#### 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]:

where *z* = (*h* − *h*_{0})/*H*; *h* is height above ground, *h*_{0} 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 can now be expanded into a MacLaurin series expansion:

Building the derivates of up to ten:

Evaluating the derivatives at *z* = 0:

Defining *γ*-dependent coefficients:

One finally obtains the MacLaurin series expansion for :

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

With *e*^{α·(1−z)} · *e*^{−γ} = *e*^{α·(1−secχ)} · *e*^{−αz} follows:

For the *TEC* integral:

The integral terms in equation (17) can be solved recursively, starting with *e*^{−αz}*dz* = − · *e*^{−αz}:

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 *e*^{−αz}*dz* = − · *e*^{−αz}, it is easy to calculate the numerical value of *z* · *e*^{−αz}*dz*. That number can then in turn be used to get the value for *z*^{2} · *e*^{−αz}*dz*, … 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 *c*_{i}. When inspecting equation (14), one recognizes that the *c*_{i} are principally sums of terms *a*_{ik} · *γ*^{k}, where *γ* = *α* · sec*χ*, i.e., *γ* is varying with *χ*, and the *a*_{ik} are nonvarying constants. For these constant coefficients *a*_{ik} a recursion formula can be set up too. To establish the recursion algorithm, the coefficients *a*_{ik} for the *c*_{i} 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 *i*th derivative, and the index *k* runs over all *k* = 1, …, *i* nonvarying constants *a*_{ik} for that derivative):

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

From the above triangle the following recursion formula can be derived for the coefficients *a*_{ik}:

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 *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).

[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 *δc*_{i} to the *γ*-dependent coefficients *c*_{i} can be obtained, where the offsets *δc*_{i} describe the deviation of the observed profile from an idealized Chapman profile:

where *x* is a fixed constant

[33] The tests show, that the *δc*_{i} can become numerically very large for high series expansion developments *i*. It seems thus to be better to estimate new constants *δc*_{i} = (*δc*_{i}/*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):

where *z*_{1} is lower *z*-border of transition, *z*_{1} = +1.5 for topside and *z*_{1} = −1.5 for bottomside, and *z*_{2} is upper *z*-border of transition, *z*_{2} = +2.0 for topside and *z*_{2} = −1.0 for bottomside

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

Setting (for topside):

Thus *u*′_{1} (*z*) · *v*_{1} (*z*) *dz* and *u*′_{2} (*z*) · *v*_{2} (*z*) *dz* result in

The last term *u*_{1} (*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 *z*^{5}:

One can recognize that the integrals (22) are also composed of terms *z*^{n} · *e*^{−αz}, whose integral values are already known from the computation of the first integral, multiplied by some factors *C*/*α*^{m}. Take as example *z*^{2} · *e*^{−αz}*dz*:

and

Thus the integral

and the coefficients *b*_{ik} for the *Q*-terms can in turn be recursively computed according to the following scheme:

When evaluating the second integral it has to be kept in mind that already the first integral has an offset *C*_{1} (integration constant) with respect to the analytical integral (where this integration constant is zero). During the second integration of the series expansion this *C*_{1}-term is integrated too, i.e.,

Thus when subtracting lower integration border from upper integration border, only the second integration constant *C*_{2} cancels, but not *C*_{1}. 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 *C*_{1}; and *C*_{1} can be removed in this way to get the correct second integral for *C*_{1} = 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*) = mirrored integral function: *ICP*_{mirr} (*x*) = . Combine both integral functions as follows:

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:

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 *h*_{0}, 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.

[38] Expressed in general form with the solar zenith distance *χ* included, the functions of the profile and the integral are with *z* = (*h* − *h*_{0})/*H*:

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

(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* = (*h* − *h*_{0})/*H*:

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

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

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*/(*e*^{x} + *e*^{−x} + *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)):

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

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

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.

#### 4.4. 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 *N*_{0}*F*_{2} of the *F*_{2} layer was estimated. The height of maximum electron density, *h*_{0}*F*_{2}, 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 *h*_{0} 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 *δc*_{i} from zero describe the offset of the observed Champ electron density profile from an ideal Chapman profile, for which *δc*_{i} = 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.

[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. *N*_{0} and *h*_{0} can always stably be estimated, while scale height, recombination coefficient, and/or combination factor should only be estimated for the *F*_{2}-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* = ) 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 *z*_{2} *z*_{3} in Figure 4. The integral value for *f*(*z*) beneath the upper curve can be computed exactly according to:

On the other hand, with an integration step width Δ*z* = *z*_{i+1} − *z*_{i} ≡ *z*_{3} − *z*_{2} some mean value (*z*) of *f*(*z*) in the interval *z*_{2} *z*_{3} can be computed as follows, corresponding to the height of the big rectangle *F* in Figure 4:

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:

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

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

Evaluation of equation (36) for each integration step *z*_{i+1} − *z*_{i} 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 *N*_{e} (*z*) is therefore expressed in terms of the profile's maximum electron density *N*_{0} and its profile function *p*(*z*, *χ*):

The slant range *TEC* is obtained by integrating *N*_{e} (*z*) along signal path:

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

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

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 Δ*z*:

where *H*_{k−1,k}, *Z*_{k−1,k}, *χ*_{k−1,k} are mean values for the respective integration step Δ*z*. Now the integral terms are replaced by corresponding terms (36) of the summation formula:

where:

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

For the computation of the partials with respect to the unknown profile parameters (the software foresees *N*_{0}, *h*_{0}, *H*_{bot}, *H*_{top}, *α*, *L*, *δc*_{i} 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.

[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*) = 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* = 0^{0}, 30^{0}, 60^{0}, 70^{0}. 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* = . 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.