Download PREDICT User`s Manual

Table 3-1.
Defining Parameters for the Impact Probability Density Functions
Mean longitude
l =
1 n
å li
n i =1
Mean latitude
d =
1 n
åd i
n i =1
Standard deviation of
longitude
sl =
Standard deviation of latitude s d =
1 n
(li - l )2
å
n - 1 i =1
1 n
å (d i - d
n - 1 i =1
)
2
å [(l -l )(d
n
i
r ld =
Correlation coefficient
i
-d
i =1
n
å (l
i =1
n
- l ) å (d
2
i
i
)]
-d
i =1
)
2
The value of the PDF function at an arbitrary longitude, l, and latitude, d, is given by the
bivariate Gaussian density function as shown in Table 3-2. This function is an
exponential function of the five defining parameters given in Table 3-1.
Table 3-2. Bivariate Gaussian Density Function
pdf (l , d ) =
e
ì
-1
ï
í
2
ïî 2 1- r ld
(
)
(
é l -l
ê
ê s l2
ë
)2 - 2 r ld (l -l )(d -d )+ (d -d )2 ùú üï
s ls d
s d2
ý
úï
ûþ
2
2ps l s d 1 - r ld
To obtain the impact probability density functions for all combinations of the time of
failure, fragment class and type of failure, the subset of PDFs is combined numerically in
the following manner.
First a grid of latitude, longitude points in the area of interest is defined. The number of
impacts per unit area is computed at each of the grid points. For each grid point and
14