Detecting Permafrost Active Layer Thickness Change From Nonlinear Baseflow Recession
Abstract
Permafrost underlies about one fifth of the global land area and affects ground stability, freshwater runoff, soil chemistry, and surfaceatmosphere gas exchange. The depth of thawed ground overlying permafrost (active layer thickness) has broadly increased across the Arctic in recent decades, coincident with a period of increased streamflow, especially the lowest flows (baseflow). Mechanistic links between active layer thickness and baseflow have recently been explored using linear reservoir theory, but most watersheds behave as nonlinear reservoirs. We derive theoretical nonlinear relationships between longterm average saturated soil thickness (proxy for active layer thickness) and longterm average baseflow. When applied to 38 years of daily streamflow data for the Kuparuk River basin on the North Slope of Alaska, the theory predicts increased cm a^{−1} between 1983 and 2020 ( cm total). The rate of increase nearly doubled to cm a^{−1} between 1990 and 2020, during which time local field measurements from Circumpolar Active Layer Monitoring sites indicate the active layer increased cm a^{−1}. The predicted rate of increase more than doubled again between 2002 and 2020, outpacing a near doubling of observed active layer thickening, consistent with trends in terrestrial water storage inferred from Gravity Recovery and Climate Experiment satellite gravimetry and ModernEra Retrospective Analysis for Research and Applications climate reanalysis. Overall, hydrologic change is accelerating in the Kuparuk River basin, and we provide a theoretical framework for estimating basinscale changes in active layer water storage from streamflow measurements.
Key Points

New relationships between active layer thickness and baseflow are developed for application to catchments underlain by permafrost

Theoretical predictions of active layer thickness trends agree with measured trends on the North Slope of Arctic Alaska

Novel methods are developed to estimate nonlinear recession flow parameters and related hydrologic signatures using generalized Pareto distributions
Plain Language Summary
Streamflow has increased in most areas of the Arctic in recent decades. This increase in streamflow has occurred along with a period of rising air temperatures and thawing permafrost. Permafrost is typically overlain by a layer of seasonally unfrozen ground referred to as active layer, which gets deeper as permafrost thaws. As active layer deepens, it can store more water. Water may also take more time to flow through a thicker active layer than it would atop frozen ground. Because there is more space to store water, thicker active layers lead to increased soil water storage which sustains streamflow during dry periods and enhances overall increased streamflow and subsurface flow. We developed an approach to measure how quickly the active layer thickness is changing using the rate of change of streamflow. This is useful because streamflow measurements are widely available and easy to obtain, whereas active layer thickness measurements are difficult to measure and less common than streamflow measurements. If the equations we developed accurately predict measured values, the pace at which active layer thickness changes can be estimated from streamflow measurements, which would expand the current knowledge of permafrost thaw rates and provide an independent way to validate computer simulations of active layer thickness.
1 Introduction
Permafrost underlies approximately 15% of the global land area (Obu, 2021) and is an important control on slope stability, shoreline erosion, water available for runoff, soil biogeochemistry, and gas exchange between the land surface and atmosphere (McKenzie et al., 2021; Walvoord & Kurylyk, 2016; Wang et al., 2022). The layer of seasonallythawed ground overlying permafrost known as active layer is a sensitive indicator of permafrost response to climate change (Kurylyk et al., 2014; Romanovsky et al., 2002). In recent decades, active layer thickness () has increased at high latitude observation sites, concurrent with observed increases in streamflow, especially the lowest flows (hereafter referred to as baseflow), which typically occur during Northern Hemisphere Fall and Winter (Déry et al., 2016; Duan et al., 2017; Durocher et al., 2019; Feng et al., 2021; Rawlins et al., 2019, 2019, 2019; Rennermalm et al., 2010; Shrestha et al., 2021; Smith et al., 2007).
The trends in and baseflow are thought to be linked via: (a) increased soil water storage capacity within a thicker active layer, (b) increased soil water residence time as flow paths lengthen within a more continuous active layer, and (c) direct contribution of melt water to streamflow, each of which may support higher baseflow in high latitude rivers (c.f., Figure 1 box IV) (Brutsaert & Hiyama, 2012; Evans et al., 2020; Jacques & Sauchyn, 2009; Lyon & Destouni, 2010; Walvoord et al., 2012; Walvoord & Striegl, 2007). In addition, heat transport via lateral subsurface flow within the active layer enhances permafrost thaw (Rowland et al., 2011; Sjöberg et al., 2021), suggesting a positive feedback. Shifts in the temporal distribution of streamflow (c.f., Figure 1) associated with changing snowmelt, river ice, dam regulation, and precipitation patterns are also thought to have contributed to broadscale changes in the timing and amount of freshwater runoff delivered to the Arctic Ocean in recent decades (Bennett, 2015; Déry et al., 2016; Durocher et al., 2019; Feng et al., 2021).
One approach to analyzing mechanistic links between change and streamflow change is baseflow recession analysis (Brutsaert & Hiyama, 2012; Evans et al., 2020), which is a classical method in hydrology that relates groundwater storage to baseflow with a power function relationship: , where is a critical storage below which the relationship does not hold, is a scale parameter related to aquifer properties via hydraulic groundwater theory, and is an order parameter indicating the degree of nonlinearity in the storagedischarge relationship. For the special case of a linear reservoir (), simple relationships between longterm change in and longterm change in can be derived under the assumption that is primarily a function of water storage in active layer, that is, the saturated (Brutsaert & Hiyama, 2012).
If these simple relationships are applicable to realworld catchments, it suggests trends can be diagnosed from streamflow measurements alone. However, streamflow recession in realworld catchments is typically not consistent with linear reservoir theory, meaning realworld recession data suggest (Aksoy & Wittenberg, 2011; Jachens et al., 2020). Although nonlinear reservoir behavior is welldocumented and widely explored in the hydrologic science literature, to our knowledge, the theoretical relationship between saturated change and baseflow has not been generalized to the nonlinear case (Hinzman et al., 2020; Sergeant et al., 2021). Doing so could open the door to retrospective analysis of change at broad scales and is necessary before such methods can be applied beyond the special case of linear reservoir theory. Exploring the nonlinear case also merits attention because the parameter can be related to aquifer properties (Rupp & Selker, 2006b), which adds another dimension along which change can be interpreted based on streamflow, which is more widely observed than .
The aim of this paper is to generalize the hydraulic groundwater theory of streamflow sensitivity to saturated change to the case of nonlinear storagedischarge behavior. Section 2 describes the background to the theory. In Section 3, we extend an earlier theory (Brutsaert & Hiyama, 2012) of saturated change for flat catchments with homogeneous soils and linear storagedischarge behavior to the case of sloped catchments with nonhomogeneous soils and nonlinear storagedischarge behavior. The nonhomogeneity considered here is vertical variation in saturated lateral hydraulic conductivity (Rupp & Selker, 2006b). These three characteristics are consistent with realworld catchment behavior, although the theory remains an effective one based on hillslopescale behavior. The main theoretical result is a set of new equations (Section 3) that relate longterm average saturated change to longterm baseflow change. In Section 4 we describe a baseflow recession algorithm that implements the theory and in Section 5 we apply the algorithm to 38 years of daily streamflow and 30 years of annual measured in the Kuparuk River basin on the North Slope of Alaska (Figure 2). Section 6 compares predictions to observations. The paper concludes with a discussion of methodological limitations, which include high sensitivity to accurate knowledge of soil properties, and the dependence on largesample streamflow data to estimate powerlaw scaling of recession flows.
2 Hydraulic Groundwater Theory
2.1 StorageDischarge Relationship
2.2 Drainable Porosity and the Effective Water Table
2.3 Characteristic Timescales
As implied by Equation 12, a critical threshold exists at , above which Equation 10 is not normalizable. In the context of Equation 4, marks a transition from small (“earlytime”) to large (“latetime”) drainage. Theoretical and numerical solutions indicate (but can approach ) during earlytime and during latetime, separated by an “intermediate” period (Rupp & Selker, 2006b; van de Giesen et al., 2005).
In general, Equation 11 will be used to derive expressions relating to baseflow, with Equations 12 and 13 providing a method to estimate and , and the basis for a discussion of the important dependence on .
3 New Equations for Change in Permafrost Active Groundwater Layer Thickness From Nonlinear Baseflow Recession Analysis
In Equation 22, is contained within the definition of , which is an advantage because is highly uncertain (Lv et al., 2021). However, Equation 22 requires a value for , which may be unavailable (Brutsaert & Hiyama, 2012; Lyon et al., 2009). In addition, Equation 22 requires a reliable estimate of , which can be difficult to obtain relative to , because is not observed. Figure 3 provides a recipe for evaluating Equation 21 from measurements of at the outlet of a river catchment.
4 Baseflow Recession Analysis
Methods to estimate drainage timescale , parameters and , and drainable porosity are required before the recipe in Figure 3 can be applied to data. We designed a baseflow recession analysis algorithm for this purpose (Cooper, 2022). The algorithm detects periods of declining flow (recession events) on qualitycontrolled streamflow timeseries, approximates , and fits Equation 3 to estimate and (Figure 4). Event detection follows recommendations in Dralle et al. (2017), is estimated with an exponential time step (Roques et al., 2017), and nonlinear leastsquares minimization is used to fit Equation 3. In addition to the recommendations in Dralle et al. (2017), we exclude recession flows on days with recorded rainfall if attains a local maximum within a 6day window centered on the date of rainfall, that is, if a rainfall response is detected. The eventbased , , and values are then used to compute for each event from Equation 9, yielding a sample population of size equal to the sample size of . Methods to estimate expected values of , , and are described in the following two sections.
4.1 Estimating Drainage Timescale and Recession Parameters and
As described in Section 2.3, the probability density function follows a Pareto distribution, unlike which can be shown to follow the exponential distribution of nonextensive statistical mechanics (Tsallis, 1988). Both distributions are associated with power law scaling. An advantage of the Pareto transformation over the exponential is that widelyused, and thoroughly vetted algorithms are available to fit the distribution, including the lowerbound (Clauset et al., 2009; Hanel et al., 2017). In particular, an expression for the maximum likelihood estimate of exists in closed form. This means can be found by a global search over all for the value that minimizes a measure of distance between the data and Pareto distribution fit to . We applied a widelyused algorithm that minimizes the KolmogorovSmirnoff distance to estimate (Clauset et al., 2009).
In practice, once was determined, the Pareto fit was repeated 1,000 times by bootstrap resampling with replacement from the underlying sample. Reported parameters are averages of the bootstrapped parameter ensemble. Uncertainties are presented as 95% confidence intervals (∼two standard deviations).
4.2 Estimating Drainable Porosity
Although the latetime RS05 solution for a powerfunction profile was derived under the assumption , the solution appears valid for , where ranges from 1 to 2 as varies from to (and at ). However, values of predict an inverted profile from Equation 5. One benefit of Equation 26, therefore, is its compatibility with any latetime value , or with if an inverted profile is acceptable.
To apply Equation 25, a line of slope was fit through the 95th percentile of the pointcloud to approximate earlytime , and a line of slope was fit through the point cloud to approximate latetime . Note that as described in Section 4.1.
4.3 Statistical Uncertainty
5 Application of Theory to Kuparuk River Basin Streamflow and Active Layer Thickness Data
5.1 SurfaceBased Observational Data
In this section, the theory proposed in Section 3 is applied to 38 years of daily streamflow data for the Kuparuk River on the North Slope of Arctic Alaska (70°16′54″N, 148°57′35″W) (Figure 2). The Kuparuk River drains an ∼8,400 km^{2} catchment area that extends from the foothills of the Brooks Range northward to the Arctic Ocean along the Central Beaufort Coastal Plain, underlain by continuous permafrost >250 m thick (McNamara et al., 1998; Osterkamp & Payne, 1981). Catchment topography is characterized by low relief with mean elevation ∼265 m a.s.l. and elevation range 10–1,500 m a.s.l.. Catchment soils are comprised of alluvial marine and floodplain deposits and aeolian sand and loess, with significant organic and mineral soils on the south side of the catchment (O'Connor et al., 2019). Active layer thickness is thought to be of the order <1 m with substantial spatial variability (Osterkamp, 2005; O'Connor et al., 2019).
Water levels were recorded at United States Geological Survey (USGS) gage 15896000 and converted to discharge by USGS personnel following USGS protocols (Rantz, 1982). Daily flows are reported to nominal 1 ft^{3} s^{−1} precision. Stage and discharge accuracy are affected by fluvial incision and landscape degradation, by river ice and aufeis during winter (Huryn et al., 2021), and by ice jams and flooding during ice breakup. Annual USGS wateryear summaries indicate automated ratings are replaced by estimated values due to aforementioned factors, and the actual precision of reported discharge varies from ∼0.1 ft^{3} s^{−1} for automated ratings to ∼1,000 ft^{3} s^{−1} for estimated values.
Stage and discharge precision together define a minimum observable which may impart bias on estimated recession parameters (Rupp & Selker, 2006a). The effect of measurement precision for Kuparuk River flows is evident in estimated flow values below ∼100 ft^{3} s^{−1}, which typically occur during October–November prior to flow cessation from ∼December–May (Figure 4a). The rapid flow increase following river ice breakup in late Spring partially masks the discretization of lowprecision estimated values. As mentioned in Section 4.2, measurement discretization was remediated by use of an exponential time step to compute (Roques et al., 2017). All flow data is converted to m^{3} d^{−1} prior to fitting Equation 3.
Active layer thickness is measured at nine locations within the Kuparuk River basin (Figure 2) (Nyland et al., 2021). The measurements used here were made by inserting smalldiameter metal probes to point of refusal at regular intervals along grids or transects of sidelength ranging from 100 to 1,000 m. Mechanical probing is supplemented by thermistors measuring soil temperature at four sites. Data are reported as endofseason averages believed to represent the annual maximum thaw depth (i.e., ). A continuous annual record from 1990 to 2020 is available for the Toolik Long Term Ecological Reserve site, from 1992 to 2020 for the Imnavait Creek site, and from 1995 to 2020 for other sites; all site data were averaged to create one continuous record for the Kuparuk River basin. The CALM program and the International Permafrost Association implemented standardized measurement protocols around 1995.
Precipitation is measured at a network of meteorological stations within and proximate to the catchment (Kane et al., 2021). Although gauge undercatch affects Arctic precipitation measurements, our goal is to determine if rainfall occurred rather than how much occurred, and undercatch is small (∼10%) during the late summer recession period when wind speeds are lower (Yang et al., 2005). Daily precipitation measurements used in this study were measured at two sites: the National Oceanic and Atmospheric Administration Cooperative Observer Program station 505136, located near the basin outlet, and the Toolik Field Station Long Term Ecological Research (LTER) site (Environmental Data Center Team, 2022), located near the south side of the catchment (Figure 2).
5.2 Topographic Data
Catchment topography was provided by USGS digital terrain models for the state of Alaska derived from interferometric synthetic aperture radar (IFSAR) (Earth Resources Observation And Science (EROS) Center, 2018). These data were provided as tiles with elevations posted at 5 m horizontal resolution and clipped to the catchment outline using the Geospatial Data Abstraction software Library (GDAL/OGR contributors, 2022).
5.3 Climate Reanalysis and Satellite Gravimetry
Climate reanalysis and satellite gravimetry data were used to close the annual water balance, which provides a method to infer permafrost thaw rate (Brutsaert & Hiyama, 2012). Climate reanalysis was provided by ModernEra Retrospective Analysis for Research and Applications, version 2 (MERRA2) (Gelaro et al., 2017). Monthly terrestrial water storage (TWS) anomalies were provided by University of Texas at Austin Center for Space Research CSR RL06 Gravity Recovery and Climate Experiment (GRACE) and GRACE FollowOn (GRACEFO) mascon solutions (http://www2.csr.utexas.edu/grace) (Save et al., 2016). GRACE data is available on a monthly timestep for the period 2002–2020. Twentytwo missing values in the GRACE timeseries and thirteen GRACEFO values were gapfilled following Yi and Sneeuw (2021).
In practice, was estimated by bringing to the lefthand side of Equation 21 and regressing the righthand side against time in years. Similarly, was estimated by regressing annual August–October minimum anomalies (proxy for catchment storage during streamflow recession) against time in years. This provides an average thaw rate over the time interval represented by , which is 19 years for (Section 5.4.2). For 16 out of 19 years, the annual minimum occurred in August–October, which coincides with the streamflow recession period. In 2018, reached a minimum in January and a second local minimum in September. In 2010 and 2019, reached local minima in November and July, respectively, with magnitudes nearly identical to adjacent October and August values. To obtain a comparable estimate of from MERRA2 reanalysis, we compute on a water year basis such that represents over a period from October 1 of year to September 30 of year .
5.4 Comparison of Theory With Data in the Kuparuk River Basin
5.4.1 Drainage Timescale, Drainable Porosity, and Expected Value of Baseflow
A total of 173 recession events were detected from 38 years of daily streamflow observations, yielding 173 individual estimates of recession parameters and . The median value was 1.7 and the interquartile range was 1.2. The maximum value was 9.2, and 28 values exceeded the theoretical earlytime value (PolubarinovaKochina, 1962). The minimum value was −0.02, and 36 values were less than the theoretical latetime minimum (Boussinesq, 1903). From these 173 events, 2,979 values were obtained, indicating an average event duration of 16 days. The Pareto distribution fit yielded a lowerbound estimate of days and expected value days (Figure 5). Out of 2,979 values, 1,494 exceeded (Figure 6, red circles). Note that Pareto distributions are typically characterized by the exponent of their complementary cumulative frequency distribution : here the estimated exponent corresponds to .
With these objective estimates of , , and provided by the Pareto fit, we turn to the point cloud diagram (Figure 6) to obtain estimates of the remaining parameters in a consistent manner. Here, a line of slope represents theoretical earlytime behavior predicted by the Boussinesq equation, and a line of slope represents empirical latetime behavior. Substitution of the respective pointcloud intercepts and into Equation 25, with reference values km^{2} and m, gives an estimate for drainable porosity , with uncertainty propagated from and . Note that implies an inverted profile in the context of Equation 5 with , which is inconsistent with field observations of in the upper Kuparuk River basin (Figure 7) (O'Connor et al., 2019).
The Pareto fit to does not provide nor , although they could be obtained from a similar modeling procedure. Instead, a value of m^{3} s^{−1} (Figure 4a, dashed line) consistent with days and was obtained from Equation 9, using the value of (m^{3} s^{−1})^{1−b} d^{−1} obtained from the point cloud. The value m^{3} s^{−1} maps to the 26th percentile of the daily flow, from which the baseflow trend was estimated by quantile regression on the 26th percentile of the annual timeseries of mean daily flow (Figure 8). From Equation 13, predicts a value of m^{3} s^{−1} (Figure 4a, dotted line), which maps to the 36th percentile of the daily flow and is similar to the mean October flow ∼13 m^{3} s^{−1} (the mean daily flow is ∼80 m^{3} s^{−1}). In this context, is an estimate of the threshold below which baseflow scales as a powerlaw, and is the trend in the flow percentile nearest the expected value of baseflow . Although we would prefer to estimate from an annual timeseries of , in practice there is insufficient data to fit Equation 12 on an annual basis.
5.4.2 Active Groundwater Layer Thickness From Recession Analysis and Observations
Trends in active layer are examined for three periods, as dictated by data availability: 1983–2020 covers the period of overlapping precipitation and discharge measurements; 1990–2020 adds CALM data; 2002–2020 adds GRACE data. From 1983 to 2020, observed mean annual flow increased cm a^{−2} and increased cm a^{−2} (Figure 8). From 1990 to 2020, these rates increased to cm a^{−2} and cm a^{−2}, respectively. During this period, measured increased cm a^{−1}, and estimated with Equation 21 increased cm a^{−1} (Figure 9). Converted to liquid water thickness using , the observed trend converts to an active layer change in storage of cm a^{−1}, in agreement with cm a^{−1} estimated from Equation 21. Water balance trends are reported in Section 5.4.3.
During 2002–2020, these trends approximately doubled. Measured increased cm a^{−1}, estimated with Equation 21 increased cm a^{−1}, and GRACE terrestrial water storage converted to soil layer thickness via increased cm a^{−1}. Converted to liquid water thickness using , the observed trend converts to an active layer change in storage capacity of cm a^{−1}, ∼30% lower than cm a^{−1} estimated with Equation 21 and ∼54% lower than cm a^{−1}. Substituting and into Equation 29 suggests cm a^{−1}, in contrast to the observed rate of active layer thickening, albeit with uncertainty dominated by that exceeds all reported trends. The observed mean annual flow increased cm a^{−2} and increased cm a^{−2} (Figure 8).
5.4.3 Interpreting Groundwater Storage Trends in Terms of Catchment Water Balance
As mentioned in Section 5.3, the trend in can only be attributed entirely to permafrost thaw if . During 1990–2020, MERRA2 reanalysis indicates catchmentmean increased cm a^{−2}, balanced by an increase in and of cm a^{−2} and cm a^{−2}, respectively, leaving a negligible residual trend in (). In this context, the acceleration of was balanced almost entirely by and a small acceleration of . However, the annual average wateryear () was ∼1.04 cm a^{−1}, well in excess of . Strictly speaking, this suggests the excess () is sufficient to explain all of the increase in and during this period, and then some. As suggested by Figure 1, precipitation appears to be an important driver of streamflow trends in the Kuparuk River basin. This is discussed further in Section 6.1.
Turning to the GRACE period (2002–2020), MERRA2 reanalysis indicates catchmentmean increased cm a^{−2}, balanced by an increase in and of cm a^{−2} and cm a^{−2}, respectively, leaving a residual acceleration in of cm a^{−2}. The annual average wateryear () was ∼0.98 cm a^{−1}, again, well in excess of . However, these values are also well in excess of cm a^{−1}, as reported in Section 5.4.2. This appears robust. For example, we tested various alternative definitions of including annual differences (rather than linear regression) on a wateryear basis ( cm a^{−1}), annual calendaryear differences ( cm a^{−1}), and annual differences of the average August–October ( cm a^{−1}). The largest values of obtained were for Northern Hemisphere winter months, but none changed the conclusion that MERRA2 () substantially exceeded both and . Consequently, substitution of MERRA2 () into Equation 29 is unlikely to provide meaningful estimates of .
6 Discussion
6.1 Active Layer Thickness and Saturated Soil Layer Thickness Change
We find that the saturated active layer thickened between 1983 and 2020 in the Kuparuk River basin, in agreement with field measurements of in the region (Nyland et al., 2021). This finding was enabled by new theoretical relationships between basin outflow during streamflow recession and the rate of change of the active groundwater layer (Section 3). Specifically, we extended an earlier linear reservoir theory (Brutsaert & Hiyama, 2012) to the nonlinear case, using the principles of hydraulic groundwater theory (Brutsaert, 2005). This provides a physical interpretation of the relationship between trends and baseflow trends for river basins with nonlinear empirical storagedischarge relationships. Although this approach dramatically simplifies realworld permafrost hydrology, it appears to provide reasonable predictions for the studied area. Pending a thorough comparison with more observations, the framework developed here may open the door to retrospective estimation of trends in data sparse Arctic catchments with short, sporadic, or even nonexistent groundbased active layer measurements.
A first application of the theory to 38 years of Kuparuk River streamflow indicates the active groundwater layer thickened ∼6.4 cm during this time. The rate of increase nearly doubled between 1990 and 2020 for a total increase of ∼9.0 cm, within 6% of the ∼9.6 cm increase indicated by direct measurements of the actual active layer thickness during this period. Both are consistent with observations of thickening active layer and a growing importance of subsurface hydrologic processes in the region (Arp et al., 2020; Luo et al., 2016; Rawlins et al., 2019; Rowland et al., 2011).
A similar picture emerged for the period 2002–2020, during which time observations of terrestrial water storage (TWS) from the GRACE and GRACEFO satellites are available to aid interpretation. First, relative to 1990–2020, the inferred rate of increase of the active groundwater layer more than doubled again, but at a rate +40% higher than direct field measurements of the actual active layer thickness. Similarly, TWS anomalies increased +36% more each year on average than predicted increases in active groundwater layer storage, albeit with uncertainties that exceed all reported trends. Although the TWS trend provides supporting evidence that basin water storage increased during this time, GRACE data has important limitations. For example, under an idealized assumption that melted ice remains stored as liquid water (or is discharged and later replaced by excess precipitation), GRACE would detect no mass change. Similarly, mass changes unrelated to groundwater storage may have contributed to TWS trends during the studied period. TWS trends are therefore not directly comparable to active groundwater layer storage trends predicted from baseflow recession, but rather can be used to indicate whether a surplus or deficit of water was available each year to support filling or draining of active layer.
Climate reanalysis and streamflow observations indicate that both precipitation and runoff are increasing in the Kuparuk River basin, at rates far exceeding inferred increases in water storage in thicker active layer over all periods examined. Focusing on 2002–2020, MERRA2 climate reanalysis indicates the annual average () was positive (∼1 cm a^{−1}), and about twice as large when computed using observed discharge rather than climate reanalysis (this holds over all periods examined). Moreover, the trend in MERRA2 outpaced the trend in leaving a residual acceleration in (). The acceleration suggests a smaller average annual increase in storage (∼0.5 cm a^{−1}) but remains an order of magnitude larger than predicted increases in active layer storage and TWS anomalies.
Taken together, the decisively positive () inferred from climate reanalysis and satellite gravimetry suggests that a surplus of precipitation was available to drive observed increases in discharge and storage in thicker active layer. This supports an interpretation that the active layer was effectively saturated throughout the studied period, and that the active groundwater layer thickness likely increased in proportion to the actual active layer thickness, at minimum. If () were decisively negative, predicted trends in active groundwater layer thickness would likely underpredict actual active layer trends owing to loss of storage to (Brutsaert & Hiyama, 2012). Although our results suggest that climate reanalysis and satellite gravimetry are too coarse to resolve <0.1 cm a^{−1} trends in active layer storage in the Kuparuk River basin, they may prove useful for interpreting recession analysis predictions at broader scales and in regions without groundbased observations.
6.2 Limitations of the Method and Suggestions for Future Research
Section 3 presents a dramatic simplification of active layer hydrology, and we discuss a few salient criticisms. First, Equation 1 acquires physical meaning from solutions to Equation 4 that assume instantaneous drawdown of an initially saturated Boussinesq aquifer discharging to a fully or partiallypenetrating channel (Brutsaert & Nieber, 1977; Rupp & Selker, 2005; van de Giesen et al., 2005). This implies negligible influence of precipitation, evaporation, channel routing, overland flow, unsaturated flow, infiltration, upwelling, and anything else that affects recession. Future work should test these assumptions. For example, frameworks that include evaporation in Equation 3 (Szilagyi et al., 2007; Zecharias & Brutsaert, 1988a) could be incorporated into the derivation that leads to Equations 19 and 20.
Transient recharge, unsaturated flow, and aquifer compressibility, the latter of which is particularly relevant to thawing permafrost (Liljedahl et al., 2016), are thought to mainly affect earlytime recession (Liang et al., 2017). This suggests possible bias when estimating from earlytime solutions using Equation 25, which is a critical uncertainty because is a small number in the denominator of Equation 19. The actual aquifercontributing area may also be smaller than the basin area. If so, the value of in Equation 25 is biased high and is biased low. This may explain why estimates obtained from the method of Equation 25 appear systematically low compared with fieldscale estimates (c.f., Equation 15 of Brutsaert & Nieber, 1977). Upwelling groundwater indicated by isolated aufeis in the lower reaches of the Kuparuk (Huryn et al., 2021) is expected to have a similar effect (Liang et al., 2017).
Precipitation was mitigated by censoring flows within a 6day window of recorded rainfall if a streamflow response was detected, whereas most prior works detected rainfall from the streamflow response alone (Brutsaert & Hiyama, 2012; Cheng et al., 2016; Dralle et al., 2017; Evans et al., 2020; many others). Using rainfall measurements to censor flows resulted in fewer detected recession events, as expected. However, it unexpectedly eliminated the smallest detected baseflow values because rainfall is common during October–November when the lowest baseflows occur in the studied area. This reduced because is inversely proportional to for , and may have contributed to the falloff at days (Figure 5). This is a specific example of a general result that depends on the sample space of , and indicates a possible low bias in from undersampling the lowest flows, which may be exacerbated if changes in extreme precipitation reduce the frequency or magnitude of low flows. The inferred magnitude days is about twothirds the canonical value 45 days obtained from linear recession analysis, whereas days is within the day characteristic uncertainty (Brutsaert, 2008; Brutsaert & Sugita, 2008; Cooper et al., 2018). A detailed study of sensitivity to methodology is needed and would likely benefit from numerical simulation of groundwater flow in addition to empirical estimation (c.f., Rupp et al., 2009).
6.3 Implications for Recession Analysis and Hydrologic Signatures
In Section 2 we showed that the nonlinear drainage timescale follows an unbounded Pareto distribution and a method to fit the distribution that provides an unbiased estimate of in the eventscale recession equation . This provides an alternative to ordinary leastsquares fitting to a bilogarithmic plot of versus . In addition to retaining largesample information, a key benefit of this method is the absence of parameter in the fitting procedure. When individual recession events are plotted on a traditional point cloud diagram, events with similar slope but different intercept produce a characteristic offset (Biswal & Kumar, 2014; Jachens et al., 2020; Zecharias & Brutsaert, 1988b). A linear fit to the point cloud systematically underestimates , in a manner analogous to the bias induced by least squares fitting to bivariate data with errors present in the independent variable (York et al., 2004). A remarkable byproduct of the procedure developed here is shown in Figure 6, where the detected sample of underlying powerlaw distributed recession flows (red circles) occupy exactly that portion of the point cloud where expert intuition would expect, far from the upper envelope, near the smallest values of for given (Brutsaert & Nieber, 1977; Rupp & Selker, 2006a).
This way of writing Equation 10 has important implications for the use of characteristic timescales as quantitative metrics of streamflow (“hydrologic signatures”) (McMillan, 2020). It reveals that a threshold exists below which theoretical latetime powerlaw scaling of baseflow is not realized. Because and together determine (the average duration of baseflow), they are directly linked to water availability. Thresholds dictating the onset of critical behavior are understood generally (Aschwanden, 2015), and in porous media contexts (Hunt & Ewing, 2009), but appear underexplored within the literature linking baseflow recession to hydrologic signatures (McMillan, 2020). At hillslope scales, can be interpreted as a timescale representing the critical transition from “intermediate” ( to “latetime” ( recession (Rupp & Selker, 2006b; van de Giesen et al., 2005). At catchment scales, represents a transition from disordered to ordered flow dictated by a combination of thresholdlike processes that remain poorly understood (Lehmann et al., 2007; Troch et al., 2009).
7 Conclusions
We developed a theoretical framework to predict the longterm average rate of change of permafrost active layer thickness using principles of hydraulic groundwater theory and nonlinear baseflow recession analysis. Our method requires measurements of streamflow recession and catchment topography, and therefore has potential to complement or extend the spatial and temporal coverage of direct active layer measurements to regions where few or none exist.
A first application of our method to 38 years of daily streamflow observations in the Kuparuk River on the North Slope of Arctic Alaska suggests that the active groundwater layer thickened by ∼0.17 cm a^{−1} between 1983 and 2020 and by ∼0.29 cm a^{−1} between 1990 and 2020, in close agreement with direct field measurements of the actual active layer thickness. The predicted rate of change more than doubled to ∼0.74 cm a^{−1} between 2002 and 2020, this time at a rate +40% higher than the observed rate of change of the actual active layer thickness. This suggests that the plotscale, siteaveraged measurements underestimate the catchmentscale rate inferred from baseflow recession, or that the groundwater layer thickness increased at a faster rate than the active layer thickness. Support for the latter interpretation was provided by satellite gravimetry and climate reanalysis, both of which indicate that terrestrial water storage increased at rates far exceeding increases in active layer storage capacity as the active layer thickened in response to permafrost thaw.
Overall, these findings suggest that both increased precipitation and permafrost thaw are playing increasingly important roles in sustaining baseflow in the Kuparuk River basin, and point to a growing importance of subsurface hydrologic processes in the region. Nonlinear baseflow recession analysis has potential to provide novel insight into these processes at the scale of river basins, and we provide a consistent analytical framework to explore them.
Glossary

 general discharge recession constants.

 earlytime discharge recession constants.

 latetime discharge recession constants.

 horizontal aquifer area, equal to .

 active layer thickness.

 aquifer breadth (distance along land surface).

 generic constant coefficients.

 aquifer thickness.

 evaporation per unit time.

 water table height.

 water table height at channel seepage face.

 recharge per unit time.

 saturated lateral hydraulic conductivity.

 saturated lateral hydraulic conductivity at top of aquifer (entire aquifer if is constant).

 channel or stream length.

 exponent of hydraulic conductivity powerfunction.

 exponent of recession constant powerfunction.

 precipitation per unit time.

 aquifer discharge per unit width of aquifer.

 aquifer discharge, assumed equal to ; measured discharge assumed to be baseflow.

 threshold aquifer discharge at onset of latetime recession.

 average aquifer discharge, assumed to equal average annual value of baseflow.

 runoff per unit time.

 aquifer storage.

 average aquifer storage.

 critical aquifer storage at which discharge equals .

 reference aquifer storage (arbitrary datum).

 thaw per unit time.

 time.

 horizontal coordinate.

 vertical coordinate.

 exponent of Pareto distribution.

 uncertainty interval halfwidth (errormargin).

 sensitivity coefficient, equal to .

 groundwater layer thickness (water table height) during aquifer drawdown (Boussinesq aquifer).

 average groundwater layer thickness during aquifer drawdown.

 drainable porosity.

 aquifer slope above horizontal base.

 aquifer drainage timescale for nonlinear reservoir model.

 threshold aquifer drainage timescale at onset of latetime recession.

 reference aquifer drainage timescale for nonlinear reservoir model.

 expected value of probability distribution.

 parameter estimate.
Acknowledgments
The Interdisciplinary Research for Arctic Coastal Environments (InteRFACE) project funded this research through the United States Department of Energy, Office of Science, Biological and Environmental Research (BER) Regional and Global Model Analysis (RGMA) program. Awarded under contract Grant # 89233218CNA000001 to Triad National Security, LLC (“Triad”).
Conflict of Interest
The authors declare no conflicts of interest relevant to this study.
Appendix A
A1 Particular Solutions to the 1D Lateral Groundwater Flow Equation
A1.1 Horizontal Aquifers
This equation can be regarded as a general solution to Equation 19, just as Equation A4 can be regarded as a general solution to Equation A2, in the sense that any value of obtained from recession analysis can be interpreted in terms of via Equation A3. This does not imply all values of have a physically meaningful interpretation (Section 4.2).
Although our interest is in latetime solutions, a similar exercise verifies that Equations A4–A6 are internally consistent for the three known earlytime () solutions, one of which is exact, but assumes and infinite aquifer width (PolubarinovaKochina, 1962). Specifically, if , then , and Equation A6 evaluates to . This does not prove, but rather demonstrates, as described in Section 3, via Equations 1620, that the definition and its relationship to via generalizes the Brutsaert and Hiyama analysis to all known functional forms for flat aquifers, or those that can be considered effectively flat.
A1.2 Sloped Aquifers
As before, we now ask if Equations A10 and A11 can be generalized to slopedaquifer solutions with a constant profile. Setting aside kinematic wave solutions (e.g., Beven, 1982), six of the 11 slopedaquifer solutions collated in Figure 3 of RS06 effectively treat as the horizontal equivalent multiplied by a dimensionless slope factor: . Note that represents the balance of gravitydriven flow via versus diffusion via .
In these cases, if is treated as a constant parameter, all linearized slopedaquifer solutions to Equation A2 conform to and therefore Equation A6, along with two solutions for which is effectively zero. The remaining three solutions include Equation A10 and two based on Equation A10, which conform to and therefore Equation A11. If is not assumed constant, then Equation 18 holds in some cases. Two examples are the latetime solution of Sanford et al. (1993) and the earlytime solution of Brutsaert (1994).
Equations A6 and A11 suggests two families of solutions are applicable. One solution family can be applied to flat (or effectively flat) aquifers, including those with a profile described by Equation A3, and to linearized approximations for sloped aquifers with a constant profile. For these solutions, leads to Equation A6. The second solution family applies to sloped aquifers with a profile described by Equation A3, for which leads to Equation A11.
Open Research
Data Availability Statement
Data and code required to reproduce all figures in this manuscript are available without restriction from https://github.com/mgcooper/baseflow. The baseflow recession analysis algorithm (v0.1.0) (Cooper, 2022) is archived and available without restriction from https://zenodo.org/record/7373924. Kuparuk River discharge data are archived at the USGS Water Data for the Nation (https://waterdata.usgs.gov/monitoringlocation/15896000/). Precipitation data are archived at the United States National Centers for Environmental Information (https://www.ncdc.noaa.gov/cdoweb/datasets/GHCND/stations/GHCND:USC00505136/detail) and the Environmental Data Center (https://www.uaf.edu/toolik/edc). MERRA2 climate reanalysis data are archived at the NASA Goddard Earth Sciences and Data Information Services Center (https://disc.gsfc.nasa.gov/datasets?project=MERRA2). GRACE and GRACEFO data are archived at the University of Texas at Austin Center for Space Research (http://www2.csr.utexas.edu/grace). Active layer thickness data provided by the Circumpolar Active Layer Monitoring program and the International Permafrost Association are archived at the Arctic Data Center (https://arcticdata.io/catalog/portals/CALM).