Homework 5* BMElib*
track

Space/time covariance models

**Date given: **10/12

**Due: **10/26
noon

** **

**Problem 1**

Consider the space/time random
field (S/TRF) *X( p)*, with

_{}

where *r* is the spatial lag
(*Km)*, *t* is the temporal
lag (day), and the covariance parameters are *c _{01}*=1.5

1) Can this field be assumed to
be homogeneous/stationary? Can you express this covariance model as a space/time
separable model? What is the variance of *X( p)*? Describe
physically the covariance model and each of its coefficients

2) Code this covariance model in *BMElib*
(i.e. write the MATLAB code for the MATLAB variables covmodel and covparam
that wold describe this covariance model in *BMElib*). See the help for modelsyntax.m if you need a reminder
about coding covariance models in *BMElib*.

covmodel =

covparam =

3) Write the
expression the covariance when the temporal lag *t *is zero:

*c** _{X}*(

4) Write the
expression the covariance when the spatial lag *r *is zero:

*c** _{X}*(

**Problem 2**

1) Consider a homogenous
space/time random field (S/TRF) *X( p)* where we know that its
covariance model verifies the following two expressions

_{}

_{}

where var* _{X}*=2.0

2) Now consider the following two expressions

_{}

_{}

where var* _{X}*=2.0

**Problem 3**

Before you start this problem, it is useful that you review some sample programs for modeling space/time covariance. For space/time data organized in the space/time grid format, the sample file is covEstimationSTg.m. For space/time data organized in the space/time vector format, the sample file is covEstimationSTv.m

For help concerning the space/time grid format and space/time vector format, use the following command

>> help stgridsyntax

When
you model the space/time covariance *c _{X}*(

To model the spatial component of the s/t covariance, select some spatial lags r=[r1 r2 ], set the temporal lag equal to zero, calculate the corresponding experimental covariance Cr=[Cr1 Cr2 ], plot Cr versus r, and find appropriate spatial parameters for the s/t covariance.

Similarly to model the temporal component of the s/t covariance, select some temporal lags t=[t1 t2 ], set the spatial lag equal to zero, calculate the experimental covariance Ct=[Ct1 Ct2 ], plot Ct versus t, and find appropriate temporal parameters for the s/t covariance.

1) Model the space/time covariance of the dataset datastg.txt provided in space/time grid format. Provide a write up describing your analysis. This write up should have a figure showing the experimental and model covariance similar to the covariance figure created in covEstimationSTg.m, it should provide the mathematical expression of the spatial and temporal components of your s/t covariance, and it should provide the expression for the full s/t covariance model. Include your code as appendix.

2) Similarly, model the space/time covariance of the dataset datastv.txt provided in space/time vector format.

**Problem 4**

Model the space/time covariance for your __project dataset__.
Provide a write up describing your covariance analysis. This write-up should
include your covariance model, it should have representative figure(s) showing
how well this covariance model fits the experimental covariance values, and it
should list your code in an Appendix. Discuss in that write up what you think
your covariance is physically representing. This write-up will serve as a part
of your final project where you present the covariance model analysis.

In order to help you with this problem, download the data file Raritan_Phos.txt, and review the estCovTutorial.m program that models the s/t covariance of Phosphorus data in the Raritan river basin.