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 p=(s,t), where s=(s1,s2) is the 2D spatial location, and t is time.  Assume that X(p) (measured in ppm) has a zero mean (over space and time), and that it has the following covariance

where r is the spatial lag (Km), t is the temporal lag (day), and the covariance parameters are c01=1.5 ppm2, ar1=1 Km and at1=3 days, c02=0.5 ppm2, ar2=30 Km and at2=700 days

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 c01, ar1, etc.

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:

cX(r,t =0) =

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

cX(r=0,t ) =

Problem 2

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

where varX=2.0 ppm2, ar1=1 Km and at1=3 days, ar2=30 Km and at2=700 days.  Find an expression for the space/time covariance model cX(r,t ) such that the above two expressions are true.  Is that a space/time separable covariance model?

2) Now consider the following two expressions

where varX=2.0 ppm2, ar1=1 Km and at1=3 days, ar2=30 Km and at2=700 days. Find an expression for the space/time covariance model cX(r,t ) such that the above two expressions are true, or if this is not possible, explain why.

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 cX(r,t ), the typical steps are to first model the spatial component cX(r=0,t ), then to model the temporal component cX(r,t =0).

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.