27
The velocity of the drifting diffraction patte rn can be dete rmined from the correlograms b y the method
discussed below. The discussion is adapted from the paper by Briggs, Phillips and Shinn (1950).
The diffraction pattern m ove s over the ground as the properties of the diffracting screen vary. Its detailed st ructure is als o sub ject to random changes.
As the ob ser�able changes are confined to the two dimen sions of the ground pl&ne , it i �· convenie�� to represent the random changes as the result of a fictitious movement in the ve rtical direction. The random variations are really due to temporal changes in the diffraction pattern so that distances in the fictitious vertical direction must be measured in units of v t ' , where t' is the actual C time scale of the random variations and v is/4 proportc - ionality constant with the dimensions of velocity. In this way the problem is reduced to the study of an un changing diffraction pattern moving in three dimensions. Distances are measured in terms of the three cartesian components x, y, and v t (see Figure C 4.1a).
The true motion of the drifting diffraction pattern is given by the vector
V = ( v , V )
X y (4 . 1 )
The proportionally constant v was labelled C the characteristic velocity by Briggs, Phillips and Shinn
(1950). They labelled the vector v ' = (v , v , v )
H C X y C
(4.2)
the fading velocity (see Figure 4.1 (b).
The signifiance of the fading velocity v ' and C its three components is best shown by considering an observer at the origin of the axes in Figure 4.1 (a)
(observations can of course, only be made in the ground (x, y) pla,ne). If the diffraction pattern has no trans- lational motion and there are no random variations with time (i.e. vc = O), the observer sees a constant amplitude.
If the diffraction pattern is the result of reflections from a moving rigid surface with randomly varying reflect ion properties, the observer sees a randomly varying amp litude and could measure a fading period (as defined in Section 3.1 ). The observed fading period (�01) and the size (60, also defined in Section 3.1) of the irregular
29 amplitude variation parallel to the direction of motion are then related by the velocity of the drifting pattern
2 2 .1. V = (v + V )2
X y
= 60 / �01
The higher the drift speed, the faster the fading.
(4.3)
In contrast we can consider reflections from a turbulent reflecting medium with no mean horizontal motion. Then v = v = X y 0. The observer measures a fading period
(�02) characteristic of the random motion and the charact eristic velocity is, by its definition, ( o0 / �02). The diffraction pattern size (q,) will, in general, depend on the direction in the (x, y) plane along which the measure ments are made.
In reality the diffraction pattern changes as it drifts past the observer and the ratio of pattern size
( o0), measured parallel to the direction of drift, to the observed fading period (�0) defines the fading velocity.
v' c = oo
I
�o (4.4)The fading velocity is a function of the direction in which o0 is measured.
The fading velocity (v' ) C comprises two parts (see equation
(4 . 2 ) )
- a component (v) due to the driftmotion of the diffraction pattern and a component ( V C ) due to random changes in the diffraction pattern. Con-
sequently if an observer moves across the ground plane,
his speed relative to the diffraction pattern is a min imum when he moves with the same velocity as the drift component of the diffraction pattern and all he sees is the random fading represented by v . C Conversely if the observer moves with such a velocity that he sees a max imum fading period, indicating a minimum fading velocity, his velocity must be equal to the drift velocity (v) • This fact provides us with a definition of the drift velocity in terms of observable quantities. This def- inition of the drift velocity (v) also provides the H
definition of the characteristic velocity (v ) because C the minimum fading velocity, as measured by the moving observer, is just V C •
4.2 . DETERMINATION OF THE VELOCITY PARAMETERS FROM THE CORRELOGRAMS.
NOTE In the succeeding sections the distinction
between t he tetrachoric correlation, L (O , O, r), and the true correlation coefficient, r, is ignored since the actual shape of the correlation function is not required in the analysis.
The observable quantities in the correlation analysis are indicated in Figure 4.2(a) which shows
31
the mean auto-correlogram and one cross-correlogram of data recorded on 15 December 1963 at a reflection height of 76 km. It will be assumed, as in the Briggs, Phillips and Shinn (1950 ) paper, that the diffraction pattern in the ground plane is statistically isotropic, i.e. the pattern size. (00) is independent of direction. It will also be assumed that the two receivers are parallel to the drift velocity vector (v ). H � The latter assumption
is invalid for the particular correlograms of Figure 4.2(a) but provides us with a simplified picture so that the
principles of the correlation analysis may be more easily explained. The observable quantities used in the analysis are
the time delay for maximum cross-correlation (r ) m
rk the cross-correlation at zero time delay
�k the time delay at which the autocorrelation has fallen to rk
and � s the time delay at which the autocorrelation has fallen to r
These parameters are shown in Figure 4.2(a).
The space and time correlation functions can be represented , to a first approximation, as ellipses of
constant correlation in the ( o , �) plane, as shown in Figure 4.2 (b). It is assumed that the time and space correlation functions have the same shape, although this shape (i.e. the exact form of the correlation functions) need not be known. The ellipses of constant correlation must therefore be concentric and have a constant axial ratio. Figure 4.2(b) can be thought of as a bird's-eye view of a hill whose horizontal cross sections are the correlation ellipses , and whose peak is the origin with a maximum correlation value of 1
.o.
The auto-correlo- gram of Figure 4.2(a) is thus a vertical cross section of the correlation hill in the plane o = O , and the cross correlogram is a vertical cross-section at O = d0 , the spacing between the two receivers whose output signals were compared. The outer ellipse in Figure 4.2(b) is the lower contour r=
rk while the inner one is r=
r • mThe position of the rk ellipse is determined by its inter cepts with the o and � axes at d0 and �k respectively while the inner ellipse must be tangential to the line
0 = d • 0 0 is the intersection of the r ellipse with m m
the 0- axis and is not an observable quantity. From the definition of v'c (equation 4-4) , and the assumption of similar space and time correlation functions, we see that