### Integrated groundwater and surface water modeling: topography effects

This is a talk highly related to my last post on spatial discretization of numerical simulation.

In groundwater modeling such as MODFLOW, topography effects are very common and yet less discussed.

There are a few distinguishable differences between low land groundwater and high land groundwater system. First, heads at high elevation are usually higher, this is observed since water table usually follow topography even not strictly. Second, variances in heads in high elevation may be insensitive to changes in forcing, For example, the low land always receive the most water (even flood) and droughts. Third, generally, water movements in high elevation are much slower due to low K values.

With consideration of these differences, it is easier to understand a few critical phenomena in groundwater modeling.

First, groundwater system is seldom in steady state. The best estimation, however, could be within winter time when stream base flow is minimal. At this stage, only limited groundwater flow and infiltration from snow melt under pressure at high elevation are driving the whole groundwater and surface water flow system. The K values are so small that the feed could last for several months even into the summer time.

Even with the steady state, it is still a challenge to produce the best initial heads after SS simulation. This is because of the sensitivity of high land groundwater system. Nearly all terms (K and infiltration rate, etc.) are small in magnitude, which means the system is highly sensitive to driving data (Thinking of climate change!!!). In order to maintain the constant storage, both in flow and out flow must be offset exactly! This is not just required for one pixel, but for all. Therefore the spatial resolution is critical here. If differences in adjacent heads exceed a certain range, it would drive the flow. For example, at high elevation region, adjacent cells have an elevation difference of 100m, which in reality surface is continuous but we have no choice to discretize it. In this case, in order to maintain the storage, the groundwater simulation may get differences in heads around 10m, which means one of them must level up or level down. So as a result, at least one of the simulated head is far below the surface elevation, which mismatch the observation.

You would think if the resolution is 1m, if might be able to resolve this issue. However, computational demand will increase exponentially as resolution increases since this is 3D simulation. Also, data availability for other inputs must also be considered. Therefore, a compromise of resolution is needed. There are a few studies which use 100m (or even less) in small catchments. It could potentially improve the simulation accuracy but care must be taken to avoid unrealistic assumptions.

Some also think, why don't we assign the initial head for all pixels? But how and what to assign?

A lot of simulations use the well measurements as initial heads, which might be a good solution. But again, these type of data are always absent in high elevation regions due to inaccessibility.

If streams are forms (even in winter), then surface runoff needs to be considered. But how? With surface water modeling. However, a typical surface water simulation starts around October, 1st,which is not exactly winter time in most regions.

So here comes the scenario, if we start simulation from October, we may be able to simulate the infiltration rate dynamics and snow accumulation storage for winter time.Without a steady simulation, it is usually encountered with the problem with initial heads as well. Since TR simulation is highly dependent upon initial heads, the initial condition could affect the low land heads.As heads at high elevations remain high throughout the year, they will constantly flow following the gradient. Even though the low K values means the recharge to low land is small, it may shape the spatial pattern through flow system.

After all, a promising way seems to be shaped:

Pay attention to the resolution, the higher you can afford, the better, when you have the topography effects.

A reasonable TR simulation starting with assumption on initial heads. The TR should run for a few water years until the surface process is close to observations (snow cover, discharge and infiltration, etc.). Then use these data to drive the groundwater model or coupled surface water-groundwater model. It is then possible to recreate the real SS simulation and others parameters.

### Spatial datasets operations: mask raster using region of interest

Climate change related studies usually involve spatial datasets extraction from a larger domain.
In this article, I will briefly discuss some potential issues and solutions.

In the most common scenario, we need to extract a raster file using a polygon based shapefile. And I will focus as an example.

In a typical desktop application such as ArcMap or ENVI, this is usually done with a tool called clip or extract using mask or ROI.

Before any analysis can be done, it is the best practice to project all datasets into the same projection.

If you are lucky enough, you may find that the polygon you will use actually matches up with the raster grid perfectly. But it rarely happens unless you created the shapefile using "fishnet" or other approaches.

What if luck is not with you? The algorithm within these tool usually will make the best estimate of the value based on the location. The nearest re-sample, but not limited to, will be used to calculate the value. But what about the outp…

### Numerical simulation: ode/pde solver and spin-up

For Earth Science model development, I inevitably have to deal with ODE and PDE equations. I also have come across some discussion related to this topic, i.e.,

https://www.researchgate.net/post/What_does_one_mean_by_Model_Spin_Up_Time

In an attempt to answer this question, as well as redefine the problem I am dealing with, I decided to organize some materials to illustrate our current state on this topic.

Models are essentially equations. In Earth Science, these equations are usually ODE or PDE. So I want to discuss this from a mathematical perspective.

Ideally, we want to solve these ODE/PDE with initial condition (IC) and boundary condition (BC) using various numerical methods.
https://en.wikipedia.org/wiki/Initial_value_problem
https://en.wikipedia.org/wiki/Boundary_value_problem

Because of the nature of geology, everything is similar to its neighbors. So we can construct a system of equations which may have multiple equation for each single grid cell. Now we have an array of equation…

### Watershed Delineation On A Hexagonal Mesh Grid: Part A

One of our recent publications is "Watershed Delineation On A Hexagonal Mesh Grid" published on Environmental Modeling and Software (link).
Here I want to provide some behind the scene details of this study.

(The figures are high resolution, you might need to zoom in to view.)

First, I'd like to introduce the motivation of this work. Many of us including me have done lots of watershed/catchment hydrology modeling. For example, one of my recent publications is a three-dimensional carbon-water cycle modeling work (link), which uses lots of watershed hydrology algorithms.
In principle, watershed hydrology should be applied to large spatial domain, even global scale. But why no one is doing it?  I will use the popular USDA SWAT model as an example. Why no one is setting up a SWAT model globally?
There are several reasons we cannot use SWAT at global scale: We cannot produce a global DEM with a desired map projection. SWAT model relies on stream network, which depends on DEM.…