TCThe CryosphereTCThe Cryosphere1994-0424Copernicus PublicationsGöttingen, Germany10.5194/tc-10-433-2016Bulk meltwater flow and liquid water content of snowpacks mapped using the
electrical self-potential (SP) methodThompsonSarah S.sarah.thompson@unis.nohttps://orcid.org/0000-0001-9112-6933KulessaBerndhttps://orcid.org/0000-0002-4830-4949EsseryRichard L. H.LüthiMartin P.https://orcid.org/0000-0003-4419-8496Department of Arctic Geology, University Centre in Svalbard (UNIS), Svalbard, NorwayDepartment of Geography, College of Science, Swansea University, Swansea, UKInstitute of Global Change, The School of Geosciences, University of Edinburgh, Edinburgh, UKDepartment of Geography, University of Zurich, Zurich, Switzerlandformerly at: Versuchsanstalt für Wasserbau, Hydrologie und Glaziology
(VAW), ETH Zürich, Zurich, SwitzerlandSarah S. Thompson (sarah.thompson@unis.no)29February20161014334448July201524August20155February201615February2016This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://tc.copernicus.org/articles/10/433/2016/tc-10-433-2016.htmlThe full text article is available as a PDF file from https://tc.copernicus.org/articles/10/433/2016/tc-10-433-2016.pdf
Our ability to measure, quantify and assimilate hydrological properties and
processes of snow in operational models is disproportionally poor compared
to the significance of seasonal snowmelt as a global water resource and
major risk factor in flood and avalanche forecasting. We show here that
strong electrical self-potential fields are generated in melting in situ
snowpacks at Rhone Glacier and Jungfraujoch Glacier, Switzerland. In
agreement with theory, the diurnal evolution of self-potential magnitudes
(∼ 60–250 mV) relates to those of bulk meltwater fluxes
(0–1.2 × 10-6 m3 s-1) principally through the
permeability and the content, electrical conductivity and pH of liquid
water. Previous work revealed that when fresh snow melts, ions are eluted in
sequence and electrical conductivity, pH and self-potential data change
diagnostically. Our snowpacks had experienced earlier stages of melt, and
complementary snow pit measurements revealed that electrical conductivity
(∼ 1–5 × 10-6 S m-1) and pH
(∼ 6.5–6.7) as well as permeabilities (respectively
∼ 9.7 × 10-5 and ∼ 4.3 × 10-5 m2
at Rhone Glacier and Jungfraujoch Glacier) were
invariant. This implies, first, that preferential elution of ions was
complete and, second, that our self-potential measurements reflect daily
changes in liquid water contents. These were calculated to increase within
the pendular regime from ∼ 1 to 5 and ∼ 3 to 5.5 %
respectively at Rhone Glacier and Jungfraujoch Glacier, as
confirmed by ground truth measurements. We conclude that the electrical
self-potential method is a promising snow and firn hydrology sensor owing to
its suitability for (1) sensing lateral and vertical liquid water flows
directly and minimally invasively, (2) complementing established
observational programs through multidimensional spatial mapping of meltwater
fluxes or liquid water content and (3) monitoring autonomously at a low cost.
Future work should focus on the development of self-potential sensor arrays
compatible with existing weather and snow monitoring technology and
observational programs, and the integration of self-potential data into
analytical frameworks.
Introduction
More than a sixth of the world's population relies on melt from seasonal
snow and glaciers for water supply (Barnett et al., 2005). Snow, and runoff
from snow, are also major resources for the hydroelectric, tourism and
inland fishery industries, and furthermore represent hazards from flooding
and avalanches (Mitterer et al., 2011). The availability of snow models
constrained by a reliable observational basis, for the forecasting of snow
hydrological properties and processes in climate, resource and hazard
applications is therefore of considerable socio-economic significance (Wever
et al., 2014). However, the parameterisation of fundamental
snow hydrological attributes, such as liquid water content and flux, is a
well-recognised major source of uncertainty in operational models used in
snow and hydrological forecasting (Livneh et al., 2010; Essery et al.,
2013). This uncertainty in operational models is rooted principally in the
inability of traditional snow hydrological techniques to provide automated
attribute measurements non-invasively and on spatial scales that match those
used in operational snow models. Relevant traditional techniques include
dielectric (Denoth, 1994) or “hand” tests (Fierz et al., 2009) of snow
liquid water contents, lysimeter measurements of discharge, temperature and
pH and electrical conductivity of bulk meltwaters (Campbell et al., 2006;
Williams et al., 2010) and manual observation or measurement of snow
density and grain size (Fierz et al., 2009). Even cutting-edge
upward-looking radar measurements of snowpack structure and liquid water
content (Heilig et al., 2010; Mitterer et al., 2011; Schmid et al., 2014)
compare unfavourably with model predictions of wetting front propagation
(Wever et al., 2014), attributed to inherent limitations of the 1-D approach in
capturing preferential flow.
By combining field measurements with a theory and model of self-potential
signals associated with unsaturated flow in melting snow (Kulessa et al.,
2012), we show here that electrical self-potential geophysical data
integrated with traditional snow measurements can address these limitations.
The self-potential technique is a passive geo-electrical method that
exploits the presence of naturally occurring electrical potentials in the
subsurface generated as a result of dipolar charge separation when water
flows through a porous matrix (“streaming potential”; Darnet et al., 2003;
Revil et al., 2006). The self-potential method has a unique ability in
delineating, monitoring and quantifying the flow of subsurface water in
groundwater aquifers and unsaturated media (e.g. Revil et al., 2006, and
references therein), and for application to numerous cold regions (e.g. French
et al., 2006; Kulessa, 2007, and references therein). This ability is due to
the fact that porewaters generally have an excess of electrical charge due
to the electrical double layer at the interface between the solid matrix (in
this case snow grains) and porewater. The advective drag of this excess of
electrical charge is responsible for a streaming current, whose divergence
generates a quasi-static electric field known as the streaming potential
(Sill, 1983; Revil et al., 2003). More recently, streaming potential theory
has been extended for unsaturated conditions (Linde et al., 2007; Revil et
al., 2007; Jougnot et al., 2012). A new theory and numerical model of
self-potential signals associated with unsaturated flow in melting snow,
along with laboratory tests, strongly promoted the technique as a
non-intrusive hydrological sensor of water fluxes (Kulessa et al., 2012) at
spatial scales intermediate between snow pits and satellite footprints or,
given independent flux measurements, of evolving physical and chemical
properties of snow and snowmelt.
We answer two fundamental questions. (1) Can the self-potential method serve
as a non-intrusive field sensor of temporally evolving bulk meltwater fluxes
and liquid water contents of snow? (2) What are the ambiguities introduced
into estimates of liquid water contents from self-potential and bulk
discharge data, by uncertainties inherent in the governing snow physical and
chemical properties? Lastly we discuss the implications and possibilities of
the technique for future snow measurement and modelling research and
practice. Our study thus takes a significant step towards the in situ
implementation of the self-potential method for improved characterisation
and monitoring of snow liquid water contents and meltwater fluxes.
Theory, field sites and methods
The Poisson equation relates the electrical field ψ to the source
current density in a partially or fully saturated snowpack:
∇⋅σ∇ψ=∇⋅js,
where σ is the bulk electrical conductivity of the porous material
(in S m-1), and js is the source current density (in A m-2).
Equation (1) applies only in the low-frequency limit of the Maxwell's
equations without external injection or retrieval of charges, or charge
storage in the snowpack. Extending the classic Helmholtz–Smoluchowski theory
for unsaturated flow in snow, the 1-D solution to Eq. (1) is
given by
ψm-ψ0=-εζησwSwHm-H0,
where ψm and Hm are respectively the electrical and hydraulic
potentials at the measurement electrode, ψ0 and H0 are the
corresponding potentials at the reference electrode, ζ is the zeta
potential (V), and ε, η, σw and Sw are
respectively the dielectric permittivity (F m-1), dynamic viscosity (in
Pa s), electrical conductivity (S m-1) and relative saturation
(dimensionless) of the melt- or rainwaters in the snowpack's pore space
(Kulessa et al., 2012). The zeta potential is the voltage across the
electrical double layer at the interface between the ice matrix and the porewaters, as controlled by these constituents' physical and electrical
properties.
To address the specific objectives set out in the introduction through
data-driven testing, we developed an experimental survey design to simulate
the geometry of the laboratory snow column of Kulessa et al. (2012) (Fig. 1b). It
was therefore our aim to characterise bulk meltwater fluxes in inclined
snowpacks at two glaciers in Valais, Switzerland, measuring all relevant
snowpack attributes for ground truth. Self-potential and traditional
snow hydrological measurements were acquired on 13, 14 and 15 June 2013 from
the ablation area snowpack at Rhone Glacier, and on 5 September 2013
from the glacial accumulation area at Jungfraujoch Glacier (Fig. 1a). At
Rhone Glacier and Jungfraujoch Glacier site elevations were respectively
2340 and 3460 m a.s.l., with surface gradients of ∼ 8
and 17∘. At the Rhone Glacier all 3 days experienced
comparable air temperature, although 15 June was noticeably cloudier with a
very low sunshine duration. Because daily average temperatures were between
5 and 15 ∘C with no fresh snowfall (MeteoSuisse), the snowpacks
would have experienced significant melting in the weeks before the surveys.
We therefore expect them to be physically mature in terms of enhanced grain
size and density due to metamorphosis, and chemically mature in terms of
invariant meltwater pH and electrical conductivity as preferential elution
of solutes has been completed (Kulessa et al., 2012, and references
therein).
(a) Example survey set-up, self-potential grid 25×25 m. The inset on the left shows the
location of both field sites. The inset on the right illustrates the self-potential
survey design; to provide each self-potential data value, a profile of 25
data points (P1, P2, etc.) was collected (Line 1, Line 2, etc.),
perpendicular to assumed bulk water flow. (b) Schematic of the
self-potential experiment developed by Kulessa et al. (2012) for the situ
snowpack surveys.
At both sites, more than 100 self-potential measurements were made at the
snow surface, and meltwater bulk discharge in a lysimeter, pH and electrical
conductivity, and snowpack characteristics including thickness, density,
grain size and liquid water content were recorded. Adopting our established
acquisition procedures (Thompson et al., 2012), we conducted all
self-potential surveys using a pair of lead/lead chloride “Petiau”
non-polarising electrodes (Petiau, 2000). The survey was carried out
following the potential amplitude method (Corry et al., 1983); this employs
a reference electrode in a fixed location and a roving electrode which is
moved through the survey area at 0.5 m intervals (Fig. 1a). Self-potential
surveys were conducted in profiles of 25 data points perpendicular to the
principal direction of water flow, where the latter was assumed to follow
the gradient indicated by snow surface topography. All self-potential
measurements were taken as differential readings relative to the reference
electrode, minimising streaming, electrochemical and thermal potentials at
the latter by grounding them outside the survey areas at the top of a local
topographic high point (Fig. 1a), submerged in a glass jar, open at the top
and filled with water-saturated local media (Kulessa et al., 2003a). The jar
was then buried upright ∼ 1 m deep to avoid exposure to
surface temperature variations. Surveys were carried out with a fixed tie-in
point (measured every second line) at the reference electrode, allowing for
correction of the effects of electrode polarisation and drift (Doherty et
al., 2010; Thompson et al., 2012).
Bulk discharge through a snowpack is preferably measured with a lysimeter
(Campbell et al., 2006; Williams et al., 2010), in this case made up of a
series of smaller (guttering) areas joined together to prevent freezing and
compaction (after Campbell et al., 2006). The lysimeter was placed at the
base of Rhone Glacier's snowpack, and at the limit of the diurnal melt
penetration depth at Jungfraujoch Glacier (determined by daily dye-tracing
experiments). Snow density (by balance) and average snow grain size (crystal
card and lens) were measured, at the start and end of each self-potential
survey to reveal any intermittent snow metamorphism, using standardised
techniques within the top and basal layers of snow pits freshly excavated at
the survey sites (Fierz et al., 2009). Liquid water content was estimated
using two different techniques, including the hand test (Colbeck et al.,
1990; Fierz et al., 2009) in the surface and base layers of Rhone Glacier's
snow pit, and the Denoth capacitance meter (Denoth, 1994) in the surface and
base layers of the snow pit at Jungfraujoch Glacier. The latter measurements were
acquired across a 2-D grid where the instrument was inserted into the
snowpack at a depth of 0.4 m following the same survey spacing as the
self-potential measurements.
Field measurement results
The drift-corrected self-potential magnitudes and meltwater bulk discharges
both increase with time through the day until a peak in late afternoon,
after which they both begin to decrease (Fig. 2). There is no
distinguishable time lag between the measured self-potential magnitude and
discharge data (Fig. 2), and the ratio between self-potential and bulk
discharge changes consistently over time (Fig. 3). Days 1 and 2 at Rhone
Glacier were characterised by higher discharges and self-potential
magnitudes compared to day 3, and intriguingly bulk discharge at
Jungfraujoch Glacier was akin to day 3 at Rhone Glacier, but self-potential
magnitudes at Jungfraujoch Glacier were much higher than days 1 and 2 at
Rhone Glacier (Fig. 2). The pH, electrical conductivity and temperature of
meltwater, recorded with each bulk discharge measurement, show no consistent
temporal or spatial variation across any of the four field surveys. Fluid
electrical conductivity values generally ranged between 1 × 10-6
and 5 × 10-6 S m-1 without spatial
or temporal consistency, while pH ranged between 6.5 and 6.9. Snow grain
size remained constant at ∼ 1.5 mm at Rhone Glacier and
∼ 1 mm at Jungfraujoch Glacier, while snow densities ranged
between 555 and 573 kg m-3 without spatial or temporal
consistency. The very small variability range of the snowpack
characteristics measured is consistent with mature snowpacks, as assumed
above with reference to prior meteorological conditions. At Rhone Glacier
the liquid water content of snow had a wetness index of 3 irrespective of
measurement time or location at the surface or base of the snow pit,
associated with a liquid water content range of 3–8 % vol. (Colbeck et
al., 1990). At Jungfraujoch Glacier liquid water content, measured using the
Denoth meter, gave profile-averaged values of 1.5 to ∼ 5.0 % vol.,
increasing consistently throughout the survey period. These
measurements and inherent uncertainties are used below for snow liquid water
content calculations, uncertainty analysis and sensitivity testing.
Time series of (a) bulk self-potential measurements and (b) bulk
discharge measurements for the three Rhone Glacier surveys and the
Jungfraujoch Glacier survey. Each self-potential data point represents the
mean value of a profile (consisting of 25 data points); the error bars
illustrate the variability over each profile. Bulk discharge was measured
over each profile by the lysimeter.
Ratio between self-potential magnitudes (V) and bulk discharge (m3 s-1)
for each of the four surveys through time, illustrating that the ratio
changes consistently over time.
Objective 1: self-potential as a snow hydrological sensor
Both survey areas were south facing, topographically inclined but otherwise
had no visibly distinguished snow surface undulations, and any snow
thickness variations were minimal. We therefore expect changes in
self-potential magnitudes to be pronounced in the downslope direction, and
minimal across-slope along any individual profile (Fig. 1a). Averaging all
25 self-potential data points acquired along any particular profile, a
1-D upslope–downslope series of self-potential magnitudes is
produced for a given survey area on a given day, together with uncertainty
estimates reflecting natural spatial and temporal variability along the
profile (Table 1). For each profile, the acquisition time of
the central data point was assigned to it, and all measurements of snowpack
and meltwater properties were averaged over the same time period
(∼ 20 min, i.e. the acquisition time of any one self-potential profile). The upslope–downslope series of average self-potential
magnitudes thus emulates measurements along a horizontally inclined version
of the 1-D snow column used in Kulessa et al. (2012) (Fig. 1b).
These authors reformulated the 1-D solution in Eq. (2) to relate
measured self-potential magnitudes and bulk discharges through their
partially saturated snow column, which we can therefore adapt here to our
field experiment.
This adaptation is dependent on four key assumptions, including that (1) water
flow within the survey areas' snowpacks is laminar and homogenous in three
dimensions, where the snowpack surface and base have constant and equal
inclination and thus maintain a spatially constant hydraulic gradient;
(2) all contributions to the measured self-potential signal from flow below the
base of the snowpack, runoff at the surface of the snowpack and flow
outside the lateral boundaries of the survey areas' snowpacks are
negligible, and all water contributing to the measured self-potential
signals is adequately captured by our bulk discharge measurements; (3) all
snow physical and chemical properties controlling the self-potential
magnitude do not vary spatially across the survey areas' snowpacks, so that
our ground-truth snow-pit data apply uniformly across them, and (4) any
spatial changes in self-potential magnitudes are dominated by temporal
changes in snow or meltwater properties, while static-elevation-
driven
spatial changes are negligible. We assess the implications of any potential
violations to these assumptions in Sect. 6.
At a given time, tn, the measured self-potential field, Ψm(tn), in our survey area is the difference between the locally
produced self-potential field, Ψl(tn), and the self-potential
field at the reference electrode, Ψ0(tn). The latter is unknown
in our field feasibility study, although our method of emplacing the
reference electrode is elaborate and designed to eliminate, or at least
minimise, any streaming potentials at the reference electrode (see Sect. 2).
Once the reference electrodes have settled in their environments, we
further expect any electrochemical or thermal potentials to be negligible.
We can therefore expect Ψ0(tn) to be close to zero, but
nonetheless apply caution and take a two-step approach. Initially we
eliminate the reference self-potential fields by considering temporal
changes in measured self-potential magnitudes before, subsequently, considering
absolute self-potential magnitudes.
Temporal changes in self-potential magnitudes
We can eliminate the reference field by differencing two self-potential
measurements acquired at two successive times:
ψm(tn)-ψm(tn-1)=Ψl(tn)-Ψl(tn-1).
Equation (3) assumes that Ψ0 and H0 are temporally invariant,
a reasonable supposition for drift-corrected self-potential data if the
reference electrode is correctly emplaced. Recognising that ψ0=H0≈0 for their snow column experiment, Kulessa et al. (2012)
reformulated Eq. (2) to show that the self-potential field at a
measurement electrode, Ψl(tn), can be approximated by
Ψl(tn)=εζσwSw(tn)Sen(tn)1kAQ(tn),
where Q (m3 s-1) is bulk discharge in the snowpack through the
cross-sectional area A (m2), k is permeability, Se is effective
saturation and n≈3.3 is the saturation exponent (after Albert and Krajeski, 1998; Kulessa et al., 2012). Assuming that any temporal changes in the
self-potential field at the reference electrodes in our field experiments
are negligible, the difference between successive field self-potential
measurements in time can be approximated by
ψm(tn)-ψm(tn-1)=εζσw1kASw(tn)Sen(tn)Q(tn)-Sw(tn-1)Sen(tn-1)Q(tn-1).
In the present case we have measured ψm(tn) and Ψm(tn-1) as well as Q(tn) and Q(tn-1). We have also measured, or can
estimate from well-established empirical relationships, all other parameters
coupling the temporal difference in self-potential fields (Ψm(tn) and ψm(tn-1)) to that of discharge (expression in
the large parentheses on the right-hand side of Eq. 5). To demonstrate the
usefulness of self-potential measurements in snow research and practice, we
can therefore evaluate Eq. (5) at successive times, tn and tn-1, to
calculate temporal changes in the liquid water content, Sw, of the
snowpacks at our field sites. This evaluation is subject to assumptions (1)
to (4) above, and is ground-truthed using snow pit measurements of liquid
water contents.
At both Rhone Glacier and Jungfraujoch Glacier, self-potential magnitude
(ψm), bulk discharge (Q), electrical conductivity (σw) and cross-sectional area (A) (survey area width × snow
depth) were measured directly. Assuming that water at 0 ∘C has a
dielectric permittivity of εr=88, the dielectric permittivity
(F m-1) of pore meltwater is ε=εrε0=7.8×10-9 F m-1, where 0=8.85×10-12 F m-1
is the dielectric permittivity of vacuum. Permeability (k) can be
derived from our snow density (ρs) and grain size (d) measurements
using Shimizu's (1970) empirical relationship:
k=0.077d2e-0.0078ρs,
where k is in m2, d is in m and ρs is in kg m-3. This commonly
used equation was derived from a fit to laboratory data collected with small
rounded grains and a starting grain diameter of ∼ 0.33 mm
(Shimizu, 1970). However, later work ascertained experimentally that
Shimizu's (1970) empirical formula does in fact apply to a much larger range
of grain diameters, as expected to be encountered in practice (less than
0.33 mm to greater than 2 mm) (Jordan et al., 1999). We can therefore expect
Eq. (6) to be robust for our purposes. Effective saturation (Se) and
Sw are related through the irreducible water saturation Swir by
Se=Sw-Swir1-Swir.
In the absence of direct measurements, we adopt the commonly used values of
Swir=0.03 and n≈3.3 (Kulessa et al., 2012), and assume
that these values are invariant in space and time at our study sites.
A significant challenge arises however in that there is one remaining
parameter, the zeta potential (ζ), which is unknown here and poorly
constrained in general. Earlier work on artificial ice samples, of fixed
bulk electrical conductivity, ascertained that the zeta potential reverses
sign from ∼+0.01 to ∼-0.02 V as
equilibrium pH increases from less than 3 to greater than 8 (Drzymala et
al., 1999; Kallay et al., 2003). The electrochemical properties of the
electrical double layer at the snow grain surfaces, and thus also the
magnitude and potentially the sign of the zeta potential, will change over
time in a fresh snowpack as the snow is affected by melt, recrystallisation
and the preferential elution of ions (Meyer and Wania, 2008; Meyer et al.,
2009; Williams et al,. 1999). Recent “natural snowmelt” laboratory
experiments were consistent with a progressive increase of pH from 4.3 to
6.3 and a simultaneous decrease in electrical conductivity from
∼ 1 × 10-1 to ∼ 6 × 10-7 S m-1,
as the elution of ions follows a well-known
sequence (Kulessa et al., 2012). Upon conclusion of the laboratory experiments of Kulessa et
al. (2012), modelled rates of change of pH and electrical
conductivity were minimal and the snow column mature. The zeta potential is
principally a function of pH and electrical conductivity, and the combined
dependency of the zeta potential on electrical conductivity (σw), meltwater pH (pHw) and the meltwater pH at the point of zero charge
(pHw) (pzc)) can be expressed as
ζσw,pH=α+βlog10σwsinπ12pHw-pHwpzc,
where α and β depend on the chemical composition of the pore
fluid and can be determined empirically (Revil et al., 1999). Kulessa et al.
(2012) inferred the zeta potential changed from ∼-7.5×10-2 V at the start of the natural snowmelt experiments to
+1.5 × 10-2 V at the end, when the rate of change of the
zeta potential was minimal.
The final values of pH and electrical conductivity that Kulessa et al. (2012)
calculated from Eq. 8 were similar to those measured at Rhone Glacier
and Jungfraujoch Glacier (respectively ∼ 6.5–6.9 and
∼ 1–5 × 10-6 S m-1), suggesting that
these in situ snowpacks were likewise mature as expected (Sect. 2). This
inference is corroborated by the absence of consistent spatial or temporal
changes in either pH or electrical conductivity throughout the survey
periods. In the laboratory study of Kulessa et al. (2012), the pH-corrected zeta
potential had values around zero for the range of electrical conductivities
(1–5 × 10-6 S m-1) measured at Rhone Glacier and
Jungfraujoch Glacier (1–5 × 10-6 S m-1), and its rate
of change became minimal along with those of pH and electrical conductivity.
We can therefore expect a small and invariant zeta potential value to apply
to the snowpacks at Rhone Glacier and Jungfraujoch Glacier. Indeed, an
excellent fit (R2≈0.85) between liquid water contents
measured at Jungfraujoch Glacier with the Denoth meter and that calculated
based on Eq. (5) is obtained when the zeta potential is assigned a value of
∼-1×10-5 V (Fig. 4). This excellent fit
suggests that in situ measurements or empirically derived estimates of the
parameters affecting coupling between measured self-potential magnitudes and
discharges in Eq. (5) are robust for practical purposes.
Temporal differences in Sw inferred from self-potential data
against temporal differences in the Denoth-measured Sw at
Jungfraujoch Glacier, according to Eq. (5).
Absolute changes in self-potential magnitudes
The same parameters affect the coupling between temporal changes in
self-potential magnitudes and discharge (Eq. 5), and absolute changes
therein as described by Eq. (4) derived by assuming that the reference
potential is zero. We are therefore encouraged to calculate absolute liquid
water contents from our self-potential data using Eq. (4). We do this
initially for Jungfraujoch Glacier because here we have detailed
ground-truth measurements of liquid water content made with a Denoth meter.
Encouragingly we find that calculated and measured ground-truth data match
each other very well (Fig. 5a), attesting to the fact that the reference
potentials at Jungfraujoch Glacier may not only be temporally invariant as
confirmed earlier, but generally have negligible magnitudes.
(a) Liquid water content calculated from Eq. (4) for the
self-potential survey carried out at Jungfraujoch Glacier, with the
corresponding Denoth measurements. The uncertainty range illustrates the
minimum and maximum model results for the range of parameters (Table 1).
(b) Liquid water content calculated from Eq. (4) for each of the three
self-potential surveys carried out at Rhone Glacier. All results are within
the range of liquid water content (% vol) estimated by the hand tests
(black dashed lines).
Best estimate of each parameter for Rhone Glacier SP (day 2) and
relative assumed uncertainty and sensitivity ranges. The sensitivity ranges
are based on the measurement accuracy of each measured parameter or the
confidence of estimates parameters. The uncertainty ranges are exaggerated
from the sensitivity values to highlight the effect of poor measurement or
estimation.
Measured/estimated parametersBest estimateSensitivity rangeUncertainty rangeSelf-potential ψm (V)Variableψm±20 %ψm±40 %Discharge Q (m3 s-1)VariableQ±20 %Q±40 %Electrical conductivity σw (S m-1)5×10-6σw±5×10-710-7–10-4Zeta potential ζ (V)-1×10-5ζ±50 %10-4–10-6Permeability fromgrain diameter d (m)0.00175d±0.0005d±0.001density ρ (kg m3)555.5ρ±70ρ±140Cross-sectional area fromwidth w (m)12.5w±5w±10depth dp (m)1.45dp±0.2dp±1
Sw calculations for a range of values for each parameter. In
each case the range is an exaggerated uncertainty range (Table 1),
highlighting the effect of each individual parameter on the calculated
Sw output, using Rhone Glacier SP day 2 as an example.
We can apply the same expectation of negligible reference self-potential
magnitudes to our surveys at Rhone Glacier on the 3 successive days. We
find that absolute liquid water contents inferred from Eq. (4) generally
fall well within the range of ∼ 3–8 % inferred from our
ground-truth hand tests. We can therefore conclude that given careful
emplacement of the reference electrode, the simple empirical relationship
between self-potential magnitudes, discharge and liquid water content is
robust not only in a laboratory setting (Kulessa et al., 2012), but also for
application to in situ snowpacks. The self-potential method therefore shows
considerable promise as a non-intrusive snow hydrological sensor.
Objective 2: self-potential sensitivity to uncertainty in snow
properties
We evaluate the sensitivity of calculated liquid water contents to both
individual and combined parameter uncertainties. For each parameter a range
of uncertainty values was created, with the respective minima and maxima
approximately twice that of the uncertainty (Table 1). Repeat water content
calculations were carried out initially by changing each parameter
individually for a range of values between the respective minima and maxima.
The results cluster broadly in three categories, including the zeta
potential (up to ∼ 20 % change in liquid water content
within the 50 % uncertainty range), followed by grain diameter, survey
area width, electrical conductivity, snow depth and snow density
(∼ 3–4 % change) and bulk discharge and self-potential
(2 % change) (Fig. 6). These three categories readily reflect our
knowledge of or ability to measure the respective parameters in situ, with
surprisingly low sensitivity to cross-sectional area despite our simplistic
calculation and significant inherent assumptions (i.e. 1–4 in Sect. 4).
Self-potential magnitudes are readily measured in the field with minimum
uncertainty (Fig. 6), although the strongly enhanced sensitivity to the zeta
potential highlights the need for focused research to tightly constrain
possible values of this parameter in in situ snowpacks.
While this gives a good indication of the parameters to which water content
calculations are most sensitive, it does not indicate possible feedbacks
between parameters. Feedbacks were therefore evaluated by calculating liquid
water contents for all possible combinations of the best estimates and
minimum and maximum parameter values (Table 1), giving over 6500 solutions
(Fig. 7). The minimum and maximum outputs were then adopted as the lower and
upper uncertainty bounds (Fig. 3). Due to the large potential uncertainty in
the zeta potential, the sensitivity range was arbitrarily set to ±50 %
for illustrative purposes (Sect. 4).
Full sensitivity analysis for each of the four data sets. Each
graph shows the full range of calculated liquid water content (Sw)
values of every combination of minimum, best estimate and maximum for each
of the parameters.
Despite our consideration of extreme potential error bounds, calculated
uncertainties in liquid water contents are restricted to a relatively small
range (∼ 20 % for large assumed uncertainty in the zeta
potential, and ∼ 3–4 % otherwise) at both Rhone Glacier
and Jungfraujoch Glacier, and absolute values remain within the pendular
regime where water bodies in the pore space remain isolated. At the latter
site the daily evolution of liquid water contents thus is well captured even
if uncertainty is taken into account (Fig. 5b), and likewise at Rhone
Glacier, calculated liquid water contents plus uncertainties still fall
within the range of field measurements (Fig. 5a). Our inferences thus not
only support the notion of Kulessa et al. (2012) that existing snow hydrological
relationships are robust for modelling purposes, but also suggest that they
may apply to in situ field surveys. These inferences can also provide an
explanation for the relatively large self-potential magnitudes generated by
relatively low bulk discharge at Jungfraujoch Glacier (Fig. 2). Because we
did not observe or infer any consistent or statistically significant
differences between Rhone Glacier and Jungfraujoch Glacier in dielectric
permittivity (ε), zeta potential (ζ), saturation
(SwSe-n), electrical conductivity (σw) or
cross-sectional area (A), the only remaining parameter that could facilitate
the observed relative difference is permeability (k). Indeed, using an
average snow density of 564 kg m-3, the differences in mean snow grain
sizes between Rhone Glacier (1.5 × 10-3 m) and Jungfraujoch
Glacier (1 × 10-3 m) translate into respective permeabilities
of 9.7 × 10-5 and 4.3 × 10-5 m2.
The relatively reduced permeability of Jungfraujoch Glacier's
accumulation-area snowpack therefore likely supported the presence of
self-potential magnitudes that were markedly elevated relative to Rhone
Glacier's ablation-area snowpack (Eq. 4). This inference emphasises the
sensitivity of the self-potential method to permeability as a fundamental
snow hydrological property, along with its observed sensitivity to bulk meltwater discharge and inferred sensitivity to liquid water content.
Synthesis and conclusions
The ability of the electrical self-potential method to sense meltwater flow
in in situ snowpacks is unique, where self-potential magnitudes scale
directly with discharge and are zero in the absence of flow. The scaling
factor (right side of Eq. 4) depends principally on the liquid water
content of the snowpack, its permeability and the water chemistry (Kulessa
et al., 2012). We have shown here that diurnal variations in the liquid
water content of in situ snowpacks can be derived from electrical
self-potential data and bulk discharge measurements with a simple lysimeter.
This derivation was subject to four key assumptions (Sect. 4) which we now
examine in turn to identify what, if any, constraints arise for future
applications.
The Reynolds number (Re) is a common measure of the mode of fluid flow through
porous media, as discussed in a relevant cryospheric context by Kulessa et
al. (2003a):
Re=ρsvLη,
where v and L are respectively characteristic fluid flow velocity (in m s-1)
and characteristic length scale of flow (in m), and ρs
and η are respectively snow density (in kg m-3) and dynamic
viscosity (in Pa s). To a first approximation the transition from laminar to
turbulent flow nominally occurs when Re≈ 10, although laminar flow
can persist at much higher values of Re (for comparison, in open channels
transition occurs at Re≈ 2300). For our purposes v can be assumed to
correspond to the average linear velocity of flow, v=QA-1n-1,
where n is effective porosity (ratio of snow and ice densities). In porous
media such as snow, L corresponds to the average pore diameter, and in the
absence of direct evidence is assumed to be equal to grain size. Where snow
is denser than ∼ 490 kg m-3, such as that at our study
sites (average ∼ 564 kg m-3), grain size is expected to
be larger than pore diameter (Schneebeli and Sokratov, 2004). This
assumption is therefore likely an overestimation of pore diameter. For the
respective snow properties and their uncertainties reported in Table 1,
values of Re between < 1 and 51 are obtained, with a best estimate
of Re≈ 1.1. These values pertain to times of highest measured
meltwater discharge when the Reynolds number is likely to be at its greatest. Despite
the unrealistically large uncertainty bounds considered in Table 1, and the
overestimation of pore diameter (L) and associated inflation of the Reynolds
number (Eq. 9), we can therefore conclude that meltwater flow in our
snowpacks was laminar. The absolute and relative inclinations of the snow
surface and base will vary to different degrees within different field
areas, thus generating differences in discharge and potentially preferential
flow. Indeed, it is an exciting attribute of self-potential measurements
that they will, in practice, aid in delineating such differences in meltwater
flow.
Persistent meltwater runoff at the snow surface is uncommon, and meltwater
flow through underlying soils or ice will normally be negligible or small
compared to flow through or at the base of snowpacks. We have also shown
that the estimation of snow properties, such as liquid water content, from
self-potential data is insensitive to the area of snowpack contributing
meltwater flow to the measured signals. Uncertainties in the area of origin
of water contributing to measured bulk discharges and thus measured
self-potential data are not therefore expected to be a major hindrance to
future applications of the self-potential method to snow problems. We have
also shown that with the exception of the zeta potential, sensitivity to
uncertainties in the snow properties governing the relationship between
self-potential data and liquid water contents is small (∼ 3–4 %
in our feasibility study). Future work must ascertain to what extent
longer term monitoring studies are affected by the preferential elution of
ions and the associated impacts on meltwater pH, electrical conductivity and
thus the zeta potential. Even if such effects were found to be of concern,
meltwater electrical conductivity and pH are readily monitored in situ with
automated probes and could be measured alongside self-potential data at a
calibration location, and subsequently be assimilated in snow models. Being
able to characterise liquid water content over significant spatial areas is limited to the
spatial distribution and density of possible electrode placement. However,
the robustness of the estimation means that in practice, self-potential measurements at
several points within the area of interest can in the future make reliable
interpolations between measurements in space and time.
The final consideration focused on the assumption that the spatial pattern
of self-potential magnitudes, measured during the day across our survey
areas, was due to temporal changes in the liquid water content of the
snowpack. This assumes that any spatial pattern due to elevation changes
between the bottom and top of our survey areas is comparatively small and
indeed negligible. Kulessa et al. (2003a) showed that elevation-driven
changes in the self-potential fields measured between upstream (Ψup) and downstream (Ψdown) locations (zup,
zdown) can be approximated by
ψup-ψdown=-εζησwSwzup-zdown,
here translated to our notation and adjusted for meltwater saturation
according to Eq. (2). Even for the maximum daily values of saturation
inferred from our measurements, the elevation-driven spatial pattern has
small magnitudes, estimated to be ∼-16.0 and -8.4 mV
respectively for Jungfraujoch Glacier and Rhone Glaciers. These values are
an order of magnitude smaller than daily changes measured at the two
glaciers (Fig. 2) and are therefore considered to be insignificant for the
purpose of the present feasibility study. In similar future applications the
relevance of such spatial changes should be assessed on a case by case
basis, and would in fact be readily incorporated into quantitative
inferences of snow properties from self-potential data where they are of
concern.
Overall our findings imply that in principle, self-potential data could be
inverted for spatial or temporal variations in any one desired parameter
(i.e. discharge, liquid water content, permeability or water chemistry), if
independent estimates of the respective remaining parameters are available.
Self-potential data are therefore well suited for assimilation in snow
models along with meteorological and snowpack observations. We have shown in
previous cryospheric applications that self-potential monitoring is readily
effected with autonomous arrays of low-cost non-polarising electrodes
connected to a high-impedance data logger (Kulessa et al., 2003a, b,
2012). In operational practice for instance, 2-D vertical arrays of
electrodes and data loggers could be installed along with meteorological
stations and upward-looking radar instrumentation, where the latter is used
to monitor snow structure and 1-D liquid water contents. Assimilation of
self-potential data along with complementary meteorological and radar data
could then facilitate unique insights into daily and longer term variations
in 2-D vertical, lateral and preferential meltwater flows, or in liquid
water contents. We conclude that the integration of self-potential
measurements into existing snow measurement and data assimilation routines
shows considerable promise in supporting a reduction of uncertainty in
quantifying snow–atmosphere energy exchanges, or in predictive modelling
used in operational snow forecasting.
Acknowledgements
This work was carried out while S. S. Thompson was working at VAW ETH Zurich within the
Swiss National Science Foundation project: Accelerated release of persistent
organic pollutants (POPs) from Alpine glaciers, research grant
200021_130083/1 BAFU, with support of the Swiss Federal
Office for the Environment (FOEN/BAFU). We would like to thank two anonymous
reviewers whose comments improved clarity of the manuscript and Martin Funk
and VAW for hosting and supporting the work and for extensive support for
fieldwork. Also thanks to Fabian Wolfsperger at WSL Institute for Snow and
Avalanche Research SLF and Ludovic Baron at UNIL Université de Lausanne
for providing equipment for fieldwork. Thanks to Jordan Mertes, Celia Lucas,
Saskia Grindreaux, Barbara Reyes-Trüssel and Moira Thompson for
invaluable help in the field.
Edited by: O. Eisen
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