Construction of Temperature Profiles by Least

Squares Polynomials$)$ Among the important inputs

in weather forecasting models are data sets consist$-$

ing of temperature values at various parts of the

atmosphere. These values are either measured dir$-$

ectly using weather balloons or inferred from remote

soundings taken by weather satellites. A typical set

of RAOB $($weather balloon$)$ data is given next. The

temperature T in kelvins may be considered as a

function of p$,$ the atmospheric pressure measured in

decibars. Pressures in the range from $1$ to $3$ decibars correspond to the top of the atmosphere, and those

in the range from $9$ to $10$ decibars correspond to the

lower part of the atmosphere.

$($a$)$ Enter the pressure values as a column vector p

by setting p$=[1$:$10]^(\text{'}),$ and enter the tem$-$

perature values as a column vector T$.$ To find

the best least squares fit to the data by a linear

function c$\_(1)$x$+$c$\_(2),$ set up an overdetermined sys$-$

tem Vc$=$T$.$ The coefficient matrix V can be

generated in MATLAB by setting

V$=[$p$,$ ones $(10,1)]$

or$,$ alternatively, by setting

A$=$vander$($p$)$;$,$V$=$A$($;$,9$:$10)$

Note For any vector x$=($x$\_(1),$x$\_(2),$dots,x$\_($n$+1))^($T$),$

the MATLAB command vander$($x$)$ generates

a full Vandermonde matrix of the form

$([$x$\_(1)^($n$),$x$\_(1)^($n$-1),$cdots,x$\_(1),1],[$x$\_(2)^($n$),$x$\_(2)^($n$-1),$cdots,x$\_(2),1],[$vdots$,],[$x$\_($n$+1)^($n$),$x$\_($n$+1)^($n$-1),$cdots,x$\_($n$+1),1])$

For a linear fit, only the last two columns of

the full Vandermonde matrix are used. More

information on the vander function can be

obtained by typing help vander. Once V

has been constructed, the least squares solu$-$

tion c of the system can be calculated using the

MATLAB $""$ operation.

$($b$)$ To see how well the linear function fits the data,

define a range of pressure values by setting

q$=1$:$0\mathrm{.}1$:$10$;

The corresponding function values can be de$-$

termined by setting

z$=$polyval$($c$,$q$)$;

We can plot the function and the data points with

the command

$($c$)$ Let us now try to obtain a better fit by us$-$

ing a cubic polynomial approximation. Again

we can calculate the coefficients of the cubic

polynomial

c$\_(1)$x$^(3)+$c$\_(2)$x$^(2)+$c$\_(3)$x$+$c$\_(4)$

that gives the best least squares fit to the data

by finding the least squares solution of an over$-$

determined system Vc$=$T$.$ The coefficient

matrix V is determined by taking the last four

columns of the matrix A$=$ vander$($p$).$ To see

the results graphically, again set

z$=$polyral$($c$,$q$)$

and plot the cubic function and data points, us$-$

ing the same plot command as before. Where do

you get the better fit, at the top or bottom of the

atmosphere?