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**The Cryosphere**
An interactive open-access journal of the European Geosciences Union

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**Research article**
15 Jan 2020

**Research article** | 15 Jan 2020

Estimation of subsurface porosities and thermal conductivities of polygonal tundra by coupled inversion of electrical resistivity, temperature, and moisture content data

^{1}Earth and Environmental Sciences Division, Los Alamos National Laboratory, Los Alamos, New Mexico, USA^{2}Climate Change Science Institute and Environmental Sciences, Oak Ridge National Laboratory, Oak Ridge, Tennessee, USA^{3}Climate and Ecosystem Division, Lawrence Berkeley National Laboratory, Berkeley, California, USA^{4}Department of Water Research Engineering and Technology, Water Research Institute, Hanoi, Vietnam

^{1}Earth and Environmental Sciences Division, Los Alamos National Laboratory, Los Alamos, New Mexico, USA^{2}Climate Change Science Institute and Environmental Sciences, Oak Ridge National Laboratory, Oak Ridge, Tennessee, USA^{3}Climate and Ecosystem Division, Lawrence Berkeley National Laboratory, Berkeley, California, USA^{4}Department of Water Research Engineering and Technology, Water Research Institute, Hanoi, Vietnam

**Correspondence**: Elchin E. Jafarov (elchin@lanl.gov)

**Correspondence**: Elchin E. Jafarov (elchin@lanl.gov)

Abstract

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Studies indicate greenhouse gas emissions following permafrost thaw will amplify current rates of atmospheric warming, a process referred to as the permafrost carbon feedback. However, large uncertainties exist regarding the timing and magnitude of the permafrost carbon feedback, in part due to uncertainties associated with subsurface permafrost parameterization and structure. Development of robust parameter estimation methods for permafrost-rich soils is becoming urgent under accelerated warming of the Arctic. Improved parameterization of the subsurface properties in land system models would lead to improved predictions and a reduction of modeling uncertainty. In this work we set the groundwork for future parameter estimation (PE) studies by developing and evaluating a joint PE algorithm that estimates soil porosities and thermal conductivities from time series of soil temperature and moisture measurements and discrete in-time electrical resistivity measurements. The algorithm utilizes the Model-Independent Parameter Estimation and Uncertainty Analysis toolbox and coupled hydrological–thermal–geophysical modeling. We test the PE algorithm against synthetic data, providing a proof of concept for the approach. We use specified subsurface porosities and thermal conductivities and coupled models to set up a synthetic state, perturb the parameters, and then verify that our PE method is able to recover the parameters and synthetic state. To evaluate the accuracy and robustness of the approach we perform multiple tests for a perturbed set of initial starting parameter combinations. In addition, we varied types and quantities of data to better understand the optimal dataset needed to improve the PE method. The results of the PE tests suggest that using multiple types of data improve the overall robustness of the method. Our numerical experiments indicate that special care needs to be taken during the field experiment setup so that (1) the vertical distance between adjacent measurement sensors allows the signal variability in space to be resolved and (2) the longer time interval between resistivity snapshots allows signal variability in time to be resolved.

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Jafarov, E. E., Harp, D. R., Coon, E. T., Dafflon, B., Tran, A. P., Atchley, A. L., Lin, Y., and Wilson, C. J.: Estimation of subsurface porosities and thermal conductivities of polygonal tundra by coupled inversion of electrical resistivity, temperature, and moisture content data, The Cryosphere, 14, 77–91, https://doi.org/10.5194/tc-14-77-2020, 2020.

1 Introduction

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Subsurface soil property parameterization contributes to a wide uncertainty range in projected active layer depth and in simulated permafrost distribution in the Northern Hemisphere when predicted using land system models (Koven et al., 2015; Harp et al., 2016). Reduction of this uncertainty is becoming urgent with recent accelerated thawing of permafrost (Biskaborn et al., 2019). Warming permafrost leads to increased infrastructure maintenance costs (Hjort et al., 2018), has a positive feedback on global climate change (McGuire et al., 2018), and increases the probability of the potential hazards for human health (Schuster et al., 2018). Better subsurface soil property parameterizations in land system models require the development of methods that can robustly estimate these soil properties including porosity and thermal conductivity of the peat and mineral layers.

Direct measurements of subsurface soil properties are labor intensive, destructive, and not always feasible (Smith and Tice, 1988; Kern, 1994; Boike and Roth, 1997; Yoshikawa et al., 2004). While soil sample analysis can provide critical information on soil properties at a fine scale, this information is limited to sparsely sampled locations. Multiple methods used in the laboratory to measure soil properties by using soil cores extracted from the field site are well summarized by Nicolsky et al. (2009), but the logistical and economic burden typically do not allow these measurements to be made in the field. Inverse modeling serves as an alternative approach to recover soil properties using a combination of indirect and direct measurements and physics-based numerical models.

Different inverse modeling frameworks have been developed to estimate soil thermal properties using physics-based models and time series of ground temperature data. Some earlier studies used heat equation models without phase change (Beck et al., 1985; Alifanov et al., 1996). More recent works include phase change, which is an important component of the energy balance in permafrost-affected soils (e.g., Nicolsky et al., 2007, 2009; Tran et al., 2017). Nicolsky et al. (2007, 2009) used an optimization-based inverse method and a variational data assimilation method to estimate soil properties. In particular, Nicolsky et al. (2007, 2009) used measured subsurface temperatures to inversely estimate thermal conductivities, porosities, freezing point temperatures, and unfrozen water coefficients, pointing out that sensitivity analyses (i.e., perturbation of the parameter values) are required in order to robustly establish a set of estimated parameters. Harp et al. (2016) used an ensemble-based method to evaluate the uncertainty of projections of permafrost conditions in a warming climate due to uncertainty in subsurface properties. Atchley et al. (2015) used data calibration to estimate hydrothermal properties of soils. All these methods used ground temperatures alone to estimate soil properties and 1-D soil columns assuming a 1-D soil structure.

Recently, Tran et al. (2017) used a coupled hydrological–thermal–geophysical modeling approach to estimate soil organic content. The approach was based on coupling the 1-D Community Land Model (CLM4.5; Oleson et al., 2013) that simulates surface–subsurface water, heat and energy exchange, and the 2-D Boundless Electrical Resistivity Tomography (BERT) forward model (Rücker et al., 2006). The simulated 1-D snapshots of the subsurface temperature, liquid water, and ice content from the CLM model were explicitly linked to soil electrical resistivities via petrophysical relationships which were then used as input to BERT's forward model to calculate apparent resistivities. Their inverse modeling framework aims to minimize the misfit between calculated and measured data, including soil temperature, liquid water content, and apparent resistivity. Here we modify and extend this approach to 2-D by using the Advanced Terrestrial Simulator (ATS) model, which was specifically developed to study fine-scale hydrothermal processes of permafrost-affected soils. In addition, instead of estimating organic content of the soil as in Tran et al. (2017), we estimate porosities and thermal conductivities of the peat (organic) and mineral layers across a 2-D transect within polygonal tundra.

Modeling the full, continuous 2-D transect allows us to simulate lateral hydrothermal fluxes not possible with individual 1-D columns known to be important in polygonal tundra (Abolt et al., 2018; Liljedahl et al., 2016). At each grid cell in the transect, a physical state develops during the ATS simulation (temperature, saturation, etc.) that is then used to calculate heterogeneous electrical resistivities via petrophysical relations. This allows more realistic simulated apparent resistivities that include the effects of lateral hydrothermal connectivity within the transect.

Through this approach, we develop a parameter estimation (PE) algorithm that aims to estimate porosities and thermal conductivities in permafrost-affected soils through joint inversion of hydrothermal and geophysical measurements, including ground temperature, saturation, and apparent resistivity. Our main objective then is to evaluate which types and number of measurements are necessary to constrain the inversion to yield a robust and accurate prediction of subsurface porosities and thermal conductivities. The inverse modeling framework couples the state-of-the-art hydrothermal permafrost simulator ATS, electrical resistivity software package BERT, and the Model-Independent Parameter Estimation and Uncertainty Analysis toolbox (PEST) software package (Doherty, 2001). We progressively test the accuracy and robustness of the method using a series of synthetic problems by (1) increasing the complexity of the meteorological data used to drive the coupled thermal–hydrological–geophysical model and (2) testing the inclusion of individual and combinations of several available measurement types on the accuracy and robustness of inversions. The results of this work can be used to better understand challenges associated with subsurface porosity and thermal conductivity estimation. Additionally, we used findings from this study to suggest how data should be collected to improve the accuracy of the estimated soil properties and to optimize the total number of measurements needed to make a robust subsurface PE.

2 Methods

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We estimate the soil properties of porosity and soil grain thermal conductivity for peat and mineral layers of a 2-D transect within polygonal tundra. Our PE approach is summarized in Fig. 1. Given specified “true” values of these parameters, we used the ATS version 0.86 model to solve for a transient, spatially distributed hydrothermal state characterized by temperature and liquid and ice saturations. ATS is a 3-D-capable coupled surface and groundwater flow and heat transport model representing the soil physics needed to capture permafrost dynamics, including flow of unfrozen water in variably saturated, partially frozen, nonhomogeneous soils (Painter et al., 2016). Given this hydrothermal state, we calculate resistivity values at every grid cell via petrophysical relationships and run the forward modeling component of the BERT software package (Rücker et al., 2006) to simulate resistance and related apparent resistivity values that would be measured with ground-coupled electrodes and an electrical resistivity tomography (ERT) acquisition system.

To set up the synthetic model, we used digital elevation data of a transect through ice-wedge polygonal tundra at the Barrow Environmental Observatory (BEO) at Utqiaġvik, Alaska (Fig. 2). Our study includes an 11 m section covering a single polygon with an ice wedge on each side. In this study we do not explicitly assign ice properties for the ice wedges. Instead, we model bulk porosities and effective thermal conductivities that can be associated with peat and mineral layers of the entire transect.

In Fig. 2a, we present the computational mesh representing the
cross section of the polygonal tundra that ATS is run on. The thickness of
the peat layer corresponds to observations at the site, with a thick peat
layer on the sides (troughs) and a thinner layer in the middle of the
low-centered polygon. A mineral layer was assigned below the peat layer
across the transect. We initially designated six synthetic direct
temperature and soil moisture measurement locations within the active layer
area, the maximum thaw layer from the ground surface to the top of the
permafrost, similar to the sensor setup at the site (Dafflon et al., 2017).
The average active layer depth is about 38 cm, as it can be seen from the
ground temperatures simulated for the synthetic model run with actual
meteorological data in Fig. 2b. The linear white region in Fig. 2b
indicates the bottom of the active layer within the transect (0 ^{∘}C).
Then we added four more synthetic direct measurement locations below the
active layer to evaluate the effect of their inclusion on PE accuracy and
robustness. All observation locations are represented as stars in Fig. 2a
corresponding to the locations of the collected daily averaged temperature
and soil moisture time series. The temperature and soil moisture time series
were recorded at depths of 5, 20, 60, and 80 cm below the surface.

The setup of the ATS model followed a standard procedure described in
several studies (Atchley et al., 2015; Painter et al., 2016; Jafarov et al.,
2018). Typically, we set up the model in several steps: (1) initialization of
the water table, (2) introduction of the energy equation to establish
antecedent permafrost, and (3) spin-up of the model with simplified and actual
meteorological data from the BEO station. We spun up the model until the
active layer achieved cyclical equilibrium. The overall depth of the
modeling domain is 50 m. We set the bottom boundary to a constant
temperature of *T*=263.55 K and set zero heat and zero mass flux boundary
conditions on the vertical sides. A seepage face was imposed at 4 cm below
the surface on each side of the domain to allow drainage to the trough
network to prevent water from pooling at the surface, as is typical of
partially degraded polygonal ground (Liljedahl et al., 2016). We use two
types of meteorological datasets as surface boundary condition drivers for
the ATS model: simplified (sinusoidal air temperature, constant
precipitation, and constant radiative forcing) and actual weather data from
the BEO site. The actual meteorological data were collected starting on
1 January 2015 and include air temperature, rain and snow precipitation,
humidity, long- and shortwave radiation, and wind speed.

We created a synthetic “truth” by designating porosities and soil grain
thermal conductivities {*ϕ*,*k*} of peat and mineral soil
as parameters in the forward model. The resulting temperature (*T*), liquid
and ice saturations (*s*_{l}, *s*_{i}), and apparent resistivities (*ρ*_{a}) were collected as the true state. Critical for these
simulations is the calculation of the thermal conductivities of the bulk
soil; calculated in ATS using Kersten numbers to interpolate between
saturated frozen, saturated unfrozen, and fully dry states (Painter et al.,
2016) where the thermal conductivities of each end-member state are
determined by the thermal conductivity of the components (soil grains, air,
water, or liquid) weighted by the relative abundance of each component in
the cell (Johansen, 1977; Peters-Lidard et al., 1998; Atchley et al., 2015).
Thermal conductivities of water, ice, and air are considered constant,
leaving soil grain thermal conductivity as the remaining parameter to be
estimated. The equation to calculate saturated, frozen thermal conductivity
(*κ*_{sat,f}) has the following form:

$$\begin{array}{}\text{(1)}& {\mathit{\kappa}}_{\text{sat,f}}={\mathit{\kappa}}_{\text{sat,uf}}\cdot {\mathit{\kappa}}_{\text{i}}^{\mathit{\varphi}}\cdot {\mathit{\kappa}}_{\text{w}}^{-\mathit{\varphi}},\end{array}$$

where *κ*_{sat,uf}, *κ*_{i}, *κ*_{w} are thermal conductivities for saturated unfrozen, ice, and liquid water, respectively; and *ϕ* is porosity.

The freezing characteristic curve is thermodynamically derived using a Clapeyron relation and the unfrozen water retention curve, as described in Painter and Karra (2014) and Painter et al. (2016). In Fig. 3 we present liquid and ice saturations for one realization of the model for winter (January) and summer (August) times of the year. The ice saturation is high below the active layer all year long and lowest within the active layer in the summer. The peat layer holds more water; therefore, ice concentration is higher than in the mineral layer in the winter. The liquid saturation plot shows that, by the end of the summer, the peat layer is drier than the mineral layer.

We sequentially couple the ATS and BERT numerical models via petrophysical
relationships used by Tran et al. (2017) and based on Archie (1942) and
Minsley et al. (2015). In that approach, the electrical resistivity (*ρ*) is determined as a function of soil characteristics, temperature,
porosity, liquid water saturation, fluid conductivity, and ice content:

$$\begin{array}{}\text{(2)}& \mathit{\rho}=\mathrm{1}/\left({\mathit{\varphi}}^{d}\left[{s}_{\text{l}}^{n}{\mathit{\sigma}}_{\text{w}}+\left({\mathit{\varphi}}^{-d}-\mathrm{1}\right){\mathit{\sigma}}_{\text{s}}\right]\cdot \left[\mathrm{1}+c\left(T-\mathrm{25}\right)\right]\right),\end{array}$$

where *σ*_{w} is the fluid electrical conductivity, *σ*_{s} is
soil/sediment electrical conduction, *n* is a saturation index, *d* is a
cementation index, and *c* is a temperature compensation factor accounting
for deviations from *T*=25 ^{∘}C.

The ice content is linked to water content through the liquid water
saturation and to *σ*_{w}, which is influenced by the concentration of Na^{+} and Cl^{−} ions and the ice–liquid fraction. Here *σ*_{w}
has the following form:

$$\begin{array}{}\text{(3)}& {\mathit{\sigma}}_{\text{w}}={\sum}_{i=\mathrm{1}}^{{n}_{\text{ion}}}{F}_{\text{c}}{\mathit{\beta}}_{\text{i}}\left|{z}_{\text{i}}\right|{C}_{i\left({S}_{{f}_{\text{i}}=\mathrm{0}}\right)}{S}_{{f}_{\text{w}}}^{-\mathit{\alpha}},\end{array}$$

where *F*_{c} is Faraday's constant, *β*_{i} and *z*_{i} are the ionic mobility and valence respectively, *C*_{i} is the concentration of the *i*th ion, and *α* is the factor influencing how the liquid water salinity increases when the fractions of liquid in ice–liquid water ${S}_{{f}_{\text{w}}}$ decrease. ${S}_{{f}_{\text{w}}}$ is defined as

$$\begin{array}{}\text{(4)}& {S}_{{f}_{\text{w}}}={s}_{\text{l}}/({s}_{\text{l}}+{s}_{\text{i}}).\end{array}$$

Both *s*_{l} and *s*_{i} are simulated by ATS. Note that *ϕ* in Eq. (2) is an estimated parameter (see Fig. 1). In this study we assume that the *n*, *d*, *σ*_{s}, *α*, *F*_{c}, ${\mathit{\beta}}_{{\mathrm{Na}}^{+}}$, ${\mathit{\beta}}_{{\mathrm{Cl}}^{-}}$, ${C}_{{\mathrm{Na}}^{+}}$, and ${C}_{{\mathrm{Cl}}^{-}}$ parameters used in
Eqs. (2) and (3) are known (see Tran et al., 2017) and focus on the
robustness of the PE algorithm in estimating porosity and thermal
conductivity.

The 2-D resistivity data inferred from ATS simulations and petrophysical
relationships get passed to BERT, which simulates resistances that are then
converted to the apparent resistivities (*ρ*_{a}). The *ρ*_{a} values correspond to an acquisition along an 11 m long transect using a 0.5 m electrode spacing and a Schlumberger configuration with a total of 138 measurements (see Fig. 2b). This configuration implies that the measurements are mostly sensitive to the electrical resistivity in the top few meters.

Since BERT and ATS operate on different unstructured meshes, we wrote a
function that interpolates the values between the two meshes. Note that the
ATS mesh is 50 m deep. We calculate *ρ* by using corresponding outputs
from the ATS model and the petrophysical relationships, and we then interpolated these values on a mesh defined in BERT and adapted to the acquisition
geometry. BERT's mesh consists of a finely resolved mesh (11 m wide by 4.5 m deep) embedded within a coarser outer mesh that is about 120 m wide and 85 m
deep. We link hydrological variables with electrical resistivities in the
fine mesh. The coarse mesh is used to reduce the effect of boundaries. It
extends until the change in the electrical resistivity between two
neighboring cells is negligible.

To test if the known soil properties can be recovered by the PE approach, we start with randomly selected initial parameter guesses. We use a Latin hypercube sampling method to generate random initial guesses of porosity and thermal conductivities around the synthetic truth (McKay et al., 1979). Each parameter combination includes four parameters: porosity and thermal conductivity for peat and mineral soil layers. These parameters were chosen due to their strong controls on both hydrologic and thermal states (Atchley et al., 2015; Nicolsky et al., 2009). The rest of the hydrothermal properties are kept fixed.

The inverse approach involves the minimization of a cost function expressed as the sum of squared differences between simulated values and synthetic measurements using the Levenberg–Marquardt (LM) algorithm (Levenberg, 1944; Marquardt 1963) implemented in the PEST software package (Doherty, 2001), which was used to handle all parameter estimation runs.

To estimate soil porosities and thermal conductivities, we minimize the cost
function (*J*), which includes calculated and synthetic *T*, *s*_{l}, and *ρ*_{a} in the following form:

$$\begin{array}{}\text{(5)}& \begin{array}{rl}J(\mathit{\varphi},k)& ={w}_{\text{T}}\sum _{i}^{{n}_{\text{sens}}}\sum _{j}^{{n}_{\text{days}}}{\left({T}_{cj}^{i}-{T}_{sj}^{i}\right)}^{\mathrm{2}}\\ & +{w}_{\text{s}}\sum _{i}^{{n}_{\text{sens}}}\sum _{j}^{{n}_{\text{days}}}{\left({s}_{\text{l}}^{{i}_{cj}}-{s}_{\text{l}}^{{i}_{sj}}\right)}^{\mathrm{2}}\\ & +{w}_{{\mathit{\rho}}_{\text{a}}}\sum _{k}^{{n}_{\text{snap}}}\sum _{m}^{{n}_{\text{meas}}}{\left({\mathit{\rho}}_{\text{a}}^{{k}_{cm}}-{\mathit{\rho}}_{\text{a}}^{{k}_{sm}}\right)}^{\mathrm{2}},\end{array}\end{array}$$

where subscripts *c* and *s* correspond to calculated and synthetic states of
the system; and *w*_{T}, *w*_{s}, and ${w}_{{\mathit{\rho}}_{\text{a}}}$ are the corresponding weights for the temperature, saturation, and apparent resistivity residuals. *n*_{sens} is the number of sensors, *n*_{days} is the number of days over which we collected the data, *n*_{snap} is the number of *ρ*_{a} snapshots, and *n*_{meas} is the number of *ρ*_{a} measurements during one snapshot. *T*_{c} and ${s}_{{\text{l}}_{\text{c}}}$ are time series from multiple sensors
collected daily from the beginning of June until the end of September. *ρ*_{a} represents the apparent resistivity data snapshots taken at a certain day. The number of apparent resistivity snapshots depends on the particular case, varying from one to eight snapshots per year. The one-snapshot case
corresponds to only one snapshot in the month of August, while the
eight-snapshot case corresponds to a snapshot taken once per month from
January until September. In addition, we tested the case where we collected
eight daily *ρ*_{a} snapshots. This was done to compare how different
time spacing would affect the estimated properties.

The weights were chosen in order to scale the contribution of each type of
residual so that contributions to the cost function are evenly distributed
across temperature, saturation, and apparent resistivity residuals. For
example, saturation residuals are on the order of a few tenths, while
apparent resistivity residuals can be tens of ohmmeters. The weights were
selected based on evaluating the individual contributions to the cost
function for each measurement type on an ensemble of simulations spanning
the parameter ranges. The apparent resistivity residual weight (${w}_{{\mathit{\rho}}_{\text{a}}}$) was set to one. The temperature and saturation residual weights
(*w*_{T} and *w*_{s}) were then modified so that each measurement type
component in the cost function had roughly equivalent magnitude over most of
the parameter space. This resulted in weights of ${w}_{{\mathit{\rho}}_{\text{a}}}=\mathrm{1}$,
${w}_{\text{T}}=\sqrt{\mathrm{2.5}\times {\mathrm{10}}^{\mathrm{3}}}$, and ${w}_{\text{s}}=\sqrt{\mathrm{3.5}\times {\mathrm{10}}^{\mathrm{5}}}$.

If the cost function satisfies a minimum criterion or the maximum allowed number of iterations, which we chose to be equal to 25, is reached, the PE terminates. The porosities and thermal conductivities corresponding to the minimum of the cost function, i.e., the parameters associated with the best fit between simulated and synthetic values, are considered the estimated parameter values as

$$\begin{array}{}\text{(6)}& \left\{\mathit{\varphi},k\right\}=J(\mathit{\varphi},k).\end{array}$$

Here {*ϕ*,*k*} are the estimated porosities and thermal
conductivities for peat and mineral soil.

Based on sensitivity analyses using simplified meteorological data, the cost function response surface was smooth and convex over the parameter ranges of interest. Therefore, we chose the LM approach because of its robust gradient-based optimization scheme that takes advantage of smooth convex response surfaces to quickly converge to minima.

To build an understanding of the inverse framework, we start with a simple
setup and then gradually add more complexity. First, we use simplified
meteorological data where we assume that air temperatures change according
to a sinusoidal function and all other terms are constants. Initially we
start with three temperature and moisture content measurement locations within
the peat layer (refer to Fig. 2a) and one ERT data snapshot. Then we
increase the number of ERT data snapshots up to eight by adding snapshots once
per month from January until August. Each ERT data snapshot calculated by
BERT uses the set of daily averaged *T* and *s*_{l} simulated by ATS and
petrophysical relations (Eqs. 2 and 3) which are varying over time. Then we
increase the number of sensors to six and add noise to the simulated data.
Introduction of noise allows us to evaluate the effect of measurement
uncertainties that will be present in the actual application of the PE
method. We added different levels of Gaussian noise to the synthetic
measurements of *T*, *s*_{l}, and *ρ*_{a} in the following way: 1 %
to *T*, 5 % to *s*_{l}, and 10 % to *ρ*_{a}. These levels of noise
for the different types of measurements are based on published literature
and our own experience (Wang et al., 2018; Dafflon et al., 2017). After that
we substitute simplified meteorological data with actual data from the BEO
site to evaluate our PE method under realistic ground surface boundary
conditions. In this case we evaluate how much and what kind of data we need
to robustly recover subsurface porosities and thermal conductivities. To do
this we test the inclusion of individual types of measurements in the cost
function (Eq. 3) as well as all possible combinations of measurement
types. We used different soil property ranges for the simplified and actual
meteorological data PE runs which are summarized in Table 1. This was done
to ensure that PE is able to recover different sets of parameters and to
test the consistency and effectiveness of the PE method. Finally, we
compared the difference between estimated parameters for eight ERT data
snapshots collected once a month versus once a day for 8 d. The notation and
a description of each run for simplified and actual meteorological data are
summarized in Table 2.

3 Results

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To evaluate the PE method performance driven by simplified meteorological
data, we ran PE experiments using 30 different random combinations of
porosity and thermal conductivity values as the initial starting point. We
used 30 PE samples of {*ϕ*,*k*} starting points in the
first experiment ((S)3*T**3**s*_{l}*1**ρ*_{a}) to illustrate the overall
performance of the parameter estimation using a large number of samples.
After that, we did only five PE runs for the simplified meteorological data
and 10 for all other runs with actual meteorological data. For all figures
after Fig. 4, for consistency and clarity, we show results for only five
PE runs per case. It is important to note that the number of samples that
one needs to run to ensure the robust convergence of the estimated
parameters depends on the specifics of the corresponding case (i.e.,
experiment-specific). If most of the LM runs converge to the same set of
parameters and have low cost function values, then that set of runs most
likely corresponds to the actual {*ϕ*,*k*}. In Fig. 4,
the red triangles represent initial guesses (parameter combinations), and the
synthetic truth is indicated by the intersection of the two dotted lines.
Yellow lines connecting red triangles with white crosses represent the path
that the LM algorithm has taken from the initial guess to the estimated
parameter combination (white crosses, Fig. 4). The yellow dots along the
yellow lines indicate the location at each LM iteration.

Figure 4 indicates that the method is able to recover porosities more
robustly than thermal conductivities, i.e., estimated porosities are similar
to their true state. According to the liquid saturation plot in Fig. 3,
liquid saturation of the mineral layer is quite dynamic and more saturated
in comparison to the peat layer. Nevertheless, thermal conductivity of the
mineral layer (*k*_{m}) corresponds to the highest mismatch. Three out of 30 inversions corresponding to *k*_{m} end up close to 1.4 W m^{−1} K^{−1} (the true value is 2 W m^{−1} K^{−1}), suggesting those
values do not correspond to the truth, since most of the estimated values
(27 cases) are concentrated around the intersection of the dotted lines. The
response surface for the corresponding cost function (Eq. 5) lies hereby in
a flat, low-gradient region. The projections of the cost function response
surfaces corresponding to porosities (Fig. 4a) have a better defined
minimum, as opposed to projections of the cost function response surfaces
corresponding to thermal conductivities (Fig. 4b), indicating
the nonuniqueness of the estimated parameters. For this experiment, we used
time series of *T* and *s*_{l} only from the first three near-surface sensors
(Fig. 2a). All of these three sensors are located in the peat layer,
suggesting that using just near-surface sensors only from one upper layer
might not be enough to recover the deeper layer thermal conductivity.

To illustrate the effect of noise on the robustness of the estimated
parameters we used cases with six near-surface sensors (6*T* and *6**s*_{l}) and a varying number of ERT snapshots driven with simplified meteorological
data. Similarly to the (S)3*T*3*s*_{l}*1**ρ*_{a} case,
(S)6*T**6**s*_{l}*1**ρ*_{a} shows good convergence for porosities and poor
convergence for thermal conductivities with an averaged error of
0.1 W m^{−1} K^{−1} (Fig. 5a, b). Adding noise to the
(S)6*T**6**s*_{l}*1**ρ*_{a}+*n* case slightly worsens the estimated porosity values and significantly worsens *k*_{m} with root-mean-squared error (RMSE)
rising from 10 % to more than 50 % (Fig. 5c, d). Figure 5e and f show that increasing the number of ERT snapshots from one to eight per year (i.e., collected
once per month from January until September) improves *k*_{m} estimates,
allowing better convergence for four out of five samples to the synthetic
truth. If we compare all three cases in Fig. 5 on how well they are able
to estimate *k*_{m}, it is clear from Fig. 5d that for the case
(S)6*T**6**s*_{l}*1**ρ*_{a}+*n* none of the *k*_{m}'s were correctly estimated,
whereas significantly improved *k*_{m} values were found by increasing the
number of monthly ERT snapshots (Fig. 5f). Moreover, all except one
estimated value showed a better match with its true value than the
(S)6*T**6**s*_{l}*1**ρ*_{a} case without any added noise.

In Fig. 6, we summarize results of the five PE runs for each of the first
six cases corresponding to simplified meteorological data listed in Table 2.
The first three matrix tables correspond to the normalized RMSE values for
each measurement type (Δ*T*, Δ*s*_{l}, and Δ*ρ*_{a}).
The last two matrix tables correspond to the normalized Euclidian distances
between the synthetic truth and estimated parameter values of *δ**ϕ*
and *δ**k*. We normalized the values in each matrix by dividing by the
maximum value from the corresponding matrix. The normalized values are
marked with tildes and range from 0 to 1, where values closer to 0
correspond to a better match and values closer to 1 correspond to a worse
match. As shown above, the method is able to accurately estimate both peat
and mineral soil porosities as well as peat layer thermal conductivity
(*k*_{p}), but it cannot always accurately estimate *k*_{m}. There is not much difference between cases (S)3*T*3*s*_{l}*1**ρ*_{a} and (S)6*T**6**s*_{l}*1**ρ*_{a} except for a slight improvement in *k*_{m},
suggesting that the small vertical distance (10 cm) between sensors 1 and 4,
2 and 5, and 3 and 6 could limit the recorded data variability, leading
to difficulties in the estimation of the *k*_{m} parameter. Since all six sensors lie within the active layer, we added additional sensors below the
active layer in the later experiments (red stars in Fig. 2a). The *ϕ*
and *k* matrix tables show that increasing the number of monthly ERT
snapshots consistently improves the estimates of *ϕ* and *k*. This
suggests that increasing the number of monthly ERT snapshots can lead to
improved convexity of the cost function (Eq. 5).

After testing the PE method for the simplified meteorological data, we applied measured meteorological data from the BEO site for the year 2015. To better understand the importance of each measurement type and their combinations within the developed PE algorithm, we tested all of the scenarios corresponding to the “actual meteorological data” column from Table 2. The results of these runs are summarized in the colored matrix tables in Fig. 7. Since there are more than twice the number of actual meteorological cases than simplified meteorological cases, it is difficult to analyze all matrix tables at once.

To compare the match between all estimated and observational values within a single plot we calculated Euclidean norms for each case independently:

$$\begin{array}{}\text{(7)}& \begin{array}{rl}& \mathrm{\Delta}{\left(\stackrel{\mathrm{\u0303}}{\mathrm{\Delta}T},\stackrel{\mathrm{\u0303}}{\mathrm{\Delta}{s}_{\text{l}}},\stackrel{\mathrm{\u0303}}{\mathrm{\Delta}{\mathit{\rho}}_{\text{a}}}\right)}_{\text{i}}=\\ & \sqrt{{\left({\displaystyle \frac{\mathrm{\Delta}{T}_{\text{i}}}{\mathrm{\Delta}{T}_{\text{max}}}}\right)}^{\mathrm{2}}+{\left({\displaystyle \frac{\mathrm{\Delta}{s}_{{\text{l}}_{\text{i}}}}{\mathrm{\Delta}{s}_{{\text{l}}_{\text{max}}}}}\right)}^{\mathrm{2}}+{\left({\displaystyle \frac{\mathrm{\Delta}{\mathit{\rho}}_{{\text{a}}_{\text{i}}}}{\mathrm{\Delta}{\mathit{\rho}}_{\text{max}}}}\right)}^{\mathrm{2}}},\end{array}\end{array}$$

$$\begin{array}{}\text{(8)}& \mathrm{\Delta}{\left(\stackrel{\mathrm{\u0303}}{\mathit{\delta}\mathit{\varphi}},\stackrel{\mathrm{\u0303}}{\mathit{\delta}k}\right)}_{\text{i}}=\sqrt{{\left({\displaystyle \frac{\mathit{\delta}{\mathit{\varphi}}_{\text{i}}}{\mathit{\delta}{\mathit{\varphi}}_{\text{max}}}}\right)}^{\mathrm{2}}+{\left({\displaystyle \frac{\mathit{\delta}{k}_{\text{i}}}{\mathit{\delta}{k}_{\text{max}}}}\right)}^{\mathrm{2}}}.\end{array}$$

The index *i* indicates the case number (see Table 2). Then we applied
*k*-means clustering analysis to identify groups of cases with a similar match
between data and estimated parameters. We divided all cases into four
classes shown in Fig. 8. Class I indicates the best cases that provide an
accurate parameter estimation as well as accurate matches with the synthetic
true measurements. Class II includes the cases that have accurate
parameter estimates and less accurate matches with the measurements. Class
III indicates all cases that have less accurate parameter estimates but
accurate matches with the measurements. Finally, class IV includes the cases
that showed the worst performance in terms of parameter estimates and the
worst matches with the measurements. We summarized the results from Fig. 8
in Table 3.

Class I (see Table 3) suggests that sensors located below the active layer as
well as increasing the number of sensors lead to more accurate parameter
estimation. In contrast, case no. 4 (corresponding to the one ERT snapshot,
1*ρ*_{a}) suggests that already one ERT data snapshot could be enough
for parameter estimation, while class II indicates that in general an
increase in the numbers of monthly ERT snapshots is important for more
accurate PE. However, increasing the number of monthly ERT snapshots leads
to a less accurate match with measurements. These results are consistent
with the results for simplified meteorological data with added noise (Fig. 5).

Class III includes six cases suggesting that if we have only soil moisture
data available for PE, then we should expect less accurate soil property
estimates. The last element in this class suggests that taking daily ERT
snapshots improves the apparent resistivity match (Fig. 7, resistivity
table) but does not improve {*ϕ*,*k*} estimates, where
monthly ERT snapshots improve thermal conductivity convergence.

Class IV once again clearly indicates that measurements obtained below the
active layer provide more accurate parameter estimates; however, they do not
improve matches to measurements. This is mainly due to significant mismatch
with *ρ*_{a}, which can be seen from the $\stackrel{\mathrm{\u0303}}{\mathrm{\Delta}{\mathit{\rho}}_{\text{a}}}$ matrix table in Fig. 8. At the actual site, the depth to the mineral soil can be deeper than 20 cm; not having sensors lower than 20 cm
therefore limits the amount of data that can help to improve the convexity of the cost function in our case.

From Fig. 8 and Table 3 we know that the *6T* case has the worst performance
in terms of matching {*ϕ*,*k*}. Similar to the experiments
with simplified meteorological data, the main difficulty for experiments
with actual meteorological data is estimating thermal conductivity. The last
matrix table ($\stackrel{\mathrm{\u0303}}{\mathit{\delta}k}$) in Fig. 7 shows that
6*T**6**s*_{l}*8**ρ*_{a}(s) has the highest maximum and mean mismatch in thermal
conductivity estimates. However, since *ϕ* estimates are a better match
with their corresponding true values, the case 6*T**6**s*_{l}*8**ρ*_{a}(s) falls into class III in Fig. 8, as opposed to case 6*T*, which
falls into class IV. The highest mismatch in thermal conductivity values for
the 6*T**6**s*_{l}*8**ρ*_{a}(s) case suggests that collecting daily ERT
snapshots improves the *ρ*_{a} match (Fig. 7, $\stackrel{\mathrm{\u0303}}{\mathrm{\Delta}{\mathit{\rho}}_{\text{a}}}$ matrix table) but does not improve estimated parameters, where
monthly ERT snapshots improve thermal conductivity estimation.

To illustrate this, we plot values of estimated parameters and the
corresponding response surfaces of the cost function for cases
10*T**10**s*_{l}*8**ρ*_{a} and 6*T**6**s*_{l}*8**ρ*_{a}(s) in Fig. 9. The PE method was able to match four out of five estimates almost perfectly and missed the
*k*_{p} for the 10*T**10**s*_{l}*8**ρ*_{a} case. The corresponding cost
function has a visible minimum and clear convexity. In contrast to this, the
6*T**6**s*_{l}*8**ρ*_{a}(s) case completely missed two estimates by converging on
values outside the boundaries, and three other estimates do not converge to the
desired cross section as well. The contour lines suggest that the
corresponding response surfaces of the cost function do not have a
well-defined global minimum.

4 Discussion

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The existence of multiple minima is common in inverse modeling and can lead to a false convergence of the PE algorithm to physically nonrealistic subsurface parameters (Nicolsky et al., 2007). This is one of the main reasons for using multiple initial guesses. If most of the inversions converge to a similar set of parameter estimates with the lowest cost function value, then that set of values is most likely the global minimum. Testing the PE algorithm using multiple starting points is a commonly used approach in evaluating the robustness of an inverse model (e.g., Hansen, 1998).

A potential strategy to improve the developed PE algorithms is to reduce the specified convergence tolerance value (i.e., minimum condition; see Fig. 1) or increase the allowable number of iterations. However, this could lead to a significant increase in computational effort. In addition, PEST provides multiple additional settings of inversion parameters to achieve a better convergence. Parameter regularization is one of them. Regularization techniques have been widely used in solving ill-posed inverse problems (Vogel, 2002). The overall idea is to constrain the objective function by imposing additional priors on the estimated model parameters. We recognize that including parameter regularization into the cost function may improve the robustness of our method. However, inclusion of the regularization would require an extensive exploration of the multiple regularization methods and values that could be applied to it, which is beyond the scope of this paper. Here we illustrate that without using regularization it is possible to achieve reasonable results by using the simple weighted cost function.

The good performance of the case with only one ERT snapshot (*1**ρ*_{a})
could be misleading due to the design of this numerical experiment; i.e., we
are using a synthetic truth produced by the same model used in the
inversion, which improves the convexity of the cost function and leads to a
well-constrained unique minimum. However, in reality, the collection of
additional information, such as organic layer thickness and temperature
data, is extremely important and is required for model calibration
(Jafarov et al., 2012; Atchley et al., 2015). In addition, real ERT surveys
can be perturbed by noise, and their interpretation may require site-specific
petrophysical relationships as opposed to the general petrophysical
relationships used in this study. Therefore, we do not suggest an inversion
based on one ERT snapshot without any additional data.

The 6*T**6**s*_{l}*8**ρ*_{a}(s) case, where all runs converge to different
values of *k*_{m}, indicates that using certain combinations of datasets
does not allow the inverse approach to properly recover *k*_{m}. It is likely that 6*T**6**s*_{l}*8**ρ*_{a}(s) does not capture much variability in soil
temperatures and soil moisture, and therefore ERT snapshots do not have much
variability as well. Once the cost function converges for one of the ERT
snapshots, it immediately converges on the other daily snapshots due to
their similarity. In fact, although 6*T**6**s*_{l}*8**ρ*_{a}(s) has a good
accuracy with observations (see the $\stackrel{\mathrm{\u0303}}{\mathrm{\Delta}{\mathit{\rho}}_{\text{a}}}$ matrix table in Fig. 7), it is unable to recover the value of *k*_{m}.

We have shown that, even in the ideal situation where we either generate observational data or use simplified meteorological data, we cannot always fit modeling results to observations. In reality, noise (e.g., the sensor's measuring resolution) influences the collected data. To investigate the impact of measurement noise, we introduced multiple levels of noise to the simplified meteorological data. The resulting PE showed that dealing with noisy data could be challenging (Fig. 5). However, our results showed that adding ERT snapshots taken monthly into the cost function improves the overall PE accuracy.

The distance between sensors could be another source of uncertainty in the
PE. As pointed out by Nicolsky et al. (2009), it is important to make sure
that a vertical difference between adjacent measurements does not introduce
additional noise that can mislead the minimization algorithm without
providing new information. If sensors are close to each other, measurements
might be the same or within the noise variability. In our setup the vertical
distance between the first two rows of sensors is about 15 cm. This could
lead to small temperature variability between sensors. Indeed, providing
greater vertical distance between sensors improved the PE accuracy. Case
no. 12 (6*T**6**s*_{l}1*ρ*_{a}) clearly illustrates this point – that by increasing the vertical distance between sensors we can improve estimated parameter accuracy.

Combining hydrothermal observations from multiple depths with monthly ERT measurements resulted in an improvement of the shape of the cost function and led to better defined minima (Fig. 9). Increasing the number of the monthly ERT snapshots improved the accuracy of the estimated parameters. In addition, we showed that having sensors below the active layer combined with ERT snapshots shows the best accuracies both in terms of estimating parameters and matching observations.

5 Conclusion

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The overarching goal of this study was to develop and validate a parameter
estimation algorithm using a synthetic setup and a 2-D coupled
thermal–hydrological–geophysical model based on a polygonal tundra site within the
Barrow Environmental Observatory. Combining hydrothermal observations from
multiple depths with monthly ERT measurements resulted in an improved shape
of the cost function and led to better defined minima and improved accuracy
of the estimated parameters. This was presented in fitness matrices for six
cases using simplified meteorological data. Similar conclusions were found
for inversion runs with actual meteorological data. It is important to note
that it was not only the number of ERT data snapshots that improved the
robustness of the PE method but rather the time frequency of the ERT data
snapshots, i.e., monthly vs. daily snapshots. In addition, collecting data
from several soil layers might improve the thermal conductivity estimates
for the corresponding soil layer. Our experiments show that robust PE can be
achieved not just by adding more sensors into the ground and increasing
the number of ERT snapshots but also by optimally distributing those sensors
within the transect (e.g., the 6*T**6**s*_{l}1*ρ*_{a} case). Overall, the
inversion runs that we investigated consistently indicated that collecting
data from multiple soils layers, providing enough vertical separation
between sensors, and collecting temporally diverse ERT data should lead to
robust parameter estimation. The exception from this conclusion is the case
*1**ρ*_{a}, which showed robust parameter estimation due to specifics of
the model setup. As discussed above, estimating porosities and thermal
conductivities based on one ERT snapshot would not be possible without
additional information on the subsurface properties.

This work developed and demonstrated the feasibility of a PE algorithm that can be used to better inform large-scale land system model subsurface parameterization. Here we demonstrated the proof of concept of the PE method. Further improvements such as the introduction of a PE regularization parameter into the cost function and leveraging additional PEST capabilities could improve method robustness. Finally, the PE method must still be tested using measured thermal–hydrological–geophysical data from the BEO site.

Data availability

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Data availability.

Forcing data used in these simulations can be downloaded from https://github.com/amanzi/ats-testsuite-arctic/tree/master/testing/Data/ (last access: 3 January 2020).

Author contributions

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Author contributions.

Numerical simulations were performed by EEJ and DRH with guidance from BD and ETC. BD and APT provided advice and guidance with ERT model and related coupling. EEJ and DRH contributed to the overall design of the research with guidance from CJW. ALA assisted with the writing process and references. YL provided expert knowledge on inverse modeling.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements

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Acknowledgements.

The authors thank Vladimir Romanovsky, Joel Rowland, and Anna Liljedahl for preparing meteorological data used in this study.

Financial support

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Financial support.

This research has been supported by the Office of Biological and Environmental Research in the DOE Office of Science (NGEE Arctic).

Review statement

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Review statement.

This paper was edited by Christian Hauck and reviewed by Christian Hauck and two anonymous referees.

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Short summary

Improved subsurface parameterization and benchmarking data are needed to reduce current uncertainty in predicting permafrost response to a warming climate. We developed a subsurface parameter estimation framework that can be used to estimate soil properties where subsurface data are available. We utilize diverse geophysical datasets such as electrical resistance data, soil moisture data, and soil temperature data to recover soil porosity and soil thermal conductivity.

Improved subsurface parameterization and benchmarking data are needed to reduce current...

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