### Integrated groundwater and surface water modeling: spatial discretization

Numerical simulations have to break time and space into increments. For example, weather predictions are usually modeled in hours or days and maps are produced with resolution.

In contrast, our real world is always behaving continuously. For example, temperature always evolves with time continuously.

We are also interested in a three dimensional simulation instead of 1D or 2D. The real world itself is 3D though we constantly forget that. It is just doesn't benefit much if we had a 3D temperature map.

For other types of simulation, however, 3D could provides much better solutions.
Groundwater flow, as an example, is best simulated with 3D numerical simulation.

The spatial discretization is always a challenge for groundwater modeling. Horizontal and vertical discretization are always coupled together. For example, in a flat grassland with homogeneous hydrologic proprieties, the horizontal discretization becomes less important. However, if the surface elevation changes drastically, the horizontal discretization matters.

Imagine a cheese cake with multi-layers, if it is placed on the table, no matter how you cut it into pieces, the cake will look exactly the same as long as you don't part the pieces.

However, if we lean the table to certain angle and consider the pieces are adjusted to sit vertically in their same positions, the system may changes dramatically: the top layer on the higher position no longer touch others; the cheese in the lower layer may mix the neighbor cake layer!

So what might be the solution? We need better space discretization to minimize the space distortion.

Let's still consider the cake example, if there is only one layer, just the cake, it might be acceptable as long as we don't flip the table. In other words, if we have thick layers, horizontal discretization becomes less important. However, when we have thin layers, fine horizontal discretization can reduce the distortion. If we cut the cake into many pieces, then top might always stick with top.

Wait, if we cut the cake to fine, what if we can't even eat it?
Like if we have fine grid, such as 0.1m, that we can't even run the simulation since we may have 1Billion grid!

Therefore, a compromise has to be made in order to make things work.

Several options are:
Bring horizontal grid into the similar scale with vertical layer to avoid distortion;
Use unstructured grid instead of uniform grid to reduce computational demand.

Besides, careful selection or definition of the layered cake and table also plays an important role!

### 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.…