The problem of geographic coordinate system in global scale simulation

Earlier I have discussed some idea about the hexagon-based grid system. Now I have provided some tests and materials to support this project.

90% percent of global maps you can find online are using the geographic coordinate system (GCS). You can try to search "global map" in Google image.

There might be 9% of them are using various map projection.
The reason why there are so many different ways to represent the Earth is that Earth is NOT flat. Google know it so they changed the Google Map recently.

So these are some basic GIS knowledge but you can also learn it from this video.

While it is generally OK to view these types of global map for daily usage, it can cause problems for large scale to global scale simulations.

In Earth science, GCS is most commonly used as the grid system for terrestrial ecosystem simulations. For example, a 1* 1 degree grid will discrete the global into a 360 * 180 matrix.

However, it will be problematic for several reasons.
For example, the area at different regions are not the same. For example. a 0.5* 0.5 grid cell is about 50*50km in Amazon whereas it is about 50*25km in Alaska, only half the size of the former one.

To illustrate the difference, let's use a classical 2D heat equation I found from here:
First, we run the simulation with the uniformly dx = dy, which present longitude and latitude in our case. The results are like this:
Then we changed to dy = 2 dx:
Last, we changed to dx = 2 dy:

If you look closer, you will see the differences. When the grid geometry is not uniform, the simulated heat distribution is also not uniform in spatial.

Now let's get back to GCS in global simulation. Because the grid geometry changes from tropic to pole regions, the simulations are likely affected as well.

Next, I will show you some results of global scale hydrologic simulation.

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.…