A revisit of spatial discretization

Discretization by definition from Wikipedia: In applied mathematicsdiscretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numerical evaluation and implementation on digital computers.
Now we add “spatial” to the term. The first intuitive definition is the discretization of functions in the spatial domain.
There are two elements in this description: functions and spatial domain.
For functions, we often refer to integral or ODEs/PDEs in numerical simulations. If these functions involve with gradient information, then they depend on spatial domain, which is how gradient is calculated.
For spatial domain, we often refer to mesh or grid. And mesh can generally be classified into structured and unstructured grid.
In practice, we have spent great effects on both aspects of the spatial discretization: mesh and corresponding function solvers.
In Earth science, lots of functions involve with gradient. For example, gas exchange depends on pressure gradient or concentration gradient. As a result, Our spatial discretization is always associated with mesh grid.
Many studies have developed methods to solve equations using the finite difference method (FDM), the finite volume method (FVM) and the finite element method (FEM). However, many of them have not considered the effects of mesh grid on these solvers. For example, most models use array to represent spatial domain. In this case, water flow at any location can only flow to 4 directions. However, in reality, water can flow to any direction because Earth has no discretization. The problem is how can we represent a direction if it is 30 degree?
We need a better way to represent the spatial domain decomposition. We can improve the resolution. We can also use unstructured grid such as TIN. Ideally, we need a mesh that matches with reality. In my opinion, adaptive hexagon grid mesh is closest.
• It can cover the sphere.
• It proves consistent connectivity.
• It provides less assumption.
• It also provide 6 flow directions.
Without a solid mesh grid, the simulation which relies on mesh may contain great uncertainty. And these uncertainty may be even larger than the uncertainty caused by FDM/FVM/FEM and algorithms.

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.…