Civil Engineering Reference
In-Depth Information
Fig. 2.8
Threshold crossing and peaks
For the type of processes covered herein it is a reasonable assumption that the joint
events
()
()
x t
x t
of
and
are
statistically
independent,
and
thus,
(
)
()
()
()
,
p
xx px px
=
⋅
. The average up crossing event that
x t
=
a
is then given
xx
x
x
by
∞
∞
()
(
)
()
()
f
ax
∫
pax x
,
pax
∫
px x
=⋅
=
⋅
⋅
(2.32)
x
xx
x
x
0
0
For each threshold up-crossing there is a corresponding down-crossing event, i.e
(
)
(
)
+= −
, although there may be several consecutive positive or negative peaks
in the process. Assuming that both
x
and
x
are Gaussian, then
f
af
a
x
x
2
2
⎡
⎤
⎡
⎤
∞
1
1
⎛ ⎞
1
1
⎛ ⎞
a
x
()
⎢
⎥
⎢
⎥
f
a
exp
∫
x
exp
dx
=
−
⋅
−
⎜ ⎟
⎜ ⎟
x
2
2
2
⎢
σ
⎥
2
⎢
σ
⎥
πσ
πσ
⎝ ⎠
⎝ ⎠
x
x
x
⎣
⎦
0
x
⎣
⎦
2
2
⎡
⎤
⎡
⎤
1
σ
πσ
1
⎛ ⎞
a
1
⎛ ⎞
a
()
()
x
fa
exp
⎢
⎥
f
0
exp
⎢
⎥
⇒
=⋅
⋅
−
=
⋅
−
(2.33)
⎜ ⎟
⎜ ⎟
x
x
2
2
2
⎢
σ
⎥
⎢
σ
⎥
⎝ ⎠
⎝ ⎠
x
x
x
⎣
⎦
⎣
⎦
1
σ
πσ
x
(2.34)
()
f
0
where :
=⋅
x
2
x
