Permafrost has been reported to be degrading at increasing rates over wide areas in northern regions of Eurasia and North America; the evidence has come mainly from in situ observations in the soil profile, which have limited spatial and invariably limited temporal coverage. Herein three methods are proposed to relate low river flows (or base flows) during the open water season with the rate of change of the active groundwater layer thickness resulting from permafrost thawing at the scale of the upstream river basin. As an example, the methods are tested with data from four gaging stations within the Lena River basin in eastern Siberia, one in the Upper Lena basin, and three in two of its tributaries, namely the Olyokma and the Aldan basins. The different results are mutually consistent and suggest that over the 1950–2008 period the active layer thickness has been increasing at average rates roughly of the order of 0.3 to 1 cm a−1 in the areas with discontinuous permafrost and at average rates about half as large in colder more eastern areas with continuous permafrost. These rates have not been steady but have been increasing; thus it appears that in the earlier years over the period 1950–1970, some large regions have not been undergoing active layer thickness increases and perhaps even decreases, whereas from the 1990s onward vast areas have experienced larger average layer thickness increases, especially those with continuous permafrost.
 There is a growing consensus that human activities related to industrialization, have globally caused additional radiative forcing accompanied by an average net warming of the environment [Hansen et al., 2006; Intergovernmental Panel on Climate Change, 2007; Doran and Zimmerman, 2009]. This warming appears to be amplified and most pronounced at northern latitudes [Serreze et al., 2000; Hinzman et al., 2005], where the temperatures have been rising at higher rates than the global mean. One of the more direct and obvious manifestations of this has been the thawing of permafrost [e.g., Romanovsky et al., 2002] which has been documented to take place on occasion at alarming rates [e.g., Ohta et al., 2008; Iijima et al., 2010]. Major harmful repercussions of this permafrost degradation have been marked subsidence of the land surface causing widespread undermining and damage to the infrastructure and built environment with increased risks for landslides; changes in ecology due to waterlogging and replacement of forest by bogs, wet sedge meadows, and ponds [e.g., Jorgenson et al., 2001] likely to impede migrating animals; positive climate change feedback by the release of potentially large amounts of greenhouse gases such as methane and carbon dioxide [e.g., Christensen et al., 2004; Goulden et al., 1998]. The latter ramification enlarges the relevance of permafrost thawing from a merely local issue to one with global implications and of global concern. There is thus a critical need, especially also in view of the various possible scenarios of global change, not just to monitor permafrost degradation in real time, but to untangle the historical evolution of this complex phenomenon over the long term as well. Unfortunately, as of now actual records, based on direct temperature measurements, cover mostly relatively short periods, and they are representative only locally for areas of very limited spatial extent.
 To alleviate this shortage of direct temperature measurements, in what follows a general conceptual methodology is developed allowing the estimation of the evolution of the thickness of the ice-free active groundwater layer at the end of the open water season in a river basin in an indirect way, namely from streamflow measurements at the outlet of the basin. The approach is an extension of a method developed earlier [Brutsaert, 2008, 2010, 2012; Brutsaert and Sugita, 2008; Sugita and Brutsaert, 2009] to determine groundwater storage changes, as manifested in available streamflow records. An initial application of these ideas was made by Lyon et al. , who assumed that the storage changes, as reflected in the drainage time scale, are due to thawing. In the implementation proposed herein three methods are developed allowing a more complete use of the streamflow data; in addition, a catchment water budget is invoked to distinguish the fraction of the storage changes due to thawing, from those due to increased precipitation and reduced evaporation and runoff. Appealing features of the proposed approach are first, that it provides permafrost changes integrated over the entire upstream basin area, rather than point values as obtained locally at temperature observation stations; and second, that it produces long-term time series that predate most direct permafrost observations, if they even exist, by many years, since routine streamflow measurements were usually started much earlier.
 As an illustration, the three resulting methods are then tested with data from four streamflow gaging stations within the Lena River basin in eastern Siberia, one in the Upper Lena basin, and three in two of its tributaries, namely the Olyokma and the Aldan basins. These four basins were selected for this purpose because of their relatively pristine nature with minimal human impact and the availability of suitable streamflow data. Basically, they are among the colder river basins on Earth, so that they can serve to illustrate some of the more extreme conditions in high-latitude continental regions. Moreover, they show some variation in permafrost conditions; the Upper Lena upstream of Solyanka (including the Olyokma subbasin) is underlain over roughly 70% of its area by permafrost and the Aldan subbasin over about 92% [e.g.,Ye et al., 2009]. Last, the Aldan subbasins are practically adjacent to the Sea of Okhotsk, a subsection of the Pacific Ocean, with possible implications for teleconnections with large-scale oceanic temperature systems [e.g.,Xie and Noguchi, 1999; Linkin and Nigam, 2008; Mochizuki et al., 2010]. Even though the primary motive for this study was an exploratory proof of concept to pave the way for broader implementation with more extensive river flow data, it will be shown that the active groundwater layer thickness trends during the past 60 years derived this way for the four Lena River subbasins are generally consistent with the trends derived from local records of permafrost depth observed in Eurasia over this period.
2. Base Flow as an Indicator of Active Groundwater Layer Thickness
 In regions underlain by permafrost, the active layer is the upper layer of the soil near the surface that undergoes thawing in the summer and freezing in the fall. The thawing starts from the surface in the spring, and the active layer reaches its maximal ice-free thickness in late summer. The lower boundary of this layer is the top of the permafrost layer. It is primarily this layer in the riparian aquifers, which produces base flow during the ice-free season.
2.1. Base Flow Parameterization and Basin-Scale Groundwater Storage
 Streamflow occurring in the absence of precipitation or upstream water inputs is referred to variously as base flow, fair weather flow, drought flow, or low flow; it can normally be assumed to consist of the cumulative outflow from all upstream phreatic aquifers along the banks. Many formulations are available to describe base flow for different types of assumed aquifer conditions [e.g., Brutsaert, 2005; Rupp and Selker, 2006]. However, the lowest of the low-flow recessions, as small perturbations of the no-flow steady state, can simply be expressed as an exponential type decay phenomenon, a trait of a linear system; this can be formulated as
where y = Q / A is the rate of flow in the stream per unit of catchment area [LT−1], Q is the volumetric rate of flow in the stream [L3T−1], A the area of the basin [L2], y0 is the value of y at the arbitrarily selected time origin t = 0, and K is the characteristic time scale of the catchment drainage process [T], also commonly referred to as the storage coefficient. Note that 0.693 Kcan be considered the storage half-life of the upstream riparian aquifers.
 The water stored in a river basin is the volume of water that has not yet flowed out of it, that is
 Integration of (2) with (1), yields the linear relationship between groundwater storage S = S(t) and base flow y = y(t)
 This storage S can be visualized as the average thickness of a layer of water [L] above the zero flow level spread over the area A of the basin. However, the thickness which this layer occupies inside the soil profile, is larger than S. In the simplest approach, commonly used in groundwater theory, capillary effects above the water table are parameterized by means of the drainable porosity ne, also known as the specific yield; this means that the water table is assumed to be a free surface and the fraction of the soil volume occupied by free and movable water is assumed to be ne. The theoretical limitations of this concept have been discussed elsewhere [Brutsaert, 2005, p. 378]. Anyway, with this assumption, the average thickness of the water layer stored in the soil profile, that is the active groundwater layer in the soil profile above permafrost, is given by
 Before using (3) with (4) to relate changes in average layer thickness η0 with observed changes in base flow y, it is necessary to determine how changes in η0 might affect the drainage time scale parameter K.
 While the validity of (1) is usually invoked on empirical grounds, it can also be derived on the basis of classical hydraulic groundwater theory, as was first shown by Boussinesq  [Brutsaert, 2005, p. 403]. Some useful insight in the physical nature of the characteristic time scale K can be gained by taking a closer look at how that derivation leads to (1). In this approach, the local long-time outflow rateq [L2T−1] from an initially saturated homogeneous unconfined aquifer underlain by a horizontal impermeable bed, (permafrost in the present case, as sketched in Figure 1) into a shallow river channel can be shown to be
 As q is expressed here as a scalar volume per unit time and per unit length of river channel, the symbol k0 is the hydraulic conductivity [L T−1] of the riparian aquifer, B is its breadth [L], that is the distance from the channel to the valley divide, ne is its drainable porosity, t is the time after the start of drainage, and as before, η0is the average vertical thickness [L] of the layer in the soil profile occupied by flowing water underlain by permafrost. The basin-wide outflow rate can be derived by integrating(5) along all upstream river channels in the basin. This integration can be simplified by assuming first, that the parameters in (5) are representative for the entire upstream basin, so that q can be considered as an average value, related to the total basin outflow by
and second, that an effective aquifer breadth can be defined as
where L is the total length of all tributary and main river sections upstream from the gaging station, where the streamflow y is measured. With (6) and (7), and with (L / A) = Dd defined as the drainage density of the basin, it is now possible to rewrite (5) in the form of (1), with a drainage time scale of the order of
2.2. Active Groundwater Layer Thickness Trend From Annual Low-Flow Changes
 Thus, if it can be assumed that ne and k0 in the aquifer and Dd of the landscape remain unaffected by changes in the underlying permafrost layer, (9) allows immediately the determination of the growth of the groundwater active layer, as follows
 If a good estimate of the drainable porosity ne and of the drainage time scale K is unavailable or impossible, but if instead an estimate of a reference or typical value of the layer thickness, say η0r, can be obtained, an alternative expression can be derived; thus, again by combining (10) with (3) and (4), and by then linearizing, one obtains
where yr is a typical or reference base flow around the time when η0r was observed. The choice between (10) and (11) has to depend on the availability of estimates of K, ne and η0r.
2.3. Active Groundwater Layer Thickness Trend From Drainage Timescale Changes
Equation (8) indicates that the drainage time scale K is inversely proportional to η0, so that the growth rate of the active groundwater layer can also be written as
again, under the assumption that ne and k0 in the aquifer and Dd of the landscape remain unaffected by changes in the underlying permafrost layer. Whenever an estimate of a reference or typical value of the active groundwater layer thickness η0r is available, (12) can be combined with (8) to allow easier computation by means of
where Kr is the value of the drainage time scale at the time of η0r. In contrast to (11), this result involves no linearization. But with linearization this can be written in the nearly equivalent form
 The idea underlying (12) is not new, but was already implemented earlier by Lyon et al. , with experimental data in northern Sweden. However, their application differed somewhat from the present one. They did not use the recessions of the lowest base flows to estimate K (see below), but the averages of all recessions; this procedure resulted in markedly lower Kvalues, which may not have represented true basin-scale base flow.
2.4. Active Groundwater Layer Thickness Trend and Permafrost Thawing
Equations (10), (11) and (13) provide three distinct methods to estimate the rate of change of the active groundwater layer thickness. Because this change represents the change in liquid water storage in the river basin (see (4)), it need not be due only to permafrost thawing, but it can be the result of changes in the other components of the water budget of the basin as well. In the presence of a permafrost thawing rate T [L T−1], the bulk equation of continuity of liquid water in the basin can be written as
where P is the precipitation rate of rain and melted snow, E the evaporation rate, and R the runoff rate. Clearly, the active groundwater layer thickness trend is due only to thawing T, if and only if precipitation is balanced by evaporation and runoff, so that (P − E − R) = 0. Moreover, as the thawing presumably takes place at the top of the underlying permafrost layer, under such conditions the trend of the average active groundwater layer thickness dη0/dt is equal to the trend of the active layer thickness as a whole, dD/dt (see Figure 1). This means that in addition to the application of (10), (11) or (13), in any detailed analysis the other terms in the water budget and their trends must also be considered.
 Finally, it should be understood that the values of (dη0/dt) calculated with the above methods represent “effective” or “basin scale,” i.e., average values for the entire upstream basin. Therefore, the values of (dη0/dt) obtained for a basin not fully covered with permafrost, that is with discontinuous permafrost, are likely to be smaller than the actual local values in those parts of the basin, where permafrost is present, and thus possibly an underestimate of these local values.
3. Example Application of the Three Methods in the Lena River Basin
3.1. Selected Subbasins and Streamflow Data
 The Lena River basin is among the largest in the world; it covers about 2.5 106 km2 and as such it ranks ninth largest. Its main stem is about 4.5 103 km long, which makes it the eleventh longest in the world. It has its headwaters at a height of some 1,640 m a.m.s.l. near Lake Baikal in the Baikal Mountains; it first flows in a general northeasterly direction but turns in a big arc roughly midway to a more northerly direction past the city of Yakutsk, to empty in the Laptev Sea, a subsection of the Arctic Ocean.
 The Russian Hydrometeorological Service has been monitoring river discharge, beside several other environmental variables, in eastern Siberia at numerous stations since the mid-1930s [e.g.,Lammers et al., 2001]. Among these, for the present example four gaging stations were selected for which records were available to us for the period 1950 through part of 2009, and which are listed in Table 1. The Lena station at Solyanka measures the upper section of the river, that is roughly 31% of the entire basin. The mouth of the Olyokma River, which joins the Lena nearly straight from the south, is just a few kilometers upstream from Solyanka Village, so that the Olyokma is nested within the Upper Lena. The same can be said about the Aldan River station at Ochotski Perevoz. This means that although four stations are analyzed, only two of them are truly independent.
Table 1. River Basins Analyzed in This Study
Upper Lena (UL)
Drainage area (km2)
Longitude, latitude (degrees)
 The data used herein cover the period 1950 through part of 2009; because the focus is on the ice-free season, only the months May through September of each year were considered. The data were obtained from two sources. The daily discharge data for the period 1950–2003 were obtained through courtesy of the Research Institute for Global Change (Japan Agency for Marine-Earth Science and Technology-JAMSTEC, in Yokosuka, Japan), whereas for the more recent period 2000–2009, they were obtained from R-ArcticNet (A Database of Pan-Arctic River Discharge developed by the Water Systems Analysis Group at the University of New Hampshire, available athttp://rims.unh.edu/data/station/list.cgi?col=1).
3.2. Estimation of Needed Variables and Parameters
 In the present context, the application of the expressions (10) and (11) derived from (1) in section 2.1requires a knowledge of the long-term changes of the base flowdy/dt.In any river basin the base flow goes through highs and lows depending on the antecedent precipitation over the region. It is thus necessary to decide which value of the low flows during the year best reflects the low-flow regime, from which to track its long-term evolution over many years. Although other choices should be possible, an objective measure for this could be the annual lowest flow rate. Indeed, this reflects the groundwater storage, which is the most sustainable in the course of that year and which can be depended upon for next year. However, because individual daily flows are normally subject to error and other uncertainties, for the present study it was decided to use a running average of 5 days, denoted here asyL5, as a more reliable measure for this purpose.
 These annual lowest 5 day flows were calculated for each of the four basins, and the resulting values were then used to calculate the changes in base flow dyL5/dt, which are needed for (10) and (11). As an example, Figure 2 shows the evolution of the annual lowest 5 day flows for the Aldan River at Verkhoyanski Perevoz. The values of dyL5/dt for all four basins are listed in Table 2.
Table 2. Trends of the Annual Lowest 5 Day Flows dyL5/dt (in mm d−1a−1) in Eastern Siberiaa
The p values indicating significance are listed in parentheses.
 Beside dy/dt, in several of the expressions derived from (1) in section 2.1 also a knowledge is required of the characteristic drainage time scale Kand of its long-term changesdK−1/dt or dK/dt. Over the past half century or so, many methods have been developed to estimate K from streamflow measurements. However, most if not all of them are somewhat subjective in their application, and there is still no agreement on an optimal method. Several likely reasons can be given for this. First, (1) is highly idealized, so that it cannot possibly encompass all the intricacies of the physical outflow processes from a natural basin. Second, in the estimation of K it is necessary to ensure that no storm runoff component is present in the measured y, and that the streamflow results only from upstream groundwater outflows; this would necessitate careful scrutiny of concurrent precipitation records. But this requirement is not easily satisfied, because rain gage networks in large basins are never dense enough to capture all events everywhere in the basin. Third, low flows in a river, when close to zero, are unavoidably subject to measurement error and often other uncertainties. In the present study a method was used, which was first proposed by Brutsaert and Nieber , and which was subsequently found to produce useful results when applied in a consistent manner.
 In brief, the procedure consists of determining the lower envelope of a logarithmic plot of appropriate −dy/dt data versus the corresponding y data, and in the present case on account of (1), it also makes use of the observation that y = −K(dy/dt). The lower envelope is used for the following reasons: (1) for a given value of y the slowest recession rate −dy/dt takes place during base flow conditions and when evaporative loss rates from the basin are minimal and (2) for a given −dy/dt this envelope yields the maximal value of y, that is, when the entire basin is fully contributing to the outflow from it, and not just some part of it. The method was implemented by plotting values of (yi − 1 − yi + 1)/2 against yi (in which the subscript i refers to the flow on the ith day) during recessions of the daily flow time series. Ideally, flows which are not strictly base flow, that is, flows during and immediately following precipitation, must be eliminated. In the absence of an adequate precipitation record, a more careful strategy must be followed to select the appropriate data points. To maximize the likelihood that the selected flows during recessions constitute “pure” drought flow, the following criteria were used in the selection process. Eliminate all data points from the streamflow record with positive and zero values of dy/dt; in a drying sequence eliminate suddenly anomalous ones, that is, points with −dy/dt which are twice or more larger than the previous one; delete at least one point after a positive or zero dy/dt (i.e., after a maximum), and all succeeding points until dy/dt stops decreasing (i.e., until the inflection point on the recession curve); during a recession, delete at least one point before a new positive or zero dy/dt (i.e., before a minimum followed by a rising flow rate); ideally, though perhaps not strictly, dy/dt should be decreasing monotonically during a recession; a recession (i.e., a succession of negative dy/dt) should be long enough to contain at least two usable points in succession. This procedure results in a cloud of points and to make some allowance for the unavoidable error even in these selected data points, the lower envelope is established by keeping roughly 5% of them below it.
 The procedure was applied, as just outlined, with the streamflow data of May through September for each decade from 1950 through August 2009 for the four basins listed in Table 1. As an example, in Figure 3 the result is shown for the Lena River for the decade 1950–1959. The resulting K values for all four basins as averages for each of the 10 year periods of their record are shown in Table 3 and Figure 4.
Table 3. Values of the Characteristic Drainage Time Scale K (in Days) for Each Decade During 1950–2009 for Each of the Four Basins Shown in Table 1a
 In the three expressions to calculate dη0/dt, namely (10), (11), and (13) (or (14)), it would seem that (11) is the most parsimonious one, because only a knowledge is required of a typical or average active groundwater layer depth η0r at the end of summer; although this parameter is relatively easy to measure and can be obtained from field surveys, not much information on it is available in the open literature. In the present case of the Lena River, on the basis of temperature measurements during 1956–1990 at 17 stations spread over the Lena River basin, Zhang et al.  found the average active layer thickness to be 1.88 m, ranging from 1.2 m to 2.3 m. This is consistent with the more local measurements near Yakutsk presented by Yang et al. [2002, Figure 11] which indicated that the active layer increased from around 2.0 m in 1950 to around 2.2 m in 1985; it is also consistent with those in that same area presented by Ohta et al.  and Iijima et al. , who found it to vary between roughly 1.3 and 1.8 m between 1998 and 2007. With the values of dyL5/dt of Table 2, and with the assumptions that η0r = 1.88 m and that , that is the average of the annual lowest 5 day flows during each period, it is now possible to implement (11). The results of this exercise are presented in Figure 5.
Equation (10), just like (11), can also be implemented by means of the values of dyL5/dt of Table 2; in addition, however, values of K (see Table 3) and of ne are required. No information is available on the drainable porosity ne in the region considered here, but some idea of its likely magnitude can be derived from other studies. In an analysis of base flows in 22 relatively small basins in the Southern Great Plains in Oklahoma, in the context of hydraulic groundwater theory, it was found [Brutsaert and Lopez, 1998, Figure 9] that ne was of the order of 0.01, ranging between 0.00260 and 0.157, with a mean of 0.0167. In the analysis of base flows in 2 large basins in Illinois by a different method, it was similarly found [Brutsaert, 2008] that ne was somewhat variable in space, with no discernible pattern, and again mostly of the order of 0.01. On the basis of these findings and in the absence of more specific information on ne, (10) was implemented with the values shown in Tables 2 and 3 and with ne = 0.01; the results are presented in Figure 6.
 As an aside, the results obtained with (10) and (11) (and shown in Figures 5 and 6) can now be compared to check how realistic the adopted value of ne is. Indeed the overall average (i.e., for all four basins and all periods) of the results obtainable by means of (10) can be made to match the average of those obtained by means of (11) by changing the value of ne from 0.01 to 0.0118; similarly, the results of the average of all four basins during the most recent period 1990–2008 from (10) can be made to match that obtained with (11) by changing ne from 0.01 to 0.0121. These are very small changes and support the use of the present rough estimate of ne = 0.01.
 Finally, for the application with (13), the values of d(1/K)/dt were calculated as finite differences between the values of K−1 at different times from Table 3; the required values of Kr were taken as the average values for each basin shown in the bottom row of Table 3, and the corresponding value of the active groundwater layer thickness was taken as before as η0r = 1.88 m. These results are presented in Figure 7.
 As a summary of the results, in Table 4 the average growth rates of the active groundwater layer thickness during 1950–2008 obtained with the three methods are compared for each of the four basins.
Table 4. Comparison Between the Growth Rates of the Active Layer Thickness (cm a−1) Over the Period 1950–2008, Obtained With the Three Different Methods
 The characteristic catchment drainage time scale K is a ubiquitous and vital parameter in the proposed approach as presented in section 2.2. The values obtained for the four basins of the present example with the method outlined in section 3.2, are presented in Table 3 and Figure 4. It can be seen that the overall average value for all six decades in all four basins is 41 ± 10 days. This is a remarkable result, because it is so close to the “standard” value of 45 ± 15days (or 1.5 ± 0.5 months) obtained in previous studies [Brutsaert, 2008, 2010, 2012; Brutsaert and Sugita, 2008] in river basins ranging between 1,000 and 100,000 km2, which are markedly smaller than the present ones. In fact, except for the Olyokma River (O) and for one decade on the Upper Aldan River (A1), all values fall within the 1 standard deviation of 15 d. This shows that the self-similarity features of the drainage process observed before in those earlier studies with smaller basins, are also present here. This in turn suggests that to a very good approximation the validity of(8), which underlies and justifies the proposed methodology, is maintained even in the larger basins of the present example application. Since ne and η0are known not to be scale-dependent at these scales, it specifically means that also here the productk0Dd2 can be assumed to be a near universal constant [cf. Brutsaert, 2008, 2010]. This can be interpreted as follows; first, within regions of homogeneous lithology with fairly uniform conductivity the drainage density Ddis a self-similar variable and independent of drainage area; second, in regions of variable lithology more permeable terrain with largerk0 requires fewer channels per unit area and thus a smaller Dd to evacuate the same amount of water, or vice versa, so that k0Dd2is close to independent of basin area, and hence also a self-similar quantity.
 The reason why the K values of the Olyokma River are somewhat smaller than in the other basins may be due to the fact that this river originates in steeper terrain. It can also be seen that the K values of the lower Aldan station (A2) are somewhat larger than those of the upper Aldan (A1); this may have to do with the steeper terrain upstream from A1 and probably with the translatory effect due to the additional length of stream channel between A1 and A2; a similar, albeit smaller, downstream increase was also observed in the K values of the Kherlen River in Mongolia [Brutsaert and Sugita, 2008].
 The main results obtained in the example application of the proposed approach in section 3.3 concern the rate of change of the active groundwater layer thickness dη0/dt. As seen in (15), this change can be due to permafrost thawing and also to the fact that precipitation may not be balanced by runoff and evaporation, that is, (P − E − R) ≠ 0. Therefore, the question as to exactly what fraction of the calculated dη0/dt is due to permafrost thawing T can only be answered by a careful hydrologic balance involving (P − E − R). Although the necessary data to check this are not available, several studies reported in the literature have looked at the hydrology of the Lena Basin in recent years, which may be helpful in clarifying this issue. Practically all cold season precipitation leaves the basin as melt runoff when the soil is still frozen, while the warm season precipitation may be recycled more inside the basin with infiltration and evaporation; moreover, the active layer develops during the summer. Therefore, it is mainly (though not only) the summer season, which is of interest here.
 Mainly owing to the sparsity of the stations and to possible undercatch, area averaged measured precipitation P in the Lena River basin has been notoriously unreliable; moreover, very little information is available on the evaporation term E [Adam and Lettenmaier, 2008]. However, in at least two studies [Fukutomi et al., 2003; Oshima and Hiyama, 2012] this difficulty was avoided by estimating the combined term (P − E) from the atmospheric water budget derived as part of reanalysis calculations. According to Fukutomi et al. [2003, Figure 5 and Table A1] during 1979–1995, the temporal and spatial mean (P − E − R) was strongly negative during summer time and about −0.7 cm a−1 on an annual basis. Similar results were presented by Oshima and Hiyama [2012, Figures 1 and 3 and Table 1]; they found that during 1980–2008 the mean of (P − E − R) in the Lena River basin was negative in the summer and early fall, and around −2.4 cm a−1 during the year; the trends were also negative, namely −0.48 mm a−2 for the summer values and −0.74 mm a−2 for the annual values.
 In summary, the results in both studies suggest that while the different estimates involve considerable error, during the past half century or so the combination (P − E − R) has been most likely negative, especially in summer; moreover, it has also been getting more negative. The present results show that ne(dη0/dt) has been positive, that is, the thickness of the movable groundwater layer η0 has been increasing; therefore, as seen in (15), this represents an underestimate of the actual permafrost thawing T, because part of this thaw water was used up to compensate for the deficit in (P − E − R) and could not manifest itself as dη0/dt. Although the deficit in (P − E − R) and its spatial and temporal distribution are not known exactly at this point, it is not unreasonable to infer that the true active layer growth values are at least as large as the present results for the active groundwater layer thickness growth rate dη0/dt. Moreover, because (P − E − R) is becoming more negative, the more recent values of ne(dη0/dt) obtained here probably underestimate the true values of the thawing rate T more than the earlier ones.
 While at first sight Figures 5–7 and Table 4 show differing and seemingly disparate results, they have several common features which allow some general observations regarding the evolution of the active layer in eastern Siberia. Table 4 shows that over the past half century the active layer thickness has increased at average rates of the order of 0.3 to 1 cm a−1 in the warmer areas of the Upper Lena River and the Olyokma River with discontinuous permafrost, and at rates of about half this much in the colder more eastern areas of the Aldan River subbasin with continuous permafrost. From Figures 5–7 it would also seem that during the fifties and sixties of the previous century, and perhaps into the seventies, the colder regions with continuous permafrost may not have undergone any net active layer growth but rather active layer shrinking, i.e., negative growth rates, perhaps with permafrost increases. However, in light of the (P − E − R) deficit, the occurrence and magnitude of such permafrost increases may be questionable. Most likely starting in the late eighties or early nineties, the active layer in the colder regions has shown dramatic growth rates of the order of at least 2 to 3 cm a−1 or probably even larger. In contrast, the areas with discontinuous permafrost of the Upper Lena and the Olyokma appear to have undergone a more sustained increase of the active layer thickness rate of change. However, as explained at the end of section 2.4, the values of (dη0/dt) obtained for these basins with discontinuous permafrost are probably smaller than the actual local values where permafrost is present.
 The results obtained with (11) and (10) (and shown in Figures 5 and 6) are fairly consistent with each other, but those obtained with (13) (and shown in Figure 7) are mostly about twice as large on average. In principle, it is difficult to judge which among the three can be expected to be closest to the true values. Still, in view of the admittedly subjective way of determining K (and a fortiori its derivatives), and the relatively straightforward way of determining dyL5/dt, it would seem that (11) and (10) should be given preference over (13). Because (11) makes use of objectively measurable data, it should probably be given preference over (10).
 How do the values obtained in the present study compare with published values derived from point observation stations? First consider large area averages. Frauenfeld et al. reviewed the changes in active layer depth at 31 stations on permafrost located throughout Russia from zero-isotherm observations, and found that during 1956–1990 it increased at an average statistically significant growth rate of 0.59 cm a−1. From a subset of 17 stations of this same data set over the same period within the Lena River basin, Zhang et al.  found an average rate of increase of 0.90 cm a−1. (As an aside, these authors also calculated a thawing index on the basis of near surface air temperature and land cover as a gridded product; interestingly, for the entire Lena River basin they showed [Zhang et al., 2005, Figure 12] that during 1950–1970 the active layer thickness must have been decreasing, which is consistent with the negative values shown in Figures 5, 6 and 7). At a more local scale, near Yakutsk during 1931–1986 the maximal annual thawing depth was found [Yang et al., 2002, Figure 11] to increase, albeit unsteadily, from about 200 cm in 1950 to about 220 cm in 1985, that is an average rate of around 0.57 cm a−1. To summarize, all these long-term point observations are of the same order as the values presented here inTable 4.
 More recent point observations in the field indicate that permafrost deterioration has been accelerating. In the same area as the Yang et al.  study some 20 km north of Yakutsk at Spasskaya Pad, during 1998–2006 the annual maximum depth of the 0°C isotherm was recorded [Ohta et al., 2008, Figure 5] to increase from around 1.27 m in 1998 to 2.0 m in 2006, that is at an average rate of about 9.1 cm a−1. On the other hand, in a parallel study at this location [Iijima et al., 2010, Figure 2] manual observations of the annual maximal thaw depth by means of a frost tube indicated that between 2000 and 2007 it increased from around 1.37 to 1.67 m, that is at a rate of about 3.3 cm a−1. Since manual measurements are considered more reliable, the latter rate of 3.3 cm a−1 is probably more representative; it is in agreement with the present results shown in Figures 5 and 6 between 1990 and 2010 for A2, the Aldan River site closest to Yakutsk, namely 2.5 and 3.2 cm a−1.
 Low flows (or base flows) in a river are fed by groundwater seepage from the upstream riparian aquifers in the basin. Permafrost thawing and permafrost growth directly affect the groundwater storage amount and mobility, which in turn control the seepage from these aquifers. Thus, long-term streamflow records are a useful but hitherto unexplored source of information on the past history of permafrost changes during the same period. On the basis of this principle and hydraulic groundwater theory, a methodology was developed producing three distinct procedures to relate low-river flows during late summer with the rate of increase of the active groundwater layer thickness resulting from permafrost thawing in the upstream river basin. Although for accuracy there is still no substitute for in situ temperature observations in the soil profile, the proposed methodology has distinct benefits. First, it provides a tool to upscale local measurements over the entire upstream river basin, possibly after calibration with such local measurements; second, it can produce long-term time series of permafrost conditions that predate most field observations by many years because streamflow measurements were often started well before permafrost observations.
 As an illustration, the three resulting methods were applied to four subbasins of the Lena River in eastern Siberia, namely a station on the Upper Lena, one on the Olyokma and two on the Aldan River. It was found that between 1950 and 2008 in the warmer areas with more discontinuous permafrost the active layer depth has been increasing at average rates of the order of 0.3 to 1 cm a−1, and in the colder areas further to the east with more continuous permafrost at average rates about half as large. These rates have not been steady, but overall they have been increasing. Early on in the period 1950–1970, some large regions, especially the colder ones with more continuous permafrost appear not to have undergone much thawing, and perhaps even active layer thickness decreases; however, from the 1990s onward they have been experiencing active layer growth rates as large as 2 cm a−1 on average, and some even larger.
 The present results are in good agreement with previously published active layer growth rate values obtained at many stations in eastern Siberia by in situ measurements within the soil profile. Nonetheless, water budget considerations in the Lena River basin, involving the observed trends in precipitation, evaporation and basin runoff, suggest that the true permafrost thawing rates are not smaller than suggested by the present results, and probably even larger.
 The overall average value of the characteristic drainage time scales was found to be 41 ± 10 days for the basins in the application of the method in eastern Siberia; this is remarkably close to the standard value of 45 ± 15 days derived in earlier studies for basins ranging in size between 1,000 and 100,000 km2. This indicates that in spite of the relatively large sizes of the present study basins, self-similarity of the groundwater outflow mechanisms, underlying and justifying the proposed methodology, was maintained. In the present context, as can be inferred from(8), this means mainly that also in the present application the product k0Dd2 can still be assumed to be a near universal constant. This can be explained as follows; first, within regions of homogeneous lithology the drainage density Ddis independent of drainage area and thus exhibits self-similarity; second, in regions of variable lithology more permeable terrain requires fewer channels per unit area to evacuate the same amount of water, or vice versa, so thatk0Dd2is close to independent of basin area, and hence also a self-similar quantity.
 Even though the value of the drainage time scale K can often be expected to lie around 45 ± 15 days [e.g., Brutsaert, 2012, Figure 1], the objective and accurate determination of its temporal rate of change dK/dt, or d (K−1)/dt still poses challenges; therefore, in future practical applications (10) and (11) should probably be preferable over (13) or (14). If no other information is available on the values of the parameters in (10) and (11), it should be possible to obtain good first approximations using η0r = 2 m, K = 45 d, and ne = 0.01.
 This study was partly supported by research project C-07 of the Research Institute for Humanity and Nature (RIHN), entitled “Global Warming and the Human-Nature Dimension in Siberia: Social Adaptation to the Changes of the Terrestrial Ecosystem, with Emphasis on Water Environments.” The authors wish to express their thanks to Hotaek Park of the Research Institute for Global Change, Japan Agency for Marine-Earth Science and Technology-JAMSTEC, at Yokosuka, Japan, for facilitating the transfer of most of the streamflow data used in this study. Most of this work was carried out while W.B. was on leave at RIHN, Kyoto, Japan; the support and hospitality making this possible are appreciated, and helpful discussions with Kazuhiro Oshima are gratefully acknowledged.