Volume 59, Issue 1 e2022WR033154
Research Article
Open Access

Detecting Permafrost Active Layer Thickness Change From Nonlinear Baseflow Recession

M. G. Cooper

Corresponding Author

M. G. Cooper

Atmospheric Sciences and Global Change Division, Pacific Northwest National Laboratory, Richland, WA, USA

Correspondence to:

M. G. Cooper,

[email protected]

Contribution: Conceptualization, Methodology, Software, Formal analysis, ​Investigation, Data curation, Writing - original draft, Writing - review & editing, Visualization

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T. Zhou

T. Zhou

Atmospheric Sciences and Global Change Division, Pacific Northwest National Laboratory, Richland, WA, USA

Contribution: ​Investigation, Resources, Writing - review & editing, Supervision

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K. E. Bennett

K. E. Bennett

Earth and Environmental Sciences Division, Los Alamos National Laboratory, Los Alamos, NM, USA

Contribution: ​Investigation, Data curation, Writing - review & editing

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W. R. Bolton

W. R. Bolton

International Arctic Research Center, University of Alaska Fairbanks, Fairbanks, AK, USA

Contribution: ​Investigation, Data curation, Writing - review & editing

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E. T. Coon

E. T. Coon

Climate Change Science Institute and Environmental Sciences Division, Oak Ridge National Laboratory, Oak Ridge, TN, USA

Contribution: ​Investigation, Writing - review & editing

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S. W. Fleming

S. W. Fleming

National Water and Climate Center, US Department of Agriculture, Natural Resources Conservation Service, Portland, OR, USA

College of Earth, Ocean, and Atmospheric Sciences, Water Resources Graduate Program, Oregon State University, Corvallis, OR, USA

Department of Earth, Ocean, and Atmospheric Science, University of British Columbia, Vancouver, BC, Canada

Contribution: ​Investigation, Writing - review & editing

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J. C. Rowland

J. C. Rowland

Earth and Environmental Sciences Division, Los Alamos National Laboratory, Los Alamos, NM, USA

Contribution: Writing - review & editing, Project administration, Funding acquisition

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J. Schwenk

J. Schwenk

Earth and Environmental Sciences Division, Los Alamos National Laboratory, Los Alamos, NM, USA

Contribution: ​Investigation, Data curation, Writing - review & editing

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First published: 04 January 2023
Citations: 2

Abstract

Permafrost underlies about one fifth of the global land area and affects ground stability, freshwater runoff, soil chemistry, and surface-atmosphere 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 long-term average saturated soil thickness urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0001 (proxy for active layer thickness) and long-term 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 urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0002 increased urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0003 cm a−1 between 1983 and 2020 (urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0004 cm total). The rate of increase nearly doubled to urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0005 cm a−1 between 1990 and 2020, during which time local field measurements from Circumpolar Active Layer Monitoring sites indicate the active layer increased urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0006 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 Modern-Era 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 basin-scale 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 seasonally-thawed 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 (urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0007) 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., 201920192019; Rennermalm et al., 2010; Shrestha et al., 2021; Smith et al., 2007).

The trends in urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0008 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 broad-scale 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).

Details are in the caption following the image

Trend in daily streamflow for each day from 15 May to 1 December for the Kuparuk River on the North Slope of Arctic Alaska. Four conceptual periods are highlighted by red box with hypothetical explanations: I) higher May–June flows driven by earlier snowmelt and/or river ice breakup, II) lower June flows driven by snowmelt deficit, III) higher June–September flows driven by increased precipitation, and IV) higher September–December flows driven by increased baseflow. River discharge data are provided by United States Geological Survey (gage 1596000).

One approach to analyzing mechanistic links between urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0009 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 urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0010 to baseflow urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0011 with a power function relationship: urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0012, where urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0013 is a critical storage below which the relationship does not hold, urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0014 is a scale parameter related to aquifer properties via hydraulic groundwater theory, and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0015 is an order parameter indicating the degree of nonlinearity in the storage-discharge relationship. For the special case of a linear reservoir (urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0016), simple relationships between long-term change in urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0017 and long-term change in urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0018 can be derived under the assumption that urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0019 is primarily a function of water storage in active layer, that is, the saturated urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0020 (Brutsaert & Hiyama, 2012).

If these simple relationships are applicable to real-world catchments, it suggests urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0021 trends can be diagnosed from streamflow measurements alone. However, streamflow recession in real-world catchments is typically not consistent with linear reservoir theory, meaning real-world recession data suggest urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0022 (Aksoy & Wittenberg, 2011; Jachens et al., 2020). Although nonlinear reservoir behavior is well-documented and widely explored in the hydrologic science literature, to our knowledge, the theoretical relationship between saturated urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0023 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 urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0024 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 urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0025 can be related to aquifer properties (Rupp & Selker, 2006b), which adds another dimension along which urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0026 change can be interpreted based on streamflow, which is more widely observed than urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0027.

The aim of this paper is to generalize the hydraulic groundwater theory of streamflow sensitivity to saturated urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0028 change to the case of nonlinear storage-discharge behavior. Section 2 describes the background to the theory. In Section 3, we extend an earlier theory (Brutsaert & Hiyama, 2012) of saturated urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0029 change for flat catchments with homogeneous soils and linear storage-discharge behavior to the case of sloped catchments with non-homogeneous soils and nonlinear storage-discharge behavior. The non-homogeneity considered here is vertical variation in saturated lateral hydraulic conductivity (Rupp & Selker, 2006b). These three characteristics are consistent with real-world catchment behavior, although the theory remains an effective one based on hillslope-scale behavior. The main theoretical result is a set of new equations (Section 3) that relate long-term average saturated urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0030 change to long-term 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 urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0031 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 large-sample streamflow data to estimate power-law scaling of recession flows.

Details are in the caption following the image

(a) Study area map showing Kuparuk basin outline, locations of United States Geological Survey (USGS) gage 1596000, Circumpolar Active Layer Monitoring (CALM) network sites, Toolik Long Term Ecological Research (LTER) weather station, and United States National Oceanic and Atmospheric Administration Cooperative Observer Program (COOP) weather station 505136. (b) Inset shows highlighted region in detail. Kuparuk basin topography is from USGS interferometric synthetic aperture radar 5 m resolution digital terrain model. Basemap credit: ©OpenTopoMap (CC-BY-SA).

2 Hydraulic Groundwater Theory

2.1 Storage-Discharge Relationship

Baseflow recession analysis relates groundwater storage urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0032 to baseflow Q with a single-valued catchment-scale storage-discharge relationship (Brutsaert & Nieber, 1977; Hall, 1968):
urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0033(1)
where baseflow urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0034 has dimensions urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0035, scale parameter urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0036 has dimensions urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0037, order parameter urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0038 is dimensionless, and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0039 is a critical storage below which (1) does not hold.
During periods when precipitation, evaporation, and any other factor that affects catchment water storage is negligible relative to streamflow, the rate of change of catchment water storage can be approximated by the conservation equation:
urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0040(2)
where urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0041 has dimension urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0042 and represents water stored in upstream catchment aquifers available to supply urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0043. With Equations 1 and 2, the rate of change of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0044 can be expressed as a power function of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0045:
urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0046(3)
where parameter urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0047 has dimensions urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0048, and the parameters in Equations 1 and 3 are related as urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0049 (Clark et al., 2009).
At hillslope scales, Equations 1 and 3 acquire physical meaning from solutions to the one-dimensional (1-D) lateral groundwater flow equation:
urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0050(4)
where urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0051 is the phreatic water surface along horizontal dimension urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0052, urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0053 is drainable porosity, urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0054 is aquifer thickness, urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0055 is a constant related to urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0056 in Equation 3, urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0057 is bed slope, urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0058 is recharge rate,
urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0059(5)
is lateral saturated hydraulic conductivity along vertical dimension urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0060, and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0061 is urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0062 at height urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0063 (Rupp & Selker, 2006b). Various approximate and exact solutions to Equation 4 for an unconfined aquifer draining into a fully- or partially-penetrating channel can be written in the same form as Equation 3 (Appendix A) (Brutsaert & Nieber, 1977; Rupp & Selker, 2006b; van de Giesen et al., 2005). At catchment scales, urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0064 and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0065 are interpreted as lumped parameters linked to catchment-effective drainage density and aquifer transmissivity, porosity, slope, and breadth (distance along the land surface from channel to catchment divide) (Brutsaert, 2005). In the linear case (urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0066), the solution to Equation 3 is an exponential urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0067 with decay constant urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0068 (Boussinesq, 1903). For urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0069, the solution to Equation 3 is a power function urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0070 with urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0071 (Section 2.3).

2.2 Drainable Porosity and the Effective Water Table

The groundwater stored in a catchment can be defined in terms of a thickness of liquid water stored in an effective catchment aquifer (Brutsaert & Hiyama, 2012):
urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0072(6)
where urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0073 is an effective groundwater layer thickness [urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0074] relative to an arbitrary reference urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0075. In this study, we assume urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0076. In general, urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0077 but it’s absolute magnitude has no bearing on the following analysis (Kirchner, 2009).
In this study and in hydraulic groundwater theory generally, the active groundwater layer is treated as a shallow, unconfined Boussinesq aquifer with negligible vertical fluxes and a free-surface water table (the Dupuit assumptions) (Brutsaert, 2005, Section 10.2.1). Capillarity is parameterized by defining urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0078 as the change in storage per unit area per unit change in effective water table:
urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0079(7)
where urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0080 is a quasi-steady (“average”) groundwater layer thickness. Equation 7 extends the usual form urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0081 to a timescale over which urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0082 is changing and assumes urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0083 is a linear function of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0084 over the timescale and soil thickness represented by urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0085. It does not assume a functional form for urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0086 during recession, but such assumptions are made in Section 4.

2.3 Characteristic Timescales

With Equations 2 and 3, a storage sensitivity function can be defined:
urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0087(8)
where:
urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0088(9)
is a nonlinear drainage timescale that carries the dimension of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0089 in Equation 3. Note that urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0090 is denoted urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0091 elsewhere (Berghuijs et al., 2016; Kirchner, 2009).
Integration of Equation 3 over some time interval urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0092, with urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0093 gives:
urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0094(10)
and integration of Equation 8 gives a mathematically identical form parameterized by urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0095:
urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0096(11)
thereby recovering Equation 1.
Although urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0097 has dimension time, Equation 9 implies non-characteristic time scaling, since urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0098 is a function of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0099. However, we can estimate an expected value urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0100 from the probability density transform of Equation 10. Evaluating urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0101, where urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0102 and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0103 normalizes the integral if urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0104, we find:
urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0105(12)
which is an unbounded Pareto distribution with shape parameter urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0106 and scale parameter urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0107.
A general feature of a power law such as Equation 12 is that the process it describes lacks a characteristic timescale for urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0108. Moreover, if urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0109, urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0110 for urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0111, where urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0112. For urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0113, however:
urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0114(13)
where urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0115 is an expected drainage timescale, urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0116 is an expected value of baseflow, and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0117 is an expected duration of baseflow. Note that urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0118 remains finite for urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0119.

As implied by Equation 12, a critical threshold exists at urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0120, above which Equation 10 is not normalizable. In the context of Equation 4, urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0121 marks a transition from small-urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0122 (“early-time”) to large-urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0123 (“late-time”) drainage. Theoretical and numerical solutions indicate urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0124 (but can approach urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0125) during early-time and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0126 during late-time, 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 urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0127 to baseflow, with Equations 12 and 13 providing a method to estimate urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0128 and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0129, and the basis for a discussion of the important dependence on urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0130.

3 New Equations for Change in Permafrost Active Groundwater Layer Thickness From Nonlinear Baseflow Recession Analysis

We first rewrite Equation 11 with urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0131 and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0132 parameterized by urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0133 (dependence on time is omitted for clarity):
urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0134(14)
where overbars indicate temporal averages over a period comparable to urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0135. The total derivative of Equation 14 in this case is:
urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0136(15)
Given Equations 69, and 14, Equation 15 can be written:
urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0145(16)
where urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0137 and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0730 are defined in Equation 7. Strictly speaking, urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0138 represents the initial saturated aquifer thickness when urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0139, and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0140 is the average saturated aquifer thickness over a period comparable to urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0141. As in prior studies (Brutsaert & Hiyama, 2012; Lyon et al., 2009), urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0142 is treated as proxy for urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0143 and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0144 as proxy for its average value.
A general result of hydraulic groundwater theory is that urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0146 can be expressed as a power function of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0147: urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0148, where urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0149 is a constant that can be related to urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0150 in Equation 5 and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0151 in Equation 3 through their mutual dependence (Rupp & Selker, 2006b) (Appendix A). For the present case, we reviewed 19 different solutions to Equation 4 collated in Figures 2 and 3 of Rupp and Selker (2006b) (hereafter RS06). Setting aside two kinematic-wave solutions (Beven, 1982), we found for the 17 remaining solutions urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0152 can be written as:
urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0153(17)
where urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0154 and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0155 are lumped constants that depend on the particular solution. For all six horizontal-aquifer solutions and six sloped-aquifer solutions, urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0156. Five sloped-aquifer solutions with urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0157 effectively treat urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0158 as the horizontal equivalent multiplied by the dimensionless slope factor urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0159. In these cases, if urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0160 is treated as a constant parameter, urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0161 is constant, and Equation 17 satisfies the following general property of a power function:
urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0162(18)
Substituting Equation 18 into Equation 16 and letting urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0163, we find the following linearized expression for urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0164 in terms of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0165:
urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0166(19)
where urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0167 is a reference urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0168 taken at a reference time, or as an average value over a reference period, and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0169 is a (linearized) sensitivity coefficient.
As an alternative to Equation 15, urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0170 can be expressed in terms of the direct dependence between urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0171 and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0172: urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0173. Rearranging in terms of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0174, substituting Equation 18, and linearizing, we find:
urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0175(20)
where urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0176 and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0177 are reference values as previously described. Noting that urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0178 for all but two sloped-aquifer solutions (Rupp & Selker, 2006b), we write particular forms of Equations 19 and 20:
urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0179(21)
and:
urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0180(22)
which recover Equation 4 of Lyon et al. (2009) and Equations 10 and 13 of Brutsaert and Hiyama (2012) for the linear case urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0181.

In Equation 22, urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0182 is contained within the definition of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0183, which is an advantage because urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0184 is highly uncertain (Lv et al., 2021). However, Equation 22 requires a value for urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0185, which may be unavailable (Brutsaert & Hiyama, 2012; Lyon et al., 2009). In addition, Equation 22 requires a reliable estimate of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0186, which can be difficult to obtain relative to urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0187, because urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0188 is not observed. Figure 3 provides a recipe for evaluating Equation 21 from measurements of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0189 at the outlet of a river catchment.

Details are in the caption following the image

Flow diagram indicating each step in the analytical determination of active groundwater layer thickness trends from baseflow recession analysis of daily streamflow data.

4 Baseflow Recession Analysis

Methods to estimate drainage timescale urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0190, parameters urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0191 and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0192, and drainable porosity urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0193 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 quality-controlled streamflow timeseries, approximates urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0194, and fits Equation 3 to estimate urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0195 and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0196 (Figure 4). Event detection follows recommendations in Dralle et al. (2017), urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0197 is estimated with an exponential time step (Roques et al., 2017), and nonlinear least-squares 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 urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0198 attains a local maximum within a 6-day window centered on the date of rainfall, that is, if a rainfall response is detected. The event-based urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0199, urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0200, and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0201 values are then used to compute urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0202 for each event from Equation 9, yielding a sample urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0203 population of size equal to the sample size of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0204. Methods to estimate expected values of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0205, urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0206, and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0207 are described in the following two sections.

Details are in the caption following the image

Example baseflow recession analysis (steps 1–3 in Figure 3): (a) daily streamflow urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0208 and two detected recession events on a logarithmic scale (left axis), and rainfall (vertical bars) on a linear scale (right axis). Horizontal lines are urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0209 (dotted) and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0210 (dashed). (b, c) urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0211 and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0212 (normalized by their initial values) estimated with an exponential time step (ETS) (Roques et al., 2017) compared to predictions from Equation 3 using best-fit parameters urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0213 and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0214. (d) Point-cloud diagram with nonlinear least squares fit to Equation 3 (dotted lines), upper-envelope, theoretical late-time fit (urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0215), and theoretical early-time fit (urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0216).

4.1 Estimating Drainage Timescale urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0217 and Recession Parameters urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0218 and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0219

As described in Section 2.3, the probability density function urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0220 follows a Pareto distribution, unlike urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0221 which can be shown to follow the urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0222-exponential distribution of non-extensive statistical mechanics (Tsallis, 1988). Both distributions are associated with power law scaling. An advantage of the Pareto transformation over the urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0223-exponential is that widely-used, and thoroughly vetted algorithms are available to fit the distribution, including the lower-bound urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0224 (Clauset et al., 2009; Hanel et al., 2017). In particular, an expression for the maximum likelihood estimate of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0225 exists in closed form. This means urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0226 can be found by a global search over all urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0227 for the value that minimizes a measure of distance between the data and Pareto distribution fit to urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0228. We applied a widely-used algorithm that minimizes the Kolmogorov-Smirnoff distance to estimate urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0229 (Clauset et al., 2009).

The Pareto distribution fit to urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0230 also provides an unbiased estimate of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0231 via the relationship:
urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0232(23)
see Equations 10-13. Equation 23 provides a method to estimate urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0233 in lieu of ordinary least-squares fitting to a bi-logarithmic plot of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0234 versus urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0235 (i.e., the “point cloud”). In contrast, the parameter urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0236 is not provided by this procedure. Although we only require urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0237 at the scale of individual recession events to estimate urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0238, it is useful to have a global estimate, call it urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0239, to maintain analytical consistency with urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0240 and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0241 when estimating quantities such as urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0242 (Section 4.2). We estimated urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0243 by fitting a line of slope urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0244 through the centroid of the urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0245 point cloud. This procedure is illustrated graphically in Section 5, but can be defined mathematically as:
urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0246(24)
where overbars indicate medians of the respective quantities within the sample space urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0247.

In practice, once urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0248 was determined, the Pareto fit was repeated 1,000 times by bootstrap resampling with replacement from the underlying urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0249 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 urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0250

An estimate for urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0251 was obtained using a method proposed by Brutsaert and Nieber (1977). In this method, an early-time expression for urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0252 is substituted into a late-time expression and common terms are eliminated, leaving urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0253 as the sole unknown. As in Section 3, we reviewed solutions to Equation 4 expressed in terms of Equation 3 collated in Figures 2 and 3 of RS06, and found the following general expression for urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0254:
urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0255(25)
where urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0256 is an effective aquifer volume, urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0257 and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0258 are the respective early- and late-time values of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0259, urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0260 and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0261 are lumped parameters that depend on the particular solution, and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0262 and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0263 are constants that can be related to urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0264 in Equation 5 and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0265 in Equation 3 through their mutual dependence. Taking the early-time solution (urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0266) from Polubarinova-Kochina (1962) and the nonlinear late-time solution (urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0267) from Boussinesq (1904), we find urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0268, urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0269, urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0270, and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0271.
The same approach with the nonlinear early- and late-time solutions for a power-function urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0272 profile (Rupp & Selker, 2005) (hereafter RS05) yields:
urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0273(26)
where urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0274 and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0275 are parameters related to the Beta function as defined by Equations 26 and 49 of RS05, respectively.

Although the late-time RS05 solution for a power-function urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0276 profile was derived under the assumption urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0277, the solution appears valid for urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0278, where urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0279 ranges from 1 to 2 as urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0280 varies from urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0281 to urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0282 (and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0283 at urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0284). However, values of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0285 predict an inverted urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0286 profile from Equation 5. One benefit of Equation 26, therefore, is its compatibility with any late-time value urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0287, or with urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0288 if an inverted urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0289 profile is acceptable.

To apply Equation 25, a line of slope urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0290 was fit through the 95th percentile of the urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0291 point-cloud to approximate early-time urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0292, and a line of slope urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0293 was fit through the urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0294 point cloud to approximate late-time urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0295. Note that urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0296 as described in Section 4.1.

4.3 Statistical Uncertainty

Unless stated otherwise, all statistical uncertainties reported in this paper correspond to 95% confidence intervals around the mean, and include second-order interaction terms because urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0297, urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0298, and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0299 are not independent (Taylor & Kuyatt, 1994). The primary quantitative prediction made in this paper is the average rate of change of the active groundwater layer, urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0300, estimated via Equation 21. To estimate the statistical uncertainty of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0301 we combined the individual uncertainty estimates of each term in Equation 21:
urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0302(27)
where urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0303 is the uncertainty of the sensitivity coefficient urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0304 (Section 3), urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0305 is the Jacobian of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0306, urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0307 is the covariance matrix of the sample populations urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0308, and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0309 is the uncertainty of the trend slope urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0310 estimated via ordinary least-squares linear regression. Recall urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0311 in Equation 21 therefore urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0312.

5 Application of Theory to Kuparuk River Basin Streamflow and Active Layer Thickness Data

5.1 Surface-Based 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 km2 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 ft3 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 water-year summaries indicate automated ratings are replaced by estimated values due to aforementioned factors, and the actual precision of reported discharge varies from ∼0.1 ft3 s−1 for automated ratings to ∼1,000 ft3 s−1 for estimated values.

Stage and discharge precision together define a minimum observable urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0313 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 ft3 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 low-precision estimated values. As mentioned in Section 4.2, measurement discretization was remediated by use of an exponential time step to compute urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0314 (Roques et al., 2017). All flow data is converted to m3 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 small-diameter metal probes to point of refusal at regular intervals along grids or transects of side-length ranging from 100 to 1,000 m. Mechanical probing is supplemented by thermistors measuring soil temperature at four sites. Data are reported as end-of-season averages believed to represent the annual maximum thaw depth (i.e., urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0315). 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 Modern-Era 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 Follow-On (GRACE-FO) 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. Twenty-two missing values in the GRACE timeseries and thirteen GRACE-FO values were gap-filled following Yi and Sneeuw (2021).

The annual liquid water balance is defined as:
urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0316(28)
where urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0317, urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0318, and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0319 (urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0320) are annual sums of precipitation, evaporation, and runoff fluxes, and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0321 (urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0322) is a source term representing the catchment-mean permafrost thaw rate. GRACE TWS anomalies do not measure urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0323 but rather urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0324 (and any other gain or loss of above- or below-ground mass). In contrast, urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0325 as predicted via Equation 21 is, in principle, comparable to Equation 28, which shows that urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0326 is only equal to urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0327 if urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0328. Therefore, we can attempt to detect urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0329 by rearranging Equation 28:
urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0330(29)
where urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0331 is storage anomaly at time urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0332 detected with baseflow recession analysis as in Equation 6, and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0333 is GRACE TWS anomaly at time urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0334.

In practice, urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0335 was estimated by bringing urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0336 to the left-hand side of Equation 21 and regressing the right-hand side urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0337 against time in years. Similarly, urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0338 was estimated by regressing annual August–October minimum urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0339 anomalies (proxy for catchment storage during streamflow recession) against time in years. This provides an average thaw rate urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0340 over the time interval represented by urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0341, which is 19 years for urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0342 (Section 5.4.2). For 16 out of 19 years, the annual minimum urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0343 occurred in August–October, which coincides with the streamflow recession period. In 2018, urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0344 reached a minimum in January and a second local minimum in September. In 2010 and 2019, urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0345 reached local minima in November and July, respectively, with magnitudes nearly identical to adjacent October and August values. To obtain a comparable estimate of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0346 from MERRA2 reanalysis, we compute urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0347 on a water year basis such that urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0348 represents urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0349 over a period from October 1 of year urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0350 to September 30 of year urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0351.

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 urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0352 and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0353. The median urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0354-value was 1.7 and the interquartile range was 1.2. The maximum urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0355-value was 9.2, and 28 urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0356-values exceeded the theoretical early-time value urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0357 (Polubarinova-Kochina, 1962). The minimum urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0358-value was −0.02, and 36 urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0359-values were less than the theoretical late-time minimum urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0360 (Boussinesq, 1903). From these 173 events, 2,979 urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0361-values were obtained, indicating an average event duration of 16 days. The Pareto distribution fit yielded a lower-bound estimate of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0362 days and expected value urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0363 days (Figure 5). Out of 2,979 urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0364-values, 1,494 exceeded urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0365 (Figure 6, red circles). Note that Pareto distributions are typically characterized by the exponent of their complementary cumulative frequency distribution urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0366: here the estimated exponent urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0367 corresponds to urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0368.

Details are in the caption following the image

Complementary cumulative frequency distribution for drainage timescale urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0369 (blue circles) and the maximum likelihood estimate (MLE) best-fit Pareto distribution (solid red line). The theoretical distribution (Equation 12) is fit to urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0370, where urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0371 is the threshold timescale beyond which baseflow scales as a power-law. The expected value urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0372 is the average drainage timescale given the best-fit parameter urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0373. Confidence intervals are two standard deviations of a bootstrapped ensemble (N = 1,000). The falloff at urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0374 days is indicative of finite system size effects which occur when the underlying population is under-sampled at extreme values but may also indicate a process transition to linear scaling as urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0375.

Details are in the caption following the image

Point-cloud diagram showing 2,979 values of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0376 and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0377 from 173 detected recession events. A line of slope urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0378.35 obtained from urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0379 (Equation 23) is compared with a line of slope urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0380 (dotted) obtained from nonlinear least-squares fit to the point cloud, and to the theoretical early-time fit (urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0381). Values of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0382 are highlighted with red circles.

With these objective estimates of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0383, urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0384, and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0385 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 urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0386 represents theoretical early-time behavior predicted by the Boussinesq equation, and a line of slope urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0387 represents empirical late-time behavior. Substitution of the respective point-cloud intercepts urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0388 and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0389 into Equation 25, with reference values urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0390 km2 and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0391 m, gives an estimate for drainable porosity urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0392, with uncertainty propagated from urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0393 and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0394. Note that urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0395 implies an inverted urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0396 profile in the context of Equation 5 with urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0397, which is inconsistent with field observations of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0398 in the upper Kuparuk River basin (Figure 7) (O'Connor et al., 2019).

Details are in the caption following the image

Saturated hydraulic conductivity (solid circles) measured at Imnavait Creek Research Station in the Kuparuk River basin (O'Connor et al., 2019) and the best-fit nonlinear least-squares power-function (solid line). The best-fit exponent urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0399 corresponds to urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0400 for both flat- and sloped-aquifer solutions (urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0401 for urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0402 for both solutions). The best-fit value obtained from recession analysis urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0403 (urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0404) predicts an inverted saturated hydraulic conductivity profile, which is inconsistent with the measurements.

The Pareto fit to urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0405 does not provide urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0406 nor urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0407, although they could be obtained from a similar modeling procedure. Instead, a value of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0408 m3 s−1 (Figure 4a, dashed line) consistent with urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0409 days and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0410 was obtained from Equation 9, using the value of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0411 (m3 s−1)1−b d−1 obtained from the point cloud. The value urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0412 m3 s−1 maps to the 26th percentile of the daily flow, from which the baseflow trend urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0413 was estimated by quantile regression on the 26th percentile of the annual timeseries of mean daily flow (Figure 8). From Equation 13, urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0414 predicts a value of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0415 m3 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 m3 s−1 (the mean daily flow is ∼80 m3 s−1). In this context, urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0416 is an estimate of the threshold below which baseflow scales as a power-law, and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0417 is the trend in the flow percentile nearest the expected value of baseflow urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0418. Although we would prefer to estimate urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0419 from an annual timeseries of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0420, in practice there is insufficient data to fit Equation 12 on an annual basis.

Details are in the caption following the image

Linear trend in the 26th percentile of daily flow on an annual basis for three periods examined in this study. The expected value of baseflow urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0421 m3 s−1 (13 cm a−1) corresponds to the 26th percentile of daily flow and is used in this study as a proxy for annual average baseflow urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0422. Error bars are 95% confidence intervals representing the relative error in urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0423. Shaded error bounds are 95% confidence intervals for the linear trend lines. Error margins printed in the legend are 95% confidence intervals for the regression slope coefficients.

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 urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0424 cm a−2 and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0425 increased urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0426 cm a−2 (Figure 8). From 1990 to 2020, these rates increased to urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0427 cm a−2 and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0428 cm a−2, respectively. During this period, measured urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0429 increased urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0430 cm a−1, and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0431 estimated with Equation 21 increased urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0432 cm a−1 (Figure 9). Converted to liquid water thickness using urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0433, the observed urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0434 trend converts to an active layer change in storage of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0435 cm a−1, in agreement with urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0436 cm a−1 estimated from Equation 21. Water balance trends are reported in Section 5.4.3.

Details are in the caption following the image

Linear trend in active layer thickness from field measurements in the Kuparuk River basin provided by the Circumpolar Active Layer Monitoring (CALM) program (Nyland et al., 2021) (blue error bars and trend lines) compared with linear trend in active groundwater layer thickness predicted with baseflow recession analysis (BFRA) (Equation 21) (red error bars and trend lines). Error bars for CALM data are two standard errors scaled by a critical t-value (95% confidence intervals); sample size varies from one site (Toolik LTER) (1990 and 1991) to nine (1995–2020) as additional monitoring sites were established. Error bars for BFRA predictions are 95% confidence intervals computed with Equation 27. Shaded error bounds on linear trend lines are 95% confidence intervals. Error margins printed in the legend are 95% confidence intervals for the regression slope coefficients.

During 2002–2020, these trends approximately doubled. Measured urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0437 increased urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0438 cm a−1, urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0439 estimated with Equation 21 increased urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0440 cm a−1, and GRACE terrestrial water storage converted to soil layer thickness via urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0441 increased urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0442 cm a−1. Converted to liquid water thickness using urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0443, the observed urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0444 trend converts to an active layer change in storage capacity of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0445 cm a−1, ∼30% lower than urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0446 cm a−1 estimated with Equation 21 and ∼54% lower than urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0447 cm a−1. Substituting urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0448 and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0449 into Equation 29 suggests urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0450 cm a−1, in contrast to the observed rate of active layer thickening, albeit with uncertainty dominated by urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0451 that exceeds all reported trends. The observed mean annual flow increased urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0452 cm a−2 and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0453 increased urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0454 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 urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0455 can only be attributed entirely to permafrost thaw if urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0456. During 1990–2020, MERRA2 reanalysis indicates catchment-mean urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0457 increased urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0458 cm a−2, balanced by an increase in urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0459 and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0460 of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0461 cm a−2 and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0462 cm a−2, respectively, leaving a negligible residual trend in (urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0463). In this context, the acceleration of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0464 was balanced almost entirely by urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0465 and a small acceleration of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0466. However, the annual average water-year (urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0467) was ∼1.04 cm a−1, well in excess of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0468. Strictly speaking, this suggests the excess (urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0469) is sufficient to explain all of the increase in urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0470 and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0471 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 catchment-mean urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0472 increased urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0473 cm a−2, balanced by an increase in urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0474 and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0475 of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0476 cm a−2 and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0477 cm a−2, respectively, leaving a residual acceleration in urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0478 of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0479 cm a−2. The annual average water-year (urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0480) was ∼0.98 cm a−1, again, well in excess of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0481. However, these values are also well in excess of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0482 cm a−1, as reported in Section 5.4.2. This appears robust. For example, we tested various alternative definitions of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0483 including annual differences (rather than linear regression) on a water-year basis (urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0484 cm a−1), annual calendar-year differences (urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0485 cm a−1), and annual differences of the average August–October urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0486 (urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0487 cm a−1). The largest values of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0488 obtained were for Northern Hemisphere winter months, but none changed the conclusion that MERRA2 (urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0489) substantially exceeded both urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0490 and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0491. Consequently, substitution of MERRA2 (urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0492) into Equation 29 is unlikely to provide meaningful estimates of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0493.

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 urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0494 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 urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0495 trends and baseflow trends for river basins with nonlinear empirical storage-discharge relationships. Although this approach dramatically simplifies real-world 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 urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0496 trends in data sparse Arctic catchments with short, sporadic, or even nonexistent ground-based 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 GRACE-FO 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 (urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0497) was positive (∼1 cm a−1), and about twice as large when computed using observed discharge urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0498 rather than climate reanalysis urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0499 (this holds over all periods examined). Moreover, the trend in MERRA2 urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0500 outpaced the trend in urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0501 leaving a residual acceleration in (urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0502). 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 (urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0503) 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 (urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0504) were decisively negative, predicted trends in active groundwater layer thickness would likely under-predict actual active layer trends owing to loss of storage to urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0505 (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 ground-based 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 partially-penetrating 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 early-time recession (Liang et al., 2017). This suggests possible bias when estimating urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0506 from early-time solutions using Equation 25, which is a critical uncertainty because urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0507 is a small number in the denominator of Equation 19. The actual aquifer-contributing area may also be smaller than the basin area. If so, the value of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0508 in Equation 25 is biased high and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0509 is biased low. This may explain why urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0510 estimates obtained from the method of Equation 25 appear systematically low compared with field-scale 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 6-day 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 urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0511 because urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0512 is inversely proportional to urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0513 for urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0514, and may have contributed to the falloff at urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0515 days (Figure 5). This is a specific example of a general result that urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0516 depends on the sample space of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0517, and indicates a possible low bias in urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0518 from under-sampling the lowest flows, which may be exacerbated if changes in extreme precipitation reduce the frequency or magnitude of low flows. The inferred magnitude urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0519 days is about two-thirds the canonical value 45 days obtained from linear recession analysis, whereas urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0520 days is within the urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0521 day characteristic uncertainty (Brutsaert, 2008; Brutsaert & Sugita, 2008; Cooper et al., 2018). A detailed study of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0522 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 urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0523 follows an unbounded Pareto distribution and a method to fit the distribution that provides an unbiased estimate of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0524 in the event-scale recession equation urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0525. This provides an alternative to ordinary least-squares fitting to a bi-logarithmic plot of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0526 versus urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0527. In addition to retaining large-sample information, a key benefit of this method is the absence of parameter urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0528 in the fitting procedure. When individual recession events are plotted on a traditional point cloud diagram, events with similar slope urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0529 but different intercept urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0530 produce a characteristic offset (Biswal & Kumar, 2014; Jachens et al., 2020; Zecharias & Brutsaert, 1988b). A linear fit to the point cloud systematically underestimates urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0531, 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 by-product of the procedure developed here is shown in Figure 6, where the detected sample of underlying power-law 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 urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0532 for given urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0533 (Brutsaert & Nieber, 1977; Rupp & Selker, 2006a).

This procedure also revealed that the drainage timescale urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0534 is equal to the threshold urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0535 plus a timescale dictated by the degree of nonlinearity encoded in urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0536:
urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0537(30)

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 late-time power-law scaling of baseflow is not realized. Because urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0538 and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0539 together determine urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0540 (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, urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0541 can be interpreted as a timescale representing the critical transition from “intermediate” (urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0542 to “late-time” (urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0543 recession (Rupp & Selker, 2006b; van de Giesen et al., 2005). At catchment scales, urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0544 represents a transition from disordered to ordered flow dictated by a combination of threshold-like processes that remain poorly understood (Lehmann et al., 2007; Troch et al., 2009).

7 Conclusions

We developed a theoretical framework to predict the long-term 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 plot-scale, site-averaged measurements underestimate the catchment-scale 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

  • urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0632
  • general discharge recession constants.
  • urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0633
  • early-time discharge recession constants.
  • urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0634
  • late-time discharge recession constants.
  • urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0635
  • horizontal aquifer area, equal to urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0636.
  • urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0637
  • active layer thickness.
  • urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0638
  • aquifer breadth (distance along land surface).
  • urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0639
  • generic constant coefficients.
  • urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0640
  • aquifer thickness.
  • urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0641
  • evaporation per unit time.
  • urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0642
  • water table height.
  • urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0643
  • water table height at channel seepage face.
  • urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0644
  • recharge per unit time.
  • urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0645
  • saturated lateral hydraulic conductivity.
  • urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0646
  • saturated lateral hydraulic conductivity at top of aquifer (entire aquifer if urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0647 is constant).
  • urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0648
  • channel or stream length.
  • urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0649
  • exponent of hydraulic conductivity power-function.
  • urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0650
  • exponent of recession constant urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0651 power-function.
  • urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0652
  • precipitation per unit time.
  • urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0653
  • aquifer discharge per unit width of aquifer.
  • urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0654
  • aquifer discharge, assumed equal to urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0655; measured discharge assumed to be baseflow.
  • urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0656
  • threshold aquifer discharge at onset of late-time recession.
  • urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0657
  • average aquifer discharge, assumed to equal average annual value of baseflow.
  • urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0658
  • runoff per unit time.
  • urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0659
  • aquifer storage.
  • urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0660
  • average aquifer storage.
  • urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0661
  • critical aquifer storage at which discharge urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0662 equals urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0663.
  • urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0664
  • reference aquifer storage (arbitrary datum).
  • urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0665
  • thaw per unit time.
  • urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0666
  • time.
  • urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0667
  • horizontal coordinate.
  • urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0668
  • vertical coordinate.
  • urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0669
  • exponent of Pareto distribution.
  • urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0670
  • uncertainty interval half-width (error-margin).
  • urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0671
  • sensitivity coefficient, equal to urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0672.
  • urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0673
  • groundwater layer thickness (water table height) during aquifer drawdown (Boussinesq aquifer).
  • urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0674
  • average groundwater layer thickness during aquifer drawdown.
  • urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0675
  • drainable porosity.
  • urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0676
  • aquifer slope above horizontal base.
  • urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0677
  • aquifer drainage timescale for nonlinear reservoir model.
  • urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0678
  • threshold aquifer drainage timescale at onset of late-time recession.
  • urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0679
  • reference aquifer drainage timescale for nonlinear reservoir model.
  • urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0680
  • expected value of probability distribution.
  • urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0681
  • 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 1-D Lateral Groundwater Flow Equation

      A1.1 Horizontal Aquifers

      Here we write particular forms of Equations 19 and 20 based on late-time solutions to the 1-D lateral flow equation for horizontal (hereafter “flat”) aquifers (Boussinesq, 19031904; Rupp & Selker, 2005). In doing so, we recover the earlier linear forms of Equations 19 and 20 given in Brutsaert and Hiyama (2012). It is helpful to first recapitulate that the parameters urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0545 and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0546 can be interpreted as solutions to the 1-D lateral flow equation for both flat and sloped aquifers:
      urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0547(A1)
      where urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0548 is the phreatic water surface along dimension urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0549, urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0550 is drainable porosity, urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0551 is lateral saturated hydraulic conductivity, urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0552 is bed slope, and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0553 is recharge rate.
      Detailed descriptions of (Equation A1) are widely available (e.g., Daly & Porporato, 2004); for our purposes, we note a particular form of (Equation A1) relevant to this analysis obtained by allowing urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0554 to vary as a power function of distance along the dimension urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0555 perpendicular to the impermeable base (Beven, 1982; Rupp & Selker, 2006b):
      urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0556(A2)
      where:
      urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0557(A3)
      is the vertical variation in lateral saturated hydraulic conductivity, urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0558 is a constant, and urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0559 (the domain of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0560 is discussed in Section 4.2). Solutions to Equations A1 and A3 take the form urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0561 and are expressed (via integration) as discharge flux at the downslope channel per unit width of aquifer: urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0562, where urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0563 is discharge at the onset of recession and catchment outlet discharge is urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0564 where urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0565 is the length of all upstream channels. Head at the downslope channel is urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0566.
      Two families of solutions to (Equations A1–A3) are considered, one for flat aquifers (Rupp & Selker, 2005) and one for sloped aquifers (Rupp & Selker, 2006b) (hereafter RS05 and RS06). Exact solutions have been obtained for urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0567 and approximations for urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0568. As mentioned in Section 3, we use the collated solutions in Figures 2 and 3 of RS06 as the basis for our generalization, and express Equation 3 in terms of Equation 17 for the case urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0569. Beginning with the flat-aquifer late-time analytical solution to Equation A2 from RS05:
      urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0570(A4)
      we write this as:
      urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0571(A5)
      which implies 𝑁 = 3 − 2𝑏, and:
      urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0573(A6)

      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 urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0574 obtained from recession analysis can be interpreted in terms of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0575 via Equation A3. This does not imply all values of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0576 have a physically meaningful interpretation (Section 4.2).

      We now ask if Equations A5 and A6 generalize to flat-aquifer solutions having a constant urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0577 profile, as suggested in Section 3 via Equations 17 and 18. Although this question can be answered by simple inspection of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0578 for each of the six early-time solutions and noting that urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0579 holds in each case, stepping through them provides an opportunity to express Equation 19 in terms of Equation A6 for each particular value of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0580, and is useful when we move to the sloped-aquifer case. For constant urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0581, urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0582, which implies urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0583, which (as derived in RS05) is consistent with the known nonlinear late-time exact solution for flat, homogeneous aquifers (Boussinesq, 1904). For that solution, urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0584, meaning urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0585, and Equation A6 evaluates to:
      urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0586(A7)
      which is consistent with Equation 18 for urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0587 and with Equation A6 for urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0588, given urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0589. Applied to the linearized late-time solution for homogeneous flat aquifers (Boussinesq, 1903), we have urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0590, urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0591, and:
      urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0592(A8)
      which is consistent with Equation 18 for urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0593 and with Equation A6 for urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0594, given urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0595. Equation A8 recovers Equation 10 of Brutsaert and Hiyama (2012), also derived from the Boussinesq (1903) linearized solution. A similar substitution of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0596 into Equation 20 recovers Equation 13 of Brutsaert and Hiyama (2012), which can be expressed in terms of urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0597 for urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0598 as:
      urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0599(A9)

      Although our interest is in late-time solutions, a similar exercise verifies that Equations A4–A6 are internally consistent for the three known early-time (urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0600) solutions, one of which is exact, but assumes urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0601 and infinite aquifer width (Polubarinova-Kochina, 1962). Specifically, if urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0602, then urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0603, and Equation A6 evaluates to urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0604. This does not prove, but rather demonstrates, as described in Section 3, via Equations 16-20, that the definition urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0605 and its relationship to urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0606 via urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0607 generalizes the Brutsaert and Hiyama urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0608 analysis to all known functional forms for flat aquifers, or those that can be considered effectively flat.

      A1.2 Sloped Aquifers

      We first note that a wider variety of solutions exist for sloped aquifers, all of which rely on linearizations of Equation A2 and assumptions beyond our scope to thoroughly evaluate. As in the previous section, we start with the general solution from RS06 for the urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0609 profile given by Equation A3, in this case for a sloped aquifer:
      urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0610(A10)
      which, using the notation urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0611, implies urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0612 and therefore:
      urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0613(A11)
      which departs from Equation A6 by a factor of two.

      As before, we now ask if Equations A10 and A11 can be generalized to sloped-aquifer solutions with a constant urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0614 profile. Setting aside kinematic wave solutions (e.g., Beven, 1982), six of the 11 sloped-aquifer solutions collated in Figure 3 of RS06 effectively treat urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0615 as the horizontal equivalent multiplied by a dimensionless slope factor: urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0616. Note that urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0617 represents the balance of gravity-driven flow via urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0618 versus diffusion via urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0619.

      In these cases, if urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0620 is treated as a constant parameter, all linearized sloped-aquifer solutions to Equation A2 conform to urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0621 and therefore Equation A6, along with two solutions for which urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0622 is effectively zero. The remaining three solutions include Equation A10 and two based on Equation A10, which conform to urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0623 and therefore Equation A11. If urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0624 is not assumed constant, then Equation 18 holds in some cases. Two examples are the late-time urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0625 solution of Sanford et al. (1993) and the early-time urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0626 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 urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0627 profile described by Equation A3, and to linearized approximations for sloped aquifers with a constant urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0628 profile. For these solutions, urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0629 leads to Equation A6. The second solution family applies to sloped aquifers with a urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0630 profile described by Equation A3, for which urn:x-wiley:00431397:media:wrcr26413:wrcr26413-math-0631 leads to Equation A11.

      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/monitoring-location/15896000/). Precipitation data are archived at the United States National Centers for Environmental Information (https://www.ncdc.noaa.gov/cdo-web/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=MERRA-2). GRACE and GRACE-FO 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).