Encyclopedia of Geodesy

Living Edition
| Editors: Erik Grafarend

• Christian Blick
• Sarah Eberle
• Willi Freeden
Living reference work entry
DOI: https://doi.org/10.1007/978-3-319-02370-0_122-1

Keywords

Normal Gravity Atmospheric Parameter Radio Occultation Beltrami Operator Atmospheric Profile
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Definition

Radio Occultation (RO) : a satellite-based measuring technique providing globally distributed datasets of vertical profiles for atmospheric parameters such as density, pressure, temperature, and water vapor.

Introduction

In order to prove or disprove research arguments used in the discussions of climate change, large globally distributed datasets over a sufficiently large time interval are essential. In this respect, the radio occultation method, a satellite-based measuring technique, first suggested by a group at Stanford University in 1962, provides a globally distributed dataset of vertical profiles of a variety of atmospheric parameters such as density, pressure, temperature, and water vapor. Several satellites equipped with measuring instruments were launched into the Earth’s atmosphere; one of them was the German CHAllenging Minisatellite Payload (CHAMP) , which also provided the data used in this work. The radio occultation method, e.g., based on CHAMP data, has several advantages over other measuring techniques, which are in use to obtain atmospheric data such as radiosondes and aircraft-based measurement techniques. Those benefits consist of weather independence, global distribution of the data from the Earth’s surface up till 40 km altitude, and high-precision data. This entry gives a short overview of the modeling aspects of radio occultation (see Smith and Weintraub, 1953; Melbourne et al., 1994; Kursinski et al., 2000; Kursinski and Hajj, 2001; Hajj et al., 2002; Melbourne, 2004; Ge, 2006; Eberle, 2010; Benzon et al., 2012; Steiner et al. 2013). In order to handle climate data observed by the radio occultation method, they have to be visualized via mathematical methods. For that purpose, a certain spline approximation method (Freeden 1998a, b; Freeden and Witte, 1982; Blick, 2011; Blick and Freeden, 2011) is introduced and applied to a CHAMP dataset, which was made available by the German Research Centre for Geosciences (GFZ) . Indeed, we deal with the modeling aspect of radio occultation as well as the visualization of the data with the help of a combined interpolation and smoothing spline illustration (Freeden and Witte, 1982). In particular, we are interested in a comparison of the density measurements taken over different years. In addition, the spline approximation method is applied to compute vertical profiles of atmospheric parameters at arbitrary positions.

Physical Background

Our point of departure is taking a look at the mathematical modeling of radio occultation (see also Blick and Freeden, 2011; Blick et al., 2015) and, more precisely, with the consideration of the physical background (see, e.g., Wickert, 2002). We deal with the examination of GPS signals (see, e.g., Wickert, 2002; Ge, 2006), where we start with a general consideration and turn to the two special cases for the neutral atmosphere and the charged ionosphere. This is done by applying the physical laws of geometrical optics to describe the propagation of the electromagnetic waves, which are emitted by the GPS satellites in two different carrier frequencies in the L-Band (radar frequency domain of 1–2 GHz) given by f 1 = 1575.42 MHz for the signal L1 and f 2 = 1227.60 MHz for the signal L2. The configuration is controlled by the atomic clock frequency (here, f 0 = 10.23 MHz) and modulated by three pseudorandom noise frequencies (PRN), where the coarse acquisition (CA) code modulates the signal L1 with a frequency of f 0 · 10−1 and the precise (P) code modulates both signals L1 and L2 with the fundamental frequency f 0. It should be observed that the airtime for every bit of the PRN sequences is exactly known relatively to the clock on board of the GPS transmitter. In order to describe the signals L1 and L2 at the position of the transmitting antenna, we recapitulate the commonly used relations (see, e.g., Kursinski et al. 2000 for more details):
$${S}_{L1}(t)=\sqrt{2{C}_{CA}}D(t)X(t) \sin \left(2\pi {f}_1t+{\theta}_1\right)+\sqrt{2{C}_{P1}}D(t)P(t) \sin \left(2\pi {f}_1t+{\theta}_1\right),$$
(1)
$${S}_{L2}(t)=\sqrt{2{C}_{P2}}D(t)P(t) \sin \left(2\pi {f}_2t+{\theta}_2\right),$$
(2)
where the relevant angles are depicted in Figure 1.

The basic idea is to measure the change of the signal emitted by the GPS satellite with a Low Earth Orbiter (LEO), e.g., CHAMP. The modification of the signal itself is based on refraction, diffraction, and scattering while passing through the atmosphere, which provides information about the atmospheric properties. In doing so, we start with the consideration of the occultation geometry.

Here, ϵ denotes the total bending angle of the ray, θ the angle between the lines which connect the center of diffraction and the satellites, Ψ C the angle between the line connecting the Earth’s center and the LEO and the ray direction, Ψ G the angle between the line connecting the Earth’s center and the GPS satellite and the ray direction, p the impact parameter, r C the distance between the LEO and the diffraction center, and r G the distance between the GPS satellite and the diffraction center.

Bending angle in the neutral atmosphere and ionosphere: We take a look at the bending angle in the neutral atmosphere (see, e.g., Kursinski et al. 2000), which is given by
$$\upepsilon =\theta +{\Psi}_C+{\Psi}_G-\pi.$$
(3)
In order to determine the ionospheric refraction, we apply the Appleton-Hartree formula. Melbourne et al. (1994) neglect terms with higher order than two and Bassiri and Hajj, (1993) make a second-order approximation. Syndergaard (1999) proposes the approximation
$${N}_k^{IO}=-C\frac{N_e}{f_k^2}-K\frac{B_{\mathrm{par}}{N}_e}{f_k^3}\approx -C\frac{N_e}{f_k^2},$$
(4)
where N k IO is the ionospheric refraction.

Retrieval of atmospheric parameters: We are interested in determining the atmospheric parameters, e.g., pressure p, density ρ, and dry temperature T (see, e.g., Kursinski et al., 2000; Hajj et al., 2002) based on RO measurements.

To explain the relation between bending angle and refraction, we utilize the equation for the optical ray and the Bouguer formula. The change in the ray direction along the ray is given by $$\frac{d\overrightarrow{s}}{ds}$$. By setting $$d\overrightarrow{r}=\overrightarrow{s}ds$$, we get
$$\frac{d\overrightarrow{s}}{ds}=\frac{1}{n}\left(\nabla n-\overrightarrow{s}\frac{dn}{ds}\right).$$
(5)
The bending angle can be determined with mathematical tools (see, e.g., Kursinski and Hajj, 2001; Hajj et al., 2002, as well as Wickert, 2002; Ge, 2006), leading to the frequently used relation for the diffraction field N(z). This equation, which connects the atmospheric pressure p, the temperature T, and the humidity e (Smith and Weintraub, 1953), is given by
$$N(z)={k}_1\frac{p(z)}{T(z)}+{k}_2\frac{e(z)}{T^2(z)},$$
(6)
where
$$\begin{array}{ll}{k}_1=77.60\frac{K}{hPa}\pm 0.05\frac{K}{hPa},\hfill & {k}_2=3.73\cdot {10}^5\frac{K^2}{hPa}.\hfill \end{array}$$
(7)
By neglecting the humidity, we obtain the identity
$$N(z)={k}_1\frac{p_{\mathrm{dry}}(z)}{T_{\mathrm{dry}}(z)}.$$
(8)
If we combine the law for ideal gas $$\rho (z)=\frac{M_{\mathrm{dry}}}{R}\frac{p(z)}{T(z)}$$ with Eq. (8), we obtain
$${\rho}_{\mathrm{dry}}(z)=\frac{M_{\mathrm{dry}}}{k_1R}N(z).$$
(9)
where ρ denotes the air density, ρ dry the dry air density, R the universal gas constant given by 8.3145 · 103 J K −1 kg 1, and M dry = 28.964 kg kmol 1 the molar mass of dry air.
For the dry pressure, we apply and integrate the hydrostatic equation. This leads us to
$${p}_{\mathrm{dry}}(z)=\frac{M_{\mathrm{dry}}}{k_1R}{\displaystyle {\int}_z^{\infty }g\left(\phi, {z}^{\prime}\right)N\left({z}^{\prime}\right)d{z}^{\prime }},$$
(10)
with the normal gravity depending on latitude ϕ and altitude z given by
$$g\left(\phi, z\right)={\left(\frac{R}{R+z}\right)}^2g\left(\phi \right)$$
(11)
and the normal gravity on the surface according to the international formula of gravity
$$g\left(\phi \right)=9.7803\left(1+0.0053{ \sin}^2\left(\phi \right)\right).$$
(12)
It follows in connection with Eq. (8) that the dry temperature reads as follows:
$${T}_{\mathrm{dry}}(z)={k}_1\frac{p_{\mathrm{dry}}(z)}{N(z)}.$$
(13)

Modeling by Splines

Next, we come to the mathematical background (based on Freeden 1981a; Freeden 1981b), which is required for the determination of the atmospheric parameters, where we adopt the idea of one-dimensional cubic splines to the sphere (as an approximation to the Earth’s surface). These splines are well known for their property, that they have minimal “bending energy”. In more detail, among all interpolating functions of the Sobolev space H (2)([a, b]), the integral taken over the second derivative $${\int}_a^b\left|{F}^{{\prime\prime} }(x)\right|dx$$ becomes minimal, where F may be physically interpreted as the deflection normal to the resting position, supposed, of course, to be horizontal. The physical model is suggested by the classical interpretation of the potential energy of a statically deflected thin beam which indeed is proportional to the integral taken over the square of the linearized curvature of the elastica of the beam. The same concept can be applied to the sphere by choosing $${\int}_{\Omega}\left|{\Delta}_{\xi}^{*}F\left(\xi \right)\right|{}^2d\omega \left(\xi \right)$$ with Δ being the Beltrami operator, where F now denotes the deflection normal to the rest position supposed to be spherical. In other words, the second derivative takes on the form of the Beltrami operator Δ*. Indeed, our interest is to state that the interpolating spline to a given dataset has minimum “bending energy” for all interpolants within the Sobolev space H (2)(Ω). Furthermore, the spline function we consider will be able to simultaneously interpolate and smooth the data. Hence, we can decide in our spline application, which knots of the input data should be strictly interpolated and which ones should be “near” the interpolating function, i.e., the points subjected to smoothing.

As is well known (see, e.g., Freeden et al. 1998), the space H (2)(Ω) equipped with the scalar product
$${\left(F,G\right)}_{H^{(2)}\left(\Omega \right)}={\left(F,G\right)}_{H_0}+{\left(F,G\right)}_{H_0^{\perp }}$$
(14)
with
$${\left(F,G\right)}_{H_0}={\displaystyle {\int}_{\Omega}F\left(\eta \right)d\omega \left(\eta \right)}{\displaystyle {\int}_{\Omega}G\left(\eta \right)d\omega \left(\eta \right)},$$
(15)
$${\left(F,G\right)}_{H_0^{\perp }}={\displaystyle {\int}_{\Omega}\left({\Delta}_{\eta}^{*}F\left(\eta \right)\Big)\left({\Delta}_{\eta}^{*}G\left(\eta \right)\right)d\omega \left(\eta \right)\right.}$$
(16)
for all F, GH (2)(Ω), constitutes a reproducing kernel Hilbert space with reproducing kernel given by
$$\begin{array}{ll}K\left(\xi, \eta \right)= \underset{={K}_0\left(\xi, \eta \right)}{\underbrace{1 }}+\underset{={K}_0^{\perp}\left(\xi, \eta \right)}{\underbrace{G\left({\left({\Delta}^{*}\right)}^2;\xi, \eta \right)}}\hfill &, \xi, \eta \in \Omega \hfill \end{array},$$
(17)
where G((Δ)2; ξ, η) is the Green function (Freeden and Schreiner, 2009) with respect to the iterated Beltrami operator (Δ)2 given by
$$G\left({\left(\Delta *\right)}^2;\xi, \eta \right)=\left\{\begin{array}{cc}\hfill \frac{1}{4\pi },\hfill & \hfill 1-\xi \cdot \eta =0\hfill \\ {}\hfill \begin{array}{l}\frac{1}{4\pi}\Big(1- ln\left(1-\xi \cdot \eta \right) \left( ln\left(1+\xi \cdot \eta \right)- ln(2)\right)\hfill \\ {}-{\mathfrak{L}}_2\left(\frac{1-\xi \cdot \eta }{2}\right)-\left( \ln {(2)}^2+ \ln (2) ln\left(1+\xi \cdot \eta \right)\right),\hfill \end{array}\hfill & \hfill 1\pm \xi \cdot \eta \ne 0\hfill \\ {}\hfill \frac{1}{4\pi }-\frac{\pi }{24},\hfill & \hfill 1+\xi \cdot \eta =0\hfill \end{array}\right.,$$
(18)
where the function $$\mathfrak{L}$$ 2 is called the dilogarithm. More explicitly, it is defined as
$${\mathfrak{L}}_2(x)=-{\displaystyle {\int}_0^x\frac{ \ln \left(1-t\right)}{t}dt}={\displaystyle \sum_{k=1}^{\infty}\frac{x^k}{k^2}}.$$
(19)
Keeping the reprostructure of H (2)(Ω) in mind, we are able to formulate the combined interpolation and smoothing method:
Suppose that δ and β 1 2 , …, β p 2 are prescribed positive weights and that (η i ,μ i ), i = 1,…, p; (ξ j , ν j ), j = 1,…, q are given data points. Let M 1,…, M p and N 1,…, N q be systems of bounded linear functionals on H (2)(Ω) such that the ((p + q) + 1) × ((p + q) + 1)-matrix
$$\left(\begin{array}{ccc}\hfill \alpha \hfill & \hfill \beta \hfill & \hfill \kappa \hfill \\ {}\hfill {\beta}^{\prime}\hfill & \hfill \gamma \hfill & \hfill \zeta \hfill \\ {}\hfill {\kappa}^{\prime}\hfill & \hfill {\zeta}^{\prime}\hfill & \hfill 0\hfill \end{array}\right)$$
(20)
is non-singular, where the matrices α, β, γ, κ, and ζ are given as follows:
$$\begin{array}{ll}\alpha ={\left({M}_i{M}_jG\left({\left({\Delta}^{*}\right)}^2;{\eta}_i,{\eta}_j\right)+\delta {\beta}_i^2{\delta}_{ij}\right)}_{\begin{array}{c}i=1,\dots, p\\ {}j=1,\dots, p\end{array}},\hfill & \left({\delta}_{ij}:\mathrm{Kronecker}\ \mathrm{symbol}\right),\hfill \end{array}$$
(21)
$$\beta ={\left({M}_i{N}_jG\left({\left({\Delta}^{*}\right)}^2;{\eta}_i,{\xi}_j\right)\right)}_{\begin{array}{c}i=1,\dots, p\\ {}j=1,\dots, q\end{array}},$$
(22)
$$\gamma ={\left({N}_i{N}_jG\left({\left({\Delta}^{*}\right)}^2;{\xi}_i,{\xi}_j\right)\right)}_{\begin{array}{c}i=1,\dots, q\\ {}j=1,\dots, q\end{array}},$$
(23)
$$\kappa ={\left({M}_i{Y}_{0,1}\left({\eta}_i\right)\right)}_{i=1,\dots, p},$$
(24)
$$\zeta ={\left({N}_j{Y}_{0,1}\left({\xi}_i\right)\right)}_{j=1,\dots, q}.$$
(25)
Then the function S (η) of the form
$$S\left(\eta \right)={c}_0+{\displaystyle \sum_{i=1}^p{a}_i{M}_iG\left({\left({\Delta}^{*}\right)}^2;\eta, {\eta}_i\right)}+{\displaystyle \sum_{j=1}^q{b}_j{N}_jG\left({\left({\Delta}^{*}\right)}^2;\eta, {\xi}_i\right)}$$
(26)
with coefficients $$a\in {\mathrm{\mathbb{R}}}^p$$, $${a}^{\prime }=\left({a}_1,\dots, {a}_p\right)$$; $$b\in {\mathrm{\mathbb{R}}}^q$$, $${b}^{\prime }=\left({b}_1,\dots, {b}_q\right)$$ and $${c}_0\in \mathrm{\mathbb{R}}$$ is uniquely determined by the linear equations
$$\begin{array}{ll}{M}_iS+\delta {\beta}_i^2{a}_i={\mu}_i,\hfill & i=1,\dots, p,\hfill \end{array}$$
(27)
$$\begin{array}{ll}{N}_jS={\nu}_j,\hfill & j=1,\dots, q,\hfill \end{array}$$
(28)
$${\displaystyle \sum_{i=1}^p{a}_i{M}_i}+{\displaystyle \sum_{j=1}^q{b}_j{N}_j=0}.$$
(29)
The function S is called the unique spline function corresponding to the data points (η i , μ i ),i = 1,…, p; (ξ j , ν j ), j = 1,…, q and represents the only element of H (2)(Ω) satisfying
$${\displaystyle \sum_{i=1}^p{\left(\frac{M_iS-{\mu}_i}{\beta}\right)}^2}+\delta {\left(S,S\right)}_{H_0^{\perp }}\le {\displaystyle \sum_{i=1}^p{\left(\frac{M_iF-{\mu}_i}{\beta_i}\right)}^2}+\delta {\left(F,F\right)}_{H_0^{\perp }}$$
(30)
for all FH (2)(Ω) with N j F = ν j ,j = 1, …, q.

The spline function SH (2)(Ω) is the only function in H (2)(Ω) having the property Eq. (30) and satisfying the prescribed constraints. As already mentioned, the approximation method can be regarded as a compromise between interpolating and smoothing (see also Freeden and Witte, 1982). These two concepts were considered separately in Freeden (1981b).

Remark 1

1. (i)

The values μ 1,…,μ p , ν 1,…,ν q , are regarded as the observed quantities, e.g., temperature, pressure, humidity, etc. for the RO case.

2. (ii)

The spline function SH (2)(Ω) satisfies that M i S is “near” μ i , i = 1, …, p and N j S is equal to ν j ,j = 1, …, q. The “nearness” of the values M i S to μ i , i = 1, …, q can be controlled by choosing the constant δ in a suitable way. A small value of δ emphasizes fidelity to the observed data at the expanse of smoothness, while a large value does the opposite.

3. (iii)

Taking δ = 0 yields M i S = μ i , i = 1,…, p, i.e., the combined smoothing and interpolation procedure leads back to strict interpolation.

4. (iv)

For numerical purposes, it is advantageous to adapt the quantities β 1 2 , …, β p 2 to the standard deviations of the measured values.

Next, we want to take a look at strict interpolation, i.e., δ = 0. This leads to the “minimum energy” property of the strict interpolating spline. Hence, interpolating n data points $${\left({\eta}_k,{\mu}_k\right)}_{k=1,\dots, n}$$, the linear equation system (27)–(29) takes on the form
$$\left(\begin{array}{cc}\hfill \alpha \hfill & \hfill \kappa \hfill \\ {}\hfill {\kappa}^{\prime}\hfill & \hfill 0\hfill \end{array}\right)\left(\begin{array}{c}\hfill a\hfill \\ {}\hfill {c}_0\hfill \end{array}\right)=\left(\begin{array}{c}\hfill \mu \hfill \\ {}\hfill 0\hfill \end{array}\right),$$
(31)
where
$$\alpha ={\left({M}_i{M}_jG\left({\left({\Delta}^{*}\right)}^2;{\eta}_i,{\eta}_j\right)\right)}_{\begin{array}{c}i=1,\dots, n\\ {}j=1,\dots, n\end{array}},$$
(32)
$$\kappa ={\left({M}_i{Y}_{0, 1}\left({\eta}_i\right)\right)}_{i=1,\dots, n}.$$
(33)
Clearly, the interpolating spline function is defined as
$$S\left(\eta \right)={c}_0+{\displaystyle \sum_{i=1}^n{a}_i{M}_iG\left({\left({\Delta}^{*}\right)}^2;\eta, {\eta}_i\right).}$$
(34)
It turns out that we obtain the following characterization of minimum norm interpolation:
Let (η 1, μ 1),…, (η n , μ n ) be n data points. Let S be the unique spline which interpolates the data points μ 1,…, μ n . Then, for all twice continuously differentiable functions F on Ω, which interpolate the data points μ 1,…, μ n , the following inequality holds true:
$${\displaystyle {\int}_{\Omega}{\left({\Delta}_{\eta}^{*}S\left(\eta \right)\right)}^2d\omega \left(\eta \right)}\le {\displaystyle {\int}_{\Omega}{\left({\Delta}_{\eta}^{*}F\left(\eta \right)\right)}^2d\omega \left(\eta \right).}$$
(35)

Remark 2

Equation ( 35 ) tells us that the “bending energy” (i.e., the integral taken over the Beltrami derivative) of the spline is minimal among all functions in H (2)(Ω) interpolating the data.

Simulations

Comparison of different years using spherical spline approximation : The spherical spline approximation method is uniquely suited for the comparison of data accumulated over different years, since the comparison involves only the subtraction of the respective spline functions. Hence, if the spline function is once computed for a given height, time period, and year, it can be compared to any other year with the same time period and height. We compare the difference in density at 20 km in Figure 2. It turns out that the density above mean sea level over Europe was lowest in 2005.
Computation of atmospheric profiles : As already described in (Blick and Freeden, 2011; Blick et al., 2015), we are also able to calculate atmospheric profiles for arbitrary locations on the Earth by use of the spherical spline approximation method. In order to compute those profiles, the spherical spline function has to be computed for several layers of the Earth’s atmosphere. The data at hand provides atmospheric profiles with measurements in 200 m intervals. By computing the spherical spline function for each of those layers and evaluating this function at the desired position on the Earth, the atmospheric profile can be computed. Exemplary, a vertical temperature profile for Kaiserslautern, Germany, was computed. The city is located at 49.424°N, 7.745°E. The nearest measurement is located at 50.514°N, 8.769°E, which corresponds to a distance of 141.23 km to the desired location. For the calculation, all measurements in July 2007 were taken into account. The smoothing parameter δ was selected as 0.01. In order to weight the measurements close to the desired location, the parameters β k were selected as 2 − η · η k . The knotes η k indicate the positions of the measurements on the unit sphere and η the position of the desired location. The results of the computations can be seen in Figure 3.

Conclusion and Outlook

In this contribution, RO first is considered from the physical as well as numerical point of view. We recapitulated the usual physically based way to handle the RO problem (for more details the reader is referred to Melbourne, 2004; Eberle, 2010; Wee et al., 2010; Benzon et al., 2012). Our numerical results show that the spherical spline approach chosen in close similarity to the one-dimensional cubic spline concept is an adequate method for the approximation of given radio occultation data. Further on, the numerical experiences assured that the spherical spline method is numerically stable even for vast linear equation systems, which contributes further to the usefulness of the method. In addition, the parameters of the spline technique demonstrate appropriate adaptivity for the adjustment to a given dataset.

Finally, we mention that there are also other institutions working on radio occultation, e.g., the Danish Meteorological Institute (DMI); EUMETSAT, Darmstadt; Jet Propulsion Laboratory (JPL), Pasadena; the University Corporation for Atmospheric Research (UCAR); and the Wegener Center, University of Graz (see, e.g., Steiner et al., 2013).

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