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A reduced order modeling approach to represent subgrid-scale hydrological dynamics for regional- and climate-scale land-surface simulations: application in a polygonal tundra landscape

01 Apr 2014-Vol. 7, Iss: 2, pp 2125-2172
TL;DR: A particular reduced-order modeling technique known as "Proper Orthogonal Decomposition mapping method" that reconstructs temporally-resolved fine- resolution solutions based on coarse-resolution solutions that produced a significant computational speedup and demonstrated that it can be used for polygonal tundra sites not included in the training dataset with relatively good accuracy.
Abstract: Existing land surface models (LSMs) describe physical and biological processes that occur over a wide range of spatial and temporal scales. For example, biogeochemical and hydrological processes responsible for carbon (CO2, CH4) exchanges with the atmosphere range from molecular scale (pore-scale O2 consumption) to tens of kilometer scale (vegetation distribution, river networks). Additionally, many processes within LSMs are nonlinearly coupled (e.g., methane production and soil moisture dynamics), and therefore simple linear upscaling techniques can result in large prediction error. In this paper we applied a particular reduced-order modeling (ROM) technique known as "Proper Orthogonal Decomposition mapping method" that reconstructs temporally-resolved fine-resolution solutions based on coarse-resolution solutions. We applied this technique to four study sites in a polygonal tundra landscape near Barrow, Alaska. Coupled surface-subsurface isothermal simulations were performed for summer months (June–September) at fine (0.25 m) and coarse (8 m) horizontal resolutions. We used simulation results from three summer seasons (1998–2000) to build ROMs of the 4-D soil moisture field for the four study sites individually (single-site) and aggregated (multi-site). The results indicate that the ROM produced a significant computational speedup (> 103) with very small relative approximation error (more » ROM. We also demonstrated that our approach: (1) efficiently corrects for coarse-resolution model bias and (2) can be used for polygonal tundra sites not included in the training dataset with relatively good accuracy ( « less

Summary (4 min read)

1 Introduction

  • The terrestrial hydrological cycle strongly impacts, and is impacted by, atmospheric processes.
  • The methods to represent spatial heterogeneity in hydrological and biogeochemical dynamics differ between watershed and regional or climate-scale models.
  • A second potential approach to account for spatial heterogeneity in soil moisture states is to relate its higher-order moments to the mean, and then apply these relationships within a model that predicts the transient coarse-resolution mean.
  • The approaches described above to capture fine-resolution spatial heterogeneity within a coarse-resolution modeling framework have some limitations.
  • In Sect. 3, these methods are used under different scenarios to develop ROMs for the polygonal tundra site that increase in generality in the following order: single-site ROMs (limited to a single site), multi-site ROMs (limited to sites included in the training data) and site-independent ROMs (applicable even for sites not included in the training data).

2.1 Site description and hydrologic simulation setup

  • The authors developed ROMs for hydrological simulations performed at four sites in the Barrow Environmental Observatory (BEO) in Barrow, Alaska (71.3◦ N, 156.5◦ W).
  • The Department of Energy (DOE) NextGeneration Ecosystem Experiments (NGEE-Arctic) project has established four intensely monitored sites (A, B, C and D, shown in Fig. 1) within the BEO in 2012 to study the impact of climate change in high-latitude regions.
  • The authors applied a version of the three-dimensional subsurface reactive transport simulator PFLOTRAN, which was modified to include surface water flows, for simulating surface– subsurface hydrologic processes at the four NGEE-Arctic study sites.
  • The simulations were carried out for four summer months (July–September) of each year between 1998 and 2006.

2.2 Development of the reduced-order modeling approach

  • The multifidelity ROM approach used in this study is based on the gappy proper orthogonal decomposition (POD) mapping approach (Robinson et al., 2012).
  • The set of parameters could include system parameters (e.g., vegetation distribution, soil types, and topography), climate forcings, time, and other quantities that have an influence on the system response.
  • The parameters that vary in the simulations that the authors have performed for each site are time (days for summer seasons in a year) and the climate forcings (precipitation and evapotranspiration rates) prescribed at that particular time.
  • F corresponded to a simulated fine-resolution three-dimensional soil moisture field, but in general,f can be any spatial quantity of interest (e.g., soil temperature or GHG emission).

2.2.1 POD method

  • The POD method is similar to the principal component analysis (Jolliffe, 2002) and the Karhunen–Loeve decomposition (Moore, 1981).
  • The authors computed the POD bases based on the kernel eigenvalue approach (Everson and Sirovich, 1995).
  • The POD projection method requires extensive modification of the existing code of the simulator, and is thus not suitable for existing LSMs.
  • To demonstrate the limit of accuracy of POD-related methods presented in subsequent subsections (Sects. 2.2.2– 2.2.5), the authors determineαPOD(p) based on Eq. (4) by evaluating f (p) explicitly and present the results in Sect.
  • In subsequent sections, the authors describe four different methods of developing a ROM that reconstructs the fine-resolution solution based on the coarse-resolution solution.

2.2.2 POD mean method (POD-mean)

  • To overcome the difficulties associated with calculating αPOD(p), the authors propose a POD-mean method (POD-mean).
  • The authors then construct a polynomial fit betweenαPOD(q) and the mean off (q) (i.e., fineresolution mean soil moisture,µf (q)), which they denote as Geosci.
  • Table 1.Summary of differences between various methods used for constructing ROM.
  • Reference ith column of the Method basis data matrix Determination ofα(p) POD f̄ f (qi) − f̄ Equation (4).
  • This particular approach works well if (1) the relationships betweenαfiti (µf ) andµf exist; and (2)µg is a good approximation ofµf .

2.2.3 POD mapping method (POD-MM)

  • In the POD-mean method, the authors only used the mean of the coarse-resolution solution,g(p), to reconstruct the fineresolution solution.
  • The POD mapping method (POD-MM) attempts to use all information ing(p) to efficiently and accurately reconstruct the fine-resolution solution.
  • The PODMM method is a modification of the gappy POD (Everson and Sirovich, 1995).
  • The POD basesζPOD-MMi can be decomposed into ζPOD-MMi = [ ζ f,POD-MM i ζ g,POD-MM i ] , (7) whereζ f,POD-MMi and ζ g,POD-MM i are components associated with the fine- and coarse-resolution models.
  • The authors note thatαPOD-MM (p) is not simply given by Eq. (4) sinceζ g,POD-MMi are not mutually orthogonal.

2.2.4 Second alternative formulation of the POD

  • The authors also introduce an alternative formulation of the PODMM method (POD-MM2) to determine whether the number of POD bases required could be reduced for a fixed approximation error threshold.
  • By using the deviation of from the mapped coarse-resolution solutiong̃, the authors remove the bias www.geosci-model-dev.net/7/2091/2014/.
  • The authors note that this alternative POD mapping formulation is possible since their coarse- and fine-resolution grids are nested (which will always be the case for the types of applications they are developing here).
  • The authors denote the resulting POD-based vector as ζPOD-MM2i = [ ζ h,POD-MM2 i ζ g,POD-MM2 i ] , (11) whereζ h,POD-MM2i are the components associated withh.

2.2.5 Third alternative formulation of the POD

  • When a solution is spatially highly correlated with a spatially varying parameterw, such as the topography, the authors may use this information in their reconstruction of the fine-resolution solution.
  • The POD-MM3 approach is developed to improve the performance of POD-MM method when one of the parameters is heterogeneous and spatially varying.
  • This method is only applicable to site-independent ROM since the surface elevation is included as a parameter in the site-independent ROM but not in the single and multi-site ROMs.

2.2.6 Error definitions

  • This error measure gives the maximum theoretical accuracy achievable using POD-related methods.
  • The authors also define¯POD as the mean ofePOD evaluated over a specified number of days.
  • For POD-X methods, where POD-X stands for PODmean, POD-MM, POD-MM2, or POD-MM3, the error measures can be constructed for each1xg, and are defined as ePOD-X1xg = ‖ f POD-X1xg − f ‖2 ‖ f ‖2 . (17) Similarly, the authors definēePOD-X1xg as the mean ofe POD-X 1xg evaluated over a specified number of days.

3 Results and discussion

  • As described in the Methods section, the authors developed the ROM models for the four NGEE-Arctic Barrow study sites chosen for detailed characterization.
  • The four sites differ in their topographic characteristics and therefore each site has a different dynamic soil moisture response to the same meteorological forcings.
  • In addition, since the parameters varied in this study are time and the magnitude of the forcing terms, historical data (prior-year simulations) can be used to construct the ROM.
  • For more general cases involving system parameters, statistical or adaptive sampling techniques are needed to generateSN (Pau et al., 2013a, b).

3.1.1 Application of POD method

  • The authors first constructed four separate ROMs, one for each site, using the POD method and the finest resolution (1xf = Geosci.
  • There is no significant difference between the error budgets as a function ofM for 2002 and 2006.
  • The above observation cannot be deduced based solely on the probability distribution functions (PDFs) of the DEM (digital elevation model) of the sites (Fig. 3) even though DEM is the only quantity that is different between the models for the four study sites.

3.1.2 Application of POD-mean method

  • To determine whether the authors can use the POD-mean method, they first examine the relationship betweenαPODi (q) andµf (q) for all q ∈ SN .
  • For all four sites, the authors foundαPOD1 to be linearly correlated toµf (Fig. 4).

3.1.3 Application of POD-MM method

  • Alternatively, the authors can determineMoptimal by examining the amount of variance represented by the firstM POD bases.
  • For site A, the maximumεPOD-MM1xg is 2.77×10−3 and the locations of large errors are not discernable from Fig. 9, indicating that large errors are only localized to small regions of the domain, resulting in small average errors,ePOD-MM1xg .

3.1.4 Application of POD-MM2 method

  • With the POD-MM2 method, the resulting error,ēPOD-MM21xg , is smaller than̄ePOD-MM1xg for smallM (Fig. 11).
  • The convergence behavior ofēPOD-MM21xg with M is less well behaved as compared to the POD-MM.
  • As a result, the minimum achievable value of ēPOD-MM21xg is larger than the minimum achievable value of ēPOD-MM1xg , especially for larger1xg.
  • The POD-MM method is thus preferred since it allows the error to be reduced systematically by increasingM, especially when1xg is large.

3.2 Multi-site ROM

  • To construct a multi-site ROM, the authors used daily snapshots from all four sites for 1998–2000 to construct a single ROM.
  • Based on the analysis performed using the POD method, the authors conclude that the POD-related methods can theoretically perform very well even when all four sites are considered in aggregate (Fig. 12).
  • The number of POD bases needed to achieve similar accuracy is greater than when separate ROMs are constructed for each site (compare Figs. 2 and 12).
  • Regions with homogeneous red color in the panels reflect the fact that large regions of the solutions are saturated.
  • Computational cost needed to construct a single multi-site ROM compared to multiple single-site ROMs.

3.3 Site-independent ROM

  • Here, the authors include the spatially heterogeneous surface elevation, as described by the DEM, in the parameter space during the construction of the ROM.
  • For the POD method, the error̄ePOD for the siteindependent ROM decreases with an increasing number of bases but not as rapidly as̄ePOD of single- or multi-site ROMs (Fig. 14).
  • For the above example, a larger number of sites needs to be included in the training data.

3.4 Application to larger-scale hydrological simulations

  • The POD mapping method shows great promise in allowing prediction of fine-resolution soil moisture dynamics using coarse-resolution simulations.
  • If the above results hold for simulations that include more sources of heterogeneity in the subsurface (e.g., conductivity) and surface properties, integration of the relevant ROMs into a land model such as CLM will allow for a much finer representation of processes than is currently possible, without a drastic increase in computational cost.
  • Thus, any bias in the coarse-resolution mean will lead to a biasedf POD-mean1xg .
  • Partitioning of the parameter space will allow us to construct multiple ROMs that are tailored to each domain.
  • As with any sampling-based technique, the POD mapping method performs well only if the snapshots of the solution used to construct the ROM form an approximation space that can reasonably represent the solution.

4 Conclusions

  • The authors describe the construction of ROMs for land surface models based on POD-related methods.
  • ROMs were built for soil moisture predictions from the PFLOTRAN model for the four NGEE-Arctic sites.
  • Both the single-site and multi-site ROMs are very accurate (< 0.1 %) with a computational speedup greater than 103.
  • The overall error magnitude is still quite low given the large topographical differences across the sites, thereby giving creditability for using ROMs in larger-scale simulations.
  • The authors provide several approaches by which they can generalize their methods to problems of larger extent and diversity in this paper.

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Geosci. Model Dev., 7, 2091–2105, 2014
www.geosci-model-dev.net/7/2091/2014/
doi:10.5194/gmd-7-2091-2014
© Author(s) 2014. CC Attribution 3.0 License.
A reduced-order modeling approach to represent subgrid-scale
hydrological dynamics for land-surface simulations:
application in a polygonal tundra landscape
G. S. H. Pau, G. Bisht, and W. J. Riley
Earth Science Division, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, California 94720, USA
Correspondence to: G. S. H. Pau (gpau@lbl.gov)
Received: 25 February 2014 Published in Geosci. Model Dev. Discuss.: 4 April 2014
Revised: 18 July 2014 Accepted: 15 August 2014 Published: 17 September 2014
Abstract. Existing land surface models (LSMs) describe
physical and biological processes that occur over a wide
range of spatial and temporal scales. For example, biogeo-
chemical and hydrological processes responsible for carbon
(CO
2
, CH
4
) exchanges with the atmosphere range from the
molecular scale (pore-scale O
2
consumption) to tens of kilo-
meters (vegetation distribution, river networks). Addition-
ally, many processes within LSMs are nonlinearly coupled
(e.g., methane production and soil moisture dynamics), and
therefore simple linear upscaling techniques can result in
large prediction error. In this paper we applied a reduced-
order modeling (ROM) technique known as “proper orthog-
onal decomposition mapping method” that reconstructs tem-
porally resolved fine-resolution solutions based on coarse-
resolution solutions. We developed four different methods
and applied them to four study sites in a polygonal tundra
landscape near Barrow, Alaska. Coupled surface–subsurface
isothermal simulations were performed for summer months
(June–September) at fine (0.25 m) and coarse (8 m) horizon-
tal resolutions. We used simulation results from three sum-
mer seasons (1998–2000) to build ROMs of the 4-D soil
moisture field for the study sites individually (single-site) and
aggregated (multi-site). The results indicate that the ROM
produced a significant computational speedup (> 10
3
) with
very small relative approximation error (< 0.1 %) for 2 vali-
dation years not used in training the ROM. We also demon-
strate that our approach: (1) efficiently corrects for coarse-
resolution model bias and (2) can be used for polygonal tun-
dra sites not included in the training data set with relatively
good accuracy (< 1.7 % relative error), thereby allowing for
the possibility of applying these ROMs across a much larger
landscape. By coupling the ROMs constructed at different
scales together hierarchically, this method has the potential to
efficiently increase the resolution of land models for coupled
climate simulations to spatial scales consistent with mecha-
nistic physical process representation.
1 Introduction
The terrestrial hydrological cycle strongly impacts, and
is impacted by, atmospheric processes. Further, a primary
control on terrestrial biogeochemical (BGC) dynamics and
greenhouse gas (GHG) emissions from soils (e.g., CO
2
, CH
4
,
N
2
O) across spatial scales is exerted by the system’s hydro-
logical state (Schuur et al., 2008). Soil moisture also im-
pacts soil temperature, which is another important controller
of GHG emissions (Torn and Chapin, 1993). Since climate
change is predicted to change the amount and temporal dis-
tribution of precipitation globally, there is a critical need for
models to not only accurately capture subgrid heterogeneity
of terrestrial hydrological processes, but also the impacts of
subgrid hydrological heterogeneity on BGC fluxes.
Terrestrial hydrological states are important for climate
prediction across a wide range of spatial scales, from soil
pores to continental. The critical spatial scale relevant to
soil moisture state and subsurface and surface fluxes may
be as small as 100m (Wood et al., 2011), although there
is vibrant disagreement about the relative increase in pre-
dictability when trying to explicitly simulate at such high
resolutions with limited observational data to constrain pa-
rameter values (Beven and Cloke, 2012). However, the im-
portance of representing fine-resolution spatial structure in
Published by Copernicus Publications on behalf of the European Geosciences Union.

2092 G. S. H. Pau et al.: Hydrological dynamics for land-surface simulations
hydrological states and fluxes has been demonstrated for sur-
face evapotranspiration budgets (Vivoni et al., 2007; Wood,
1997), runoff and streamflow (Arrigo and Salvucci, 2005;
Barrios and Francés, 2012; Vivoni et al., 2007), and atmo-
spheric feedbacks (Nykanen and Foufoula-Georgiou, 2001).
It remains unclear what the critical spatial scale is for bio-
geochemical dynamics, but it has been shown that “hot spot”
formation is important for wetland biogeochemistry at scales
O(10cm) (Frei et al., 2012) and for nitrogen cycle varia-
tions at O(m) (McClain et al., 2003). In contrast, the cur-
rent suite of land surface models applicable at watershed
(e.g., PAWS, Riley and Shen, 2014; Shen, 2009; regional,
Maxwell et al., 2012; or climate, Koven et al., 2013; Tang et
al., 2013) scales typically represent hydrological or biogeo-
chemical cycles at O(100m–km) scales.
The methods to represent spatial heterogeneity in hydro-
logical and biogeochemical dynamics differ between wa-
tershed and regional or climate-scale models. While many
current watershed-scale models explicitly represent lateral
inter-connectivity for subsurface and surface fluxes, regional-
and climate-scale models currently rely on a non-spatially-
explicit tiling approach. For example, CLM4.5 (Koven et al.,
2013; Lawrence et al., 2012; Tang et al., 2013), the land
model integrated in the Community Earth System Model
(Hurrell et al., 2013), represents land-surface grid cells with
the same horizontal extent as the atmospheric grid cells
(which can range from 1
× 1
for climate change sim-
ulations to 0.25
× 0.25
for relatively short simulations;
Bacmeister et al., 2014; Wehner et al., 2014). These grid
cells are disaggregated into a subgrid hierarchy of non-
spatially-explicit land units (e.g., vegetated, lakes, glacier,
urban), columns (with variability in hydrological, snow, and
crop management), and plant functional types (accounting
for variations in broad categories of plants and bare ground).
Therefore, we contend that representing the much smaller
spatial scales now recognized to control hydrological and
biogeochemical dynamics in regional and global-scale mod-
els will require a reformulation of the overall design of these
models.
One potential approach to represent spatial heterogeneity
in soil moisture fields at resolutions finer than represented
in a particular modeling framework is to relate the statistical
properties of the soil moisture field with the spatial scale. Hu
et al. (1997) showed that the variance (σ
2
θ
) of the soil mois-
ture (θ) field at different spatial averaging areas (A) can be
related to the ratio of those areas raised to a scaling expo-
nent (γ ). They also showed that γ is related to the spatial
correlation structure of the soil moisture field and that it de-
creases as soils dry. Observational studies have described a
power law decay of variance as a function of the observa-
tion scale (Rodriguez-Iturbe et al., 1995; Wood, 1998), and
several investigators have demonstrated that the relationship
between σ
2
θ
and spatial scale is not “simple” (i.e., not log-log
linear across all spatial scales; e.g., Das and Mohanty, 2008;
Famiglietti et al., 1999; Joshi and Mohanty, 2010; Mascaro
et al., 2010, 2011; Nykanen and Foufoula-Georgiou, 2001).
A second potential approach to account for spatial hetero-
geneity in soil moisture states is to relate its higher-order mo-
ments to the mean, and then apply these relationships within
a model that predicts the transient coarse-resolution mean. In
many observationally based studies, an upward convex rela-
tionship between the mean and variance has been reported
(e.g., Brocca et al., 2010, 2012; Choi and Jacobs, 2011;
Famiglietti et al., 2008; Lawrence and Hornberger, 2007; Li
and Rodell, 2013; Pan and Peters-Lidard, 2008; Rosenbaum
et al., 2012; Tague et al., 2010; Teuling et al., 2007; Teuling
and Troch, 2005). Theoretical analyses have also indicated
that an upward convex relationship is consistent with current
understanding of soil moisture dynamics (e.g., Vereecken et
al., 2007). However, as discussed in Brocca et al. (2007), the
relationships between soil moisture mean and statistical mo-
ments have been reported to depend on many factors, includ-
ing lateral redistribution, radiation, soil characteristics, veg-
etation characteristics, elevation above the drainage channel,
downslope gradient, bedrock topography, and specific ups-
lope area. These large number of observed controllers and
the lack of an accepted set of dominant factors argue that
substantial work remains before this type of information can
be integrated with land models to represent subgrid spatial
heterogeneity.
Modeling studies have also been performed to investigate
spatial scaling properties of moisture and how these prop-
erties relate to ecosystem properties. For example, Ivanov
et al. (2010) studied spatial heterogeneity in moisture on an
idealized small hill slope, and found hysteretic patterns dur-
ing the wetting–drying cycle and that the system response
depends on precipitation magnitude. Riley and Shen (2014)
used a distributed modeling framework to analyze relation-
ships between mean and higher-order moments of soil mois-
ture and ecosystem properties in a watershed in Michigan.
They concluded that the strongest relationship between the
observed declines in variance and increases in mean mois-
ture (past a peak in this relationship) was with the gradient
convolved with mean evapotranspiration. Other studies have
focused on upscaling fine-resolution model parameters to ef-
fective coarser-resolution parameters. For example, Jana and
Mohanty (2012) showed that power-law scaling of hydraulic
parameters was able to capture subgrid topographic effects
for four different hill slope configurations.
Theoretical work to explicitly include spatial heterogene-
ity in the hydrological governing equations has also been ap-
plied to this problem. Albertson and Montaldo (2003) and
Montaldo and Albertson (2003) developed a relationship for
the time rate of change of soil moisture variance based on the
mean moisture and spatial covariances between soil mois-
ture, infiltration, drainage, and ET. Teuling and Troch (2005)
applied a similar approach to study the impacts of vegetation,
soil properties, and topography on the controls of soil mois-
ture variance. Kumar (2004) applied a Reynolds averaging
Geosci. Model Dev., 7, 2091–2105, 2014 www.geosci-model-dev.net/7/2091/2014/

G. S. H. Pau et al.: Hydrological dynamics for land-surface simulations 2093
approach, and ignoring second- and higher-order terms, de-
rived a relationship for the time rate of change of the mean
moisture field that depends on the moisture variance. Choi et
al. (2007) applied the model to a 25 000 km
2
Appalachian
Mountain region for the summer months of 1 year and found
that subgrid variability significantly affected the prediction
of mean soil moisture.
The approaches described above to capture fine-resolution
spatial heterogeneity within a coarse-resolution modeling
framework have some limitations. First, the soil moisture
probability density function is often very non-normal (Ryu
and Famiglietti, 2005), making the sole use of variance as a
descriptor of moisture heterogeneity insufficient. A similar
problem arises with the Reynolds averaging approach that
does not include higher-order terms. This approach also re-
quires a method to “close” the solution (i.e., relate the higher-
order terms to the mean moisture), and there is no gener-
ally accepted method to perform this closure. Perhaps the
largest constraint of these approaches in the context of cli-
mate change and atmospheric interactions is that they can-
not account for the temporal memory in the system that im-
pacts biogeochemical transformations. In particular, the bio-
geochemical dynamics at a particular point in time depend
on the state and dynamics that occurred in the past, and just
knowing the statistical distribution of moisture at a particular
time may not maintain the continuity required for accurate
prediction. Therefore, for applications related to regional- to
global-scale interactions with the atmosphere, a method is
required that allows for (1) computationally tractable simu-
lations (i.e., relatively coarser resolution); (2) spatially ex-
plicit prediction of the temporal evolution of soil moisture
at relatively finer resolutions; and (3) integration of the rel-
atively finer resolution soil moisture predictions with repre-
sentations of the relevant biogeochemical dynamics.
To that end, we describe a generally applicable reduced-
order modeling technique to reconstruct a fine-resolution
heterogeneous 4-D soil moisture solution from a coarse-
resolution simulation, thereby resulting in significant com-
putational savings. In this study, we built ROMs based on the
proper orthogonal decomposition mapping method (Robin-
son et al., 2012), which first involved training the ROMs
using fine- and coarse-resolution simulations over multiple
years. Hydrologic simulations of coupled surface and subsur-
face processes for an Alaska polygonal tundra system were
performed using the PFLOTRAN model (Bisht and Riley,
2014; Hammond et al., 2012). Simulations were performed
for four study sites in Alaska with distinct polygonal surface
characteristics and individual ROMs were built for each site.
The resulting ROMs were then applied over periods outside
of the ROM training period.
In Sect. 2 we describe the polygonal tundra site used for
our simulations, the PFLOTRAN hydrological simulations
configuration, and the methods used to develop and evaluate
the ROMs. In Sect. 3, these methods are used under differ-
ent scenarios to develop ROMs for the polygonal tundra site
that increase in generality in the following order: single-site
ROMs (limited to a single site), multi-site ROMs (limited
to sites included in the training data) and site-independent
ROMs (applicable even for sites not included in the train-
ing data). For each of the above scenarios, different ROMs
can be developed using methods that we propose in Sect. 2;
the applicability of a method to a given scenario is discussed
in Sect. 2. We then compare the accuracy of the different
ROMs and end with a discussion of limitations of the ap-
proach, possible improvements, and methods to incorporate
the proposed ROM approach within a global-scale hydrolog-
ical and biogeochemical model.
2 Methods
2.1 Site description and hydrologic simulation setup
In this study, we developed ROMs for hydrological simula-
tions performed at four sites in the Barrow Environmental
Observatory (BEO) in Barrow, Alaska (71.3
N, 156.5
W).
The BEO lies within the Alaskan Arctic costal plain, which
is a relatively flat region, characterized by thaw lakes and
drained basins (Hinkel et al., 2003; Sellmann et al., 1975)
and polygonal ground features (Hinkel et al., 2001; Hubbard
et al., 2013). The Department of Energy (DOE) Next-
Generation Ecosystem Experiments (NGEE-Arctic) project
has established four intensely monitored sites (A, B, C and D,
shown in Fig. 1) within the BEO in 2012 to study the impact
of climate change in high-latitude regions. The four NGEE-
Arctic study sites have distinct micro-topographic features,
which include low-centered (A), high-centered (B), and tran-
sitional polygons (C, and D). The mean annual air tempera-
ture for our study sites is approximately 13
C (Walker et
al., 2005) and the mean annual precipitation is 106mm, with
the majority of precipitation falling during the summer sea-
son (Wu et al., 2013). The study site is underlain with contin-
uous permafrost and the seasonally active layer depth ranges
between 30 and 90cm (Hinkel et al., 2003).
We applied a version of the three-dimensional subsurface
reactive transport simulator PFLOTRAN, which was modi-
fied to include surface water flows, for simulating surface–
subsurface hydrologic processes at the four NGEE-Arctic
study sites. The subsurface flows in PFLOTRAN are solved
with a finite volume and an implicit time integration scheme,
and are sequentially coupled to a finite-volume-based sur-
face flow solution that is solved explicitly in time. Simula-
tions at the four study sites were conducted using meshes at
horizontal resolutions of 0.25, 0.5, 1.0, 2.0, 4.0, and 8.0 m.
A constant vertical resolution of 5 cm with a total depth of
50 cm was used for all simulations. The simulations were car-
ried out for four summer months (July–September) of each
year between 1998 and 2006. Evapotranspiration and effec-
tive precipitation boundary conditions for the PFLOTRAN
simulations were obtained from offline simulations of the
www.geosci-model-dev.net/7/2091/2014/ Geosci. Model Dev., 7, 2091–2105, 2014

2094 G. S. H. Pau et al.: Hydrological dynamics for land-surface simulations
Community Land Model (CLM4.5; Oleson et al., 2013). Ver-
tical heterogeneity in soil properties was prescribed using
data from Hinzman et al. (1991). A static active layer depth
of 50 cm, corresponding approximately to the maximum sea-
sonal value, was assumed for all simulations. Details of the
model setup are provided in Bisht and Riley (2014). In the
current study, the ROM was trained on 3 years of data (1998–
2000), and the ROM predictions for 2002 and 2006 were
compared against fine-resolution simulations.
2.2 Development of the reduced-order
modeling approach
The multifidelity ROM approach used in this study is based
on the gappy proper orthogonal decomposition (POD) map-
ping approach (Robinson et al., 2012). Let p be a set of
parameters that defines a particular solution or observation.
The set of parameters could include system parameters (e.g.,
vegetation distribution, soil types, and topography), climate
forcings, time, and other quantities that have an influence
on the system response. In this paper, the parameters that
vary in the simulations that we have performed for each
site are time (days for summer seasons in a year) and the
climate forcings (precipitation and evapotranspiration rates)
prescribed at that particular time. Then, given a sample set
S
N
= {q
1
,...,q
N
}, where q
i
is a set of parameters p, and N
is the number of samples, we can compute the corresponding
solution {f (q
1
),...,f (q
N
)}. In this paper, f corresponded
to a simulated fine-resolution three-dimensional soil mois-
ture field, but in general, f can be any spatial quantity of
interest (e.g., soil temperature or GHG emission).
2.2.1 POD method
The POD approximation of f , f
POD
, is given by
f (p) f
POD
(p) = f
ref
+
M
X
i=1
α
i
(p)ζ
POD
i
, (1)
where M N N, N is the degree of freedom of f , f
ref
is the reference basis (here, f
ref
=
¯
f =
1
N
N
P
i=1
f (q
i
)), ζ
POD
i
are the POD bases and M is the number of POD bases.
The POD bases are determined through a singular value de-
composition (SVD) of the data matrix given by W
POD
=
f (q
1
)
¯
f ,...,f (q
N
)
¯
f
:
W
POD
= UDV
T
, (2)
where U R
N ×N
are the left eigenvectors, V R
N×N
are
the right eigenvectors, and D = diag
1
,...,λ
N
) R
N ×N
,
with λ
1
λ
2
. . . λ
N
> 0. The POD bases ζ
POD
i
,1 i
N are thus given by W
POD
V, and λ
i
are the associated eigen-
values with each POD basis. The POD method is similar
to the principal component analysis (Jolliffe, 2002) and the
Karhunen–Loeve decomposition (Moore, 1981). We com-
puted the POD bases based on the kernel eigenvalue ap-
proach (Everson and Sirovich, 1995).
The number of POD bases (denoted by M) used to recon-
struct the approximate solution to a certain level of error (ε
λ
)
can be determined by finding M that satisfies
e
λ
M
= 1
M
X
i=1
λ
i
T
ε
λ
, (3)
where λ
T
=
N
P
i=1
λ
i
. As mentioned in Wilkinson (2011), the
dimensional reduction afforded by the POD method depends
on the extent to which the components of f are correlated.
We note that Eq. (1) only states how f is represented in a
linear space spanned by the POD bases, but there are mul-
tiple approaches of determining α(p) =
{
α
1
(p),...,α
M
(p)
}
for a given p. One optimal solution of α that minimizes the
least squares error between f (p) and f
POD
(p), denoted by
α
POD
(p), is given by
α
POD
i
(p) = ζ
POD,T
i
(f (p )
¯
f ), i = 1, . . ., M. (4)
However, α
POD
(p), determined using Eq. (4), does not lead
to any computational savings since f (p) is the quantity we
would like to approximate. Determination of f (p) can be
avoided by using the POD projection method (Willcox and
Peraire, 2002), which discretizes the governing equations us-
ing the linear space spanned by ζ
POD
i
and solves the resulting
algebraic equations for α
POD
(p). However, the POD projec-
tion method requires extensive modification of the existing
code of the simulator, and is thus not suitable for existing
LSMs. To demonstrate the limit of accuracy of POD-related
methods presented in subsequent subsections (Sects. 2.2.2–
2.2.5), we determine α
POD
(p) based on Eq. (4) by evaluating
f (p) explicitly and present the results in Sect. 3.
In subsequent sections, we describe four different methods
of developing a ROM that reconstructs the fine-resolution so-
lution based on the coarse-resolution solution. Each of the
methods is a modification of the basic POD method, but uses
a different reference basis, data matrix, or method to com-
pute α(p). The differences among the various methods for
developing a ROM are summarized in Table 1.
2.2.2 POD mean method (POD-mean)
To overcome the difficulties associated with calculating
α
POD
(p), we propose a POD-mean method (POD-mean).
We first determine α
POD
(q),q S
N
using Eq. (4); this step
requires negligible computational overhead since construc-
tion of ROM based on the POD method already requires the
determination of f (q),q S
N
. We then construct a poly-
nomial fit between α
POD
(q) and the mean of f (q) (i.e., fine-
resolution mean soil moisture, µ
f
(q)), which we denote as
Geosci. Model Dev., 7, 2091–2105, 2014 www.geosci-model-dev.net/7/2091/2014/

G. S. H. Pau et al.: Hydrological dynamics for land-surface simulations 2095
Table 1. Summary of differences between various methods used for constructing ROM. In the table below, f is the fine-resolution solution;
g is the coarse resolution solution; h is given by Eq. (10);
ˆ
f and
ˆ
g are given by Eq. (15); p is any given parameter set; q
i
is the ith parameter
set in S
N
; µ
f
(p) and µ
g
(p) are spatially averaged f (p) and g(p); and ζ
g,POD-MM
i
, ζ
g,POD-MM2
i
and ζ
g,POD-MM3
i
are POD bases for
POD-MM, POD-MM2 and POD-MM3 methods, respectively. (Please refer to each method’s subsection in Sect. 2 for more details on the
above variables.)
Reference ith column of the
Method basis data matrix Determination of α(p)
POD
¯
f f (q
i
)
¯
f Equation (4).
POD-mean
¯
f f (q
i
)
¯
f Approximated by α
fit
g
(p)), where α
fit
is a polynomial fit between α(q) and µ
f
(q).
POD-MM
¯
f
¯
g
f (q
i
)
¯
f
g(q
i
)
¯
g
Equation (8) using g(p).
POD-MM2
¯
h
¯
g
h(q
i
)
¯
h
¯
g
Equation (8), by substituting ζ
g,POD-MM
i
by ζ
g,POD-MM2
i
.
POD-MM3
ˆ
f
ˆ
g
f (q
i
)
ˆ
f
g(q
i
)
ˆ
g
Equation (8), by substituting ζ
g,POD-MM
i
by ζ
g,POD-MM3
i
and
¯
g by
ˆ
g.
α
fit
f
). Then, for any given p, we approximate f by
f
POD-mean
1x
g
(p) =
¯
f +
N
X
i=1
α
fit
i
g
(p))ζ
POD
i
, (5)
where µ
g
(p) is the mean of g(p), a coarse-resolution solu-
tion simulated at resolution 1x
g
> 1x
f
. This particular ap-
proach works well if (1) the relationships between α
fit
i
f
)
and µ
f
exist; and (2) µ
g
is a good approximation of µ
f
. For
the Arctic tundra study sites, we will show that these condi-
tions hold true for i = 1, and f
POD-mean
1x
g
is a good approxi-
mation of f .
2.2.3 POD mapping method (POD-MM)
In the POD-mean method, we only used the mean of the
coarse-resolution solution, g(p), to reconstruct the fine-
resolution solution. The POD mapping method (POD-MM)
attempts to use all information in g(p) to efficiently and ac-
curately reconstruct the fine-resolution solution. The POD-
MM method is a modification of the gappy POD (Everson
and Sirovich, 1995). For the same sample set S
N
, we deter-
mine {g(q
1
),...,g(q
N
)}, where q
i
S
N
. As in Robinson et
al. (2012), the multifidelity POD bases, ζ
POD-MM
i
, are then
determined through a SVD of the data matrix W
POD-MM
:
W
POD-MM
=
f (q
1
)
¯
f
g(q
1
)
¯
g
...
f (q
N
)
¯
f
g(q
N
)
¯
g
, (6)
where
¯
f is as defined before and
¯
g =
1
N
N
P
i=1
g(q
i
). The POD
bases ζ
POD-MM
i
can be decomposed into
ζ
POD-MM
i
=
"
ζ
f,POD-MM
i
ζ
g,POD-MM
i
#
, (7)
where ζ
f,POD-MM
i
and ζ
g,POD-MM
i
are components associ-
ated with the fine- and coarse-resolution models. Given a
coarse-resolution solution g(p), we first determine
α
POD-MM
(p) = argmin
γ
k g(p)
¯
g (8)
M
X
i=1
γ
i
ζ
g,POD-MM
i
k
2
,
where k · k
2
is the L
2
norm. We note that α
POD-MM
(p) is not
simply given by Eq. (4) since ζ
g,POD-MM
i
are not mutually
orthogonal. The approximate solution, f
POD-MM
1x
g
(p), is then
given by f
POD-MM
1x
g
(p) =
¯
f +
M
P
i=1
α
POD-MM
i
(p)ζ
f,POD-MM
i
,
where 1x
g
is the resolution at which g(p) is computed.
2.2.4 Second alternative formulation of the POD
mapping method (POD-MM2)
We also introduce an alternative formulation of the POD-
MM method (POD-MM2) to determine whether the number
of POD bases required could be reduced for a fixed approx-
imation error threshold. Instead of applying Eq. (6), we per-
form a SVD of the data matrix W
POD-MM2
:
W
POD-MM2
=
h(q
1
)
¯
h
g(q
1
)
¯
g
...
h(q
N
)
¯
h
g(q
N
)
¯
g
, (9)
where
h(p) = f (p)
˜
g(p), (10)
and
˜
g is the solution obtained from a piecewise constant
mapping of g from the coarse-resolution grid of g onto the
fine-resolution grid of f . By using the deviation of f from
the mapped coarse-resolution solution
˜
g, we remove the bias
www.geosci-model-dev.net/7/2091/2014/ Geosci. Model Dev., 7, 2091–2105, 2014

Citations
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Posted ContentDOI
TL;DR: In this paper, the authors analyze the hypothesis that microtopography is a dominant controller of soil moisture in polygonal landscapes and develop two spatially-explicit methods to downscale coarse-resolution simulations of the soil moisture.
Abstract: Microtopographic features, such as polygonal ground, are characteristic sources of landscape heterogeneity in the Alaskan Arctic coastal plain. Here, we analyze the hypothesis that microtopography is a dominant controller of soil moisture in polygonal landscapes. We perform multi-year surface–subsurface isothermal flow simulations using the PFLOTRAN model for summer months at six spatial resolutions (0.25–8 m, in increments of a factor of 2). Simulations are performed for four study sites near Barrow, Alaska that are part of the NGEE-Arctic project. Results indicate a non-linear scaling relationship for statistical moments of soil moisture. Mean soil moisture for all study sites is accurately captured in coarser resolution simulations, but soil moisture variance is significantly under-estimated in coarser resolution simulations. The decrease in soil moisture variance in coarser resolution simulations is greater than the decrease in soil moisture variance obtained by coarsening out the fine resolution simulations. We also develop relationships to estimate the fine-resolution soil moisture probability distribution function (PDF) using coarse resolution simulations and topography. Although the estimated soil moisture PDF is underestimated during very wet conditions, the moments computed from the inferred soil moisture PDF had good agreement with the full model solutions (bias 0.99) for all four sites. Lastly, we develop two spatially-explicit methods to downscale coarse-resolution simulations of soil moisture. The first downscaling method requires simulation of soil moisture at fine and coarse resolution, while the second downscaling approach uses only topographical information at the two resolutions. Both downscaling approaches are able to accurately estimate fine-resolution soil moisture spatial patterns when compared to fine-resolution simulations (mean error for all study sites are < ± 1 %), but the first downscaling method more accurately estimates soil moisture variance.

2 citations


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TL;DR: In this article, the authors present a graphical representation of data using Principal Component Analysis (PCA) for time series and other non-independent data, as well as a generalization and adaptation of principal component analysis.
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TL;DR: In this paper, it is shown that principal component analysis (PCA) is a powerful tool for coping with structural instability in dynamic systems, and it is proposed that the first step in model reduction is to apply the mechanics of minimal realization using these working subspaces.
Abstract: Kalman's minimal realization theory involves geometric objects (controllable, unobservable subspaces) which are subject to structural instability. Specifically, arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces. This situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization. Structural instability associated with geometric theories is not unique to control; it arises in the theory of linear equations as well. In this setting, the computational problems have been studied for decades and excellent tools have been developed for coping with the situation. One of the main goals of this paper is to call attention to principal component analysis (Hotelling, 1933), and an algorithm (Golub and Reinsch, 1970) for computing the singular value decompositon of a matrix. Together they form a powerful tool for coping with structural instability in dynamic systems. As developed in this paper, principal component analysis is a technique for analyzing signals. (Singular value decomposition provides the computational machinery.) For this reason, Kalman's minimal realization theory is recast in terms of responses to injected signals. Application of the signal analysis to controllability and observability leads to a coordinate system in which the "internally balanced" model has special properties. For asymptotically stable systems, this yields working approximations of X_{c}, X_{\bar{o}} , the controllable and unobservable subspaces. It is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces.

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TL;DR: In this paper, the authors define biogeochemical hot spots as patches that show disproportionately high reaction rates relative to the surrounding matrix, whereas hot moments occur when episodic hydrological flowpaths reactivate and/or mobilize accumulated reactants.
Abstract: Rates and reactions of biogeochemical processes vary in space and time to produce both hot spots and hot moments of elemental cycling. We define biogeochemical hot spots as patches that show disproportionately high reaction rates relative to the surrounding matrix, whereas hot moments are defined as short periods of time that exhibit disproportionately high reaction rates relative to longer intervening time periods. As has been appreciated by ecologists for decades, hot spot and hot moment activity is often enhanced at terrestrial-aquatic interfaces. Using examples from the carbon (C) and nitrogen (N) cycles, we show that hot spots occur where hydrological flowpaths converge with substrates or other flowpaths containing complementary or missing reactants. Hot moments occur when episodic hydrological flowpaths reactivate and/or mobilize accumulated reactants. By focusing on the delivery of specific missing reactants via hydrologic flowpaths, we can forge a better mechanistic understanding of the factors that create hot spots and hot moments. Such a mechanistic understanding is necessary so that biogeochemical hot spots can be identified at broader spatiotemporal scales and factored into quantitative models. We specifically recommend that resource managers incorporate both natural and artificially created biogeochemical hot spots into their plans for water quality management. Finally, we emphasize the needs for further research to assess the potential importance of hot spot and hot moment phenomena in the cycling of different bioactive elements, improve our ability to predict their occurrence, assess their importance in landscape biogeochemistry, and evaluate their utility as tools for resource management.

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  • ..., 2012) and for nitrogen cycle variations at ∼O(m) (McClain et al., 2003)....

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  • ...It remains unclear what the critical spatial scale is for biogeochemical dynamics, although it has been shown that “hot spot” formation is important for wetland biogeochemistry occurs at scales ∼O(10 cm) (Frei et al., 2012) and for nitrogen cycle variations at ∼O(m) (McClain et al., 2003)....

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TL;DR: The Community Earth System Model (CESM) as discussed by the authors is a community tool used to investigate a diverse set of Earth system interactions across multiple time and space scales, including biogeochemical cycles, a variety of atmospheric chemistry options, the Greenland Ice Sheet, and an atmosphere that extends to the lower thermosphere.
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Frequently Asked Questions (2)
Q1. What are the contributions in "A reduced-order modeling approach to represent subgrid-scale hydrological dynamics for land-surface simulations: application in a polygonal tundra landscape" ?

In this paper the authors applied a reducedorder modeling ( ROM ) technique known as “ proper orthogonal decomposition mapping method ” that reconstructs temporally resolved fine-resolution solutions based on coarseresolution solutions. The authors developed four different methods and applied them to four study sites in a polygonal tundra landscape near Barrow, Alaska. The authors used simulation results from three summer seasons ( 1998–2000 ) to build ROMs of the 4-D soil moisture field for the study sites individually ( single-site ) and aggregated ( multi-site ). The authors also demonstrate that their approach: ( 1 ) efficiently corrects for coarseresolution model bias and ( 2 ) can be used for polygonal tundra sites not included in the training data set with relatively good accuracy ( < 1. 7 % relative error ), thereby allowing for the possibility of applying these ROMs across a much larger landscape. By coupling the ROMs constructed at different scales together hierarchically, this method has the potential to efficiently increase the resolution of land models for coupled climate simulations to spatial scales consistent with mechanistic physical process representation. 

The resulting ROM is subsequently used to predict future responses.