Letter from the Editor

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Letter from the Editor

UCTEA

CHAMBER OF SURVEYING AND CADASTRE ENGINEERS

Journal of Geodesy and Geoinformation

It is my great pleasure to introduce the readers to the first issue of the “Journal of Geodesy and Geoinformation” formerly called “HKM Journal of Geodesy, Geoinformation and Land Management” with its new facet and structure. The journal will cover not only the traditional topics in Geodesy and Geoinformation but also the earth sciences and earth oriented space sciences and technologies. At the same time we are pleased to announce the launch of our new website which can be seen at http://www.hkmodergi.org including the Electronic Submission and Peer-Review System. The journal aims to create an international forum for discussions where the exchange of ideas between researchers around the world will become possible with the purpose to meet global, regional and local challenges in all topics mentioned. Therefore we strongly encourage everyone to participate in this new process to improve the quality of papers and technical notes to be published. Additionally I would like to extend my thanks in advance to our excellent staff and national and international editorial board members for their efforts to raise the standards of the journal. At this opportunity I also have to thank our previous editors Prof. Dr.

Ahmet Aksoy and Prof. Dr. Ahmet Yaşayan who worked tirelessly for the journal in the last years. My special thanks, too, go to the Chamber of Surveying and Cadastre Engineers as publishers for bearing the costs of all these new steps and additional financial supports for the authors whose papers are accepted to be published in the Journal of Geodesy and Geoinformation.

Thanks a lot and best wishes.

Mahmut Onur Karslıoğlu

Editor in Chief

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Editörden Mektup

TMMOB

HARİTA VE KADASTRO MÜHENDİSLERİ ODASI

Jeodezi ve Jeoinformasyon Dergisi

Eski adıyla “HKM Jeodezi, Jeoinformasyon ve Arazi Yönetimi” değişen adıyla “Jeodezi ve Jeoinformasyon” dergisini, yeni görünüşü ve yapısıyla okuyuculara tanıtmaktan büyük bir mutluluk duymaktayım. Dergimiz sadece Jeodezi ve Jeoinformasyon alanındaki geleneksel konuları değil, yer bilimleri ve yere yönelik uzay bilimleri ve teknolojilerini de kapsayacaktır.

Aynı anda, yeni web sayfasının hizmete başladığını da sizlere bildirmekten dolayı çok mutluyuz.

Yeni web sayfası, Elektronik Başvuru ve Hakemli Değerlendirme Sistemini de içermekte ve http://www.hkmodergi.org bağlantısından erişilebilmektedir. Dergi, bilimsel tartışmalar çerçevesinde uluslarası bir forum yaratmayı amaçlamaktadır. Bu forumda, tüm dünyadaki araştırmacılar arasında, düşüncelerin karşılıklı olarak alışverişi mümkün kılınacak ve daha önce belirtilen konulardaki küresel, bölgesel ve lokal problemlerin çözümlerine ulaşılabilecektir. Bu amaç doğrultusunda, yayınlanabilecek makale ve teknik notların kalitesinin yükseltilmesi için herkesin bu yeni sürece katılımını tüm gücümüzle teşvik etmekteyiz. Bunların yanısıra, derginin standartlarını yükseltmek için gösterdikleri çabadan ötürü, görevlerini mükemmel bir biçimde yürüten çalışma arkadaşlarıma, ulusal ve uluslarası editöryayı oluşturan araştırmacılara şimdiden teşekkür ederim. Daha önceki editörler Prof. Dr. Ahmet Aksoy ve Prof. Dr. Ahmet Yaşayan’a son yıllardaki özverili ve bıkmak bilmeyen çalışmaları için yine bu fırsatta teşekkürlerimi bir borç bilirim. Tüm bu yeni adımların finansal yükünü taşıdığı ve ayrıca Jeodezi ve Jeoinformasyon Dergisi’nde yayına kabul edilmiş makalelerin yazarlarına ek ekonomik destek sağladığı için yayıncı kuruluş olarak Harita ve Kadastro Mühendisleri Odamıza özel olarak teşekkürlerimi sunarım.

En iyi dileklerim ve Selamlarımla.

Mahmut Onur Karslıoğlu

Editör

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UCTEA Chamber of Surveying and Cadastre Engineers

Journal of Geodesy and Geoinformation

TMMOB

Harita ve Kadastro Mühendisleri Odası

Jeodezi ve Jeoinformasyon Dergisi

www.hkmodergi.org

Regional spatio-temporal modeling of the ionospheric Vertical Total Electron Content (VTEC) using

Multivariate Adaptive Regression B-Splines (BMARS)

Mahmut Onur Karslıoğlu^1,2,*, Murat Durmaz²

1 Middle East Technical University, Civil Engineering Department, Geomatics Engineering Division, 06800, Ankara, Turkey,

2 Middle East Technical University, Department of Geodetic and Geographic Information Technologies, 06800, Ankara, Turkey,

Accepted: 10 May 2012 Received: 02 April 2012 Pub. Online: 10 July 2012 Volume: 1

Number: 1 Page: 9 - 16 May 2012

Abstract

Spatio‒temporal Regional modeling of the ionosphere in terms of the vertical total electron content (VTEC) is accomplished using a non‒parametric Multivariate Adaptive Regression B‒ Spline (BMARS) algorithm on the basis of Global Positioning System (GPS) observations. The basis functions are constructed as compactly supported tensor products of quadratic B‒Splines which are derived from the observations automatically. A smooth approximation is achieved by scale‒by‒scale model building strategy which searches for best fitting B Spline to the data at each scale. The real data set processed is gathered from ground based GPS stations in Europe and falls within the time interval of the geomagnetic storm on 15 February, 2011. The result of BMARS modeling apparently demonstrates the efficiency and the potential of the method. It is also compared both numerically and visually with a well‒known global and regional VTEC modeling based on spherical harmonics and B‒Splines respectively.

Keywords

Ionosphere, GPS, MARS, B‒spline, BMARS, Regional modeling

Kabul: 10 Mayıs 2012 Alındı: 02 April 2012 Web Yayın: 10 Temmuz 2012

Cilt: 1 Sayı: 1 Sayfa: 9 - 16 Mayıs 2012

Özet

Bölgesel iyonosferik Düşey Toplam Elektron İçeriğinin (VTEC) mekansal ve zamansal boyutlarda Çok Değişkenli Uyabilen B-Spline Regresyonu (BMARS) kullanılarak belirlenmesi

Mekansal ve zamansal bölgesel İyonosferin Düşey Toplam Elektron İçeriği (VTEC) cinsinden modellenmesi parametrik olmayan Çok Değişkenli Uyabilen B-Spline fonksiyonlarına Dayalı Regresyon (BMARS) kullanı- larak gerçekleştirilmektedir. Gözlemlerden otomatik olarak üretilen Baz Fonksiyonları, karesel (ikinci dere- ce) B-Spline fonksiyonlarının sıkılaştırılmış destekli tensör çarpımlarından oluşturulmaktadır. Yumuşatılmış bir yaklaştırıma sıralı ölçeklendirmeye dayalı bir model kurma stratejisiyle ulaşılmaktadır. Bu strateji veriye her ölçekte en iyi uyan B-Spline fonksiyonunu aramaktadır. İşlenen veri grubu Avrupa’ daki yersel GPS istasyonlarından toplanmış olup, 15 Şubat 2011 tarihinde meydana gelen jeomanyetik bir fırtınayı da içer- mektedir. BMARS modellemesinin sonucu bu yöntemin etkinliğini ve potansiyelini açıkca göstermektedir.

Hesaplanan sonuç, aynı zamanda gerek nümerik gerekse görsel olarak, tanınmış küresel ve bölgesel model- lerle karşılaştırılmıştır. Küresel model küresel harmonik fonksiyonlara, bölgesel model de B-Spline fonksi- yonlarına dayanmaktadır.

Anahtar Sözcükler

İyonosfer, GPS, MARS, B-spline, BMARS, Bölgesel modelleme

*Corresponding Author: Tel: +90 (312) 2102440 Fax: +90 (312) 2105401

E‒mail: [email protected] (Karslıoğlu M.O.), [email protected] (Durmaz M.)

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10 Regional spatio - temporal modeling of the ionospheric Vertical Total Electron Content (VTEC) using Multivariate Adaptive Regression B-Splines (BMARS)

1. Introduction

Observations obtained by dual frequency GPS receivers which can be installed on terrestrial and space‒borne plat- forms are widely used for the determination of the Total Electron Content (TEC) of the ionosphere. The distribution of the TEC data has a major impact on the propagation of radio waves in the ionized atmosphere, which is crucial for terrestrial and Earth‒space communications including navigation satellite systems such as GNSS (Liu and Gao 2004;

Policy Workshop Report 2011). The number of electrons in a column of one meter squared cross‒section along the ray‒

path of the signal between the satellite and the receiver are integrated to find the so‒called Slant Total Electron Content (STEC). But, in this work Vertical Total Electron Content (VTEC) is preferred for the modeling; because ground based GPS observations have low sensitivity to the vertical structure of the ionosphere. In this approach the ionosphere is represented by a spherical layer of infinitesimal thick- ness at a certain height where all electrons are assumed to be concentrated even though some errors are introduced by the mapping function which transforms STEC into VTEC (Dettmering 2003; Jin et al. 2006, 2008; Brunini et al. 2010).

Spatio‒temporal modeling of VTEC can be carried out by a three dimensional function depending on geographic latitude, geographic longitude and time (Mannucci et al. 1998;

Hernandez - Pajares and Sanz 1999; Schaer 1999; Yuan and Ou 2002; Brunini et al. 2004; Jin et al. 2004). Two dimensional modeling of VTEC in terms of geographic latitude and geographic longitude is traditionally represented by spherical harmonic functions as global functions over the en- tire globe with regularly distributed data. In case of regional and local modeling of both the electron density and the VTEC for the spatio‒temporal variations of the ionosphere, a multi‒dimensional approach on the basis of Euclidian quadratic B‒splines and tensor products is preferred to the global functions by Schmidt (2007), Schmidt et al. (2007a, 2007b), Zeilhofer (2008) and Nohutcu et al. (2007, 2010).

The details of the multi‒dimensional approaches can be found in Garcia‒Fernandez (2004). Another kind of local and regional modeling concerns a non‒parametric modeling using Multivariate Adaptive Regression Splines (MARS) based on piecewise‒linear functions. MARS can handle very large sets and be used for both linear and non‒linear modeling in different time series (Lewis and Stevens 1991; Ekman and Kubin 1999; Yang et al. 2004; Crino and Brown 2007).

MARS is an adaptive method requiring no gridding of the input space. Durmaz et al. (2010) applied MARS to regional VTEC modeling over a large part of Europe successfully.

Subsequently Multivariate Adaptive Regression B‒Splines (BMARS) was presented by Durmaz and Karslioglu (2011) with tensor product B‒splines as basis functions obtained from observation locations. The difference between MARS and BMARS algorithm is not confined only to the kind of basis functions. BMARS uses also a scale‒by‒scale model building strategy where the basis functions are selected from large scales to smaller ones. This approach results in a significant amount of reduction in the number of basis functions. The result leads to a smoother model than the original MARS. Since B‒splines are compactly supported better lo- calization is achieved for small scales (Bakin et al. 1997;

Bakin et al. 2000).

In this work BMARS is applied to regional VTEC modeling over Europe within the time interval of the geomagnetic storm on February 15th, 2011. To assess the performance of

the algorithm the result of BMARS is compared both numerically and visually with another regional tensor product quadratic B‒spline modeling presented by Nohutcu et al. (2010) and the modeling using spherical harmonics. This paper is organized as follows: In the second section the principles of VTEC calculation are given using ground‒based GPS observations. The third section treats mathematical fundamentals of BMARS. Section 4 deals with the application and analysis of the results over Europe and focuses on comparisons of different VTEC modeling strategies. In the last section the conclusion is presented.

2. VTEC Modeling from Ground‒based GPS Observations

The range delay on electromagnetic signals caused by the ionosphere is a function of the total number of electrons along the ray‒path between the satellite and the receiver (STEC) and the signal frequency. The ionospheric range de- lay, I_i can be expressed in meters as

I_i= ±40.3

f_i² STEC

₍₁₎

where f_i is the signal frequency of the carrier L_i with i= 1, 2 (Liu and Gao 2003; Hofmann - Wellenhof et al. 2008). This delay is positive for pseudorange and negative for carrier phase measurements.

The ionospheric observable also called geometry‒free linear combination is formed by subtracting simultaneous pseudorange or carrier phase observations. Using this combination makes it possible to remove the satellite‒receiver geometrical range and all frequency independent biases (Ciraolo et al. 2007). For pseudorange measurements the ionospheric observable, P₄ is given by

P₄= P₁! P₂= I₁! I₂+ br + bs +"_p

⁽²⁾

where P₁ is the pseudorange measurement for L₁, br and bs are the so‒called inter‒frequency biases (IFBs) on pseudorange measurement due to hardware delays of the receiver and the satellite, respectively, and ε_p is the combination of observational noise and multi‒path effects in P₁ and P₂. For carrier phase measurements the ionospheric observable, Φ₄ can be written as follows:

!⁴=!₁"!₂= L₂" L₁+#₁N₁"#₂N₂+ Br + Bs +$_L

(3) where Φ_i is the carrier‒phase measurement in meter for L_i, λ_i is the wavelength of the L_i carrier, N_i is the integer carrier‒

phase ambiguity, Br and Bs are IFBs on carrier‒phase mea- surement. ε_L is the combination of observational noise and multi‒path effects in Φ₁ and Φ₂. STEC can be calculated by combining Eq. (1) with the Eqs. (2) ‒ or ‒ (3), respectively.

Although carrier phase measurements show low noise behavior in comparison with pseudo range measurements it is not favored here for calculating STEC due to the estimation of the ambiguity term in the preprocessing. Instead a carrier phase smoothing technique (carrier phase leveling) is used both to reduce the effect of pseude range noise and carrier‒

phase ambiguity terms from the data (Ciraolo et al. 2007).

This procedure begins by adding Eqs. (2) and (3) as follows P₄+Φ₄=λ₁N₁– λ₂N₂+Br+Bs+br+bs+ ε_p (4) where P₄ and Φ₄ are obtained from GPS observations. The

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Mahmut Onur Karslıoğlu, Murat Durmaz / Vol.1  No.1  2012 11

ambiguity terms N₁ and N₂ are constant for every continuous arc of carrier‒phase observations without cycle slips. The IFBs can be considered as stable for days to months so they can be taken as constant values for a continuous arc (Gao et al. 1994). In Eq. (1) noise and multi‒path terms for carrier‒

phase observations have been neglected, since the both terms are lower than the corresponding ones in pseudorange observations. Thus, an average value

P₄+!_{4 arc} can be computed for a continuous arc

P₄+!_{4 arc}=1

n (P₄+!₄)_i

i=1

"

n ⁼ #₁N₁$#₂N_{2 arc}+ Br + Bs + br + bs + %_{P arc}

P₄+!_{4 arc}=1

n (P₄+!₄)_i

i=1

"

n ⁼ ^#1N₁$#₂N_{2 arc}+ Br + Bs + br + bs + %_{P arc}

(5)

where n is the number of measurements in the continuous arc. Subtracting Eq. (3) from Eq. (5), the ambiguity terms can be eliminated

!P₄= P₄+!_{4 arc}"!₄# I₁" I₂+ br + bs + $_{P arc}"$_L (6) where

P!₄

is the pseudorange ionospheric observable smoothed with the carrier‒phase ionospheric observable.

Inserting ionospheric range delays from Eq. (1) into Eq. (6), STEC can be calculated in TECU as:

STEC= ( !P₄! b_r! b_s! "_{P arc}+"_L) ( f₁²f₂²)

40.3 ( f₂²! f₁²)

(7) For height independent two‒dimensional or three dimensional ionosphere models where the time can be taken as the third dimension STEC values are usually converted to Vertical Total Electron Content (VTEC) by introducing a mapping function in context with a single layer model. The single layer model (Fig.1) is interpreted as an infinitesimally thick shell where all electrons in the ionosphere are assumed to be contained. The single layer model mapping function F_I which relates VTEC and STEC is given by:

F_I=STEC

VTEC= 1

cos z ' with

sinz '= R

R+ Hsin z

⁽⁸⁾

where R is the mean Earth radius, z and z' are the satellite zenith angles of the receiver and the ionospheric pierce point (IPP), H is the height of this idealized layer or mean altitude approximately corresponds to the altitude of the maximum electron

density and its height can vary between 350 and 450 kilometers (Schaer 1999; Seeber 2003; Hofmann‒Wellenhof et al. 2008) 3. Multivariate Adaptive Regression B‒Splines (BMARS)

Multivariate Adaptive Regression B‒Splines called (BMARS) is an algorithm which is based on the similar principles of the original Multivariate Adaptive Regression Spline (MARS) (Friedman 1991). On the contrary to MARS where only truncated power basis functions are used BMARS can take advantage of B‒Splines of any order as basis functions (Bakin et al. 1997; Bakin et al. 2000). B‒Splines are compactly supported functions meaning that the function values are zero outside the compact interval. As an example scaling functions of 2D quadratic B‒Spline for level 2 and 3 can be seen in Fig.2. The length of the support interval can be determined by the selection of the knot location. This leads to a new kind of model building strategy in a scale‒by‒scale approach which is another difference apart from the kind of basis functions between MARS and BMARS algorithm. The scale here corresponds to the length of the support interval.

Let y be an observation vector containing N observations with related locations x_k=[x_k,1, x_k,2,..., x_k,p]^T,k=1,2,...,N and p which is the dimension of the observation locations, the observation equation can be arranged in the following way

y =

y₁ y₂

! y_k

! yN

!

"

##

$

%

&

=

f (x₁) f (x₂)

! f (x_k)

! f (x_N)

!

"

##

#

$

%

&

+ e₁ e₂

! e_k

! e_N

!

"

##

#

$

%

&

⁽⁹⁾

where f(x) is the unknown multivariate function, e=[e₁,e₂,...,e_N]^T is the vector of measurement errors with zero mean and finite variance. Least squares regression proce- dure is performed to find an approximate (x) of f(x) that fits the observations best (Durmaz and Karslioglu 2011). The BMARS algorithm builds the following regression function, f(x)

f (x) =

"

_i=0^M!1^#ih_i(x)

⁽¹⁰⁾

Figure 1: Single Layer Model for the ionosphere (Schaer 1999) Figure 2: Quadratic B‒Spline scaling functions for levels 2 and 3. The higher the level the narrower becomes the function.

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by automatically generating and fitting basis functions h_i(x) which are directly obtained from the observations. In Eq.

(10), M is the number of basis functions, β_i is the coefficient of the basis functions which are redefined as

h_i(x) = !_l=1^LⁱB_d,t

l ,i(x_v(l,i)) with i > 0 (11) where L_iis the interaction degree of the i^th basis function, v(l,i) is the variable associated with the B‒Spline with cor- responding knot location t_l,i. The multivariate B‒Spline ba- sis functions h_i(x)are constructed from tensor product of univariate B‒splines. In order to use such basis functions a slight modification of the model building and basis forma- tion are required. First of all BMARS algorithm builds the regression function just like in MARS in two stages, namely forward and backward. In the forward stage a relatively large model that possibly overfits the given data is established. In the backward or elimination stage, the least significant terms as suboptimal basis functions are removed from the model to improve the quality of the fit. This means that Generalized Cross Validation (GCV) as a prediction error of the current model can be made minimal.

The B‒spline basis functions are produced from nested sets of knot locationsC_s^j

where s is defined as the scale index and j is the component index of the multivariate variable x.

For every variable x_j the knot locations are sorted in ascend- ing order, x_k,j< x_k+1,j, k=1,2,...,N‒1 where N is the number of observations. Then, set C_s^j

is defined as

C_s^j={x_{1, j},t_{r, j}^s ,x_{N , j}}, r= 1,2,...,2^s!1with t_{r, j}^s = x_{[r 2}!sN+0.5], j

⁽¹²⁾

where the rounded integers [r2^‒SN+0.5], correspond to the rank of r 2^‒S 100 percentile of the sorted knot locations x_k,j(Durmaz and Karslioglu 2011; Bakin et al. 1997). For example, for scale index s=1 the set C₁^j

can be described as {x_1,j,x[N/2+0.5] ,j ,x_N,j}(Durmaz and Karslioglu 2011). For s=2,

C₂^j={x_{1, j},x[N /4+0.5], j,x[2N /4+0.5], j,x[3N /4+0.5], j,x_{N , j}}

^As

the scale s index increases, the scale (support interval) of B‒Splines decreases leading to more localized functions. For every set C_s^j

a different set of B‒Splines can be produced by using endpoint interpolation which modifies the first and the last two scaling functions at the interval boundaries (Durmaz and Karslioglu 2011).

3.1 Forward Stage

Forward stage starts with a constant basis function of h₀(x)=1 and estimates its coefficient with least square estimation procedure. Then new tensor product basis functions are added to the model starting with the scale index s=1. A new tensor product B-Spline h_m(x) is searched at each iteration over all available univariate B‒Spline basis functions defined by the knot location sets C_s^j

h_m(x) = h_i(x)B_d,t

n, js (x_j) (13)

where h_i(x) is an already selected basis function in the model (0 ! i < m)

^and

B_d,t

n, j

s (x_j) is a univariate B‒Spline of de- gree d that is constructed by the knot location

t_{n, j}^s

^{from the}

set C_s^j related to the variable x_j. This requires that h_i(x) does not contain the variable x_j. The basis function h_m(x) that decreases the residual most is added to the model (Bakin et al.

1997; Bakin et al. 2000). If the GCV score of the resulting model tends to increase, then the scale index is increased or the length of the support interval is decreased. This means that the algorithm has already included enough large‒scale components into the model so that smaller scale components should be searched (Durmaz and Karslioglu 2011).

3.2 Backward Stage

The backward stage is similar to the MARS backward stage.

The large model produced in the forward stage is pruned by removing the terms, which make least contributions to the re- siduals, so that final model has a good prediction performance in terms of GCV score. This iterative procedure is continued until an optimal effective number of terms are present in the final model which minimizes the following GCV score:

GCV (!) = ^k=1

"

N ^%_&^yk#

"

_i=0^M#1ˆ$_ih_i(x_k)^'₍²

(1# k(!) / N )² (14)

In Eq. (14), λ is the tuning parameter which can also be in- terpreted as a regularization parameter, K(λ) is a measure of effective number of terms in the model which may be defined simply as a linear function of the number of terms in the model.

BMARS algorithm uses a scale‒by‒scale model building strategy (switching from scale to scale using Eq. 13) which results in a significant number of reductions in the number of candidate basis functions to be searched for small s or equiv- alently large scales. Another advantage of the algorithm is that it separates the small‒scale features from the large scale ones and results in a smooth approximation of the function (Bakin et al. 1997).

4. Regional VTEC Modeling using BMARS For the VTEC modeling observation equation (9) can be re- arranged in the following form

y₁ y₂

! y_k

! y_N

!

"

##

#

$

%

&

=

VTEC (x₁) VTEC (x₂)

! VTEC (x_k)

! VTEC (x_N)

!

"

##

#

$

%

&

+ e₁ e₂

! e_k

! e_N

!

"

##

#

$

%

&

(15)

where VTEC(x) is the VTEC model function which is estimat- ed by the BMARS algorithm in 3D in an earth fixed reference frame with x_k=[λ,φ,t]^T, where λ is the geographic longitude, φ is the geographic latitude, t is the time in UT (Universal Time).

N is the number of VTEC observations, y_k is the k^th VTEC measurement at location x_k=[λ_kφ_kt_k]^T, k=1,2,...,Nand e_k is the random measurement error. The BMARS models in this study utilize second degree or quadratic B‒Splines as basis functions because of their good numerical properties. Quadratic

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B‒Splines are widely used in ionosphere modeling as shown by Nohutcu et al. (2010) and Schmidt et al. (2007a, b). For a better interpretation, the resulting BMARS model (see Eq.

(15)) is written in the following open form:

+ !_m

m₃=1 3

M₃

"

^B_2,t

nt (m3)

st (m3)(t)+ !_m

m₄=1 4

M₄

"

^B_2,_#

n# (m4 )s# (m4 )(#)B

2,$

n$ (m4 ) s$ (m4 )($)

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VTEC(!,",t) = #₀+ #_m

m₁=1 1

M₁

$

^B_2,!

n! (m1)s! (m1)(!) + #_m

m₂=1 2

M₂

$

^B_2,"

n" (m2 )s" (m2 )(")

+ _m !_m₅

5=1 M₅

" B

2,#

n# (m5) s# (m5)(#)B_2,t

nt (m5)

st (m5)(t)+ _m !_m₆

6=1 M₆

" B

2,tnt (m6 )st (m6 )(t)B

2,$_{n$ (m6 )}^{s$ (m6 )}($)

+ !_m

m₅=1 5

M₅

" B

2,#

n# (m5) s# (m5)(#)B_2,t

nt (m5)

st (m5)(t)+ !_m

m₆=1 6

M₆

" B

2,tnt (m6 )st (m6 )(t)B

2,$_{n$ (m6 )}^{s$ (m6 )}($)

+ !_m

m₇=1 7

M₇

" B

2,#_{n# (m7 )}^{s# (m7 )}(#)B

2,$_{n$ (m7 )}^{s$ (m7 )}($)B_2,t

nt (m7 )st (m7 )(t)

where the total number of terms, M is defined as M= 1+!_i=1⁷ M_i All univariate B‒Splines

B2,!_{n! (mi )}^{s! (mi )}

B2,"

n" (mi )s" (mi )

B2,t

nt (mi )st (mi )

(λ)

B2,!_{n! (mi )}^{s! (mi )}

B2,"n" (mi )s" (mi )

B2,t

nt (mi ) st (mi )

(φ) and

B2,!_{n! (mi )}^{s! (mi )}

B2,"

n" (mi )s" (mi )

B2,t

nt (mi ) st (mi )

(t) are generated from the sets

C_s

!(m_i)

! ,C_s

"(m_i)

"

C_s

t(m_i) t

and

C_s

!(m_i)

! ,C_s

"(m_i)

"

C_s

t(m_i)

t respectively, as described in Section 3 in which s_λ(m_i), s_φ(m_i) and s_t(m_i) define the scale of B‒splines for variables λ, φ and t at the m_i^th term. β₀ is the coefficient of the constant function while β_mi are the coefficients of the corresponding tensor product B‒splines. As can be seen from Eq. (16), BMARS builds a regression function based on tensor product B‒splines with different scales and interactions.

In this study, the BMARS algorithm which is imple- mented as a MATLAB routine by the authors is applied to the VTEC modeling over Europe during the geomagnetic storm caused by an X‒class solar flare on February 15, 2011.

Two datasets were collected from 29 ground based GPS stations belonging to International GNSS Service (IGS) and/

or Reference Frame Sub‒commission for Europe (EUREF) networks for the days 17^th and 18^th February 2011. This time interval is selected intentionally in order to model the VTEC distribution which is strongly influenced by the geomagnetic storm. One hour subset of the 24h dataset is used for each day to generate the VTEC map. Each subset is centered on 10:00 AM UT at local noon for the selected area where the ionosphere is highly variable.

The IFB values for the satellites are obtained from Center of Orbit Determination in Europe (CODE) solution whereas the IFB values for the receivers are calculated by our own software. VTEC values in TECU (1 Total Electron Content Unit (TECU) = 10¹⁶ electrons⁄m² ) for each observation are computed as described in Section 2 taking elevation cut‒off angle as 15° and the height of the single layer as 400 km.

The observations are assumed to have the same accuracy and quality. Satellite positions are interpolated from the precise orbit files provided by several IGS analysis centers.

VTEC maps generated by the BMARS algorithm in an earth fixed reference frame using quadratic B‒Splines for the first subset from the day 17 February 2011 defined as S1 and the second subset from 18 February 2011 defined as S2 are presented in Fig.3(a) and Fig.3(b). Basis Functions Allowed (MBFA) equals 300. The Root Mean Square Error (RMSE) which is calculated from the difference between the estimated observations and the observations equals 0.3825 TECU for S1 with the number of selected terms 279 and the resulting scale index 3. For S2, RMSE is calculated as 0.4601 TECU with the number of selected terms 289 and the resulting scale index 4.

The result of the modeling based on the spherical har- monic expansion with degree, n and order m, 8 (n = m = 8) in the earth fixed reference frame is given in Fig.4(a) and Fig.4(b) for the subsets S1 and S2. For S1 the RMSE turned out to be 0.562 TECU and for S2, the RMSE is equal to 0.893 TECU.

Fig. 5 shows the results of another regional modeling using second degree or normalized quadratic B‒spline modeling in the earth fixed reference frame. To be compatible with BMARS modeling the level of B‒Spline (see Fig. 2), which corresponds to the scale in BMARS, is taken as 3 for each dimension, λ, φ and t. This means that 1000(= [2³ + 2]³) coefficients are calculated in the procedure. The RMSE value of the B‒Spline VTEC modeling is 0.387 TECU for S1 (Fig. 5(a)) and 0.502 TECU for S2 (Fig. 5(b)). Since no large VTEC variations are present for (S1) similar modeling results are observed in all methods in terms of low RMSE values.

Nevertheless, visual comparison and numerical results indicate that BMARS can provide smoother VTEC maps with smaller RMSE values than the spherical harmonic (SH) and B‒Spline modeling of VTEC. For the theoretical detail of SH and B‒Spline modeling of VTEC see Nohutcu et al.

(2010). The effect of geomagnetic activity is observable in all VTEC maps approximately up to the 30‒35 TECU.

Additionally, difference VTEC maps between SH, B‒spline and BMARS modeling with the Root Mean Square (RMS) of the differences are also presented in Fig. 6, 7 and 8.

Fig. 6 shows the difference VTEC maps between BMARS and SH modeling which delivers RMS values for S1, RMS=

0.8251 TECU and for S2, RMS= 1.7009 while in Fig. 7 the difference maps between B‒Spline and SH modeling are demonstrated with corresponding RMS=0.6516 TECU for S1 and RMS=1.078 TECU for S2 respectively.

Finally, the difference VTEC maps are produced to em- phasize the difference between BMARS and B‒Spline modeling on the basis of RMS values RMS= 0.7675 TECU for S1 and RMS= 1.0215 TECU for S2 plotted in Fig. 8.

Due to the low variations in S1 all the modeling results show similar patterns with small RMSE values (see Fig.

3(a), 4(a) and 5(a)). It can be seen from the difference VTEC maps that the variations between BMARS and SH modeling (Fig. 6(a)) are larger than the variations between B‒Spline and SH modeling (Fig.7 (a)). That is because BMARS fits the data better than the B‒Spline. Processing S1, BMARS and B‒Spline modeling deliver similar results on the basis of RMS values; RMS= 0.8251 TECU and RMS= 0.7675 TECU in Fig. 6(a) and Fig. 7(a). VTEC variations in S2 are

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higher than S1 which causes a high RMSE value in spherical harmonic modeling (Fig. 4(b)). In this time span VTEC achieved also high values up to the 30‒35 TECU due to the geomagnetic storm which was mentioned in previous chap- ters. Consequently, BMARS increases scale index to 4. This indicates that more localized basis functions are generated by BMARS than by B‒Spline level 3. These differences can also clearly be seen from the variations in difference

VTEC maps. Although the models produce similar VTEC maps for S2 (Fig. 3(b), 4(b), and 5(b)) the difference par- ticularly between BMARS and SH modeling (Fig. 6(b)) becomes obvious. The comparison of numerical results in difference VTEC maps shows that the difference between B‒Spline and SH modeling is lower than the difference between BMARS and SH modeling in terms of RMS values as shown in Fig. 6(b) and Fig. 7(b). This can be thought of

Figure 5: VTEC maps obtained by quadratic 3D B‒splines with level 3 (number of coefficients = 1000) for S1 and S2, 10:00 AM, UT. (a) RMSE = 0.387 TECU; (b) RMSE = 0.502 TECU.

(a) 17 Feb. 2011, RMSE = 0.387 TECU (b) 18 Feb. 2011, RMSE = 0.502 TECU

Figure 4: VTEC maps obtained by Spherical Harmonic Expansion degree, n = 8 and order, m = 8 for S1 and S2, 10:00 AM UT . (a) RMSE = 0.562 TECU; (b) RMSE = 0.893.

(a) 17 Feb. 2011, RMSE = 0.562 TECU (b) 18 Feb. 2011, RMSE = 0.893 TECU

Figure 3: VTEC maps obtained by BMARS for S1 and S2, 10:00 AM . (a) RMSE = 0.3825; (b) RMSE = 0.4601.

(a) 17 Feb. 2011, RMSE = 0.3825 TECU, number of selected terms = 279, GCV = 0.1691,

scale index = 3

(b) 18 Feb. 2011, RMSE = 0.4601 TECU, number of selected terms = 289, GCV = 0.2913,

scale index = 4

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as having a BMARS model behavior which includes more localized basis functions than that of B‒Spline. The dissimi- larity in modeling results between BMARS and B‒Spline can be referred to the distinct production method of basis functions and model building strategy of BMARS which can obviously be observed for S2 in the VTEC maps in Fig.8. In view of all the figures generated and their numerical analysis one can conclude that BMARS algorithm is able to generate smaller and even better RMSE values than both SH and

B‒spline modeling, Because BMARS algorithm fits better to the observations and adds smaller scale localized features to the model as can be seen from all difference VTEC maps.

5. Conclusion

In this work Multivariate Adaptive Regression B‒

Splines (BMARS) has been applied to regional modeling of the VTEC. The performance and adaptivity of the BMARS algorithm have been demonstrated using data

Figure 6: Difference VTEC maps between BMARS and SH modeling for S1 and S2, 10:00 AM,UT . (a) RMS = 0.8251 TECU; (b) RMS = 1.7009 TECU.

(a) 17 Feb. 2011; RMS = 0.8251 TECU (b) 18 Feb. 2011; RMS = 1.7009 TECU

Figure 7: Difference VTEC maps between 3D quadratic B‒spline and SH modeling for S1 and S2, 10:00 AM,UT. (a) RMS = 0.6516 TECU; (b) RMS = 1.078 TECU.

Figure 8: Difference VTEC maps between BMARS and B‒spline for S1 and S2, 10:00 AM,UT. (a) RMS = 0.7675 TECU, (b) RMS = 1.0215 TECU

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sets of real GPS observations over Europe during the geomagnetic storm caused by an X‒class solar flare on February 15, 2011. From the tests containing two different data subsets in terms of S1 and S2 can be resulted that the BMARS algorithm possesses the capability to efficiently model the regional ionospheric VTEC distri- butions. An overall consistency can be observed between the VTEC maps from the SH, B‒Spline and BMARS modeling. However, numerical and visual comparisons of BMARS with SH and B‒Spline algorithm suggest that the BMARS algorithm gives smoother VTEC maps with smaller RMSE values. With these results it can be con- cluded that the BMARS algorithm is more suitable for regional VTEC modeling than SH modeling which is in fact based on global functions. Another consequence is that BMARS can also be a viable alternative to regional B‒Spline based methods.

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UCTEA Chamber of Surveying and Cadastre Engineers

Journal of Geodesy and Geoinformation

TMMOB

Harita ve Kadastro Mühendisleri Odası

Jeodezi ve Jeoinformasyon Dergisi

www.hkmodergi.org

Modelling Very Long Baseline Interferometry (VLBI) observations

Kamil Teke^1,*, Emine Tanır Kayıkçı², Johannes Böhm³ and Harald Schuh³

1Hacettepe University, Faculty of Engineering, Geomatics Engineering Department, 06800, Ankara, Turkey

2Karadeniz Technical University, Faculty of Engineering, Geomatics Engineering Department, 61080, Trabzon, Turkey

3Vienna University of Technology, Institute of Geodesy and Geophysics, 1040, Vienna, Austria

Accepted: 12 May 2012 Received: 19 March 2012 Pub. Online: 10 July 2012 Volume: 1

Number: 1 Page: 17 - 26 May 2012

Abstract

The main objective of this study is to provide in detail the theoretical basis of the Very Long Baseline Interferometry (VLBI) delay model, mainly according to the International Earth Rotation and Reference Systems Service (IERS) Conventions 2010. This goes along with introducing the concept of continuous piece‒wise linear offset (CPWLO) functions for estimating sub‒daily geodetic parameters at pre‒defined epochs, e.g. at Universal Time (UT) integer hours or at integer fractions or multiples of integer hours. The geodetic parameters can be simultaneously and accurately estimated from VLBI observations in sub‒daily resolution if enough observations within each estimation interval are carried out from homogenously dis- tributed Earth‒fixed VLBI antennas to space‒fixed radio sources. After providing the basic VLBI model of the geometric delay including clock synchronization and tropospheric effects, the partial derivatives of VLBI observation equation with respect to the most important geodetic parameters are given and some typical VLBI results are shown.

Keywords

Very Long Baseline Interferometry (VLBI), VLBI delay model, continuous piece‒wise linear offsets, CPWLO.

Kabul: 12 Mayıs 2012 Alındı: 19 Mart 2012 Web Yayın: 10 Temmuz 2012 Cilt: 1

Sayı: 1 Sayfa: 17 - 26 Mayıs 2012

Özet

Çok Uzun Baz Enterferometrisi (VLBI) ölçülerinin modellenmesi

Bu çalışmanın temel amacı Çok Uzun Baz Enterferometrisi (VLBI) sinyal gecikme modelinin teorik detayla- rını Uluslararası Yer Dönüklük ve Referans Sistemleri Servisi (IERS) 2010 Konvansiyonları temelinde sun- maktır. Bu kapsamda sürekli parçalı lineer offset (CPWLO) fonksiyonları ile gün‒içi zamansal çözünürlükte önceden belirlenmiş epoklarda, örneğin: Evrensel Zaman (UT) tam saatleri, tam saatlerin katları veya bö- lümlerinde jeodezik parametrelerin kestirimi ayrıca açıklanmıştır. Yerküre’deki homojen dağılımlı Yer‒sabit VLBI antenlerinden gök küredeki homojen dağılımlı uzay‒sabit radyo kaynaklarına her parametre kestirim epoğu aralığında yapılacak yeterli sayıda VLBI ölçülerinden, jeodezik parametreler gün‒içi zamansal çö- zünürlükte eşzamanlı ve duyarlığı yüksek olarak kestirilebilir. Saat senkronizasyon ve troposfer gecikme modellerini içeren temel VLBI gecikme modeli sunulduktan sonra VLBI ölçü denkleminin başlıca jeodezik parametrelere göre kısmi türevleri verilmiş ve tipik VLBI sonuçları gösterilmiştir.

Anahtar Sözcükler

Çok Uzun Baz Enterferometrisi (VLBI), VLBI gecikme modeli, sürekli parçalı lineer offsetler, CPWLO.

*Corresponding Author: Phone: +90 (312) 2976990 Fax: +90 (312) 2976167

E‒mail: [email protected] (Teke K.), [email protected] (Tanır E.T), [email protected] (Böhm J.), [email protected] (Schuh H.)

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18 Modelling Very Long Baseline Interferometry (VLBI) observations

1. Introduction

Very Long Baseline Interferometry (VLBI) is a part of geodesy for nearly 40 years. Since 1999 VLBI is coordinated by the International VLBI Service for Geodesy and Astrometry (IVS) – a Service of the International Association of Geodesy (IAG) and International Astronomical Union (IAU) (Schlüter and Behrend 2007).

The main data and products that VLBI technique pro- vides for scientific community are; Celestial Intermediate Pole (CIP) position in Earth‒fixed reference system i.e.

International Terrestrial Reference System (ITRS) (polar motion coordinates), the Earth axis absolute rotation angle w.r.t. space‒fixed reference system i.e. Geocentric Celestial Reference System (GCRS), the CIP position in ICRS (nu- tation offsets), source coordinates in Barycentric Celestial Reference System (BCRS), antenna Cartesian coordinates in ITRF, ionosphere delays, elevation angle dependent troposphere delays, delays due to azimuthal asymmetric parts of the troposphere (gradients), clock errors, special geo- physical parameters such as tidal Love and Shida numbers and several technique specific parameters.

Very Long Baseline Interferometry (VLBI) is a geometric space geodetic technique: it measures the time difference between the arrival of a radio wavefront emitted by a distant quasi‒stellar radio source (quasar) at two Earth‒based antennas (Figure 1).

Using large numbers of time difference measurements from many quasars observed with a global network of antennas, VLBI determines the inertial reference frame (space‒

fixed frame, e.g. ICRF2 (Fey et al. 2009) defined by quasar coordinates and simultaneously the precise positions of the antennas in the terrestrial frame (Earth‒fixed frame, e.g.

VTRF08 (Böckmann et al. 2010). Further information about VLBI technology we refer to Campbell (1979, 2004), Schuh (1987), Nothnagel (1991), Sovers et al. (1998), Takahashi et al. (2000), Böhm et al. (2011a), Teke et al. (2009), Teke (2011).

In Section 2, the VLBI delay model is introduced which is known as “consensus model” and proposed in the International Earth Rotation and Reference Systems Service (IERS) Conventions 2010. We basically follow the IERS Conventions 2010 for the transformation of antenna coordinates from the International Terrestrial Reference System (ITRS) to the Geocentric Celestial Reference System (GCRS) and for the VLBI delay model (Petit and Luzum 2010). Subsection 2.3 and 2.4 reviews the delay models in VLBI analysis due to clock and troposphere errors, respectively. Subsection 2.5 presents the partial derivatives of the VLBI delay model with respect to the estimated parameters are mainly based on Sovers et al. (1998), Böhm et al. (2011a) and Teke (2011). In Section 3, we introduce continuous piece‒wise linear offset (CPWLO) functions for sub‒daily VLBI parameter estimation.

2. VLBI Delay Model

The VLBI observation equation (delay model) can be simply written as

‒c.τ= b^⊕. k^e+Δτ_clock+Δτ_trop+Δτ_iono+Δτrelativistic... (1) where c is the velocity of light in vacuum environment, is the unit source (quasar) vector defined in a space‒fixed, barycentric and equatorial celestial system, e.g. ICRF2, is the baseline vector of the VLBI antennas defined in an Earth‒fixed, geocentric, equatorial terrestrial coordinate system, e.g. VTRF08. Δτ_clock is the delay correction due to the synchronization and frequency discrepancies of atomic clocks relative to a fixed clock, Δτ_trop is the troposphere delay correction, Δτ_ionois the ionosphere delay correction and Δτrelativistic are the delay corrections due to the relativistic effects.

The VLBI delay model is designed primarily for the analysis of VLBI observations to extra‒galactic objects from the surface of the Earth. The observable quantities of the VLBI space geodetic technique are recorded signals mea- sured in proper time by the station clocks. The VLBI clocks are synchronized to Universal Time Coordinated (UTC). The ICRF catalogue coordinates of the sources are defined in the Barycentric Celestial Reference System (BCRS) (space‒

fixed, origin is the center of the mass of the Solar system, equatorial) and the VLBI antennas coordinates are defined in the International Terrestrial Reference System (ITRS) (Earth‒fixed, origin is the center of the mass of the Earth).

There is another coordinate system defined as Geocentric Celestial Reference System (GCRS) which is kinematically non‒rotating w.r.t. BCRS and the origin is at the center of the mass of the Earth. The calculated delay is the time of arrival at station 2, t₂, minus the time of the arrival at station 1, t₁, in GCRS (Petit and Luzum 2010). The UTC time tag of the scan at the first station, i.e. t₁, serves as the time reference for the measurement and all scalar and vector quantities are assumed to be calculated at t₁, including e.g. the troposphere delay, tidal corrections on antenna TRF coordinates, the barycentric velocity of the geocenter. However, t₁ with time scale UTC should be transformed to the appropriate time scale corresponding to the time argument to be used

Figure 1: Principle of VLBI observation: The primary observable is the difference in the arrival times of a plane wavefront emitted by an extragalactic radio source at two VLBI telescopes