# Rain Gage Network Linear Algebra?

• July 26th 2005, 12:17 PM
turten
Linear Algebra?
Hi Math Experts,

I have a 22 Rain-Gauge network, I have bulit 22 nominal models of each rain gauge from correlation analysis, and I have something like that:

RG1 = a1*RG2+a2*RG7+a3*RG15
RG2 = b1*RG1+b2*RG9+b3*RG19
...
RG22 = v1*RG3+v2*RG5+v3*RG9

where RGn is the rain income in the last 5 minutes in the RainGauge number n.

now I have to generate a "simulated consistent rain" with a computer program that I have to build. This means that I can fix a rain in RG1, and I have to obtain the rain in the 21 other RainGauges according to the relations.

For me this problem is similar to solving a Linear Equation System like Ax=b, but this system is something like Ax=x and I don't know how to deal with. I've read Gaussian Reduction and LU Decompossition, but I think this is a wrong way.

Can anyone give me some light to see the correct path to solve this.

Jaume.
• August 17th 2005, 12:59 AM
TheFarmer42
It looks like you have a linear regression model.

It doesn't look like you have taken auto (serial)correlation into account there.

I'm not the top statistician in the world, but i bet you a tenner there is correlation in rainfall data.

Depending on your purpose and the information you have, i'd say there is a better way to do it. Find a statistician or do a google search for simulated rainfall data or the like.

Cheers,
TF
• August 17th 2005, 09:23 AM
rgep
Briefly, to turn an equation of the form Ax = x into one of the form Ax = b, consider the matrix A-I: we have Ax = x iff (A-I)x = 0.
• September 7th 2005, 05:10 PM
hpe
Quote:

Originally Posted by turten
Hi Math Experts,

I have a 22 Rain-Gauge network, I have bulit 22 nominal models of each rain gauge from correlation analysis, and I have something like that:

RG1 = a1*RG2+a2*RG7+a3*RG15
RG2 = b1*RG1+b2*RG9+b3*RG19
...
RG22 = v1*RG3+v2*RG5+v3*RG9

This is (in matrix notation) a model of the form Ax = x (see the other posts), where A is a 22 x 22 matrix. It appears from what you write that the matrix is quite sparse (it has only about three non-zero entries in each row).

For this to be self-consistent, the matrix A should have the eigenvalue $\lambda = 1$, that is there should be a vector y such that Ay = y. Find that vector y (it should have all positive entries assuming all a1, ..., v3 are positive). It contains the ratios of rainfall data that is consistent with your model, i.e. $y_k/y_l = RGk/RGl$ for all k,l. You can therefore predict all rain data from RG1 with the formula $RGk = RG1 \cdot y_k/y_1$.