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Keywords:

  • B splines;
  • VTEC;
  • ionosphere;
  • spherical harmonics

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Fundamentals
  5. 3. VTEC Model Approaches
  6. 4. Comparisons
  7. 5. Conclusion and Outlook
  8. References
  9. Supporting Information

[1] Most of the geodetic ionosphere models describe the electron density of the Earth's atmosphere by global maps of the vertical total electron content (VTEC) for fixed time intervals (e.g., 2 h) assuming a single-layer model and using a spherical harmonic expansion up to a specific degree n. However, it is well known that spherical harmonic models are not suited for representing data of heterogeneous density and quality. As a consequence, data gaps cannot be handled appropriately. In this paper we present a different approach for modeling VTEC generally depending on space and time defining the 3-D system of base functions as tensor products of trigonometric B spline functions for the longitude and two sets of endpoint-interpolating B spline functions for latitude and time, respectively. We compare this approach to a spherical harmonic expansion with similar resolution and show how data gaps influence the accuracy of VTEC maps even in areas with good data coverage.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Fundamentals
  5. 3. VTEC Model Approaches
  6. 4. Comparisons
  7. 5. Conclusion and Outlook
  8. References
  9. Supporting Information

[2] Modern single frequency satellite navigation and positioning systems require the use of precise and high resolution ionosphere correction models. On the other hand, precise measurements of modern space-geodetic techniques, such as the Global Navigation Satellite Systems (GNSS) allow the study of ionosphere variations with an unprecedented accuracy. Usually the ionosphere is assumed to be a single layer in a constant height and the vertical total electron content (VTEC) is modeled globally in most cases by a spherical harmonic expansion up to specific degree for fixed time intervals. The VTEC maps from Center of Orbit Determination in Berne (CODE) [Schaer, 1999] computed as one of five VTEC solutions within the International GNSS Service (IGS) [Hernández-Pajarez et al., 2009] are a prominent example. However, spherical harmonic models are not optimal for representing data of heterogeneous density and quality. As an alternative we introduce a global VTEC model which is based on compactly supported localizing B spline functions allowing the handling of data gaps in case of unevenly distributed data, such as data from terrestrial GNSS stations. This allows us to work within an Earth-fixed reference system (in contrast to Sun-fixed reference systems [Dettmering, 2003] used by IGS) and to parameterize not only the horizontal position but also the time dependency of VTEC.

[3] In this paper we will compare two different model approaches, especially the ability to compensate and handle areas without data; that is, data gaps will be investigated. We distinguish between (1) a classical approach which is based on a system of three-dimensional (3-D) base functions defined as the tensor product of spherical harmonics for longitude and latitude as well as polynomial B spline functions for the time (section 3.1) and (2) an alternative approach which is constructed as a system of 3-D base functions defined as tensor products of trigonometric B spline functions for the longitude and two sets of polynomial B spline functions for latitude and time, respectively (section 3.2).

[4] For providing the reader with the theoretical background, in section 2 the basic formulae for 1-D and 2-D expansions are introduced; this is done briefly for the spherical harmonic expansion and the polynomial B spline expansion, but more detailed for the trigonometric splines since the latter are so far not used very often in ionospheric modeling. Based on these foundations, we construct in section 3 our two approaches mentioned above in detail and apply them to two different simulated data sets in section 4. The main objective of the paper is the comparison of the two approaches with respect to a global approximation of VTEC and the handling of data gaps. These two issues are discussed in section 4.2.

2. Fundamentals

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Fundamentals
  5. 3. VTEC Model Approaches
  6. 4. Comparisons
  7. 5. Conclusion and Outlook
  8. References
  9. Supporting Information

[5] Let Σ be a geocentric coordinate system with third axis coinciding with the rotational axis of the Earth. Let further f(x, t) be a square-integrable function (signal) given in points P with position vector x = R [cos ϕ cos λ, cos ϕ sin λ, sin ϕ]T (λ = longitude, ϕ = latitude) on a sphere ΩR (R = radius) at times tT (T = [tmin,tmax] = (observation) time interval).

[6] To derive the two approaches for modeling the spatiotemporal function f(x, t), we start in section 2.1 with the classical 2-D spherical harmonic expansion. Since we are aiming on 3-D models we discuss 1-D series expansions in terms of polynomial B splines and trigonometric B splines in sections 2.2 and 2.3. The tensor product approach [Schmidt, 2007] is used to derive the 2-D spline model in Subsection 2.4 which means an alternative to the spherical harmonic approach.

2.1. Spherical Harmonic Expansion

[7] Assuming that a square-integrable 2-D spatial function f(x) = f(λ, ϕ) is given on a sphere ΩR, it can be represented as a series expansion

  • equation image

in terms of spherical harmonics Yn,m (λ, ϕ) calculable via the relations

  • equation image

from the associated Legendre function Pn,m(sin ϕ) of degree n and order m; the quantities cn,m are the spherical harmonic coefficients. In numerical applications we have to truncate the expansion (1) at a finite degree value n = N, such that double sum consists of all N′ = (N + 1)2 terms [see, e.g., Schmidt et al., 2007].

[8] Although technically the expansion (1) could be applied even for very high degree values, it is well known that spherical harmonic models cannot represent data of heterogeneous density and quality in a proper way [Schmidt et al., 2007]. Since ionosphere input data from ground- and space-based GNSS measurements, from altimetry missions and other data sources is mostly far from having an even distribution over the globe, the application of a spherical harmonic expansion is restricted. In practice the processing centers truncate the expansion mostly at N = 15 and choose a Sun-fixed coordinate system [Dettmering, 2003]. The series coefficients cn,m are usually estimated by parameter estimation from given signal values f(xi) in discrete points Pi ∈ ΩR with position vector xi with i = 1,…,I and I > N′. The maximum degree N depends on the sampling interval of the position vectors xi.

[9] The truncated spherical harmonic expansion of the function f(xi) = f(λi, ϕi) ≕ fi can be written as the scalar product

  • equation image

of the two N′ × 1 vectors ai = a(xi) = (Yn,m(xi)) of the spherical harmonics (equation (2)) and c = [c0,0, c1, −1,…, cN,N]T of the coefficients cn,m.

2.2. Polynomial B Spline Expansion

[10] We assume now that a square-integrable 1-D function g(t) is given within the interval T = [tmin, tmax ] with tequation imageT. Then it can be represented as series expansion

  • equation image

in terms of normalized quadratic polynomial B splinesequation imagekJ(t) = NJ,k2(t) depending on the resolution level Jequation imageequation image0 with unknown series coefficients ckJ. There exist a lot of textbooks and papers [e.g., Stollnitz et al., 1995a, 1995b; Lyche and Schumaker, 2000] treating B spline modeling. We do not repeat the basics here, but refer to Schmidt [2007] for an introduction into B spline modeling in ionosphere research; another introductory report on spline modeling is given by Jekeli [2005]. Note that for applying the B spline approach (equation (4)) a normalization has to be performed by introducing the variable x = (ttmin/(tmaxtmin), i.e., xequation image [0, 1] = equation image. But in order to avoid confusion we do not distinguish between these two representations in the following and use always the notation according to equation (4); for more details, see Schmidt [2007]. As can be seen from Figure 1, B splines are characterized by their compact support; that is, the functions are different from zero only within a small subinterval equation imagekJequation imageequation image. The position of equation imagekJ within the unit interval equation image is defined by the index k ∈ {0,…, KJ − 1}. The total number KJ = 2J + 2 of B splines within equation image is defined by the level value J. If finer structures shall be modeled the level J has to be increased. However, the chosen level value J depends on the sampling interval of the input data. Assuming a constant sampling interval Δt = titi−1 for all positions ti with i = 2,…,I observations g(ti) are available, the relation

  • equation image

has to be fulfilled, wherein dJ means approximately the distance between two adjacent level J B spline functions [see Mößmer, 2009]. Since the length of the subintervals equation imagekJ with respect to the variable t is around 3dJ [Schmidt, 2007], condition (5) states that each B spline coefficient ckJ is affected by at least three adjacent observations g(ti).

image

Figure 1. Set of KJ = 2J + 2 = 10 polynomial B splines equation imagekJ(t) for J = 3 within the unit interval equation image. The subinterval equation image43 of the spline function equation image43(t) (indicated by bold black curve) has a length of 3d3 = 3 × 1/9 = 0.33.

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2.3. Trigonometric B Spline Expansion

[11] We assume now that a square-integrable 1-D function h(λ) is given on a circle, i.e., within the closed interval I = [0, 2π] with λequation imageI under the constraint h(0) = h(2π). Thus, analogous to equation (4) the function h(λ) can be represented as a series expansion

  • equation image

in terms of the base functions equation imagekJ(λ) depending on the resolution level Jequation imageequation image0 with unknown series coefficients equation imagekJ. As shown by Lyche and Schumaker [2000] in detail and discussed briefly in the following, the functions equation imagekJ(λ) are defined by the normalized periodic trigonometric B splines TJ,k3(λ) of order 3. A recurrence relation for calculating the periodic trigonometric B splines TJ,k3 with k = 0,…, equation imageJ − 1 from the initial trigonometric B splines TJ,k1 of order 1 is given, e.g., by Schumaker and Traas [1991]. For that purpose a sequence of nondecreasing values (knots)

  • equation image

is introduced and due to the periodicity extended by

  • equation image

As can be seen from Figure 2, periodic trigonometric B splines are like the polynomial B splines (compare Figure 1) characterized by their compact support; that is, subintervals equation imagekJ = [λkJ, λk+3J] exist within the interval I with nonzero values. The position of these subintervals equation imagekJ is defined by the index kequation image {0,…, equation imageJ − 1. In the sequel we assume equally spaced knots λkJ according to

  • equation image

with spacing interval hJ = 2π/equation imageJ and total number equation imageJ = 3 × 2J. Following Lyche and Schumaker [2000], we define the functions

  • equation image

Setting hJh and λλkJequation image, respectively, the functions Tequation image(λλkJ) = Th3(θ) can be calculated via

  • equation image

Finally, the base functions equation imagekJ(λ) introduced in equation (6) are defined as

  • equation image

(compare Figure 2). Assuming a constant sampling interval Δλ = λiλi−1 for all positions λi with i = 2,…,I of the observations h(λi) the relation (5) holds also for periodic trigonometric B splines, i.e.,

  • equation image

Since the length of the subintervals equation imagekJ is around 3 · equation imageJ, each B spline coefficient equation imagekJ is affected by at least three adjacent observations h(λi).

image

Figure 2. Set of equation imageJ = 3 × 2J = 12 trigonometric B splines equation imagekJ(λ) according to equation (12) for J = 2. The subinterval equation image22 of the spline function equation image22(λ) (solid curve) (indicated by bold black curve) has a length of 3equation image2 = 2π/4 ≈ 1.57. The bold black dashed curves mean the nonzero part of equation image112(λ) and show the “wraparound” effect.

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2.4. Spherical B Spline Expansion

[12] As an alternative to the spherical harmonic expansion (1), we introduce a 2-D series expansion in terms of polynomial and trigonometric B spline functions. For that purpose we rewrite equation (6) for the function h(λ) ≕ f(λ, equation image) = f(x) as introduced in equation (1) and obtain by omitting the “tilde”:

  • equation image

For modeling the latitude-dependent coefficients cequation image(ϕ) by polynomial B splines we identify the variable t in equation (4) with ϕ and obtain for each coefficient cequation image(ϕ) an expansion

  • equation image

with ϕ ∈ [−π/2, π/2]. Combining equations (14) and (15) yields the spherical B spline expansion

  • equation image

with the K = equation image · equation image unknown series coefficients dequation image for Jiequation image0, ki = 0,…, equation image − 1 with i = 1, 2 and the 2-D tensor product base functions

  • equation image

[see, e.g, Schumaker and Traas, 1991]. Analogous to equation (3) we rewrite the B spline expansion (16) of the function f(xi) = f(λi, ϕi) ≕ fi as the scalar product

  • equation image

wherein bi = b(xi) = (Φequation image(xi)) is the K × 1 vector of the 2-D tensor product base functions (equation (17)) with position vector x = xi and

  • equation image

the K × 1 vector of the coefficients dequation image.

[13] Note, due to the localizing feature of the B spline base functions (compare Figures 1 and 2) the series coefficients dequation image are related to specific spatial positions defined by the integer values k1 and k2 with respect to longitude λ and latitude ϕ. Recall, the spherical harmonics coefficients cn,m are global parameters.

[14] Whereas the spherical harmonics (equation (2)) are defined on the sphere and directly related to longitude and latitude via the associated Legendre polynomials Pn,m(sin equation image) the base functions (equation (17)) are basically defined in the Euclidean space equation image2. Thus, for applying the expansion (16) to functions given on a sphere ΩR additional constraints have to be considered. Since at the south and the north pole the conditions f(λ, ϕ = −π/2) = fs and f(λ, ϕ = π/2) = fn for any longitude λ ∈ [0, 2π] hold, the first sets of constraints for the coefficients dequation image and dequation image−1 with k1 = 0,…,equation image − 1 can be derived [see, e.g., Schumaker and Traas, 1991]. These constraints in combination with the postulation of continuity conditions for the spline representation at the poles leads finally to a linear equation system

  • equation image

of altogether R constraints affecting on the coefficient vector (equation (19)) with R < K; in equation (20), H means the given R × K matrix of the constraints, 0 is the R × 1 zero vector. For more details concerning the constraints we refer to Schumaker and Traas [1991] and Lyche and Schumaker [2000] [see also [Jekeli, 2005]].

3. VTEC Model Approaches

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Fundamentals
  5. 3. VTEC Model Approaches
  6. 4. Comparisons
  7. 5. Conclusion and Outlook
  8. References
  9. Supporting Information

[15] In this section we combine the spatial representations (1) and (16) with the temporal expansion (4) in order to model a spatiotemporal, i.e., 3-D function f(λ, ϕ, t) = f(x, t). To be more specific, we identify f(λ, ϕ, t) with the difference ΔVTEC of VTEC values, e.g., derived from observations of space-geodetic techniques, and the corresponding values VTEC0 of a given background (initial) ionosphere model, i.e.,

  • equation image

In this study we assume that ΔVTEC values are given as

  • equation image

i.e., as the sum of the observations yi(tq) and the related measurement errors ei(tq) at discrete spatial, i.e., 2-D positions Pi = P(λ = λi, ϕ = ϕi) = P(xi) with i = 1,…, I on the sphere ΩR at discrete times tq with q = 1,…, Q. Although we do not introduce requirements to the spatial distribution of the observation sites Pi, we choose a constant sampling interval ΔT for the observation times tq = t1 + (q − 1) · ΔT within the interval T = [t1, tQ].

[16] For the construction of our two approaches in sections 3.1 and 3.2 we introduce the I × 1 vectors yq = [y1(tq), y2(tq),…,yI(tq)]T and eq = [e1(tq)e2(tq),…,eI(tq)]T of the observations yi(tq) and the measurement errors ei(tq), respectively.

3.1. Time-Dependent Spherical Harmonic Expansion

[17] For modeling the 3-D function f(xi, tq) = fi(tq) by means of the spherical harmonic expansion, we rewrite equation (3) and obtain

  • equation image

for q = 1,…,Q with c(tq) ≕ cq. Writing equation (23) for i = 1,…,I and introducing the decomposition equation (22), the linear equation system

  • equation image

follows, wherein the I × K matrix A is defined as A = [a1,a2,…,aI]T. Since the time dependency is propagated into the series coefficients cn,m(tq), i.e., the components of the vector cq, we introduce the polynomial B spline expansion (4) with g(tq) ≕ cn,m(tq) and ckJcn,m,kJ as

  • equation image

Writing equation (25) for each (n, m) combination, we obtain

  • equation image

wherein the N′ × KJ matrix C is defined as

  • equation image

The KJ × 1 vector uq = u(tq) reads

  • equation image

Inserting equation (26) into equation (24) yields yq + eq = A C uq. With the I × Q observation matrix Y = [y1,y2,…,yQ], the error matrix E = [e1,e2,…,eQ] and the KJ × Q matrix U = [u1u2,…,uQ] we obtain the relation

  • equation image

which can be rearranged as

  • equation image

[see, e.g., Koch, 1999]. Herein “vec” symbolizes an operator which orders the columns of a matrix one below the other as a vector; circled cross means the Kronecker product. Defining the (I · Q) × (N′ · KJ) matrix X1UTA as well as the (N′ · KJ) × 1 vector β1 ≔ vecC the Gauss-Markov model

  • equation image

is established, wherein σ2 is an unknown variance factor and P the given positive definite (I · Q) × (I · Q) weight matrix of the observations; E(·) and D(·) denote the expectation and the covariance matrix of a matrix or vector, respectively [see, e.g., Koch, 1999].

[18] Provided that the distribution of the observation sites Pi is appropriate, i.e., globally, the matrix X1 is of full column rank and the least squares method yields the estimation

  • equation image

for the vector β1. The corresponding covariance matrix reads D(equation image1) = σ2(X1TPX1)−1; formulae for the estimation of the variance factor σ2 are given, e.g., by Koch [1999]. With equation image1 the estimation equation image of the matrix equation (27) is given, and thus, the estimation

  • equation image

can be calculated in any spatial point P at any time tT according to equations (23) and (26).

3.2. Time-Dependent B Spline Expansion

[19] Considering the time dependency analogous to equation (23), the expansion (18) of the function fi(tq) reads

  • equation image

for q = 1,…,Q with d(tq) = dq. Following the procedure of section 3.1, we obtain the linear model

  • equation image

with the I × K matrix B = [b1,b2,…,bI]T. The K × 1 vector dq can be modeled as

  • equation image

wherein the K × equation image matrix D is defined as

  • equation image

With KJequation image the equation image × 1 vector uq was already defined in equation (28). Analog to equation (30) we obtain

  • equation image

Introducing the (I · Q) × (K · equation image) matrix X2 = UTB as well as the (K · equation image) × 1 vector β2 = vecD the Gauss-Markov model

  • equation image

is established. However, opposite to the spherical harmonic approach described in section 3.1, the constraints in equation (20) have to be considered in the B spline case for estimating β2. Equation (20) reads with equation (36) and setting q = 1,…, Q

  • equation image

The equation image × Q matrix U was already introduced in the context of equation (29). The R × Q matrix V has to be introduced in equation (40), since condition (5) requires equation image < Q. In other words, the equation H D U = 0 cannot be fulfilled strictly and the matrix V can be interpreted as an error matrix with respect to the “observed” zero-matrix 0. Thus, after again applying the rules for the vec operator and the Kronecker product we obtain the linear model

  • equation image

Defining the (R · Q) × (K · equation image) matrix H2 = UTH the Gauss-Markov model equation (39) can be extended to

  • equation image

The least squares method yields the estimation

  • equation image

for the vector β2. The corresponding covariance matrix reads D(equation image2) = σ2(X2TPX2 + H2TH2)−1. With equation image2 the estimator equation image of the matrix equation (37) is given. Thus, the estimation

  • equation image

can be calculated in any spatial point P ∈ ΩR at any time tT according to equations (34) and (36).

4. Comparisons

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Fundamentals
  5. 3. VTEC Model Approaches
  6. 4. Comparisons
  7. 5. Conclusion and Outlook
  8. References
  9. Supporting Information

[20] The two approaches described in sections 3.1 and 3.2 will be applied to two different data sets in order to study the influence of data gaps on the results globally and locally.

4.1. Data Generation

[21] For our investigations we define grid points P(λv, ϕw) on the sphere ΩR and calculate VTEC “observations” VTEC(xv,w, tq) from the ionosphere model IRI2007 [Bilitza and Reinisch, 2008] for 21 July 2006 in the Earth-fixed geographical coordinate system as mentioned in the Introduction. The coordinates λv and ϕw are defined on a regular grid as λv = (v − 1)Δλ for v = 1,…, V and ϕw = −π/2 + (w − 1)Δϕ with w = 1,…, W with spatial sampling intervals Δλ = 5° and Δϕ = 2.5°. The discrete times tq were already introduced in the context of equation (22); as sampling interval we choose ΔT = 2 hours. For the background model VTEC0 (xv,w, tq) we use a smoothed version of the IRI2007 VTEC values with the same sampling intervals. From these altogether around 68,000 observations we select two data sets:

[22] 1. In data set 1 we extract 20,400 observations yi(tq) randomly from the spatiotemporal regular grid; that is, we set the numbers II1 and Q introduced in equation (22) to I1 = 1700 and Q = 12. Figure 3 shows exemplarily the input data for tq = t7 = 12 UT. The black crosses indicate the randomly distributed observation sites Pi with i = 1,…,I1. As can be seen, the data are characterized by a more or less equal distribution, obviously no data gaps exist.

image

Figure 3. Input signal f(x, tq) = ΔVTEC(x, tq) at 21 July 2006 at time tq = t7 = 1200 UT. The black crosses indicate the sites Pi of data set 1, where the I1 observations yi(t7) are given; ΔVTEC is color coded in TECU (1 TECU = 1016 electrons/m2).

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[23] 2. In data set 2 we use the same input data as for data set 1, but we construct an artificial data gap; that is, we exclude all observation sites within the region [160°, 250°] × [−60°, 0°] (white box in Figure 5), altogether around 18,288 observations remain; that is, we set II2 = 1524 and Q = 12.

[24] For 21 July 2006, VTEC varies globally between 0 and 35 TECU, whereas the input data values ΔVTEC(x, tq) are within the range of ±15 TECU (1 TECU = 1016 electrons/m2); compare Figure 3.

4.2 Results

[25] In case of the classical model approach (equation (23)) we choose N = 15 for the spatial part and J = 3, i.e., K3 = 10 for the temporal part; thus, the number of unknowns amounts N′ · K3 = 2560. Furthermore, in the Gauss-Markov model (equation (31)) we identify the weight matrix P with the unit matrix I. Evaluating equation (32), we obtain the estimation equation image and calculate equation image from equation (33). Figure 4 (left) shows exemplarily the result for t7 = 12 UT. The root-mean-square (RMS) value

  • equation image

of the residuals equation imagei(tq) = equation imagei(tq) − fi(tq) at time tq = t7 = 12 UT amounts 0.56 TECU. RMS values for other selected times tq are listed in Table 1.

image

Figure 4. Estimations equation image(x, tq) = equation image(x, tq) at 21 July 2006 at time tq = t7 = 1200 UT calculated from (left) the classical approach and (middle) the alternative approach as well as (right) the differences between the two estimations; all data in TECU.

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Table 1. RMS Values σ1ja
qtqσ11σ12σ11–2σ1–21σ1–22
  • a

    According to equation (45) of data set 1 calculated by the classical approach j = 1 or the alternative approach j = 2 at selected times tq with q ∈ {1,…, 12}; RMS values σ11–2 of the differences between the classical and the alternative approach with respect to data set 1; RMS values σ1–2j of the differences between the two data sets 1 and data set 2 with respect to the two approaches j; and average RMS values over the corresponding Q = 12 RMS values for the whole day (bottom row); all data in TECU.

220.250.280.170.0960.017
460.520.530.230.1190.028
6100.540.530.340.1010.015
7120.560.530.320.0920.009
8140.520.540.300.0960.019
10180.510.490.330.0890.054
11200.430.460.330.1020.059
       
 day0.440.400.280.0990.032

[26] For the alternative B spline approach equation (34) we select J1 = J2 = J3 = 3, i.e., equation image = 24 and equation image = equation image = 10, respectively; altogether we have to solve with K = equation image · KJ2 = 240 for 2400 unknowns. Note, our second approach needs 160 coefficients less than the first approach. Again we choose P = I for the weight matrix P in the extended Gauss-Markov model (equation (42)). We obtain the estimated coefficient matrix equation image from the solution (43) and calculate equation image values from equation (44). Figure 4 (middle) depicts exemplarily the result for t7 = 12 UT. The RMS value σ12(t7) of the residuals amounts 0.53 TECU. Again RMS values for other selected times are listed in Table 1.

[27] Figure 4 (right) shows the differences between the two approaches (classical minus alternative) for 12 UT which are in the range of ±2.0 TECU. As can be noticed from Figure 4 the differences are mainly caused due to the oscillations of the spherical harmonics and partly due to the peaks of the B splines. The corresponding RMS value σ11−2(t7) reads 0.32 TECU and is listed in Table 1. The average RMS value for the whole day of investigation is around 0.28 TECU. To summarize the results listed in Table 1, we have to state that in case of a “good” data coverage; that is, a global distribution with no data gaps and a uniform accuracy both approaches are well suited for modeling VTEC. However, the RMS values of the differences between the two approaches indicate significant systematical deviations. As can be seen, for instance, from Figure 4 (left) the spherical harmonic approach causes typical oscillating structures following the geomagnetic equator. This artificial erroneous feature could probably be reduced by introducing geomagnetic coordinates. Using the so-called “modified dip” (modip) latitude as second spatial coordinate, which is according to Azpilicueta et al. [2006] adapted to the real magnetic inclination, will align the ionospheric structures to the coordinate system and may improve the modeling accuracy. But since this paper deals with the influence of data gaps on the results of our two approaches, our investigations are based on the same initial conditions; that is, we use for both approaches the same Earth-fixed coordinate system with the same set of variables.

[28] Depending on the sampling intervals of the input data the model resolution, defined by the numbers N and J for the classical case as well as J1, J2 and J3 for the alternative approach, could be increased. Usually a model with a higher resolution provides a better approximation of the input data.

[29] In the same manner as we exerted before the classical and the alternative approach to data set 1 we now apply the procedures to data set 2 (results are not shown here). It is worth to be mentioned that for the alternative approach we exclude in the extended Gauss-Markov model (equation (42)) all terms of the expansion (16) with coefficients located within the data gap. Thus, the number of unknowns is further reduced. Since we want to study the influence of the data gap in data set 2, we compute the differences between the estimations from the two data sets for both approaches outside the gap. Figure 5 (left) shows these differences (data set 1 minus data set 2) for the classical approach, and Figure 5 (right) shows the corresponding differences from the alternative approach. Significant deviations for the classical approach are around ±5 TECU. It can be seen that in particular oscillating structures are disturbing the VTEC representation everywhere on the globe and at any time of the day. For the alternative approach, however, significant deviations can only be noticed close to the gap.

image

Figure 5. Differences between the estimations from the data sets 1 and 2 using (left) the classical approach and (right) the alternative approach for 21 July 2006 at time tq = t7 = 1200 UT. The white box indicates the area for which no data are available (data gap), and the red box indicates the limit for the RMS calculations; all data in TECU.

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[30] Table 1 shows the RMS values of the differences discussed before. These numbers are related to the area outside the red box in Figure 5 in order to avoid huge influences at the edges of the data gap. For the classical approach we obtain for 12 UT an RMS value of 0.092 TECU, the corresponding value of the alternative approach is just one tenth, i.e., 0.009 TECU. For the whole day we obtain from the averaged RMS values an quotient of around 0.1 to 0.03 TECU. Thus, the RMS value of the alternative approach is about 30% of the RMS value for the classical approach. Even if the absolute improvement is small and sometimes larger errors from other sources may overlay the quality of the mathematical approach, we can state that our investigations show that the alternative approach is much better suited for modeling VTEC data with gaps than the classical approach.

5. Conclusion and Outlook

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Fundamentals
  5. 3. VTEC Model Approaches
  6. 4. Comparisons
  7. 5. Conclusion and Outlook
  8. References
  9. Supporting Information

[31] In this paper we derived two approaches for modeling a spatiotemporal function (signal) such as VTEC globally. The first, the classical approach is based on a series expansion in spherical harmonics; the time dependency of the spherical harmonic coefficients is modeled by a series expansion in terms of polynomial B splines. In the alternative approach we model the spatial part as a series expansion in trigonometric and polynomial B spline functions; the temporal part is again expanded in polynomial B splines. Thus, both approaches are based on tensor product base functions. Since the first approach is set up on global base functions, namely, the spherical harmonics, and the second on localizing B spline functions both approaches are well suited for modeling global data without gaps. However, significant differences occur and have to be studied in more detail.

[32] In case of data gaps the alternative expansion provides significant better results as the classical approach since the B spline functions have compact support. Thus, systematic artificial structures occur only close to the edges of the gap.

[33] We established for both approaches a Gauss-Markov model and applied them to two simulated data sets derived from IRI2007. In future investigations the approaches will be used for evaluating real data, e.g., from GNSS (terrestrial and spaceborne), altimetry, very long baseline interferometry (VLBI), etc.; that is, the unknown model parameters will be estimated from various space-geodetic techniques. The authors already derived a combination strategy based on variance component estimation and applied the procedure to regional VTEC modeling [Dettmering et al., 2011].

[34] Furthermore, due to the tensor product approach the B spline expansion can be generalized easily to the 4-D case, i.e., to modeling the electron density. As shown by Lyche and Schumaker [2000] and partly by Stollnitz et al. [1995a, 1995b] and Schmidt [2007], the applied B spline functions (trigonometric and polynomial) can be used for generating a multiscale representation, which allows for the decomposition of an ionospheric target function (VTEC, electron density) into a certain number of detail signals by successive low-pass filtering. Since many parts of the detail signals are usually very small and, therefore can be neglected without a significant loss of information. These items providing data compression (usually with a very high compression rate) and denoising will be studied by the authors in future investigations.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Fundamentals
  5. 3. VTEC Model Approaches
  6. 4. Comparisons
  7. 5. Conclusion and Outlook
  8. References
  9. Supporting Information

Supporting Information

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Fundamentals
  5. 3. VTEC Model Approaches
  6. 4. Comparisons
  7. 5. Conclusion and Outlook
  8. References
  9. Supporting Information
FilenameFormatSizeDescription
rds5846-sup-0001-t01.txtplain text document1KTab-delimited Table 1.

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