from the InSAR-measured relative phase difference that is the 2π modulus of the absolute phase difference) requires a priori information such as a few ground tie points with known absolute deformation or an assumption of a stable area. To our knowledge, no systematic measurements of ground surface deformation were made in this area during the InSAR time span.
 Without any ground calibration data, we use a relatively stable area as a reference for InSAR measurements. From soil samples collected in the Beaufort Coastal Plain, Pullman et al.  found a high potential of thaw settlement in alluvial and marine deposits and almost zero or little potential in sandy soils with floodplain deposits, due to the absence of water/ice in sand. In general, coarse gravels are not susceptible to frost heave or thaw settlement in repeated freeze-thaw cycles. Moreover, all SAR images used for this study were taken in the mid or late summer, long after the snowmelt and peak discharge time, so that the impact of spring high-level streamflow is at a minimum. In addition, due to heat source from streamflow and spring flooding, taliks (unfrozen ground) might have formed and the permafrost surface may be several or even tens of meters under the riverbed and floodplain areas. Ice-rich permafrost may have been thawed due to talik formation, implying that there is little or no long-term thaw settlement over floodplain areas. For these reasons, we assume little surface deformation in floodplain areas, compared with areas away from the rivers covered with exposed tundra (hereinafter referred to as tundra areas). Practically, we choose a point in the Sagavanirktok floodplain (148°18′46.2″W, 70°13′56.4″N, marked as a red star in Figure 1a; and note that it is not located in the active river channel) as the reference for phase unwrapping and assume that the displacement at this point is zero in every interferogram, independent of the time interval between the two SAR images used in interferometry. In the remainder of this paper, we refer to the surface deformation relative to this reference point as surface deformation, usually without reaffirming that it is actually relative deformation.
 Second, InSAR requires invariant surface conditions to maintain phase coherence for robust differential phase measurements. However, temporal variations in surface dielectric properties are common over permafrost areas due to changes in vegetation, soil moisture, and snow cover conditions. Such variations can easily cause loss of coherence (or decorrelation) that corrupts interferometry signals [Zebker and Villasenor, 1992]. To avoid severe decorrelation problems, we only use SAR data collected during the thawing seasons from June to September, when the ground surface is snow-free most of the time (see discussion on snow cover in section 6.2). The nearly barren floodplain areas show high coherence in most of the interferograms. This is a benefit of choosing a floodplain point as our reference point. The river channels themselves are narrow (tens of meters) and show no coherence because water bodies reflect radar waves away from the satellites. Therefore, on an interferogram with a spatial resolution of about 60 meters and deformation maps shown in section 4, decorrelation patterns over river channels are too narrow to be seen.
 Third, InSAR measures only one-dimensional surface displacements in the LOS direction. Some advanced techniques, such as combining interferograms in both ascending and descending tracks [e.g., Fialko et al., 2001], can provide three-dimensional displacement information. In this study, however, we assume that permafrost-related deformation is predominantly vertical, because the study area is relatively flat and horizontal movement is constrained by surrounding soils. Therefore, we simply map the observed LOS deformation to the vertical direction using the varying radar incidence angles.
 Using the ROI_PAC software package [Rosen et al., 2004], we apply conventional InSAR processing methods to 14 ERS-1/2 SAR scenes (Frame 2817, Track 315; see Table 1 for a list of SAR scenes). Except for 1994, the ERS-1/2 satellites collected SAR images over the study area at least once each thawing season during 1992–2000. We use (i) the orbit products provided by the Delft University of Technology [Scharroo and Visser, 1998] and (ii) a USGS 2-arc-second digital elevation model (Figure 1a) with vertical accuracy better than 15 m [Gesch et al., 2002] to remove topographic contributions to the interferograms. From these 14 ERS SAR images, we are able to make 31 interconnected interferograms (i.e., interferograms sharing common SAR images) that show high coherence and that are free of obvious ionospheric artifacts [Gray et al., 2000]. See Table 2 for a detailed description of the interferograms.
Names of interferograms are in the format “yyyymmdd-yyyymmdd.” The dates before and after the hyphen are the master and slave scenes, respectively. Column “B perp” lists the perpendicular satellite baselines between the two SAR scenes.
 After applying a power-spectrum filter to interferometric phases [Goldstein and Werner, 1998], we unwrap individual interferograms with respect to the Sagavanirktok floodplain reference point and remove all ground points that display low coherence, using the branch-cut algorithm [Goldstein et al., 1988]. Next, we adopt a baseline refinement process, implemented in ROI_PAC, to remove slope-like long-wavelength signals that are caused mainly by InSAR orbital errors. Finally, we co-register the unwrapped interferograms into a geographic grid with 60-m spacing.
3. Models of InSAR Observations and Time Series of Surface Deformation
 At each point in each interferogram, we model the InSAR-measured change in vertical radar-ground target distance U, as
where λ is the radar wavelength, which is 5.6 cm for ERS C-band radar; θ is the radar incidence angle, which varies across each SAR image from 19° to 27°; ϕ is the unwrapped interferometric phase in radians; D is the net vertical displacement between the dates of the two SAR scenes used to construct the interferogram; ɛtopo is the residual topographic contribution due to errors in the digital elevation model (DEM); and ɛother includes all other errors such as atmospheric delay errors, residual InSAR orbital errors after baseline refinement, and random observational noise, all of which are randomly distributed from one interferogram to another.
 We model the vertical displacement between two thawing-season dates t1 and t2 as the summation of a secular term and a seasonal term:
where R is the secular rate, and A is the amplitude coefficient of the seasonal displacement. In the seasonal term, t1thaw and t2thaw are the numbers of thawing days prior to t1 and t2, respectively (i.e., if t1 is in the thawing season of year yr1, then t1thaw is the time interval between the onset of thawing in year yr1 and the SAR acquisition date t1). The square-root-of-thawing-day relation is based on the simplified Stefan equation, which is commonly used in analytic estimates of active layer thawing depth [e.g., Nelson et al., 1997]. Thawing in the Prudhoe Bay area starts on different dates in different years, ranging from mid May to mid June [Romanovsky and Osterkamp, 1997]. This range of dates is also supported by in situ active-layer temperature measurements made at 5 cm below the surface at a meteorological station at Franklin Bluffs in our study area [Kane and Hinzman, 1997] (re-plotted in Figure 2). These measurements show that the active-layer temperature rose to 0°C on around 1 June in different years. Here, we assume the same thawing onset date of 1 June in every year in our model. We also assume a constant long-term rate R and a constant seasonal coefficient A over the entire eight-year period (1992–2000). It is equivalent to assuming no inter-annual variability either in the long-term rate or in the seasonal cycle. The impacts of these assumptions on the inversion for the secular rates and seasonal coefficients will be discussed in section 6.1.
where B is the length of the perpendicular satellite baseline (the component of the baseline vector perpendicular to the LOS direction); r is the radar-target distance; θ′ is the radar look angle (23° for the ERS satellites); and ɛDEM is the DEM error.
 Using the 31 interferograms and equations (1)–(3), we invert for R, A, ɛDEM, and their variances, by applying the network inversion strategy described by Biggs et al. . For simplicity, we assume the same InSAR-measurement error at every point in every interferogram. Many ground points appear in fewer than 31 interferograms because of poor coherence, which affects the variances of the fitted R and A at those points. We choose an empirical threshold for the variance of R and only keep points that have R variances lower than that threshold. Results for R and A will be shown in section 4.
 We also use the interferograms to solve for displacement time series at individual points. We examine closely spaced points, using the singular value decomposition (SVD) approach described by Berardino et al. . Studying two adjacent points takes advantage of the fact that the phase gradient between these two points is dominated by the differential deformation signal, because (i) the gradient is free of unwrapping errors associated with errors in the a priori information used for unwrapping, and (ii) atmospheric and orbital errors are highly correlated between adjacent points and so largely cancel out in the difference.
 Two of the 31 interferograms are shown in Figure 3. The observed LOS deformation has been converted into the vertical direction (see section 2 and equation (1)). Figure 3a spans a time interval of only 35 days (between 25 June and 30 July of 1996), and is thus dominated by relative seasonal subsidence. The gray areas are mostly places where no reliable InSAR measurements can be made due to the phase decorrelation problem. Most of the thermokarst ponds show no coherence in interferograms, as standing water bodies reflect radar waves away from the satellites. Figure 3a shows that tundra areas subsided by 10–20 mm relative to floodplain areas during those 35 days.
 In contrast, Figure 3b, which uses two scenes taken at almost the same date but two years apart (18 September 1995 and 23 September 1997), is dominated by relatively long-term subsidence, as the seasonal signals mostly cancel out. Figure 3b shows that tundra areas subsided relative to floodplain areas with a spatial pattern similar to that of the seasonal deformation shown in Figure 3a, though with a smaller magnitude of 3–10 mm. The area near Prudhoe Bay (outlined by a dashed black circle in Figure 3b) underwent relative uplift of at least 6 mm with respect to the floodplain area during those two years. Because we have applied a stronger power-spectrum filter (filter strength α equals 0.6, see Goldstein and Werner ) on this two-year interferogram than on the one-month interferogram (α equals 0.4), many of the small gray areas (i.e., decorrelation features) present in Figure 3a have disappeared, suggesting that they are filter artifacts, caused by the small thermokarst ponds.
 Individual deformation maps such as the two shown in Figure 3 are contaminated by several error sources, including unwrapping errors, an inaccurate DEM, and atmospheric delay artifacts [e.g., Zebker et al., 1997]. When we reanalyze the SAR data using an independent DEM of the Kuparuk River watershed [Nolan, 2003], we find similar differential signals between tundra and floodplain areas. Therefore, we conclude that such differential deformation signals are not caused by errors in the USGS DEM used in this study. In any case, when we use all interferograms to solve for the secular and seasonal coefficients, we simultaneously fit DEM errors so that they do not contaminate our estimates of geophysical signals.
 It is also unlikely that atmospheric delay artifacts cause the differential deformation signals. A uniform decrease in water vapor across the region between two SAR acquisition dates would cause tundra areas to appear to subside relative to floodplain areas, because the radar waves travel through more atmosphere to reach the low-lying floodplains than to reach the higher-lying tundra areas. Such a correlation pattern between InSAR measurements and topography is consistent with the results shown in Figure 3. However, the following estimate indicates that the magnitudes in Figure 3 are too large to be explained by such atmospheric artifacts.
 Atmospheric delay L is the integrated refractivity along radar-target path r from the ground surface s to the SAR antenna a, given by
where N is the dimensionless refractivity, which can be expressed as a function of pressure, temperature, relative humidity, cloud water content, and electron density (see detailed expressions for N from Bevis et al.  and Hanssen ). At two nearby ground points sfloodplain and stundra with an elevation difference Δh, radar waves travel along similar paths. Therefore, we can assume the same N along the two paths. We further project the LOS direction into the vertical direction. Then the differential atmospheric delay ΔLz in the vertical direction z can be written as
ΔLz is always positive as radar waves travel longer to reach sfloodplain than stundra.
 To assess ΔLz quantitatively, we approximate this integration from sfloodplain to stundra as the integration from a surface point ssurface to a point Δh higher, as
According to the USGS DEM (Figure 1a), the topographic difference between floodplain and nearby tundra areas is less than 20 m. Thus, to obtain an upper bound, we set Δh as 20 m. Near the surface, N is a function of only pressure, temperature, and relative humidity. To calculate N, we use NCEP North American Regional Reanalysis (NARR) [Mesinger et al., 2006] daily atmospheric products at four levels (the surface, 2 m above the surface, 10 m above the surface, and 30 m above the surface) at one point (148.3°W, 70.2°N) at the time (21:00 UTC) closest to the acquisition time of SAR data in a day (around 21:28 UTC). Among these four levels, however, NARR only provides relative humidity at 2 m above the surface. For a first-order estimate, we simply assume that relative humidity is the same at these four levels. Next, we linearly interpolate the calculated N at the four levels and integrate it from the surface to Δh above the surface, using equation (6). Figure 4 shows the time series of ΔLz between June and September during 1995–1997. The atmospheric signal shown in one interferogram corresponds to the difference of the differential delay at the two SAR acquisition dates. Figure 4 shows that the peak-to-peak amplitude of ΔLz is less than 1 mm, which means that even in the worst case there would be no more than an apparent 1 mm differential deformation due to atmospheric delay, a number far too small to explain the signals shown in Figure 3. This first-order analysis does not rule out possible contamination from short-scale atmospheric anomalies due to turbulent mixing in the troposphere. But it is unlikely that those anomalies would show the pattern of floodplain/tundra dichotomy shown in Figure 3. The atmospheric errors would be even smaller after we fit for R and A using 31 interferograms, because these artifacts are likely to be uncorrelated in time.
Figures 5a and 5b show maps of the secular rates of relative subsidence R and seasonal subsidence coefficients A, respectively, obtained by fitting to all 31 interferograms. Figure 5a shows that most tundra areas subsided relative to the floodplain areas at rates near 2 cm/decade, though some areas show rates as small as 1 cm/decade or as large as 4 cm/decade. In addition, there are two regions with outlying values: the Prudhoe Bay area (outlined by a dashed circle in Figure 5a) uplifted, and at rates larger than 3 cm/decade; and there is an area of relative subsidence with rates of 4–7 cm/decade located west of Prudhoe Bay between the Kuparuk River and the Sagavanirktok River (outlined by a dashed ellipse in Figure 5a). Figure 5b shows a general pattern of seasonal subsidence of 1–4 cm over four months over the tundra areas relative to the floodplain areas. The magnitude of seasonal subsidence agrees with the GPS measurements made by Little et al. .
 These inversion results are potentially sensitive to the low temporal sampling rate (about two samples per season) of the available ERS SAR data. Nevertheless, we apply the seasonal Mann-Kendall trend test [Hirsch and Slack, 1984] to a number of arbitrarily chosen points, and conclude that the differential secular trends between the tundra and floodplain areas are statistically significant with a probability level of greater than 95%, and that they are thus likely to be real geophysical signals.
Figure 6 shows a time series of relative vertical displacements between the two points shown in Figure 5a as a black star (a tundra location) and a black triangle (a floodplain location). The displacements are all referenced to the earliest date of SAR acquisition (1 August 1992). This time series (plus signs in Figure 6) shows consistent relative seasonal subsidence in all years (1995–1998) that have more than one SAR image. The relative secular rate of subsidence fit over the entire time series is 1.3 cm/decade. But the time series suggests that the subsidence was larger in 1998 than in the other years. This is consistent with the fact that during the 1990s across the North Slope of Alaska, the deepest thaw depth [Osterkamp, 2005] and the highest summer temperatures and heaviest precipitation [Hinkel and Nelson, 2003] occurred in 1998. Similarly, Wolfe et al.  observed that ground subsidence increased by 1 to 7 cm in 1998, compared with previous maximum subsidence, on Richards Island and the Yukon coast in Canada.
 We construct a similar time series for relative displacement between two nearby points, one located in the Prudhoe Bay uplift area shown in Figure 5a and one outside this area. This time series (not shown here) shows seasonal subsidence during 1992–1995, but strong seasonal uplift during 1996–1997, and then seasonal subsidence in 1998. These inconsistent trends in seasonal deformation imply a different forcing mechanism than what causes ground deformation over other tundra areas.
5. Causes of Observed Surface Subsidence
 In this study, we mainly focus on a regional synopsis of ground-surface deformation, i.e., tundra areas subsided seasonally and secularly relative to floodplain areas. We discuss the mechanisms for the observed seasonal and secular deformation in this section. At local scales (hundreds of meters to a few kilometers), ground deformation is controlled by local or even micro-scale surface vegetation, soil deposits, water/ice content in the active layer, active layer thickness, surface roughness, hydrological settings, and geomorphological processes. The temporal variation of each local factor is inhomogeneous as well. These factors could be responsible for the short-scale non-systematic spatial variability evident in the InSAR-observed deformation. Future InSAR work that focuses on obtaining in a higher spatial resolution over a smaller region could address the correlation between surface deformation and local environments.
 We suggest that the seasonal thawing-season subsidence is caused mainly by thaw settlement in the active layer. During the thawing season, surface subsidence occurs as the ice in the active layer melts into liquid water, resulting in a volume decrease. Due to volume expansion when transforming back from water to ice in the freezing season, frost heave occurs as the inverse process of thaw settlement. In general, the surface undergoes similar amounts of subsidence and uplift during the annual thaw-freeze cycle. On the North Slope of Alaska, the maximum thaw depth usually occurs in September and reaches 30–70 cm [Romanovsky and Osterkamp, 1997; Zhang et al., 1997; Hinkel and Nelson, 2003]. Given this range, we can estimate the magnitude of the seasonal thaw settlement. If we assume, to first order, a soil porosity of 40% and a frozen active layer with saturated ice before thawing, then the thaw settlement caused by volume contraction (corresponding to a density change from 0.9 g/cm3 of ice to 1 g/cm3 of water) is 1.2–2.8 cm within one thawing season. Over some regions, thaw settlement could be larger due to higher ice content in the active layer, such as in a thick peat layer with a potential porosity of up to 70%. Overall, this analytic estimate and the InSAR-measured seasonal subsidence of a few cm are of the same order of magnitude.
 The observed secular subsidence, on the other hand, is unlikely to be due to thaw settlement of the active layer. In principle, if the ALT increased during the 1990s, it would have caused a corresponding increase in the amplitude of the seasonal thaw cycle, which would affect our estimate of the secular rates because we simultaneously fit R and A to the InSAR data. If we assume an active layer porosity of 40%, we find that a 2 cm/decade subsidence rate would require an impossibly large ALT increase within one decade of 50 cm, which is the same magnitude as the ALT itself on the North Slope, and so would imply a doubling of ALT within a single decade. But field measurements of ALT on the North Slope show no evidence of a significant long-term trend during the 1990s [Brown et al., 2000; Osterkamp, 2007]. We conclude that even if there was a small secular increase of ALT on the North Slope, small enough to go undetected in field measurements, it would have contributed only slightly to the observed secular trend in surface subsidence, due to the fact that the phase change from ice to water causes only a ∼9% volume change.
 Instead, we hypothesize that the secular surface subsidence is likely due to thawing of ice-rich permafrost directly beneath the active layer. If enough heat transfers through the active layer to the underlying permafrost, ice-rich permafrost thaws and ground ice melts into liquid water. Meltwater drains into lowlands, river channels, and thaw lakes, resulting in surface subsidence over areas underlain by ice-rich permafrost.
 Thawing of ice-rich permafrost near the permafrost table offers a possible explanation both for the InSAR-measured secular surface subsidence and for why ground-based measurements on the North Slope reveal negligible trends in ALT despite an observed secular increase in permafrost temperatures. Permafrost temperature increase is mainly controlled by winter-time changes in air temperature and snow conditions [Zhang, 2005], while ALT is mainly influenced by air temperature in summer months [Nelson et al., 1997; Zhang et al., 1997; Brown et al., 2000; Hinkel and Nelson, 2003]. However, there would be only a small increase in ALT due to thawing of ice-rich permafrost. Similarly, thaw-tube measurements on ice-rich soils in the Mackenzie Valley, Canada, found a progression of thawing into the ground, but little change in ALT [Nixon and Taylor, 1998]. Several other studies in the same area [e.g., Brown et al., 2000; Atkinson et al., 2006; Smith et al., 2009] observed significant thaw penetration in 1998 but little change in ALT.
 Our InSAR results show an average surface subsidence rate of 2 cm/decade with a range of 1–4 cm/decade (except for the two anomalous areas mentioned in section 4), implying a secular net loss of about 2 cm of ice per decade in the study area. To quantitatively understand the implications of such a net loss of ice (represented by S), we study a system consisting of an active layer with a porosity of 40% (represented by e) and an underlying layer of ice-rich permafrost that, by volume, contains 70% (represented by ei) ice and 30% (i.e., 1 − ei) dry sediments.
 Upon thawing of a layer of ice-rich permafrost of thickness H, all the ice (thickness Hice = eiH) in this layer melts. Some of this meltwater runs off, but the rest remains and refreezes into the dry sediments and adds to the overlying active layer. The thickness of the dry sediments in the thawed ice-rich permafrost, Hsediment, can be expressed by Hsediment = (1 − ei)H. The refrozen ice thickness H′ice satisfies H′ice = e (Hsediment + H′ice), as the post-thaw porosity is the same as that of the overlying active layer (e). So,
In this process, the ALT increases by ΔHAL, given by
The remaining meltwater drains away since it cannot be held by the saturated active layer. This part of the melted ice corresponds to a net ice loss in this system, and thus has a thickness of S, which can be expressed by
Based on equations (10) and (11) as well as the assumed values of e and ei, a net loss in ice of 2 cm/decade (S) implies a melting of 2.8 cm/decade of ice from the ice-rich permafrost (Hice), and an increase in ALT of 2 cm/decade (ΔHAL). A rate of change of ALT this small would be difficult to detect in mechanical probe data, because of measurement uncertainties and the spatial averaging that is performed over the measurement grids of 100 × 100 m to 1000 × 1000 m.
 From an energy budget perspective, thawing of ice-rich permafrost on Alaskan North Slope is comparable to the 5 cm/decade increase in ALT in Siberia (represented by ΔHALSiberia). To show that, we assume (i) that changes in ALT are at constant rates every year in both regions, and (ii) that the active layer porosity in Siberia is also 40% (also represented by e). According to the above calculation (equations (7), (10), and (11)), on Alaskan North Slope, from the end of one thawing season to the end of the next thawing season, 2.8 mm (Hice) of ice melts in the ice-rich permafrost. Of that, 0.8 mm (H′ice) becomes part of a thickened active layer and must be melted again in each subsequent thawing season. Therefore, in the ith thawing season after the climate began warming, the total melted ice is Hice + (i − 1) H′ice greater than the ice that melted in the years prior to warming. Let HtotalAlaska be the total thickness of melted ice in the active layer and permafrost over a period of n years, relative to the beginning of warming. It is given by
In Siberia, thawing of permafrost results in a thickening active layer, but with negligible net loss of ice in the active layer-permafrost system. Therefore, in the ith thawing season, the ALT increases by i ΔHALSiberia relative to the beginning of warming. The amount of ice that thaws during the ith thawing season is larger by i ΔHALSiberia e than the amount of ice that melted in the years prior to warming. Let HtotalSiberia be the total thickness of thawed ice in the active layer over a period of n years, relative to the beginning of warming. It is given by
Based on equations (12) and (13), we calculate that over a decade, HtotalAlaska and HtotalSiberia are 64 mm and 110 mm, respectively. Thus, the increases in the energy budget required to thaw such amounts of ice over a decade are comparable in both regions.
 It is difficult to meaningfully compare our observed seasonal and secular ground deformation with thaw settlements measured in other permafrost areas, since the amount and rate of thaw settlement are related to many factors including ALT, soil density and porosity, water content, changes in climatic conditions, and changes in surface conditions including snow cover. Many ground-based studies on thaw settlement have focused on significant ground subsidence due to natural or artificial disturbances [e.g., Mackay, 1970; Burgess and Smith, 2003] or on subsidence of man-made structures due to the thawing of permafrost [e.g., Hanna et al., 1990; Nixon, 1990; Jin et al., 2000]. Our results in general show smaller subsidence than those measurements.
 An ongoing investigation using oil well data (Alaska Oil and Gas Conservation Commission, http://www.aogcc.alaska.gov) suggests that the secular uplift evident in the Prudhoe Bay area might be associated with the fact that a larger volume of water and gas was injected into the wells than the production volume of oil during 1995–2000. Likewise, the anomalously large secular subsidence to the west of the Bay area could also be associated with oil-field activity. Alternatively, the uplift in the Prudhoe Bay area could be due to aggradation of new ice near the permafrost surface. However, no in situ data are available to support this hypothesis.
6.1. Interannual Variability of Surface Deformation
 The inversion results shown in section 4 ignore the possibility of inter-annual variations in surface deformation. The simple model (equation (2)) used in the inversion does not account for possible inter-annual variations in the yearly averaged subsidence, in the thawing onset date, or in the seasonal amplitude. Some insight into those variations can be obtained by using the time series of relative vertical displacements shown in Figure 6, which indicates a larger seasonal subsidence in 1998 than in the other years.
 To test the effects of the apparent larger subsidence in 1998, we use only 21 interferograms that span 1992–1997 for the same inversion. We find similar coefficients of seasonal subsidence as those shown in Figure 5b but generally smaller long-term rates of about 1–3 cm/decade, compared with those shown in Figure 5a. Given the short time span (i.e., five years) and the smaller number (i.e., 21) of interferograms used in this test, as well as the fact that our entire set of data lasts only two additional years to 2000, we can conclude only that it is possible that the increased ground subsidence in 1998 might be causing our 1992–2000 fitted rates to overestimate the secular rate over a longer period.
 We find that the inversion results shown in section 4 are insensitive to the choice of thawing onset date. We vary the thawing onset date from 15 May to 16 June (i.e., ±15 days relative to 1 June, the assumed thawing onset date, see section 3) and conduct the same inversion using the same 31 interferograms. Then we calculate the range of fitted secular rates (or seasonal subsidence) at each point, which is the difference between the maximum and minimum fitted secular rates (or seasonal subsidence) with varying thawing onset dates. As shown in Figure 7, these ranges are at least one order of magnitude smaller than those shown in Figure 5, indicating that the inversion for the secular rate and seasonal subsidence is relatively insensitive to the assumed thawing onset date.
6.2. Secondary Effects on Surface Subsidence Estimates
 In addition to permafrost-related deformation, other processes could also result in seasonal or secular signals in our interferograms. Since InSAR can only detect relative deformation, we ignore any nearly spatially uniform deformation that might exist at regional scales (100 km wide), such as post-glacial rebound (about 2 mm/yr of uplift in this region, according to the ICE-5G model of Peltier ). In this subsection, we discuss contributions to InSAR signals from hydrological, geomorphological process, changes in soil moisture and snow cover.
 It is possible that hydrological loading and unloading could cause surface deformation. Some of the SAR data were collected as late as the end of September, so there is the possibility of snow loads in and around the study area. We model such snow loading effects by convolving the snow water equivalent products of the Global Land Data Assimilation System (GLDAS) [Rodell et al., 2004] with Farrell's Green's functions for surface displacements [Farrell, 1972]. Our model predicts that the seasonal displacements due to snow loading have amplitudes less than 2 mm. The time series of our modeled loading-induced deformation shows no significant secular trend (less than 1 mm/decade). Moreover, the modeled displacements due to hydrological loading are reasonably uniform across an image, and so would largely be removed in any case by the InSAR baseline refinement process that reduces InSAR orbital errors.
 During the early melting season, the hydrological loading effects could conceivably be opposite in sign in the floodplain and the tundra areas, because of possibly increasing water loading in the former and snowmelt unloading in the latter. Any such difference would not be present in output from the GLDAS model, because that model does not include a river routing scheme. The subsidence induced by this difference in loading would have short-scales and so would not be reduced by the baseline refinement process. It could cause a phase lag in surface deformation between the floodplain and tundra areas. However, because our inversion results appear to be insensitive to the onset time of deformation (see section 6.1), we conclude that the effects of this possible difference in hydrological loading is small.
 Similarly, the ground surface could deform due to possible sedimentation loading and erosion unloading from active river channels. These geomorphological deformation signals are small, due to the following reasons: First, the fact that floodplain areas show high coherence in most of our interferograms indicates that the sedimentation and erosion processes are weak, since otherwise they would be likely to reshape the surface and change the dielectric characteristics of surface scatterers, and thus cause InSAR temporal decorrelation. For example, several studies [e.g., Wegmüller et al., 2000; Smith, 2002, and references therein] use temporal decorrelation patterns to detect areas of active deposition and erosion. Second, since the isostatic adjustment to loading has a larger spatial wavelength than that of the loading itself, this process cannot explain the sharp contrast in deformation signature between floodplain and tundra areas. Furthermore our InSAR results show no obvious deformation patterns centered over river deltas, indicating a non-significant impact of sedimentation on surface deformation.
 Another minor effect on InSAR measurements is temporal variability of liquid/solid water content in the active layer. Due to the impact of water molecules on the soil permittivity, changes in soil moisture can result in changes in radar penetration depth and thus in InSAR LOS ranges. For instance, according to the study of Wegmüller , a change from frozen soil, which is completely dry, to thawed soil could cause a decreasing penetration depth of about 3 cm for radar waves at 4.6 GHz frequency. From the active layer temperature records at Franklin Bluffs (see Figure 2, temperature was slightly above 0°C on 25 September 29 1993 and September 1999) and records of soil moisture at 10 cm below the surface at Betty Pingo during 1996–2000 (148°53′36.5″W, 70°16′57.3″N, Natural Resources Conservation Service, http://soils.usda.gov/survey/scan/alaska/BettyPingo/), it is reasonable to assume that the top 5 cm of the active layer — the maximum penetration depth of C-band radar waves into dry soils — were completely thawed on our SAR acquisition dates. Therefore, we are only concerned with seasonal and secular changes in soil moisture when the active layer was wet. Modeling penetration depth variability is difficult due to limited quantitative knowledge of the surface media's dielectric properties and their temporal variations. Here we use the theoretical study of Nolan and Fatland , specifically the “C” and “CT” curves shown in their Figure 2. We discuss the artifacts in InSAR signals due to three types of temporal changes in soil moisture: (i) seasonal drying due to evaporation, (ii) abrupt surface wetting due to precipitation, and (iii) possible secular changes in soil moisture.
 First, the top 5 cm of soil becomes drier due to 24-hr evaporation during the thawing season. To assess the resulting change in radar penetration depth, we need soil moisture values at the beginning and the end of each thawing season. However, there were no systematic measurements of the moisture content in the top 5 cm of soil in our study area during 1992–2000. According to the soil moisture measurements in the Imnavait Creek watershed (149°17′W, 68°37′N, to the south of our study area) made in the late 1980s [Hinzman et al., 1991], we assume (i) a soil moisture of about 40–60% of volumetric water content (VWC) on the top 5 cm in the active layer, at the beginning of June right after spring melt and shallow thawing of the active layer and (ii) a drying process that could induce a seasonal decrease in VWC to 20–30% at the end of September. This decrease in soil moisture could lead to a 3–8 mm increase in radar penetration depth (or apparent surface subsidence), according to Nolan and Fatland . Note that our study area is in general wetter than the Imnavait Creek watershed in the entire thawing season. It is possible that the actual soil moisture artifacts are smaller than the 3–8 mm increase in radar penetration depth given here [Nolan and Fatland, 2003]. Several other factors neglected here, including the 23° incidence angle of the radar waves, the surface roughness, and different soil types in our study area from those used by Nolan and Fatland , could modify these values. Additionally, a decrease in VWC could possibly cause a real surface subsidence due to soil consolidation.
 Second, rain storms in the late thawing season on the North Slope of Alaska can rapidly and dramatically increase the soil moisture, causing an apparent ground uplift signal on an interferogram. For instance, measurements in the Imnavait Creek watershed [Hinzman et al., 1991] showed an abrupt increase in soil moisture in the top 5 cm from ∼40% to ∼90% within a few days in the July of 1987. Meteorological measurements at Betty Pingo [Kane and Hinzman, 1997] recorded 7 mm and 13 mm daily precipitation on 25 June 1996 and 14 July 1997 (one day prior to the SAR acquisition date), respectively, but no significant precipitation on the other SAR dates or on the day prior to those measurements. In the time series sampled by the acquisition dates of our ERS SAR images, soil moisture appears to vary non-monotonically in the thawing seasons. Because these soil moisture artifacts in InSAR results are not strongly systematic and do not obey the rule of square-root-of-thawing-days (equation (2)), we expect relatively smaller impacts on the fitted seasonal subsidence, compared with those on individual interferograms.
 Third, changes in soil moisture cannot explain our observed secular ground subsidence. A secular subsidence rate of even 1 cm/decade, which is on the low end of our results, would require an unrealistically high secular decrease in VWC of 20% over one decade.
 Since we use a floodplain point as our InSAR reference, our results are sensitive to differential penetration depth change due to soil moisture changes between a tundra point and the floodplain point. In general, sandy floodplain points are drier than silty tundra points. Even a small decrease in soil moisture can cause an increase of penetration depth of a few cm on the floodplain points if they are dry enough, and thus an apparent uplift signature over tundra areas. That is opposite to our observed subsidence. Therefore, it is possible that our InSAR measurements are underestimates of the actual seasonal thaw settlement of active layer.
 According to several remote-sensing products (including National Snow and Ice Data Center , Armstrong and Brodzik , and AVHRR imagery) in our study region on the SAR acquisition dates, the ground surface was snow covered only in one SAR image taken on 28 September 1999. Since this SAR image is only used once in interferometry as a slave scene, snow cover might cause a false subsidence signal due to the delay caused by a shallow snow layer. Nevertheless, it has little impact on our inversion results, which are made from 31 interferograms.
 Applying the InSAR technique to ERS-1/2 SAR data spanning 1992–2000, we are able to detect seasonal and secular surface subsidence over the tundra areas relative to the floodplain areas on the North Slope of Alaska, in individual interferograms and in inversion results. The seasonal subsidence of 1–4 cm is likely caused by seasonal thaw settlement related to melting of ice in the active layer. We postulate that the secular subsidence of 1–4 cm/decade is due to thawing of ice-rich permafrost near the permafrost table in response to warming permafrost temperatures in the 1990s. Such mechanisms of long-term net loss of ice in permafrost are consistent with independent measurements that suggest there has been no significant secular increase in ALT on the North Slope of Alaska during this time period. From an energy budget perspective, the InSAR-measured rates of ∼2 cm/decade of secular subsidence, if indeed caused by a net loss of ice, are consistent with the ∼5 cm/decade secular ALT increase observed in Siberia, as similar amounts of ice melts in both regions over one decade.
 As demonstrated in this paper, InSAR-observed secular subsidence provides information that complements long-term in situ ALT measurements to obtain a better understanding of permafrost changes under warming temperatures. InSAR is uniquely suited for monitoring surface deformation at high spatial resolution over large permafrost areas. Combinations of these measurement types will become increasingly valuable for studying polar permafrost, as more in situ sites are established by the CALM program and as more SAR data (especially L-band data) from various satellites become available.
 The ERS SAR data are copyrighted by the European Space Agency and provided by the Alaska Satellite Facility. We thank Matthew Pritchard for providing patches for ROI_PAC to read SAR data in AKCEOS format. We thank Anahita Tikku for providing helpful discussion. We also thank three anonymous reviewers and the Associate Editor for their constructive and insightful comments. DEM data and Landsat data are available from the U.S. Geological Survey, EROS Data Center, Sioux Falls, South Dakota. This work was partially supported by NASA grants NNX06AH37G and NNX06AE65G to the University of Colorado, by NASA Headquarters under the Earth and Space Science Fellowship Program - NNX08AU85H, by the NSF cooperative agreement number OPP-0327664 and the NSF grant ARC-0901962 to the University of Colorado.