# Let Z(t) be a stationary TRF with mean mZ(t) and covariance CZ(*)

```ENVR 765 Exam 1
Date given: 02/18/2014 Due: 2/25/2014 noon
This is an open book take home exam. You may use course notes and books, but you cannot use help from
somebody (e.g. no help from classmates or students who have already taken the course). Your answer may
be typed or hand written, which ever you like better. The exam is due at noon. You may send it by email
(scan your answer if it is hand written) or drop it in the instructor’s mailbox. If you can’t do that, then you
may hand it in at the beginning of class the day the exam is due.
Let Z(t) be a TRF with a known covariance CZ(), and with a mean trend mZ(t) that may
or may not be known. Let tk be the estimation time, and let th==[ t1, …, tn]T be the times
at which the hard data is available. Then the random variable zk=Z(tk) and the random
vector zh=[Z(t1), …, Z(tn)]T are correlated with known covariance, so that
Ckk=cov(zk,zk)=var(zk), Chh=cov(zh, zh) and Chk=cov(zh, zk) have known covariance
values. Assume that the mean trend is given by mZ(t)=o+t+t2 where  =[o, ,
]T are parameters that may or may not be known.
1) First, assume that  is known and given by  =(X TChh-1X) -1XT Chh-1 zh,meas, where X =
 1 t1 t12 


 1 t 2 t 22 
2

 , X0= 1 t k t k , and zh,meas are the measured value for zh. In other words,
 ... ... ... 
1 t t2 
n
n 

you assume that there is no uncertainty in , and as a result the mean trend is known.
Derive the best linear unbiased estimate ẑ k for zk given measured values for zh. Show all
the steps of your derivation so that the grader can follow how you go from one step to the
other. Your answer may be typed or hand written, which ever you like better.
Hints: This is a type of kriging estimator covered in class; it is up to you to find which
one. In order to show all the steps of your derivation, the best is that you derive the steps
yourself and write each step of the derivation as needed for you to be able to go from one
step to the next.


2) Now assume that  is unknown. Derive what is in this case the best linear unbiased
estimate ẑ k for zk given measured values for zh.
Hints: This is a different type of kriging estimator, which we also covered in class. Again
make sure to show all the steps for your derivation of this estimator.
3) Let’s assume that CX()=20 CFU2 exp(-3/5 hr), and that at times t1=2 hr, t2=5 hr ,
t3=10 hr, t4=15 hr and t5=16 hr we have the measured values X(t1)=29 CFU, X(t2)=17
CFU, X(t3)=12 CFU, X(t4)=16 CFU and X(t5)=10 CFU. For both kriging estimators
derived above:
a. Calculate at time tk=8hr the mean trend value, the kriging estimate, and the
corresponding estimation error variance.
b. Plot the mean trend and kriging estimate +/- its corresponding standard deviation
from times 0 to 20 hr.
c. Explain the difference, or lack thereof, in the results between these two kriging
estimators.
```