SEARCH

SEARCH BY CITATION

Keywords:

  • 3-D mathematical modeling;
  • GPS;
  • ionosphere

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Some General Aspects of the ESOC 3-D Mathematical Ionosphere Model at the Beginning
  5. 3. Accounting for the High Time Variability of the Ionosphere
  6. 4. Different Aspects of the ESOC 3-D Mathematical Ionosphere Model
  7. 5. Conclusions
  8. References

[1] For several years a clear trend from the application of classical so-called “single layer” models to attempts to model the ionosphere in accordance to its real three-dimensional nature can be observed (see, e.g., Scherliess et al., 2003; Hernández-Pajares et al., 1999). European Space Agency (ESA)/European Space Operations Centre (ESOC) commenced in 1998 employing a three-dimensional (3-D) model for ionosphere processing (Feltens, 1998; Feltens et al., 1998). This first version of a 3-D model at ESOC model was based on a simple Chapman profile approach, assuming that the vertical component of the ionosphere could be mathematically expressed in terms of a single β-layer Chapman profile function. The profile function's parameters, maximum electron density N0 and its height h0, were in turn expressed as surface functions of geomagnetic latitude and local time whose coefficients were estimated. In this way a horizontal variation of N0 and h0 was modeled, and the profile function varied vertically, depending on the actual N0 and h0 values at a certain location. The ionosphere, however, consists of several layers. Additionally, the plasmasphere on top of the ionosphere must be accounted for, and the scale height, needed to compute the profile function z-parameter, is height-dependent. Furthermore, some of the ionosphere layers are so called α-layers and some parts of the ionosphere show special behavior. All these effects must be accounted for in a proper 3-D mathematical modeling. It is the intent of this paper to give a substantial overview over the 3-D ionosphere models developed at ESOC and their current testing and implementation status.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Some General Aspects of the ESOC 3-D Mathematical Ionosphere Model at the Beginning
  5. 3. Accounting for the High Time Variability of the Ionosphere
  6. 4. Different Aspects of the ESOC 3-D Mathematical Ionosphere Model
  7. 5. Conclusions
  8. References

[2] According to Ratcliffe [1972], the ionosphere is “the part of the atmosphere above about 60 km, where free electrons exist in numbers sufficient to influence the travel of radio waves.” The ionosphere extends from an altitude of about 50 km up to a height of about 1000 km where it transits smoothly into the plasmasphere [see, e.g., Komjathy, 1997]. There is no clear upper border between ionosphere and plasmasphere, but a smooth transition rather in the form of a steady decay of electron content. The plasmasphere extends up to about 30,000 km, again with a smooth transition instead of a clear border to the interplanetary plasma.

[3] The Sun's extreme ultraviolet (EUV) and X-ray radiation can impart enough photoionization energy on atoms and molecules in the upper atmosphere to produce electrically positive charged ions and negative charged free electrons. A secondary ionizing force of lesser importance is cosmic radiation. Counteracting these ionization processes is the recombination of ions and electrons to neutral atoms and molecules. In the lower regions of the ionosphere, free electrons can simply attach to neutral atoms to produce negatively charged ions. The absorption of EUV radiation increases as altitude decreases; the net effect of this process and the downward increasing density of neutral molecules is the formation of a bulge-formed layer, whereby at the bulge's peak the maximum electron density is reached and from which on downwards the electron density decays strongly. However, owing to different kinds of atoms and molecules residing in different altitudes of the atmosphere and owing to their distinct absorption rates, the ionosphere is composed not only of one layer but of several layers, denoted with the letters D, E, F1, F2, whereby D, the lowest layer, is sometimes further subdivided into D2 and D1 (or D and C). The highest and most dominant layer is F2.

[4] The ionosphere is strongly coupled with the solar wind and the magnetosphere [see, e.g., Lester, 2003]. Observations of ionospheric convection can help to reconstruct the coupling of the energy submitted by the Sun, via the solar wind, into the magnetosphere and ionosphere. In turn, observations made in the ionosphere may allow for a prediction and forecast of space weather: Convection responds to the changing interplanetary magnetic field (IMF) and solar wind pressure. It thus provides a measure of energy transfer into the magnetosphere. This energy is then stored in the magnetotail before being released, normally in a sudden event, as part of a substorm process which then also drives ionospheric convection. Enhanced flows in the ionosphere result in a heating of ionospheric plasma. The increased temperatures enhance recombination rates, and plasma density in the F region decreases, especially at high latitudes. Upwelling of neutral gas during large magnetic storms results in changes in the neutral composition at F region peak densities, which can also cause changes in the F region plasma density. In addition, electric fields associated with the convection process drive currents in the E region.

[5] Satellite navigation systems, like GPS, GLONASS, and Galileo, can be used to measure two basic quantities related to the ionosphere [see, e.g., Komjathy, 1997]:

[6] 1. The delay due to the integrated total electron content (TEC) a radio signal is experiencing during its transit from spacecraft to receiver through the ionosphere.

[7] 2. Profiles of observed electron density in dependence on altitude can be derived from occultation observations of rising and setting GNSS satellites as viewed on board of a low-orbiting spacecraft equipped with a GNSS receiver. Examples are Champ (http://www.gfz-potsdam.de/pb1/op/champ/index_CHAMP.html) and COSMIC (http://www.cosmic.ucar.edu).

[8] Several models have been developed since around the beginning of the 1990s to mathematically describe the ionospheric layers. Some of the most important of these models are in short overview:

[9] 1. The Bent model [Llewellyn and Bent, 1973] is an empirical model for the global ionosphere. The ionosphere's topside (i.e., the region above the F2 peak) electron density is represented by a composition of three exponential functions, a parabola describes the bulge region, and the bottomside (i.e., the region below the F2 peak) is mathematically described with biparabolic functions. The Bent model is valid for a height range extending from 150 km to 1000 km. The driving parameters for the Bent model are the critical frequency of the F2 layer, f0F2, and the maximal usable frequency M(3000)F2. The f0F2 and M(3000)F2 are in turn computed from a set of coefficients. These coefficients were derived from Alouette-1 (1962–1966) and Ariel-3 (1967–1968) satellite data for the topside and from ground data collected at 14 sites on the American continent (1962–1969) for the bottomside.

[10] 2. The International Reference Ionosphere (IRI) [see, e.g., Bilitza, 2001] is an empirical global model describing electron density, electron temperature, ion temperature, and ion composition in the altitude range from about 50 km to 2000 km. The major data sources are the global ionosonde network, incoherent scatter radars, the ISIS and Alouette topside sounders, and in situ measurements on several satellites and rockets. In IRI, the topside electron density description is based on the Bent model. The bottomside is composed of five height regions which are modeled with hyperbolic secant, linear and exponential functions. The IRI model coefficients are provided as functions of f0F2, geomagnetic latitude and solar activity (F10.7). There are considerations to employ the NeQuick model for IRI topside modeling in future [Bilitza et al., 2006].

[11] 3. The NeQuick model [see, e.g., Radicella and Leitinger, 2001; Di Giovanni and Radicella, 1990] is an empirical model and consists of a composition of Epstein functions to represent the ionosphere's electron density. The topside is modeled as one semi-Epstein layer with a height-dependent thickness parameter. The bottomside is modeled as a sum of three Epstein layers, one for E, F1, F2. The model parameters are computed from the critical frequencies of the different layers, M(3000)F2, and geomagnetic dip at location. For further details, see the above references.

[12] 4. The model of Ching and Chiu [1973] and Chiu [1975] is an empirical model describing the ionospheric electron density in the height range between 110 km and 1000 km. The model is based on data collected at 50 ionosonde stations in the years 1957–1970. Input parameters into the model are altitude, season, geographic and geomagnetic latitude, local time, and monthly smoothed Zürich sunspot numbers. The electron density at a requested location/time is then computed as the sum of three Chapman layers, one for E, F1, F2.

[13] The three-dimensional (3-D) ionosphere model which is currently under development at ESOC follows basically the concept of the Ching and Chiu model in that it models the ionosphere as a sum of empirical profile functions. The ESOC model will use as only input TEC observables computed from ground-based GNSS data and observed electron density profiles derived from occultation data. The only exception is the height-dependent scale height function (see section 4.3), which will need the Zürich sunspot number as input.

[14] Developments in 3-D ionosphere modeling at ESOC have been made into several directions. In the following sections the different aspects will be summarized. Some aspects will be described down to a certain level of detail, while others can only be shortly mentioned: (1) sequential estimate algorithm for the ionosphere's high time variability, (2) new profile functions, (3) height-dependent scale height, (4) plasmasphere, (5) dedicated TEC integrator, (6) special behavior of the D-layer.

[15] The new mathematical models have been coded in the form of about 90 new Fortran subroutines and unit tested. Prior to implementation into the operational software, further tests, validations and tuning (e.g., of the TEC integrator) are currently (May 2007) under work. The results presented in this paper are thus restricted to the tests performed so far. Further model upgrades may follow. It is hoped to have the new software operational by the end of 2007.

2. Some General Aspects of the ESOC 3-D Mathematical Ionosphere Model at the Beginning

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Some General Aspects of the ESOC 3-D Mathematical Ionosphere Model at the Beginning
  5. 3. Accounting for the High Time Variability of the Ionosphere
  6. 4. Different Aspects of the ESOC 3-D Mathematical Ionosphere Model
  7. 5. Conclusions
  8. References

[16] In the ESOC model the 3-D height distribution of free electrons is mathematically described as sum of profile functions, one for each layer, analogously to the concept of Ching and Chiu [1973]. This sum of profile functions can be validated with observed electron density profiles obtained from occultation data. A dedicated TEC integrator (see section 4.5) will allow for the computation of the integrated TEC for the processing of TEC data derived for GNSS dual-frequency measurements.

[17] The software foresees the handling of five layers (F2, F1, E, D2, D1) at maximum. Additionally, distinct behavior of the different layers can be accounted for, e.g., E and F1 are depending on solar zenith angle χ and are so-called α layers, while others are not. For each layer the scale height can be handled in a different way, either as empirical height-dependent function, or as constant (one single for the whole profile function or one separately for the profile function's bottomside and topside) which can be estimated or kept fixed. The plasmasphere can be accounted for either by adding an empirical exponential correction term on the topside of the highest layer's (F2) profile function or by using the height-dependent scale height function, providing big values at huge altitudes.

[18] According to the explanations above, the electron density at a certain altitude is expressed as the sum of the electron densities provided by the profile functions of all layers at that altitude, plus the plasmaspheric exponential function (if the plasmasphere is not accounted for through the height-dependent scale height function):

  • equation image

where:

Ne (h)

total ionospheric electron density at altitude h;

Ni (h)

electron density of the layers i = D1, D2, E, F1, F2 at altitude h;

N0i

maximum electron density of the layers i = D1, D2, E, F1, F2 (scales the layer's profile function);

pi (h)

profile function describing the layer i's electron density as function of altitude h;

plasmasp(hh0F2)

exponential correction to the topside part of the highest layer's profile function for the plasmasphere, for hh0F2.

[19] This approach follows basically the concept of Ching and Chiu [1973], Chiu [1975], and Zhang and Berkey [1999], and the TEC is the sum of the integrals of the profile functions of all layers along signal path s:

  • equation image

where Pi (h) is profile function integral of layer i.

[20] The combination of TEC measurements with observed electron density profiles can make the estimation of profile shape parameters, like the height of maximum electron density h0, easier, since the TEC represents the integral over electron density along signal path, i.e., from mathematical point of view the area enclosed by the electron density profile. Generally, electron density information can be extracted from pure TEC observables, requiring proper observation geometry. However, the combined evaluation of TEC with observed electron density data, e.g., profiles derived from occultation data (CHAMP, COSMIC, SWARM) and/or from ionosondes [Galkin et al., 1999] will insure more reliable fits. The new 3-D model will thus process TEC data in combination with observed electron density data.

3. Accounting for the High Time Variability of the Ionosphere

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Some General Aspects of the ESOC 3-D Mathematical Ionosphere Model at the Beginning
  5. 3. Accounting for the High Time Variability of the Ionosphere
  6. 4. Different Aspects of the ESOC 3-D Mathematical Ionosphere Model
  7. 5. Conclusions
  8. References

[21] The ionosphere is a highly varying medium. Therefore a sequential estimation scheme has been developed, working according to the following principle:

[22] With a certain time resolution, currently set to 2 hours but could also be set to shorter time steps, normal equation systems are established for i time intervals:

  • equation image

where:

Ai

design matrix for the observations of the update i;

Pi

observations weight matrix for update i;

li

vector of observed-minus-computed residuals for update i.

This archive of normal equation systems is then used in two different ways for ionosphere model parameters estimation and for Differential Code Biases (DCBs) estimation:

3.1. Ionosphere Model Parameters

[23] To make an update for the time interval i + 1, the following pseudo-observation equations will be attached to the normal equations Ai+1TPi+1Ai+1Ai+1TPi+1li+1li+1TPi+1li+1:

  • equation image
  • equation image

where:

B

design matrix for these pseudo-observations, i.e., a unit matrix;

Δequation imagei+1

vector of the corrections to the unknowns for update i + 1;

xk

vector of the values of the unknowns in the actual iteration step;

xk0

vector of the values of the unknowns from the previous update i to which the xk shall be constrained to; in the first iteration Xi+1 is a zero-vector.

Squaring out these pseudo-observation equations into a normal equation system and using a proper weight matrix Wi+1, one obtains:

  • equation image

Or, since B is a unit matrix:

  • equation image

The normal equations (6) for constraining must then be added to the normal equations Ai+1TPi+1Ai+1Ai+1TPi+1li+1li+1TPi+1li+1 of update i + 1:

  • equation image

The estimated corrections for the unknowns are then:

  • equation image

The weight matrix Wi+1 in turn is computed from the normal equation systems of the previous updates i, i − 1, i − 2, … as follows:

  • equation image

where:

F

appropriate scaling factor;

rmsi2

RMS squared of previous update i;

fk

scaling factor for the normal matrix of update k; equation image

ξ

appropriate scaling factor;

t

time of current update i + 1;

tk

time of update k (in this way the influence of older updates is exponentially reduced in the weight matrix).

Once the weight matrix is established with equation (9), the normal equation system (8) can be solved to obtain the corrections Δequation imagei+1 for update i + 1 of the unknown ionosphere parameters. The RMS and the mean errors of the unknowns for update i + 1 are finally calculated as:

  • equation image
  • equation image

where:

equation imagei+1TPi+1equation imagei+1

squared sum of residuals for update i + 1;

rmsi+1

RMS of update i + 1;

n

number of observation equations (including the pseudo-observation equations),

u

number of unknowns (ionosphere model parameters and DCBs);

mx

mean error of an unknown x.

The normal equations Ai+1TPi+1Ai+1Ai+1TPi+1li+1li+1TPi+1li+1rmsi+1 of update i + 1 enter then in turn into the establishment of the weight matrix Wi+2 to constrain update i + 2, and so on … Ionosphere model parameters as well as DCB values will be estimated during this fit, but only the ionosphere model parameters are of interest here.

3.2. DCBs

[24] DCBs (for satellites and receivers) fits are done in daily batch estimates. To realize this, the normal equations are put together from the archive (3) as follows from all d normal equations of 1 day:

  • equation image

The normal equation system (11) is then solved to obtain the estimated corrections Δequation image for the unknown DCB values for that day. The RMS and the mean errors of the unknown DCBs are finally calculated as:

  • equation image
  • equation image

where:

equation imageTPequation image

squared sum of residuals;

rms

RMS;

n

number of observation equations, i.e., the total number of all observations of that day;

u

number of unknowns (DCB values and TEC parameters);

mD

mean error of an unknown D.

The solution of the normal equations (11) provides the DCB values for the satellites and the receivers as well as the ionosphere model parameters, but only the DCB values are of interest in this fit.

4. Different Aspects of the ESOC 3-D Mathematical Ionosphere Model

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Some General Aspects of the ESOC 3-D Mathematical Ionosphere Model at the Beginning
  5. 3. Accounting for the High Time Variability of the Ionosphere
  6. 4. Different Aspects of the ESOC 3-D Mathematical Ionosphere Model
  7. 5. Conclusions
  8. References

[25] Mathematical developments for the 3-D ionosphere model have been made into several directions, which will be summarized in the successive sections.

4.1. Profile Functions

[26] Several profile functions were worked out, of which three will exemplarily be presented in this paper. One selection criterion of candidates to become a profile function was the existence of an analytical integral of the respective candidate. The first one is the MacLaurin series expansion of the Chapman profile function: The development of the Chapman profile function into a MacLaurin series expansion offers the possibility to fit observed electron density profile data to such a series expansion, i.e. estimating corrections to the series expansion coefficients, and the estimated offsets with respect to the initial coefficient values of the series expansion can be interpreted as deviation of the observed profile from an ideal Chapman profile (see also section 4.4). As will also be shown below, the series expansion offers an alternative method to compute the Chapman profile integral value, which is essential for TEC observations processing. However, as can be seen from the subsequent formulae too, the series expansion requires increased computer processing load.

[27] The mathematical developments for the MacLaurin series expansion commence with the α–layer Chapman profile function [see, e.g., Kelso, 1964]:

  • equation image

where z = (hh0)/H; h is height above ground, h0 is height of maximum electron density, and H is scale height.

[28] Concerning TEC integration, sec χ and thus γ is assumed to be constant in equation (13), i.e., not varying with altitude. The variation of the solar zenith angle χ is then accounted for in the summation formula of the TEC integrator (see section 4.5).  The term equation image can now be expanded into a MacLaurin series expansion:

  • equation image

Building the derivates of equation image up to ten:

  • equation image

Evaluating the derivatives at z = 0:

  • equation image

Defining γ-dependent coefficients:

  • equation image

One finally obtains the MacLaurin series expansion for equation image:

  • equation image

Replacing equation image in equation (13) now by its series expansion (15), one gets the following profile function:

  • equation image

With eα·(1−z) · eγ = eα·(1−secχ) · eαz follows:

  • equation image

For the TEC integral:

  • equation image

The integral terms in equation (17) can be solved recursively, starting with equation imageeαzdz = −equation image · eαz:

  • equation image

The scheme (18) can also be applied purely numerically for a given z-value, i.e., starting with the knowledge of the numerical value for equation imageeαzdz = −equation image · eαz, it is easy to calculate the numerical value of equation imagez · eαzdz. That number can then in turn be used to get the value for equation imagez2 · eαzdz, … and so on. This means that all these integrals can be evaluated purely numerically in a loop up to the desired series expansion degree n, without explicit knowledge of their individual analytical formulae.

[29] Finally, a recursion formula must be set up for the computation of the γ-dependent coefficients ci. When inspecting equation (14), one recognizes that the ci are principally sums of terms aik · γk, where γ = α · secχ, i.e., γ is varying with χ, and the aik are nonvarying constants. For these constant coefficients aik a recursion formula can be set up too. To establish the recursion algorithm, the coefficients aik for the ci of the different derivatives f(i) (z) are extracted from equation (14) and put together in triangular form as follows (where i stands for the ith derivative, and the index k runs over all k = 1, …, i nonvarying constants aik for that derivative):

  • equation image

Taking the numbers from equation (14), one gets thus the following triangle for the first ten derivatives:

  • equation image

From the above triangle the following recursion formula can be derived for the coefficients aik:

  • equation image

In this way an α-layer Chapman profile can be replaced by a MacLaurin series expansion, which can be evaluated with a combination of recursion formulae, and the integral as well.

[30] The top left part of Figure 1 shows for α = 1/2 the curve of the Chapman profile formula versus the series expansion of degree n = 18 for the solar zenith angles χ = 0° (red), 30° (green), 60° (blue) (the dark colors are the curves of the closed formula, the bright colors are the curves of the series expansion). Since the curves of the closed formula are almost hidden behind the corresponding series expansion curves (indicating the good coincidence of the series expansion with the closed formula), the closed curves are also shown separately in the top right part of Figure 1. The bottom left part of Figure 1 displays the curves of the first integral of the Chapman profile series expansion for the same solar zenith angles in the same colors, and the bottom right part of Figure 1 the second integral (for details see below). It can be seen that the convergence region of the series expansion extends from z ≈ −1.5 [LEFT RIGHT ARROW] z ≈ +2.5, depending on solar zenith angle χ, i.e., the central part of the profile is covered by the series expansion. For the modeling of the parts outside convergence region, the series expansion is replaced by auxiliary exponential functions (for details see below). Concerning the evaluation of the integral from the series expansion, the integral values obtained are biased with respect to the ones obtained from the analytical formulae by big integration constants (this can be checked for α = 1, where an analytical integral is available), but when differencing between successive integral values, these constants cancel, and the integral curves in Figure 1 were obtained by subtracting from all series expansion integral values the corresponding integral value of the lowest possible z within the convergence region (where the values of the analytical integral converge to zero). The amount of the integration constant depends on degree n of series expansion development, solar zenith angle χ, and α-value. The corresponding first integral curves for α = 1 and their coincidence with the respective analytical integral curves can be seen in the work of Feltens and Dow [2007] (a PDF version of this paper can be obtained upon request from Joachim.Feltens@esa.int).

image

Figure 1. Chapman profile function versus MacLaurin series expansion and first and second integrals.

Download figure to PowerPoint

[31] Usually ionosphere profiles are shown with altitude on ordinate and electron density on abscissa. For this paper it was, however, decided to show the profile functions on ordinate and the z-values on abscissa, as is common practice in mathematics, since the investigations in this paper are primarily of mathematical nature (of course for ionosphere modeling).

[32] From fits of observed electron density profiles to this series expansion profile function, corrections δci to the γ-dependent coefficients ci can be obtained, where the offsets δci describe the deviation of the observed profile from an idealized Chapman profile:

  • equation image

where x is a fixed constant

[33] The tests show, that the δci can become numerically very large for high series expansion developments i. It seems thus to be better to estimate new constants δci = (δci/i!) instead.

[34] Outside convergence region, the series expansion CPser is replaced by simple exponential approximation functions APfct at the bottom and at the top. In order to achieve a smooth transition of the series expansion to the exponential approximation functions, electron density and TEC are computed as follows within a certain z-range (the formulae here are for topside transition, for bottomside analogous):

  • equation image
  • equation image

where z1 is lower z-border of transition, z1 = +1.5 for topside and z1 = −1.5 for bottomside, and z2 is upper z-border of transition, z2 = +2.0 for topside and z2 = −1.0 for bottomside

[35] Basically, the TEC integral in equation (21) is the sum of two integral terms of the kind:

  • equation image

Setting (for topside):

  • equation image
  • equation image
  • equation image

Thus equation imageu1 (z) · v1 (z) dz and equation imageu2 (z) · v2 (z) dz result in

  • equation image

The last term equation imageu1 (z) dz in the upper expression corresponds to the second integral of the series expansion over z. This second integral can be computed recursively too, making use of the terms of the first integral (17). From equation (18), analytical formulae for the integral terms of the first integral are established up to z5:

  • equation image

One can recognize that the integrals (22) are also composed of terms zn · eαz, whose integral values are already known from the computation of the first integral, multiplied by some factors C/αm. Take as example equation imagez2 · eαzdz:

  • equation image

and

  • equation image

Thus the integral

  • equation image

and the coefficients bik for the Q-terms can in turn be recursively computed according to the following scheme:

  • equation image

When evaluating the second integral it has to be kept in mind that already the first integral has an offset C1 (integration constant) with respect to the analytical integral (where this integration constant is zero). During the second integration of the series expansion this C1-term is integrated too, i.e.,

  • equation image

Thus when subtracting lower integration border from upper integration border, only the second integration constant C2 cancels, but not C1. Since at z = −2 the first integral is very close to zero (see bottom left part of Figure 1) the series expansion's first integral evaluated at z = −2 can be used as approximate value of C1; and C1 can be removed in this way to get the correct second integral for C1 = 0.

[36] Plots show that there are discontinuities between the exponential approximation function/transition function/series expansion integral curves (again different integration constants). TEC integration must thus be performed within the five sections, topside exponential approximation function, topside transition function, series expansion, bottomside transition function, and bottomside exponential approximation function, separately and thereafter summed up for the TEC of the whole profile.

[37] Two further examples of newly developed profile functions can only be shortly mentioned: The first one is a superimposition of the Chapman profile function with its mirrored counterpart: The Chapman profile has a fixed ratio of topside electron density with respect to bottomside electron density. One way to change this ratio is to combine the Chapman profile with its mirrored counterpart. The formula development commences with the analytical integral function of the β-layer Chapman profile [Feltens, 1998]:Chapman profile integral: ICP(x) = equation image [RIGHTWARDS ARROW] mirrored integral function: ICPmirr (x) = equation image.  Combine both integral functions as follows:

  • equation image

where P(x) is the profile function integral of the combined curve, and L is a combination factor 0 ≤ L ≤ 1 telling at which percentual ratio the Chapman profile and its mirrored counterpart shall be combined. The first derivative of this profile function integral with respect to x is the profile function itself:

  • equation image

The left part of Figure 2 shows the profile functions resulting from combinations of the Chapman profile with its mirrored counterpart for L = 0.0, 0.3, 0.5, 0.7, 1.0. Here L = 0.0 (dark blue curve) corresponds to a pure Chapman profile, L = 0.5 (orange curve) corresponds to the mean of the Chapman profile and its mirrored counterpart and is a symmetric profile with respect to the height of maximum electron density h0, and L = 1.0 (dark red curve) would correspond to a pure mirrored Chapman profile. For the real ionosphere only L values in the range of 0 ≤ L < 0.5 might be of relevance. The right part of Figure 2 displays the curves of the integral functions in correspondence to the profile curves at the left. The fact that for L > 0.0 parts of the integral curves are lying below zero is no problem, since when subtracting lower from upper integration border, these negative constants cancel.

image

Figure 2. Profile functions obtained by combining the Chapman profile function with its mirrored counterpart for 0 ≤ L ≤ 1 and the related integral functions.

Download figure to PowerPoint

[38] Expressed in general form with the solar zenith distance χ included, the functions of the profile and the integral are with z = (hh0)/H:

  • equation image
  • equation image

The factor L can be determined from a profile fit to observed electron density data. The observed profile can then be interpreted as being composed by (1 − L)% of a Chapman profile and by L% of the Chapman profile's mirrored counterpart.

[39] The second one is a profile function derived from the hyperbolic secant: The basic function is

  • equation image
  • equation image

(There exist by the way at least two further analytical integral functions for the hyperbolic secant [Gradshteyn and Ryzhik, 1965]. Mathematical checks showed that they have all the same curvature and are only differing by constant offsets. It is out of scope of this paper to go into more detail about this mathematically interesting aspect.)

[40] From this basic function the following profile function and integral can be derived, depending on solar zenith distance χ and z = (hh0)/H:

  • equation image
  • equation image

There is a close similarity between the hyperbolic secant and the commonly used Epstein function [see, e.g., Radicella and Leitinger, 2001]:

  • equation image

Reduced to its basic form, the Epstein function becomes, and can be transformed to:

  • equation image

When comparing the rightmost expression of equation (29) of the Epstein function with the basic function (27a) of the hyperbolic secant, one recognizes that both functions are special cases of a general function of the type K/(ex + ex + C).

[41] According to Bilitza et al. [2006], the Epstein function can also be expressed in terms of a squared hyperbolic secant. The following relation can easily be proven to be correct (compare with equation (29)):

  • equation image

On the other hand, the integral of the hyperbolic secant squared is [see, e.g., Gradshteyn and Ryzhik, 1965]:

  • equation image

The TEC could thus be computed with the following analytical integral of the Epstein function:

  • equation image

However, for ESOC ionosphere modeling the hyperbolic secant has been selected. Beyond the profile functions presented, some further ones were worked out for 3-D ionosphere processing, but it is out of scope to present them all here.

4.2. Plasmasphere

[42] Typical electron densities in the plasmasphere are about 1010electrons/m3, dropping by about 1 or 2 orders of magnitude at its upper border, the plasmapause [Komjathy, 1997]. ESOC 3-D ionosphere modeling will allow basically for two ways to account for the plasmasphere. (1) Usage of the empirical height-dependent scale height function (see section 4.3); owing to the large scale height values introduced by this function at huge altitudes z = (hh0)/H changes only very slowly in these high regions, thus spreading the curvature of the profile function of the highest profile. (2) For hh0F2 an empirical exponential correction function with a big scale height can be added on top of the profile function of the highest profile (F2), equation (1), thus causing a very slow decay of the topside's electron content. This correction function is quite simple and it was thus decided to renounce on further details in this paper.

4.3. Other Aspects

[43] An empirical function has been set up to describe the height variation of the scale height. The coefficients of this empirical function are fitted to standard tables of scale height values for solar maximum und minimum. The actual height-dependent scale heights are then computed as combination of the solar maximum and minimum functions, depending on actual Zürich sunspot number, i.e., at this point the ESOC 3-D model needs non-GNSS data as additional input. This height-dependent scale height function can be used as alternative to constant scale heights (one constant per profile function, or per profile function separately constants for topside and bottomside, either kept fixed or estimated). For comparison, Fox [1994] for instance allows for a linear variation of the scale height in his approach.

[44] Ionization of the lowest parts (D1) of the D-layer is driven by cosmic rays, causing a sudden increase of electron density at sunrise and a rapid decrease at sunset [Ratcliffe, 1972]. To account for this special behavior, the solar zenith angle term sec χ, which is normally used in the profile functions, has been replaced by a Fourier series representation for lowest D layer parts modeling.

4.4. Some Preliminary Test Results for the Profile Functions

[45] Figure 3 shows some examples of Chapman profile MacLaurin series expansion fits to Champ profiles (http://w3swaci.dlr.de/). The fitted series expansion curves are shown in Green and the Champ electron densities as red dots in Figure 3. To be consistent with Figures 1 and 2, also Figure 3 shows the altitude on ordinate and the electron density on abscissa. One series expansion, equation (16), was fitted to each Champ profile, i.e., the estimated series expansion coefficients cover all ionosphere layers visible in the Champ profile, and the whole profile with all its layers is thus represented by one series expansion function. Depending on the individual characteristics of a Champ profile, series expansion developments of degree n = 8…14 were fitted. As additional parameter the maximum electron density N0F2 of the F2 layer was estimated. The height of maximum electron density, h0F2, was kept fixed to the height value associated to the highest electron density value in the Champ profile table, and the scale heights were kept fixed so, that the resulting z range did not exceed the series expansion convergence region. Sometimes unwanted oscillations of the fitted curve can be observed at the ends, like at the right end of PRN30's curve, top left in Figure 3, which may be caused due to sparse observation data at the ends. Also the top left PRN30 electron density profile shows a very strong E layer. Champ profiles recorded at the beginning of the mission provided electron density values in the height range between 60 and 600 km. Owing to Champ's steady orbital decay, the upper border of this height range went in the meantime down to about 350 km. For this height range it made no sense to try to fit approximation function parameters, equation (21), to these Champ data. Also it must be said that these tests were the very first test runs for series expansion fits. Further tests, also trying to estimate h0 and scale heights, will follow. Keeping in mind that Figure 3 shows the very first test results, the series expansion seems to be very flexible to adapt to very different profile shapes, and that in form of a closed analytical function with an analytical integral (17) for the TEC. The deviation of the estimated coefficients δci from zero describe the offset of the observed Champ electron density profile from an ideal Chapman profile, for which δci = 0. The systematic evaluation of a large amount of observed electron density data (Champ, COSMIC, ionosonde, …) and the analysis of estimated series expansion coefficients, e.g., by sorting them to solar cycle, geomagnetic latitude, season, etc., may allow for some interpretation of ionospheric behavior, in a similar way as is done for low degree and order spherical harmonic coefficients for the earth gravity field for example.

image

Figure 3. Chapman profile MacLaurin series expansions fitted to Champ profiles.

Download figure to PowerPoint

[46] Also some tests have been performed for the other profile functions, e.g., equations (25) and (28a). Each ionospheric layer has to be modeled with such a profile function, i.e., the complete electron density profile is then mathematically described as a sum of such profile functions, following the same concept as Ching and Chiu [1973]. First tests show that only two of scale height, recombination coefficient α, and combination factor L should be estimated together because all three have similar effects on a profile's shape and it is difficult to separate them properly in the estimation process. N0 and h0 can always stably be estimated, while scale height, recombination coefficient, and/or combination factor should only be estimated for the F2-layer. However, these are only preliminary statements; further test must of cause follow.

4.5. TEC Integrator

[47] TEC integration is done with the help of a special summation formula, whose development will be described in the following. The basic principle is that all the profile functions (section 4.1) are referred to a so called β-layer (α = 1). Deviations of α from one (e.g. α-layer, α = equation image) are then accounted for in the summation formula (tested for 1.1 ≥ α ≥ 0.4). In this way the analytical integrals of the profile functions for the β-layer may allow for some physical interpretations.

[48] Principle: Each given profile function f(z) and its analytical integral I(z) are valid for α = 1. Compute now by using this analytical integral an integral value for a power α of the profile function g(z) = {f(z)}α. The development of the summation formula is based on the classical “rectangle approach” for approximate integral computation (Figure 4). Here, however, not the convergence of the upper and lower sum to the exact integral value is used, but the exact value for α = 1 can be computed with the analytical integral of f(z) and an integral value g(z) = {f(z)}α for α ≠ 1 is searched for. To solve the problem, consider for instance the interval z2 [RIGHTWARDS ARROW] z3 in Figure 4. The integral value for f(z) beneath the upper curve can be computed exactly according to:

  • equation image

On the other hand, with an integration step width Δz = zi+1ziz3z2 some mean value equation image(z) of f(z) in the interval z2 [RIGHTWARDS ARROW] z3 can be computed as follows, corresponding to the height of the big rectangle F in Figure 4:

  • equation image

The area G of the small rectangle in Figure 4, corresponding to the approximate integral value of g(z) = {f(z)}α, can then be expressed as follows:

  • equation image

Putting equation (34) into equation (35), one obtains:

  • equation image

For the special case α = 1, equation (36) reduces to equation (33) and for α = equation image, equation (36) becomes:

  • equation image

Evaluation of equation (36) for each integration step zi+1zi and summing up provides the total integral value, i.e., the summation formula consists of the sum of terms (36) over the whole z range to be integrated. This summation formula must now be applied to TEC integration. Electron density Ne (z) is therefore expressed in terms of the profile's maximum electron density N0 and its profile function p(z, χ):

  • equation image

The slant range TEC is obtained by integrating Ne (z) along signal path:

  • equation image

To integrate equation (39), the slant range element ds must be converted into a corresponding element dz [Feltens, 1998]:

  • equation image

Putting equation (38) and equation (40) into equation (39):

  • equation image

For practical computation, integral (41) is now expressed in terms of the summation formula. Since also satellite zenith distance Z, solar zenith distance χ, and scale height H are varying along signal path, and this circumstance was not accounted for in the analytical integral, it is now included in the summation formula too. First of all, integral (41) is split up into subintegrals, one for each integration step Δz:

  • equation image

where Hk−1,k, Zk−1,k, χk−1,k are mean values for the respective integration step Δz. Now the integral terms are replaced by corresponding terms (36) of the summation formula:

  • equation image

where:

  • equation image

Thus the final summation formula for TEC integration can be set up:

  • equation image
  • equation image
  • equation image

For the computation of the partials with respect to the unknown profile parameters (the software foresees N0, h0, Hbot, Htop, α, L, δci per layer, tests must show which of these parameters are finally estimable) the summation formula is differentiated with respect to the respective parameter and then also summed up (the principle is similar to that of the variational equations in orbit determination). Per chain rule, the partials of the respective surface function coefficients to represent the parameter globally are then attached. The TEC integral (42) is computed separately for each layer. The whole TEC is finally obtained as sum of the TEC values of all layers.

image

Figure 4. Principle of the TEC integrator scheme.

Download figure to PowerPoint

[49] If observed electron densities shall be processed, e.g., from COSMIC, observation equations will be set up with equation (38).

[50] The principal integration scheme has successfully been tested: Simple integral functions I(z) of f(z) were used for numerical integrations with different α ≠ 1 with the summation formula, and the results obtained were compared with the values of the corresponding analytical integrals for g(z) = {f(z)}α. In the same way, the analytical integral I(z) = equation image of the β-layer Chapman profile was used to get integral values for different α ≠ 1 with the summation formula, which were then compared with the corresponding numbers computed with the Chapman profile series expansion for these α values, equation (17). Tests were made for α = 1.1, 1.0, 0.9, 0.8, 0.7, 0.6, 0.5, 0.4, and χ = 00, 300, 600, 700. Since the series expansion was used for the reference integral computation, this test was by the way also another validation of the series expansion. Apart from two exceptions, the differences between integral value obtained from summation formula and from analytical integral were less than 0.002, in most cases less than 0.001, with a tendency to be more accurate for the lower α-values, e.g., α = equation image. The integration step width was 0.1 in terms of z. Compared with conventional numerical integration schemes, this might be not so accurate, but for current ionosphere applications it seems to be sufficient, and the availability of the analytical integrals for the basic profile functions may allow for some physical interpretation of results. While this paper is written (May 2007), the tests with the final summation formulae (42) are still outstanding. These tests are currently under preparation.

5. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Some General Aspects of the ESOC 3-D Mathematical Ionosphere Model at the Beginning
  5. 3. Accounting for the High Time Variability of the Ionosphere
  6. 4. Different Aspects of the ESOC 3-D Mathematical Ionosphere Model
  7. 5. Conclusions
  8. References

[51] On the basis of a first version of a 3-D ionosphere model [Feltens, 1998, 2002] (a PDF version of this paper can be obtained upon request from Joachim.Feltens@esa.int) significant progress could be made in the development of more sophisticated 3-D algorithms at ESOC. Improvements have been made in several directions: Consideration of different ionospheric layers and their special features, height-dependent scale height, plasmasphere, new profile functions, and a dedicated TEC integrator. This paper presented an overview over all these different aspects, while special attention has been paid to the latter two. Preliminary test results are promising.

[52] “Classical” TEC data can be combined with observed electron densities originating from different kind of sources (occultation, ionosonde). In spite of that electron density information can be extracted from pure TEC data, assuming proper geometry conditions, the inclusion of electron density data will make 3-D model fitting easier. A lot of tests of these new algorithms are still outstanding while this paper is written (May 2007). It is hoped to have the new 3-D modeling operational by the end of 2007.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Some General Aspects of the ESOC 3-D Mathematical Ionosphere Model at the Beginning
  5. 3. Accounting for the High Time Variability of the Ionosphere
  6. 4. Different Aspects of the ESOC 3-D Mathematical Ionosphere Model
  7. 5. Conclusions
  8. References
  • Bilitza, D. (2001), International Reference Ionosphere 2000, Radio Sci., 36, 261275.
  • Bilitza, D., B. W. Reinisch, S. M. Radicella, S. Pulinets, T. Gulyaeva, and L. Triskova (2006), Improvements of the International Reference Ionosphere model for the topside electron density profile, Radio Sci., 41, RS5S15, doi:10.1029/2005RS003370.
  • Ching, B. K., and Y. T. Chiu (1973), A phenomenological model of global ionospheric electron density in the E-, F1- and F2-regions, J. Atmos. Terr. Phys., 35, 16151630.
  • Chiu, Y. T. (1975), An improved phenomenological model of ionospheric density, J. Atmos. Terr. Phys., 37, 15631570.
  • Di Giovanni, G., and S. M. Radicella (1990), An analytical model of the electron density profile in the ionosphere, Adv. Space Res., 10, 2730.
  • Feltens, J. (1998), Chapman profile approach for 3-D global TEC representation, paper presented at 1998 IGS Analysis Centers Workshop, Eur. Space Op. Cent., Darmstadt, Germany.
  • Feltens, J. (2002), Current status of ESOC ionosphere modeling and planned improvements, paper presented at IGS/IAACs Ionosphere Workshop, Eur. Space Op. Cent., Darmstadt, Germany.
  • Feltens, J., and J. M. Dow (2007), Realized and planned improvements in ESA/ESOC ionosphere modeling, IGS presentation, in Proceedings of the 2006 IGS Workshop, ESOC, Darmstadt, Germany, in press.
  • Feltens, J., J. M. Dow, T. J. Martín-Mur, C. García Martínez, and P. Bernedo (1998), Routine production of ionosphere TEC maps at ESOC: First results, paper presented at 1998 IGS Analysis Centers Workshop, Eur. Space Op. Cent., Darmstadt, Germany.
  • Fox, M. W. (1994), A simple, convenient formalism for electron density profiles, Radio Sci., 29, 14731491.
  • Galkin, I. A., B. W. Reinisch, and D. F. Kitrosser (1999), Advances in digisonde networking, paper presented at 1999 International Ionospheric Effects Symposium (IES99), Off. of Naval Res., Alexandria, Va.
  • Gradshteyn, I. S., and I. M. Ryzhik (1965), Table of Integrals, Series, and Products, Academic Press, New York.
  • Hernández-Pajares, H., J. M. Juan, and J. Sanz (1999), New approaches in global ionospheric determination using ground GPS data, J. Atmos. Terr. Phys., 61, 12371247.
  • Kelso, J. M. (1964), Radio Ray Propagation in the Ionosphere, Electr. Sci. Ser., McGraw-Hill, New York.
  • Komjathy, A. (1997), Global ionospheric total electron content mapping using the Global Positioning System, Ph.D. dissertation, Univ. of New Brunswick, Fredericton, N. B., Canada.
  • Lester, M. (2003), Ionospheric convection and its relevance for space weather, Adv. Space Res., 31, 941950.
  • Llewellyn, S. K., and R. B. Bent (1973), Documentation and description of the Bent ionospheric model, Rep. AFCRL-TR-73-0657, Air Force Cambridge Res. Lab., Hanscom AFB, Bedford, Mass.
  • Radicella, S. M., and R. Leitinger (2001), The evolution of the DGR approach to model electron density profiles, Adv. Space Res., 27, 3540.
  • Ratcliffe, J. A. (1972), An Introduction Into the Ionosphere and Magnetosphere, Cambridge Univ. Press, Cambridge, U. K.
  • Scherliess, L., R. W. Schunk, J. J. Sojka, and D. C. Thompson (2003), The USU GAIM data assimilation model for the ionosphere, Eos. Trans. AGU, 84(46), Fall Meet. Suppl., Abstract SM42E-04.
  • Zhang, Y. H., and F. T. Berkey (1999), On the simulation of ionograms in the presence of travelling ionospheric disturbances using ionospheric ray-tracing, paper presented at 1999 International Ionospheric Effects Symposium (IES99), Off. of Naval Res., Alexandria, Va.