# 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

## 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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### Cites background or methods from "A reduced order modeling approach t..."

...We are presently exploring additional methods to infer the spatial structure of soil moisture at fine resolution from coarse-resolution simulations using a Princi-15 pal Orthogonal Decomposition method (Pau et al., 2014)....

[...]

...…models; i.e., developing relationships (i.e., reduced order models (ROMs)) between the mean properties (which could be estimated with a coarser-resolution model) and either the statistical (Riley and Shen, 2014) or spatially explicit (Pau et al., 2014) properties of the field of interest (here θ)....

[...]

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### "A reduced order modeling approach t..." refers methods in this paper

...The POD method is thus similar to principal component analysis (Jolliffe, 2002) and Karhunen Loeve decomposition (Moore, 1981)....

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### "A reduced order modeling approach t..." refers background in this paper

..., 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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### "A reduced order modeling approach t..." refers background in this paper

...5 (Koven et al., 2013; Lawrence5 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 simulations to ∼ 0.25◦×0.25◦ for relatively short simulations, Bacmeister et al., 2013; Wehner et al., 2014)....

[...]

...5 (Koven et al., 2013; Lawrence5 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…...

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###### Q2. What are the future works mentioned in the paper "A reduced-order modeling approach to represent subgrid-scale hydrological dynamics for land-surface simulations: application in a polygonal tundra landscape" ?

The resulting ROM is subsequently used to predict future responses.