UAG-93: Chapters 1 to 5 - URSI

The time taken by an analysis depends primarily on the number of group refractive index
calculations, and hence on the number of virtual-height calculations. Each cross in Fig. 1 represents
a frequency at which the virtual-height integral must be evaluated, to advance the analysis one step.
The time depends only slightly on the number of terms used jn the real-height expansion, since (for a
given i) all the terms B(i,j) where j = 1 to NT are obtained from the same set of group indjces
(Appendix D.1). The overall time for each mode js, therefore, roughly proportional to the value of
NV, divided by the value of NH (the number of new real heights calculated at each step). This gives
the approximate relative times NV/NH Iisted for each va.lue of l"lODE in Table 2. Following co.lumni
give the measured times for the analysis of an ionogram with 20, 40, 60 or 90 scaled data points, on a
Burroughs 86700 computer. The fr'rst set of times is obtajned by specifying a value of l,4ODE from 1 to
i0. This gives five-point integrais (except for Modes 9 and 10) which are generally adequate at low
and medium latjtudes. As shown at the bottom of Table 2 the mean times for modes 1 to B are given (to
with.in a fewpercent, at N > 20) bytheexpression T = 60N +0.BN2msec, where N is the numberof
virtual-height data points.
For max.imum accuracy 12-point integrals are specified, by calling P0LAN with the value of M0DE
increased by 10 (Appendix D.1). l'4odes 9 and 10 aiways use 12-point integrals, since they include a
large number of data points at each step, so times and results are the same whether l4ode 9, 10 or Mode
19,20 is specified. A zero value for l40DE uses the default value M0DE = 5 at djp angles up to 60o;
this changes automatically to MODE = 15 (for" 12-point integrals) at dip angles of 60'or more.
l''1odes 9 to 20 also shift automatically to 17-point integration when required, to maintain accurdcy at
dip angles up to 90'(Appendix 8.3). Using 12-point integrals the mean times in Table 2 are
approximately equal to T = BON + 0.BNl msec. Thus for a normal mix of 5 and 12-point integrals,
changes in the number N of scaled virtual heights causes the time required for a real-height
calculation to vary approximately as N(1+N/90).
5.3 Curve Fitting Procedures
The accuracy of real-height calculatjons depends on the dccuracy with which we represent the
shape of the profile, between the frequencies at whjch virtual heights are known. All hjgh-order
modes of polynomial analysis are designed to fit one or more virtual heights beyond the point at
which the next real height is to be calculated. Thus the shape of a calculated profile segment is
determined using data from both sides of the segment, rather than from only the known (lower) part
of the profile. This corresponds to the use of central rather than one-sided interpolation, for
estimat.in gradient changes in the calculated section, and is the main reason for the increased
accuracy and flexibility of overlapping-polynomial methods.
The different modes in P0LAN correspond to different select.ions of the points to be fitted
at each step of the calculation. A polynomial expression, with NT terms, is fitted to data at a
succession of given frequencies F;. The origin of the polynomiaJ is taken at the frequency Fo,
and the fitted frequencies correspond to values of i from 0 or -1 up to NV. Over approximate)y the
first half of this range, fitting is to previously-calculated real heights. At all values of i from
I to NV the polynomial is fitted to values of virtual height H'i (corrected for group retardatjon
due to ionisation at plasma frequencies less than F6). Thus in ahe first half (approximately) of the
range the polynomial is required to have the same real height and the same slope as the previouslycalculated
sectjons. In the second half of the range only virtual-height data are known. This does,
however, contain all the information which is available about the profile, so by fitting the
virtual-height data we are matching all available information.
For modes 1 to 4 the number of polynomial terms which define each real-height section is equal to
the number of heights (real and virtual) included in the analysis. The result is therefore an exact
fit to the NV virtual heights and NR known real heights. |,,/ith the higher order modes the number
of defining quantities (flV + 1Xp1) is greater than the number of terms NT used jn the real height
expansion (9). This gives a least-squares fit, with some smoothing of the virtual-height data.
Real-height equations are given a larger weight in the analysis, so that the known reaj heiqhts are
fitLed almost exactly.
_Virtual-height
points at each end of a calculated profile segment (the heavy lines in Fig. 1)
should be fitted accurately. Points at higher and lower frequencies are required only to jndicate
the general trend of the profile, and are given less weight in least-squares modes of analysis. This
causes the gradients at the ends of the calculated segment to be defined more closeJy, at the expense
of points further away. Individual data points also enter and pass out of the calculation reqion more
gradualJy, when the weights are varied, so that spurious fluctuatjons are minimised.
The effective weights given to the real and vjrtual-height data points are shown in Table 3,
for each of the least-squares modes of analysis. For real-height points the weight is effectively
infinite at Fo, corresponding to the origin (FA,HA) in (9). The weight is laige at known real
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