Lab 6- Mixed Model and Kalman Filter

Given : 11/8/00 due: 11/28/00

1a. In a class presentation a student mistakenly tried to develop the mixed model using the following approach :

Mathematical model:

(1)

Where:

y denotes the vector of observations,

A₁ is the design matrix

A₂ is the design matrix

x is vector of unknown fixed parameters

e vector representing the random observational errors

X is vector of unknown stochastic (random) variables

–The dispersion of random observations

(2)

We assume statistically independentence between e and X, and obviously that the model error vector e is considerably smaller then the A₁X vector

1. .

We compute the value of x by plugging in the values of X, writing our model as a Gauss-Markov model:

(3)

Thus using LESS of GMM we get.

(4)

Derive the random part of our model by writing (1) as :

(5)

Using condition equation model ( w = Be with w = and B=).

(6)

n -number of observations, r - number of unknown random parameters

and the predicted random vector is

(7)

Rewrite the steps in which the student did mistakes in the derivation.

Explain the differences between the student model and the correct (given in class) mixed model, what problem will the student have when he tries to compute the dispersion.

b. If we suspect that in the mixed model ( given in equation 1) b₀ is affected by a certain scale error, we may replace the above prior information by

x₁ = b₀×w-e₀

with the additional unknown parameter w. Derive an equation system from which the LESS for x and x and w can be recovered directly, possibly with the corresponding Mean -Square Error or dispersion matrices, respectively.

c. To determine magnetic susceptibilities –k- of three bodies in the earth crust, measurements of the vertical magnetic field intensity - i -are taken on a profile and are given in the following table (S is the running distance in meters)

S(meters)	100	150	200	250	300	350
i (nT)	52.7	58.8	64.5	71.1	76.7	82.8

The standard deviation of the magnetic field intensity measurement i is ± 0.1nT.

The magnetic field intensity - i – is a summation of a fixed trend ( from the earth magnetic field ) and a random signal due to random magnetic sources in the crust,

The effect of the strong trend (the earth magnetic field) in the data will be modeled using a first order polynomial (y=ax+b); the coefficients of the polynomial a,b are unknown and to be estimated.

The random signal part of the magnetic field intensity - i – is computed by

where H_i the induced magnetic field in nT is computed as a function of the distance to the source; and is given in the following matrix

nT

Magnetic susceptibilities k_i are unitless and considered to be a random vector k with E(k)=0, and D(k)=s²×I. (s² = 0.09 ) .

Figure 1: Profile of the data and its modeling using a combination of a trend (y=ax+b) and random signal H×k.

Compute the polynomial coefficients with their dispersion and the three susceptibilities of the point sources and their dispersion.

(Hint use the mixed model of (1) where A₁=H; x=k)

(Note: We simplified the problem and skipped many of the details related to magnetism theory e.g. the computation of H. In reality the average earth magnetic field is 60000 nT (=b) and its change rate varies from place to place. The magnetic susceptibilities, however correspond to true values of rocks and minerals in SI units; olivine-Diabase ~ 25; Pyroxenite ~125 and Pyrrhotite ~1-6000, Guess which one is which,)

2.

a. Explain the difference and the equivalency between dynamic linear model (Kalman filter) and sequential Gauss Markov Model. Namely: for each write the model, the resulted prediction, its dispersion and explain what are the advantages and disadvantages of one model with respect to the other and what are the reasons for that.

b. An atmospheric measurement device was launched upward with initial velocity 51 m/sec from the earth surface. The distance y_k above the earth at time t_k was observed. The observations are listed in the following table:

T(sec)	0	1	2	3	4	5	6	7	8	9
Y(meter)	0	45.3	80.1	105.8	121.7	127.2	123.9	109.5	85.5	52.3

Write the state equation of the vector

Where S_k is the distance and V_k the Velocity (Hint: use g=9.81 m/sec² and physical laws to describe this vector in terms of S_k-1, V_k-1 and Dt=(t_k - t_k-1)).

Assume there is no system noise in the state equation.

Write the observation equation of y_k as a function of the X_k and assume a white noise for the observation error with a distribution of : e_k ~ (o, 12(meter)²)

Use this model to estimate S_k and V_k and their dispersions at each observation epoch. Given the initial value of the state vector to be:

with