Modelling the Bias of Abel Inversion Algorithm for Ionospheric Vertical Electron Density (VED) Profile Retrieve Gary Ouyang and Jinling Wang School of Surveying and Spatial Information Systems University of New South Wales, Sydney NSW 2052, Australia (Tel: +61-2-9385 4185, E-mail:
[email protected]) Abstract:
When GPS signals are transmitted to the receivers installed in the LEO (Low Earth Orbit) satellites, these signals are bent by the atmosphere and the ionosphere. The vertical electron density (VED) profile of the ionosphere can be retrieved by calculating the signal bending angles by the Abel inversion algorithm. But this retrieved VED includes a big bias (about 20% ~ 40% relative RMS in average) because this method is based on the two unrealistic assumptions: 1) electron density spherical symmetry; and 2) electron density even distribution in each layer. Therefore, the retrieved electron density only roughly represents the ionospheric general distribution in a wide region, and is not accurate for a specific location. To overcome the above problem, this paper has proposed a new method. Firstly, the bias from the Abel inversion algorithm is numerically modelled; and secondly, the accurate VED at a target location is generated by removing above solved bias. The validation of this technique is also discussed through the data comparisons between the proposed model and the ionosonde profile (as “real” one).
Keywords: Vertical electron density, Abel inversion, Bias model
1. Introduction
an error autocorrelation function with location dependence.
The radio occultation (RO) technique has been developed since 1960s by JPL (Jet Propulsion Laboratory) and Standford University. Initially, the purpose was to research other planets. From 1995, the RO concept has been combined with GPS technology to study the Earth’s atmosphere and the ionosphere. The GPS receivers are on board LEO satellites (i.e. COSMIC, CHAMP, SAC-C and GRACE etc) with orbit altitudes around 300km~1000km and forward and backward real time sounding GPS signals (Hajj, et al., 2002). The radio signal from a GPS satellite to a LEO satellite is called as radio occultation (RO). The GPS RO system consists of GPS receivers, high-gain occultation antennas and precise orbit determination (POD) antennas. About 2,500 globally distributed RO are sounded every day. When GPS radio signals pass through the ionosphere, they are refracted and/or diffracted. The result is that the bending angles and the phase variation are generated. Through calculating these bending angles and excess Doppler frequency shift by the Abel inversion algorithm, the vertical electron density profile response to different layers are generated.
2. Methodology
In the Abel inversion technology, the ionosphere is regarded as having the property of local spherical symmetry, even distribution and electron density variation only with regard to altitude. Of course, such assumption is not realistic. Therefore, the retrieved VED profile only roughly describes the ionospheric variation in a wide area in stead of a positive position. But the advantages of this method is that only small amount of observed data are processed by the simply algorithm compared with a large amount of data in the tomographic reconstruct technique. If the retrieved VED profile can be further revised to generate the VED response to a given position by a given algorithm, it will be valuable (Hajj et al., 2002; Hajj et al., 2004; Garcia-Fernandez, 2004). In this Chapter, an improved Abel inversion algorithm is proposed. This improved algorithm can make the retrieved VED best fit for the conditions of the target position through introducing an Abel inversion bias model that is expressed using
2.1 The Abel inversion Based on the assumption of electron density spherical symmetry and even distribution in every layer, VED profile can be retrieved by the Abel inversion algorithm using RO data. The method is achieved through three steps: 1) obtaining the bending angle; 2) generating the refractivity index profile; and 3) retrieving the vertical electron density profile. In fact, when the RO signal passes the ionosphere, the bending angle is about 0.030 in the ionospheric F layer and could be ignored. Although the bending angles in D and E layers are a little bigger, they can still be ignored because the bias caused from this is much smaller than the bias from the ionospheric spherical symmetry assumption. Therefore, the RO is regarded as traveling in straight line (Hajj et al., 2002; Hajj et al., 2004; Garcia-Fernandez, 2004).
l2,j
l2,j
…
2 li,j
GPS
…
LEO
1 2
2 l1,j
Layer …
i
…
n
…
i
…
2 1
Fig. 1: The RO and n layers ionosphere Fig. 1 describes the ionosphere divided into n layers from the bottom boundary (i.e. 50km altitude) to the upper boundary (i.e. 2500km). The space above the orbit high of the LEO satellite is assumed as that there is only one layer (or layer 1). If the LEO orbit height is H km, the layer 1 is from H km ~2500km. Then layer 2 is under layer 1, layer 3 is under layer 2, and so on, till layer n. In this paper, the LEO COSMIC data are used, and the LEO orbit height is defined as 400km. Each layer is 5km and the total layer number is 71. Fig. 1 also shows the jth RO signal
passes ith ionospheric layer.
2l1 j is the length from the LEO
satellite to the ionospheric upper boundary. 2l ij is the intercept length in ith (the most inner layer of this RO signal).
where k is the number of the layers in the ionosphere; ltop is the total layer number above the h m F2 which is from the retrieved Abel inversion; εTOPk is the accumulated bias in the
l kj ( k = 1, " ,( n - 1)) is the gap distance between the kth and the
kth layer (above the h m F2 ); 1 / β top is the correlation layer
(k+1)th layer boundaries, which is along the jth RO signal ray path.
which is equal to 1 / ltop when there are total ltop layers above the h m F2 ; N m F2 is the retrieved peak electron
Therefore, the VED can be retrieved from layer 1 to layer n by the “peeling onion” method according to the following formula (Kaplan, 1996; Garcia-Fernandez, 2004; Gaussiran et al., 2004; Liu et al., 2005):
density from the Abel inversion; ζ top is the bias factor related to top layers and need to be solved. Equation (3) means the accumulated bias in the topside will
Ni ≅
i −1 STEC j − 2 ⋅ ∑ N k ⋅ l kj k =1 2 ⋅ l ij
gradually decrease from the maximum ζ top ⋅ N m F2 (in the (1)
th
where N i is the i layer electron density; STEC j is the total electron content along the jth RO signal ray path from the GPS to the LEO satellites, which can be calculated as:
layer of h m F2 ) to zeros in layer 1 following the autocorrelation function e
− β top ⋅(ltop − k )
.
2) Bottom side part --- below the h m F2 :
εBOTk = e
2 2 f1 f 2 STEC = ( P − P ) + Bias j 1 2 2 2 40.3( f 2 − f1 )
− β bot ⋅ (lbot − ( k − ltop ))
⋅ (ζ bot ⋅ N m F2 )
k = (ltop + 1), (ltop + 2), " , (ltop + lbot )
(2)
(4)
where f1 and f2 are the GPS signal frequencies (unit: MH), P1 and P2 are the smoothed pseudo-ranges.
where k is the number of the layers in the ionosphere; lbot is
2.2 Modelling the bias from the Abel inversion algorithm
retrieved Abel inversion; εBOTk is the accumulated bias in the
Some prior assumptions of the Abel inversion algorithm (Section 2.1) cause the big bias of the retrieved VED. This bias has a major character of gradually increasing (or accumulation) from the top layer (i.e. 400km altitude) to bottom layer (i.e. 50km) because of using the “peeling onion” method for the solution. When the electron densities are solved by Equation (1), the layer 1 electron density is calculated first, and then second layer is generated based on the layer 1 result. After that, layer 3 electron density is computed by the results of layers 1 and 2, this procedure will continue till all layer electron densities are retrieved. Therefore, the bias of the retrieved electron density will become bigger when the altitude decreases. Based on the above bias character, an autocorrelation function can be arbitrarily used for modelling this bias accumulation. According to the results of some empirical models (i.e. Chapman profile), the VED profiles can be approximately expressed by the topside (above h m F2 ) and bottom side (below h m F2 ) exponent functions of the altitude. Therefore, we can still model the bias accumulation as two exponent functions of the altitude (topside and bottom side) as follows:
εTOPk = e
kth layer (below the h m F2 ); 1 / β bot is the correlation layer which is equal to 1 / lbot when there are total lbot layers below the h m F2 ; N m F2 is the peak electron density from the retrieved Abel inversion; ζ bot is the bias factor related to bottom layers and need to be solved.
Equation (4) means the accumulated bias in the bottom side will gradually increases from the layer of h m F2 , and reaches to the maximum value ζ bot ⋅ N m F2 in the layer n (total n following the autocorrelation function
layers)
e
− β bot ⋅ (lbot − ( k − ltop )) .
The VED profile, which is retrieved by the Abel inversion, only represents the VED rough distribution in a big area. Each of the specific locations among this big area has its own VED in distribution. We can regard that each pair of ζ top and ζ bot Equations (3) and (4) are related to a given location or they are the function of the location. After a pair of ζ
1) Topside part --- above h m F2 : − β top ⋅(ltop − k )
the total layer number below the h m F2 which is from the
top
and ζ
bot
,
which are related to a specific location, are solved, εTOP and k
⋅ (ζ top ⋅ N m F2 )
εBOTk can be calculated by Equations (3) and (4), and then
k = 1, 2, " , ltop
(3)
they are removed from the VED which is retrieved from the Abel inversion. Finally, the VED related to this given location can be
generated as below: nVED top − Ionosonde top
nVEDtop = VEDtop ± εTOPk = VEDtop ± e
− β top ⋅ (ltop − k )
2
= minimum (11)
⋅ (ζ top ⋅ N m F2 )
nVED bot − Ionosonde bot = minimum 2
k = 1, 2, " , ltop (5)
(12)
where nVEDtop is the final revised VED profile above the h m F2 ; VED top is the VED profile above the h m F2 from the Abel inversion. nVED bot = VED bot ± εBOT k − β bot ⋅ (lbot − ( k − ltop )) = VED bot ± e ⋅ (ζ bot ⋅ N m F2 )
or
VED top + e
− β top ⋅ (ltop − k )
− Ionosonde top = minimum 2 k = 1, 2, " , ltop (13)
k = (ltop + 1), (ltop + 2 ), " , (ltop + lbot )
(6) where nVEDbot is the final revised VED profile below the h m F2 ;
VED bot is the VED profile below the h m F2 from
VED bot + e
− β bot ⋅ (lbot − ( k − ltop )) ' ⋅ (ζ bot ⋅ N m F2 )
− Ionosonde bot = minimum 2
the Abel inversion;
k = (ltop + 1), (ltop + 2 ), " , (ltop + lbot ) (14)
Let '
ζ top = ±ζ top
(7)
'
ζ bot = ±ζ bot
(8)
Then Equations (5) and (6) are rewritten as:
nVEDtop = VEDtop + e
− β top ⋅ (ltop − k )
' ⋅ (ζ top ⋅ N m F2 ) k = 1, 2, " , ltop
(9)
nVED − β e
bot
bot
' ⋅ (ζ top ⋅ N m F2 )
= VED
bot
+
⋅ ( lbot − ( k − ltop ))
⋅ (ζ
' ⋅ N F ) bot m 2
where
⋅
2
is 2-norm; Ionosonde top is the ionosonde profile
in the topside; and Ionosonde bot is the ionosonde profile in the bottom side. Equations (13) and (14) are non-linear. They can be solved by iteration and search algorithms. The iterative algorithm includes the Gauss-Newton method, the Levenberg-Marquardt Method, the Powell’s Dog Leg Method, A Hybrid Method: L–M and Quasi–Newton, A Secant Version of the L–M Method and A Secant Version of the Dog Leg Method etc. The search algorithm is achieved by the procedures: firstly, a group of unknown parameters is chosen based on a given rule, and then the values of the target function response to these parameters are calculated; Secondly, through comparing the calculated values of the target function, some (one) of the unknown parameters, which are related to the calculated target function values that are closest to the measurement data, are preserved, and the rest unknown parameters are discarded; thirdly, some new unknown parameters are added based on the given rules, and the step one is repeated until no more better unknown parameters are chosen. In this paper, the search algorithm is adopted (Kelley, 1999; Madsen et al., 2004). Generally speaking, the Abel inversion has the bias between 20% and 40% in electron density (Garcia-Fernandez, 2004). Because
k = ( ltop + 1), ( ltop + 2 ), " , ( ltop + lbot ) (10) ' ' The estimation of ζ top and ζ bot (or ζ top and ζ bot ) is discussed below.
N m F2 is the maximum of VED profile, we can conclude that the absolute of the accumulated biases in each layer should be
Although the bias from the Abel inversion algorithm can not be
use a bigger N m F2 to replace the above value of 0.5 N m F2 .
fully modelled the best solution of ζ top and ζ bot should satisfy the following conditions:
less than 0.5 N m F2 . To include all other bias impacts, we may Based on Equation (3), in the topside, the absolute of the maximum accumulated bias is ζ top ⋅ N m F2 (in the layer of
h m F2 ). Also according to Equation (4), in the bottom side, the absolute of the maximum accumulated bias is ζ bot ⋅ N m F2 (in the layer n or maximum layer). Hence, we conclude that the magnitudes of ζ
top
and ζ
bot
are in the range from between 0
' and 1. According to Equations (7) and (8), we have -1< ζ top
