It is favorable to use a 3D Richards’ equation to develop the state and parameter estimation algorithm. However, a 3D model with fine discretization leads to increased number of states, which MHE may not be able to handle in an online fashion. One of the benefits of the proposed subsystem guidelines is that the states on a horizontal layer within a subsystem are very similar. Therefore, it is possible to use, for example, the states at the center column (a 1D system) of a subsystem to approximate the vertical capillary potential dynamics of different nodes within the same subsystem. Based on this, we further propose to use a 1D Richards’ equation to approximate the dynamics of each 3D subsystem. This can significantly reduce the computational demand for each decentralized estimator. Specifically, we propose to use the center column of each subsystem to approximate the vertical water dynamcis of a subsystem.

Based on this approximation, we study the significance of interaction between subsystems and motivate the use of a decentralized estimation framework. First, according to the FD model of Equation (

10), the state at time instant

$t+1$,

${h}_{i,j,k}(t+1)$ is dependent on itself and its adjacent states at time instant

t.

Figure 2 shows an illustration of the neighboring states of

${h}_{i,j,k}$. Let us assume that the states

${h}_{i,j,k}$,

${h}_{i,j,k-1}$ and

${h}_{i,j,k+1}$ belong to (a 1D approximation of) one subsystem, and

${h}_{i-1,j,k}$,

${h}_{i+1,j,k}$,

${h}_{i,j-1,k}$ and

${h}_{i,j+1,k}$ belong to the neighbouring subsystems. Let us examine the following term on the right-hand side of Equation (

10):

Suppose that the discretization along

x direction is equally spaced, the above term can be further simplified into the following:

This term quantitatively measures how the neighboring states along

x direction (

${h}_{i-1,j,k}$ and

${h}_{i+1,j,k}$) contribute to the evolution of

${h}_{i,j,k}$. Similarly, the following two terms quantify how neighboring states in

y and

z directions affect the evolution of

${h}_{i,j,k}$, respectively:

According to Equation (

10), the summation of the above three terms contributes to the evolution of

${h}_{i,j,k}$. As mentioned before,

${\Delta}x$ and

${\Delta}y$ represent the horizontal distances between two subsystems. Their values can be a few meters or even larger.

${\Delta}z$ is the interval beween two vertical discretization nodes and its value typically is about 1 to 25 centimeters. For example, if

${\Delta}x={\Delta}y=10\phantom{\rule{3.33333pt}{0ex}}m$ and

${\Delta}z=1\phantom{\rule{3.33333pt}{0ex}}cm$, then, the denominator of term (

18) is

${10}^{6}$ times smaller than those of terms (

16) and (

17). Because the seven states shown in

Figure 2 have values of similar magnitudes, the numerators of the above three terms are of similar magnitudes. Hence, the term (

18) is around

${10}^{6}$ times greater than terms (

16) and (

17), which implies that the contribution of the states in

z direction (

${h}_{i,j,k-1}$ and

${h}_{i,j,k+1}$) to the evolution of

${h}_{i,j,k}$ is significantly larger than those in

x and

y directions. In other words, the horizontal interaction between the states is notably smaller, as compared to the vertical interaction between the states. This justifies the use a decentralized estimation framework.