Interpretation of elevation-scan HF backscatter data from Losquet Island radar

Interpretation of elevation-scan HF backscatter data from Losquet Island radar N. RUELLE and T. LANIIEAU Fr~unce Telecon~Crntre

National

d‘Etudes drs T~l~communication4.

1X11

Lannion.

France

Abstract Most methods using HF ground hackscatter radar data to cstimvte the ionospheric bottomside electron density prolile rely upon mul~l-frcqucncy measurements of the minimum group delay. However. inli?rmation of the same nature cun also be extracted at a single frequency if the elcvatmn angle can hc precisely controlled. We outline the analysis of this technique. known as elevation-scan hackscatter wunding. The relevant parameter estimlrtion problem is studied using ;I Baycsian approach. We report on ;,n experiment using the Losquet Island radar to illustrate this method. The performance is compared to ionosondc datn. This tcchniquc provides u method of teledetection of the bottomside F-region electron density prolilc hundreds of km from the radar site however. further devclopmrnt is needed to protide increahed reliahilit> of the ebliatrs.

I. INTRODCCTION

recent paper by DYSON (1991) shows that the simulation of sweep-frequency backscatter soundings can be useful in interpreting experimental results from a backscattcr radar, in this case the Jindalce system. The cxpcrimental data may then be used to deduct the parameters of an ‘equivalent’ ionosphere. which Dyson dcfincs as reproducing the propagation charactcristics of the ionosphere accurately, though not necessarily describing the ionosphere itself. Even this is still a difficult task due to the strongly nonlinear nature of the associated inverse problem. Since 1990 we have been able to use the Losquet backscatter radar in a new cxpcrimcntal mode, elcvation-scan, which was not available at Jindalcc. It was then likcwisc ncccssary for us to simulate the backscattcr sounding process in order to understand the experimental results. We are able to obscrvc the same morphology and main characteristics in cxpcrimcnt and simulation of elevation-scan soundings. This encourages us to bclicvc that WC understand (in a preliminary fashion) the nonlinear action of the ionosphere on the sounding data. We may then also use the quantitative cxpcrimcntal data to deduce the parameters of the -equivalent’ ionosphere. Following the custom of most workers in this field. WC USC a quasi-parabolic (QP) ionospheric model. WC arc able, by using a Bayesian inversion technique. to show that there exists a unique set 01 QP ‘equivalent’ paramctcrs which is the solution to the inverse problem. for data values derived from an experimental elevation-scan sounding. We then prcscnt QP ‘cquivalcnt‘ paramctcrs dcrivcd from ;I

.4

series of cxpcrimcntal data. They arc compared to actual F-region ionospheric paramctcrs determined independently using ionosonde data.

2. THE

(‘NET

HF

BACKSCATTEK

LOSQLET

RADAR

AT

ISLAND

This instrument. also described by GALTHIEK c’f (1991) is situated near Lannion, on the coast of Brittany. It is operated by the Ccntre National d’Etudes des T&Ii-communications (CNET). Its main points of interest arc the antenna arrays. The transmitting array is circular. with 32 antennas and a diameter of 94 m. The rcccibing array is circular. conccntric. with 64 antennas and a diameter of I40 m. Art-a) directii’ity patterns arc controlled through the use of phase shifters on all antennas. This allows beam formation and steering both for transmission and reccption. and both in azimuth and elevation. The 3 dB bcamwidth is rather large compared to similar instruments. being about 5 in azimuth. and about IO in clcvation at 20 MH7. Beam steering. however. is possible over 360 in azimuth and from 0 to 90 in elevation. d.

3. EIXVZTION-SC’AN

BACKSCATTt:R

SOI’NDING

The Losquct instrument is capable of recording the received power P, from ground-backscattcrcd cchocs refracted by the F-region of the ionosphere. with discrimination in operating frequency f; group delay T,. ;izimuth and elevation pointing (main beam borcsight) angles A,, and E,,. When Gmuth and frequency

arc held fixed and the elevation pointing ;rngle E,, is stepped between 0 and 90 in discrete steps, an elevation-scan backscatter sounding is obtained. Roccivcd power Icvel P, is rcprescnted its it function ofelevation angle E,, and group delay T,, as in Fig. I. It is interesting to note the morphology, which is quite typical, exhibiting a single peak in the elevation anglegroup delay plane. with B signal-to-noise mtio of IO 15dB. An example of this type of data obtained using another radar systcln can bc found in tk raiw papet by (‘KO~I‘ (1972). It is known that for a single hlycr ionosphere. and for operating frequencies above the critical frequency ofthe ionosphere, oblique rays leaving the transmitter with increasing elevation angles exhibit decreasing 3“roup delay. reaching a minimum just before penctration of the ionospheric layer by the radio w;tves. Near this minimum group delay T,,,,,,. the so-called skip-zone focusing occurs. corresponding to :tn elev;ttion :mgle which WCcall the focusing angle E,,,, and resulting in an enhancement of the received power level. In this arca of the elevation angle~group delay plant (in the caxc of the Losquct radar) the signal-tonoise ratio hccomej large enough for a peak to hccome visible in the cupcrimental data. At operating frcquencies well ;tbovc the critical frequency of the ionosphcrc. the approximation that focusing actually occurs at 7;,,,,, is v:tlid. Thus the morphology of Losquct elevation-scan soundings is due to the combination of an ionospheric focusing process and the ;rngular filtering effect of the radar beam. WC call this morphology the focusing peak. The following simulation process :rllows us to confirm this.

1.

1.

P, * G, .G‘,. I’ . i’. dS (4-n)‘.LI

.

(1)

where P, is the transmitted power. G,(A, E) the transmit gain, G,(A. E) the receive gain, I the ionospheric attenuation. i. the wavelength, dS the backscattering cross-section of the Earth’s surface enclosed by the flux tube defined by the solid angle da, and L the

0.

olhcrwise

where r is the distance from the centre of the Earth. IV,,,the maximum electron density. I’,,,the KIIUC ol‘r at the peak of the layer, Y,,is the value of I’ itt the base of the layer. J‘,,, = I’,,,--,, is the layer semi-thickness. WC also note r,, the radius of the Earth, /I,,, = I’,,,- r,, the layer height, a&f:. the critical frequency of the QP layer. The expression for 7, is then analytical in 1; .L. II,,,, j’,,, and E. which is why WC USC the QP model. We wish to calculate the rcceivcd power within a (small) group delay gate dT,. The solid :mgle dR is computed so that its extent in clcvation dE corresponds to a group delay span of d.Tg. This provides the value of dS. E and 7, arc treated :IS constant within this group delay gate. Then by integration of dP, over 360 in azimuth we obmin the received power within a group delay gate Pa :

SlMC’L.\TION

The simulation of the experimental elevation-scan backscatter process is accomplished in a simple fashion by using ;I form of the radar equation derived from that adapted for backscatter by SH~ARMAN (1987) where the rcceivcd power df, IS expressed for 21small solid angle dR = d/l cos E dE around a ray with azimuth ;md elevation angles .4 and E. rcspeclively. rclativc to the radar site : dP, =

propagation distance, here held cqtutl to (c’. T_) fat simplicity. Array sizes and shapes. ;IS well 3s antenna types, were taken into account when modclling transmit and receive antenna array gains G, and G, at Losquct, as functions of fi A, E and pointing angles _4,, and E,,. C:tlculation of group dcluy 7, for the central ray of dR is :tccomplishcd using :I three-parameter quasip;tr;tboIlc model of the ionosphcr-e ;is defined by (‘K0t.I Lllld f-fOOC;ASlA;c (196X) giving the dtXtl_011 density :V,

s ihll

P,(/: 7,, A o. Ed =

I 0

dP,.

(3)

By giving ,f’and A,, fixed values, and having E,, vary. we are able to simulate the elevation-scan mode. as illustrated in Fig. 2. which shows P, in the 7,-E,, plane. The morphology of the simulation is similar to experimental scans. in that a clear peak is generally seen in both. This can be seen by comparing the examples of Figs I and 2. In Fig. I. the higher r&rtive values :rt high and low clcvations within the peak group delay gate are probably due to higher sidelobe levels in the elevation plane than those of the theoretical diagram used in the simulation of Fig 2. Numerical values of T,,,,, and E,,,, c;tn be computed by solving :

Elevation-scan

28

1992

Jan

10:24

UT

f =20.034

HF backscatter

MHz

105

data

cIzo= 1100

1r=0.30

IIS Pr


10.5

37.4 35.4

i ”

33.4

9.0

31.4 29.4

t 7.5

27.4

h

25.4

9

6.0

23.4

-

21.4

0

19.4

4.5

-

17.4 15.4

.-

13.4

15.6

1.5

0.30 3

15

27 ELEUCITION

Fig.

1. Example of Losquet elevation-scan

39 Eo

51


backscatter ionogram, showing 1992, 1024 UT, /= 20.0 MHz.

the focusing

peak. 28 January

106

fc=

N. RUELLE~~~ T. LANDEAU

14.5

MHz

hn=325

kn

yn=i!N

kn

f=20.0

HHr

Tr=tS,SD

ns

-81.0 -83.0 -S.U

-87.0 -8P.O -Pl.O -93.01 -55.0 -PT.0 -PP.0 -LcJl.O -103.0 -f06.0 -107.0

Fig. 2. Example of devotion-scan backscatter ionogram simulation, showing the focusing peak. f, = l4.s MI&, h, = 325 km, ym = I50 km, f = 20.0 MHz.

107

Elevation-scan HF backscatter data

lo:24

28/01/92

UT

hn



450 425 400 375

275 250 225 200

s’o 7 1’00’

1.50.

75 100 125 150 175 yn

0

10

20

30

40

50

60

Fig. 3(a).

70

so

Y” km

50

(km>

90

100

%

108

N. RUELLE and

2s/o1/92

lo:24

T.

LANDEAU

UT

Ihn

(km>

450 425 400 375 350 325 300 275 250 22s 200 I

fc

19

18

17

16

15

14

13

12

11

10

9

8

50

lb0

li0.

75 100 125 160 175 yn

0

10

20

30-

40

50

60

70

SO

Ym

kn

50


90

100

.%

Fig. 3. Example of inversion result : upin percentage normalized to peak value, in the h-h,,,, h,-y, and f,-y, planes, (a) cuts through peak region, (b) projections of maximum values. 28 January 1992, 1024 UT, f= 20.0 MHz (TeXP= 7.5 ms, Ecrp = 30”).

The coordinates of this peak are then the solution the inverse This

allows us to cheek that the position

lated peak in the Tg-E,, and E,,,,.

plane is in fact given by r,,,,,,

as explained qualitatively

Following

of the simu-

this interpretation.

in paragraph 3.

cxpcrimcntul

data arc

made to yield quantitative values of 7;,,,,, and E,,,,

: T,,,,

problem.

giving

the ‘cquivalcnt’

spheric parameters. Figure 3 actually shows the values ofthc trpa~t~iori probability

density, normalized

ii particular

to the peak value. for

numerical cuample. first in cuts through

three

orthagonal

[FIB.

3(a)]

xnd

passing

planes

through

also in projections

and EC,,,.These quantitati\c valuc~ of 7;,,,,, and E,,,,

\alucs upon these three plants

tigures give ;III idea of the morpholog)

for ;I given model prolilc. l‘tmctional relations

It may hc noted that the

thus obtained arc vcr! nonlinear.

rcmolc sensing.

which cnahlc us to tackle the

or invcrsc,

[Fig.

problem of dctcrmining

of the

method typicall!

shown to b’crather badly detcrmincd by this invcrzion taint!

tours in the f, /I,,, plant [Fig. i(b)].

marifcstcd

This INVERSE

Thcsc

method. f; and Ir,,, exhibit ;I dcgrcc of coupled unccr-

ionospheric ‘cqtiivalcnt’ parameter5 within the sound-

5.

3(b)].

dctcrmlnes the model parameters. The parameter I‘,,, is

ing zone. from apcrimuntal

data.

pxk

ol‘~~. which is

they allow an estimation

accuracy with which the inversion

Thcsc two steps of data reduction and simulation 1a) the foundations

typical. In particular

this

of the maximum

can also bc simulated ~15functions off,.

/I,,,. .I.,,, and,/

to

iono-

by the elongated shape of the ca-

well-posedness is in contrast with the situation

of the ‘Icading-cdyc‘ inverse problem, as pointed out

PROHI.EZl

by Dusou

The in\,erse problem of determining ,I;. It,,, and J’,,,

This

(1091 ). and R~l.1 1.1.and GAl!IHn:i<

(1990).

is generally ill-posed and must bc constrained bq

kno\\ing /. 7 ,,,,,, and E,,,, was the subject of a Ph.D.

cithcr rcduciny tlil\ number of model paramctcrs IN

thesis

by RLI:I.I.I.

providing

mcnr.

;\s

(I99 I).

described

in

A formal

Hayesian

the Appendix.

II-cat-

was chosen

more information.

(1991) the constraint

In the cast of DJN)~\;

is given b! normalizing

the data

because we wished to characterize better the strongly

using the muaimum values of frcqucncy and group

nonlinear

delay on the Lading edge and (potcntiallq)

relationship

between the data values and

the model parameters. important bcforc

question

resorting

and in particular

(II‘ the uniqueness

study

the

plcmcntar;,

01‘thc solution.

to 21more usual iterative leai;~ squares

In thl5 ~a),

information

information. ij

applying to this ln\crsc problem the

“cncr;11 I’ormulation of T.WAMOI.A ( 19X7). wc obtain cc the complctc solution to the invcrsc problem by cvaluating the (I ~IO.S~~V%J~~ probability parameter \pacc (1;.

/7,,,. .I‘,,,).

In our cast the additional

provided

mcasurcmcnt E,,,,. This

inbcrsion scheme.

using the

trailing cd!:e of the sweep-frequency ionogram as supby

the

elevation

angle

constraint is sufficient to regu-

larize the invcrsc problem. allowing the dctormination of numerical values for QP ‘cqui\alcnt’ parameters lbr an! set al‘cxpcrimental

data.

density u,, over the

This

is given by (A 1 I)

the Appendix :

6. l3PERlMENT/~I.

As

a

RESULTS

trial of these techniques. the Losquet

radar

was used to acquire cxperimcntal data for a 7 h period on 2X .lanuary Azimuth

lY92.

pointing

bctwccn 0830 and 1530 l.lT.

angle .,I,, was

scatter from the pround (Alpine quencics. For

ii given set of expcrimcntal

the evaluation of 0,) over

a

values TL,,, and EL.,,,

grid of points covering the

MHz

and tMent!-two

paramctcr hpace (I,. /I,,,. ,I.,,,) is necessary. A complete

20.0 MH7.

cxploratlon

tocol and

of pi,, values in three dimensions

shows that. for R QP ionospheric formation

model.

then

the trans-

:

16.7 and 30.0

MHz.

scvcn clcvation-scan soundings

1IO . giving back-

region) and two l’rcwere used. Twenthwcrc rccordcd at 1h.,-7

clcvation-scan

soundings

We now describe the experimental

;II pro-

I IIC protocol for data analysis. I. Durillg each elevation-scan sounding. nine suc-

cessive elevation scans were performed. For each scan

( 1, ~w 4#>,1--t ( L. h,,,.j,,,,)

(6)

the elevation angle E,, \aricd in 3 steps between 0 and 60 (ciE,, = 3 ). For each of the 21 E,, values

of a

scan.

is generally a well posed inverse problem in the sense

the radar was operated during a coherent integration

that the LI po~,rcriori

time of 0.3 3. and the resulting

probability

single peak region within

density (J,’ exhibits

the three-dimensional

a

space.

to yield rctreived power

signal was processed

as ;i function

or group

delay

:

I IO

N. RUELLE and T. LANDEAL 12

1

Twin

11


10

9

8

7

J

6

5

4

3

2

1

4

(a)

8:00

O

LO

1

lo:oo

9:oo

Efoc

ll:oo

12:oo TIME

13: 00

14:oo

15:oo

14:OO



57 54 51 48 45 42 39 36 33 30 27 24 21 18 15 12 9 6 3 (b)

O

4

8:00

9:oo

Fig. 4. Experimental

1o:oo

11:oo

12:oo TIME

13: 00

14: 00

results from 28 January 1992. (a) group delay and (b) elevation ,f = 16.7 MHr. x : / = 20.0 MHz.

15:oo

focusing

lb:00

angle, l

:

Elevation-scan HF backscatter data P,( T,), the group delay gate size ST, being 0.3ms. The total scan time was 12.8 s. Then the nine received power values were averaged within each TX-E,, gate to eliminate rapid fluctuations mainly due to unresolved multipath. This gave the elevation-scan sounding : P,( T,. E,,). The total sounding time was 2 min. Soundings were recorded every I5 min for each frequency. 2. After examination of the sounding results. a moving average within each Tg-E,, gate over three elevation scan soundings (45 min) was performed, to yield averaged soundings with better signal-to-noise ratio. An example of an averaged sounding is given in Fig. 1. 3. The coordinates ( Tcyp.EC,,) of the focusing peak in the T,-E,, plane were recorded for each averaged sounding. The resulting time series are represented in Fig. 4(a) and 4(b). Uncertainties arc: 6T,,,,,, = 0.3 ms and iiEioL= 3 For each frequency. data pairs are obtained on average every I5 min for 7 h. The results show stable values for r,,,,. resulting in averages 01 6.5 ms at 16.7 MHz, and 7.5 ms at 20.0 MHz. &.,p is less stable, with occasional rapid variations. Averages arc 39 at 16.7 MHz and 33 at 20.0 MHz.

20

fc

III

4. Inversion was then performed by calculating cr,& !I,,,. r,,,) for each data pair according to equation (5). A solution (single peak region) was found in every case. Figure 3(a) and 3(b) gives an example of results from such an inversion. The coordinates (./i. h,,y,,,) of the maximum value of CJ,,were then recorded. The resulting time series are shown in Fig. 5(a), 5(b) and 5(c). Occasional rapid variations of,/: and II,,,are seen. associated with the variations of the focusing angle mentioned in 3. indicating that this inversion method is very scnsitivc to variations of E,,,. J’,,,exhibits great instability. 5. WC compared the ‘equivalent’ ionospheric parametcrs thus obtained with independent ionospheric data in the form of electron density profiles. Due to the simplicity of the QP model a perfect match could not be expected although some similarities could. In order to do this, vertical incidence ionograms from the ionosonde at Poiticrs (roughly 200 km from the ionospheric region sounded by backscatter) were processed using the real-height program POLAN [see TITHERIDGE (1985) for details] to obtain values of f;,, ?. /c,,, 2 and J,,,~ 2 every 15 min during the study

(MHz>

19 1 1s 17 16

1 I

15 14 13

1

12

(a)

11

-

10

-

9

-

8

-

7

-

6

-

5

-

4

-

3

-

2

-

’ O

s:oo

9:oo

1o:oo

11:oo

12:oo TIME
Fig. 5(a)

13:

00

14: 00

15:oo

1.4:00

II?

N. RWLLE and T. LANDFAI~

450

hn



425 400 375 350 325 300 275 250 225 200 17s 150 125 100 75 50 25 ‘T

Cb)

’

’

I: 00

200

k ,m

9:‘oo

IOiOO

lliO0

12:‘oo TIME

13:‘oo

14:‘oo

tsioo

16iOO



175

150

125

100

75

50

25

Cc)

0

s

::0 IO

9:oo

10: 00

Fig. S. Invcrslon results from 2~ January

ll:oo

12:oo TIME

13: 00

14: 00

1992. (a) f;. Ch) h,. CC)J‘,,,. 0 : f = 16.7 MHz. + : vertical incidence.

15:oo

x

14:oo

: f= 20.0MHf.

period.

These

values are also shown in Fig. 5(a), S(b)

and 5(c). They show quiescent ionospheric conditions during

the whole

7 h period.

with

slowly

varying

values of/,,, ‘. h ,,,, 2 and J’,,,~>.

This

7 h stud)

period

shows

that

‘cquivalcnt’

parameters 1; and /I,,, dctcrmined by elevation-scan hackscattcr are reasonably

similar

for the two fre-

qucncics. and reasonably close to ‘real’ values. It is.

6. The differences between the two sets of values:

howcvcr.

too short

a period to give a statisticall!

(/,p/;,, ?). (/I,,,-/I,,,, >). (.I,,,,--\‘,,,, ?) WCKanalysed:

significant c\,aluation of the difference between ‘equi-

0 For ( f;----/i,, ?) all ~alucs u-c in the interval [- I MFix. +I? MHz]. Iiowc~ci, Mhilc at 70.0 Mb17

method.

the mean error

ix close to U-O.

Mhich would stem

to indicate that the ‘cquivalcnt’ I, Fcnerally unbiased indication

of the value of,f;,,

balcnt’

an
‘real‘ to

ionospheric

parameters

by this

gi\c any prccisc indication

01‘ the

rcason’r foi the diicrcpancics which occu-.

gives an

T within the

\olumc of the ionosphere sounded by the backscattc~ radar. at 16.7 MHL

there is ;I mean error

of about

-t-Z MH/. 0

[ --3

I-or

[

2) all calues arc’ in the interval

h-m. t- 100 km]. and thcrc arc mean crrorb

of‘ ahout N.0

(if,,,-//,,,,

+ SO km at IO.7 MHr

and +15

km at

hlH/.

0 For (J’,,, - J’,,,~2) all \alucs arc in the interval 75 km. -1 125 km]. Thcrc are mean errors ot‘about

t 75 km :lt 16.7 MH/

and 0 km at X.0

Mf-l;l.

Wc have demonstrated

acquisition

scan backacattcr data. extraction

of elevation-

01‘ focusing

coordinates ‘T,,,,,, and E,,,,. and determination

calcnt’ ionospheric paramctcrs using a QP model. We arc thub able to provide tclcdctcction of ‘equivalent’ parameter‘;. at ;I given I’rcqucnc) and in a given dircction. This

teledctection applies to ;I well dcfncd

of the i~~nojpherc (sounding zone) bccansc.

Consider-inp the total range of!,,,,. which ib 125 km.

IS due to ;I cone of rays which is narrow

docy not appear to be reliably determined. and com-

arimuth

parisons ~.ith .I’,,,, - arc

of no real

valiic.

For f; and /2,,).

points

may be due to horizontal

vari-

ations within

the ionosphcrc such as travellinp

iono-

spheric disturhunccs.

~ (rradicnts or tilts

clcvation angle dctcrmination. nature 01‘ the propagation.

to which the

because of the oblique

would be mnrc scnsitivo

and elevation. This

‘leading-cdg’

methods

me:lsurements.

and

bc rotated over 360

USC multi-frcqucncy

Also. bccausc ~hc hcam can

in Cmuth,

.AnaI>sis ofc\pcrimcntal \alcnt’

parameters

actual

me;in

01‘ the ‘equi-

which

I‘uturc cxpcrimcnth

will enable us to claminc azimuthal

than the \crtical incidcncc sounding and L\hich would

both in

is in contrast uith most

thus sample the ionosphere O\‘CI

an e~tendetl ;lrca radially.

dri\c the occ:!sional

rapid variations

arca

since axi-

muth and frcqucncy arc kept tixcd. the skip-zone echo

and the larpc \,ariations seen at both frequencies. .i‘,,,

the outlying

peak

of ‘cqu-

variations.

data shows that the ‘cqui-

;~rc not \;cry diKei-cnt from

ionobphcric

clcctroii

density

the

profile

valent parameters which al-c:not rctlcctod in the vcr-

bvithin the :,ounding miic.

tical incidence data. On the other hand at 16.7 MH;/

imcntal data to obtain statistically

the cxistcncc 01‘ a mean error for all thrco paramctcrs

on the ditTcrcncc between ‘equivalent’ and ‘r-cal’ iono-

compared

to vertical incidcncc

occur at 20.0

data. which dots not

M HL. may indicate a defect in the model

\\hich could bc corrected cmpiricallq.

We now need more cxpcrsignificant bounds

spheric parameters and to study reasons for the discrcpancies, inc!udlng the ctrect of ionospheric layers and horizontal

gradients.

lower

APPENDIX In this appendix we derive an expression for the ~rposfc~iori probability density op. for the inverse problem defined in Section 4. using the Bay&an methodology described by TARANTOLA (1987) and using his notation. This cz p~s~rir)ri probability density is defined over the parameter space P f f;. h,. J,,,,) so that iIlt~gr~iti(~n of CT!,over any area of P provides the probability that a solution of the inverse problem lies within that arcit. In f:tct (iI, embodies the state of information on the inverse problem. for it $c.en set ofcwperimental d;lt;z. after we have interpreted those d;ltn in the light ofour knou,ledge of the direct (modclling) problem.

errors on data values q ,,,,, and E,,, are Gaussian. independent. with variances (ST,,,,,, and 6&,,,. We then have:

0 represents the itlform~ition from tbc model. This inform:trion is contained in the operator I:: p + rl which solves the direci problem. We know that the probzrhitit) density 0\c1- D 2 P is :

Ntl.p) = O,,(tllp)“l,,(/‘l

of,(djp) = ri(d- Fipj). (Al) Since /J reprcscnts WC may write :

(Ah)

\+hcrc O,,(C/]/I)is a conditional probability density describing the information and uncertainties wc’hare on the values ol the data, for each set ofv:tlues of the modct p~r~~rn~ters, and i!,, is the uniform prohabitity density otcr P. If we suppose that we have i!n eY:+ct model tlcscribcd by /? then we hnvc t hc point distribution : (A7)

TT,‘, which wc wish to calculate. is the marginal probability density given h) :

the state of no inf~~rt~l~iti(~n over If x P.

P((l.P)= P.,(‘I)‘/J,,(P)

(Al)

where /tz, and /L,, are uniform probability densities. Then J’ is the N p&ri probability density, that is. it represents the jnform~tion that we ha\c before wc begin to interpret our experimental data. Assuj~iing that the prior inform~ti(~n about data and pnr;rmoterz is obtained by independent processes, MC may then uritc:

p(tl.p) = (l,,(d)‘1J,,t/‘1.

(Al)

and by using (.A71

and (A6) into (AX) wc then hliv;e:

:

(A31

In our cast the only prior information that we assume on model parnmeter values is that they arc bounded within 21 domain P’ of I’. Thus : P,,(P). /‘E P /J,,(P)= 0. pd:P’. i case. P’ is detincd by: 5 MHz < /; < < h,,, < 450 km : 50 km < j‘,,;< 175 km.

By substituting

so that by taking into account (A4) mod expression :

(95)we have a final

(A4)

In our 20 MHz: 200 itm assuming F-region parameters. Let the experiti7~llt~~l data set be called : (T,,,, EC,,).There is an uncertainty attached to the measurement process. We wilt suppose in this cast to simplify matters that measurement

(A I I) is then numerically evaluted over P’ to yield the complete state of o pt~,s~~rjffrj inforln~iti~~il associated with a given set of experimental values. including the existence and tmiquencss of ii solution.