### Groundwater hydrology modeling: relation between steady state and transient simulation in MODFLOW simulation

Standard groundwater modeling usually uses the three-dimensional Richard's equation to describe the groundwater flow. Solution to the Richard equation usually requires numerical methods such as finite-difference equation used in USGS MODFLOW.
To solute the array of equations within MODFLOW, an initial head is required regardless of steady state (SS) or transient (TR) simulation.
As the MODFLOW manual states that in a steady state simulation, the storage terms are ignored since storage of each grid cell is constant with time.
One of the common problems to run the MODFLOW is  that we don't have the initial head data for the simulation. Therefore it is usually suggested that a SS simulation could be run at first to provide the head for the TR simulation. This is because SS is independent with initial head, but TR isn't.
However, whether placing the SS simulation ahead of TR simulations is doubtful in many practices! Here are the explanations:
First, the SS flow equation should only be simulated when the system is in equilibrium state.By definition, the steady state flow equation means for each grid cell, the flow and storage don't change with time. In most natural scenarios, these requirements are rarely met. However, the hydrologic system may be very close to equilibrium under certain circumstances. For instance, during winter time, the water flow is ignorable if most hydrologic processes are impaired due to the snow cover and frozen ground.

Second. even if the hydrologic system is close to equilibrium, inappropriate settings for the SS simulation will produce unrealistic results. In order to achieve the steady state for each grid cell, numerical method solver will redistribute the water within the hydrologic system through the head redistribution. After the SS simulation, head distribution may be rather different compared with reality. Several aspects may account for the discrepancy: 1) source/sink terms including pumping, infiltration are not accurate enough; 2) boundary condition is not accurate or missing; 3) model structure, layer settings have flaws which can't match up the reality.

Essentially, the head distribution at any given stress period is sufficient enough to start a simulation. However, approach to establish the head distribution remains the challenge. For cases started with SS, dedicated considerations of the above aspects may provide reliable results. Or else, reasonable assumption of initial head may yield even better results for TR simulation.

On the other side, model calibration could provide an approach to further reduce the uncertainty in initial head distribution even though these related variable are not best described as 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.…