5.5 Assessment of the in the vTEC and Issess e t o t e t e ... · f ll i fi titi i i id dfollowing...
1
A A s s s s e e s s s s m m e e n n t t o o f f S S I I R R G G A A S S I I o o n n o o s s p p h h e e r r i i c c M M a a p p s s e e r r r r o o r r s s b b a a s s e e d d o o n n a a A A s s s s e e s s s s m m e e n n t t o o f f S S I I R R G G A A S S I I o o n n o o s s p p h h e e r r i i c c M M a a p p s s e e r r r r o o r r s s b b a a s s e e d d o o n n a a A A s s s s e e s s s s m m e e n n t t o o f f S S I I R R G G A A S S I I o o n n o o s s p p h h e e r r i i c c M M a a p p s s e e r r r r o o r r s s b b a a s s e e d d o o n n a a i l i l ti n n u u m m e e r r i i c c a a l l s s i i m m u u l l a a t t i i o o n n n n u u m m e e r r i i c c a a l l s s i i m m u u l l a a t t i i o o n n Claudio Brunini Emilio Camilion Francisco Azpilicueta Claudio Brunini, Emilio Camilion, Francisco Azpilicueta Facultad de Ciencias Astronómicas y Geofísicas Facultad de Ciencias Astronómicas y Geofísicas Universidad Nacional de La Plata Universidad Nacional de La Plata and CONICET Argentina and CONICET, Argentina claudiobrunini@yahoo com [email protected] Fi Fig. cuts dens 1 Introduction dens comp 1. Introduction comp Q As responsible of the International Terrestrial Reference evaluates the errors that are produced when NeQ As responsible of the International Terrestrial Reference Frame (ITRF) densification in Latin America SIRGAS the model is used to reproduce those along Frame (ITRF) densification in Latin America, SIRGAS (Si t d R f i G éti l A éi ) the model is used to reproduce those ionospheric scenarios along merid (Sistema de Referencia Geocéntrico para las Américas) ionospheric scenarios. merid t 10 manages a continuously operational GNSS network with to 10 manages a continuously operational GNSS network with more than 200 receivers SIRGAS uses this network to heigh more than 200 receivers. SIRGAS uses this network to t i l f ti l T t l El t C t t Marc compute regional maps of vertical Total Electron Content Marc Septe (TEC)(Fif. 1) that are released to the community through Septe the SIRGAS web page (www sirgas org) June the SIRGAS web page (www.sirgas.org ). Dece As other similar products (e.g.: Global Ionospheric Maps Dece and f computed by the International GNSS Service) SIRGAS and f d h computed by the International GNSS Service), SIRGAS maps are based on a thin layer ionospheric mode in which and h maps are based on a thin layer ionospheric mode in which solar the whole ionosphere is represented by one spherical simu layer of infinitesimal thickness at a fixed height and a simu NeQ layer of infinitesimal thickness at a fixed height and a geometrical mapping function that relates the vertical to NeQ geometrical mapping function that relates the vertical to th l t TEC dens the slant TEC. solid This contribution aims to characterize the errors solid repre This contribution aims to characterize the errors i t d t th thi l i h i d Th k repre i associated to the thin layer ionospheric mode. The work ionos relies upon a numerical simulation performed with an 350 k relies upon a numerical simulation performed with an empirical model of the Earth’s ionosphere which allows empirical model of the Earth s ionosphere, which allows ti li ti bt t ll d i h i i Fig 1 SIRGAS vTEC for day 205 2005 0 UT creating a realistic but controlled ionospheric scenario, Fig.1. SIRGAS vTEC for day 205, 2005, 0 UT. and then 4 Assessment of the mapping function error 4. Assessment of the mapping function error 2 Thin layer ionospheric model The vTEC error due to the use of the mapping function is: 2. Thin layer ionospheric model r h r h The whole ionosphere is represented by one spherical layer of infinitesimal thickness at a fixed height, L h . 2 2 ( ) E E r h r h dr h TEC TEC h N N d L Th l t TEC ( C ) l i t llit t i li f i ht (LOS) i l t d t th ti l TEC cos , (, ) cos L L L L L h vTEC vTEC z h Nr Nr dr z . The slant TEC ( sTEC ) along a given satellite- to-receiver line-of-sight (LOS) is related to the vertical TEC 1 1 cos E E r h r h z ( L vTEC ) at the ionospheric penetration point (IPP) by: The systematic component of this error is characterized by the average of all the observed L ( L vTEC ) at the ionospheric penetration point (IPP) by: The systematic component of this error is characterized by the average of all the observed L to ±90º) from the 7 GNSS receivers: sec L L L sTEC z h vTEC , (1) to ±90º) from the 7 GNSS receivers: sec L L L sTEC z h vTEC , (1) 90º 1 1 E z 1/2 2 2 1 1 L L E h h dz , where ( ) L L z h is the LOS E z zenith angle at the IPP and 2 sec 1 sin L L E E E L z h r z r h 90º 7 180º L L E h h dz , L L E L L E E E L 90º R E z ( E z is the zenith angle at the observing point and E r the Earth’s radius) is the geometrical mapping function). and the varying component is characterized by the average of the (absolute value) deviations: E E 90º z The L vTEC is parameterized as a function of time, t , and the geographic latitude, L , and longitude, L , of 90 1 1 E z h h h d L L L the IPP: 7 180º L L L E h h h dz , the IPP: 90º 7 180 R E z 0 , , ; ,..., L L L L vTEC f t x x . (2) The error assessment is performed for 5 levels of solar activity: low (F10 7 = 70 SFU) interme 0 , , ; ,..., L L L L n vTEC f t x x . (2) The error assessment is performed for 5 levels of solar activity: low (F10.7 = 70 SFU), interme SFU) i t di t (130 SFU) i t di t hi h (160 SFU) d hi h (190 SFU) f th 12 The parameters, i x , 0,..., i n , and the satellite + receiver inter-frequency biases (IFB), R S , are SFU), intermediate (130 SFU), intermediate-high (160 SFU), and high (190 SFU); for the 12 The parameters, i x , 0,..., i n , and the satellite receiver inter frequency biases (IFB), R S , are ti t d f th GNSS b ti b d th f ll i ti f b ti year; and for 11 heights of the ionospheric layer, from 350 km to 550 km, with steps of 25 km. estimated from the GNSS observations based on the following equation of observation: Th lt f thi t i d i Fi 4 d 5 sec ; L z h f t x x (3) The results of this assessment are summarized in Figs. 4 and 5. 0 sec , , ; ,..., I I L L L L L n L z h f t x x , (3) where L is the dual-frequency GNSS ionospheric observable and v is the associated observational error where I L is the dual-frequency GNSS ionospheric observable and I v is the associated observational error. 5 070 / 03 5 070 / 06 3 N i l i l ti Fi 2 G t f th i l i l ti 3. Numerical simulation Fig. 2. Geometry for the numerical simulation. The vTEC distribution in the SIRGAS region is The vTEC distribution in the SIRGAS region is 0 h L )> 0 h L )> characterized by large horizontal gradients 0 < (h 0 < (h caused by the Appleton Anomaly (Fig 3) caused by the Appleton Anomaly (Fig. 3). These gradients are particularly noticeable during daytime and particularly in the North-South 300 350 400 450 500 550 -5 h 300 350 400 450 500 -5 h daytime and particularly in the North South direction h L h L 190 / 03 190 / 06 direction. 5 190 / 03 5 190 / 06 In order to study the errors caused by these In order to study the errors caused by these gradients in the thin layer ionospheric model the gradients in the thin layer ionospheric model, the f ll i fi titi i i id d (Fi 2) following fictitious scenario is considered (Fig.2): 0 (h L )> 0 (h L )> 14 LT; < ( < ( 14 LT; A GNSS satellite moving along the 300º A GNSS satellite moving along the 300 meridian; Pl f th fi 300º idi 5 5 meridian; Plane of the figure ≡ 300º meridian; 300 350 400 450 500 550 -5 h L 300 350 400 450 500 -5 h L An array of 7 GNSS receivers along the same G ≡ geocentre; h L h L An array of 7 GNSS receivers along the same meridian one every 10º of latitude from 40º to G ≡ geocentre; Fig. 4. Variation with the height of the ionospheric layer, L h (km), of L h (points) and meridian, one every 10º of latitude from –40º to º GQ ≡ projection of the Equator on the meridian plane; L L TECu for low (F10 7 = 70 SFu) and high (F10 7 = 190 SFu) solar activity and for March (03) a 20º; GQ ≡ projection of the Equator on the meridian plane; TECu, for low (F10.7 = 70 SFu) and high (F10.7 = 190 SFu) solar activity, and for March (03) a LOS zenith angles from every receiver R ≡ fictitious GNSS receive at geography latitude R ; The systematic effect of the mapping function error i e : h cancels at an ionospheric la LOS zenith angles from every receiver R fictitious GNSS receive at geography latitude R ; The systematic effect of the mapping function error, i.e.: L h , cancels at an ionospheric la covering all zenith angles, from 0º to ±90º S ≡ fictitious GNSS satellite on the meridian plane; that varies with the solar activity and month. (North and South of the observing point) h ≡ position of a point P along the LOS (North and South of the observing point). E r r h , ≡ position of a point, P , along the LOS. 190 450 190 The NeQuick electron density model is used to simulate the electron density distribution 190 450 190 The NeQuick electron density model is used to simulate the electron density distribution. For this particular geometry NeQuick reduces to a two-dimensional function, , eNQ N r , so that: 160 160 eNQ h 160 400 160 2 E r h ( ) 400 (, ) eNQ L vTEC N r dr , (4.a) 130 10.7 130 10.7 1 E eNQ L r h F1 F1 1 E r h h 350 2 E r h dr TEC N d (4 b) 100 100 , eNQ sTEC N r , and (4.b) 1 cos E r h z 1 E TEC h TEC (4 ) 1 2 3 4 5 6 7 8 9 10 11 12 70 300 1 2 3 4 5 6 7 8 70 cos L L L vTEC z h sTEC , (4.c) 1 2 3 4 5 6 7 8 9 10 11 12 month 1 2 3 4 5 6 7 8 month L L L h h 100 k d h 1000 k th i l b d f th i h Fig 5 Variation with the solar activity and month of h (km) (left) and h (TECu) (right) where 1 h ~100 km and 2 h ~1000 km are the nominal bounds of the ionosphere. Fig. 5. Variation with the solar activity and month of ,0 L h (km) (left) and ,0 L h (TECu) (right). European Geophysical Union, 2010 General Assembly, Vienna, May 2-7, 2010. 5. Assessment of the in the vTEC and IFB estimation errors Th ( ) h di d i S ti 4 t t th k f E (3) h th i l The ( ) L h error discussed in Section 4 propagates to the unknowns of Eq. (3) when their values are estimated from the GNSS observations and causes an incorrect estimation of both the x x parameters estimated from the GNSS observations and causes an incorrect estimation of both, the 0 x ,…, n x parameters f th TEC f ti d th IFB of the L vTEC function and the IFB, . In order to study the error propagation a Legendre’s functions expansion is used to represent the latitudinal In order to study the error propagation, a Legendre s functions expansion is used to represent the latitudinal i ti f th TEC l th 300º idi f 50º t 30º f hi l tit d t 14 LT variation of the vTEC along the 300º meridian, from –50º to +30º of geographic latitude, at 14 LT: 12 12 0 0 ; sin l l vTEC f x x x P (6a) 0 ,0 0 ; ,..., sin n l l l vTEC f x x x P . (6.a) 3 L tit di l 0 l Th fi t i t it i t fit b L t S th i (6 ) t th N Q i k TEC 3. Latitudinal The first experiment consists in to fit by Least Squares the expansion (6.a) to the NeQuick vTEC: of the electron 2 12 E r h sity distribution 2 12 sin ( ) E vTEC x P Nr dr v (6 b) sity distribution puted with the ,0 0 sin (, ) l l L l vTEC x P Nr dr v . (6.b) puted with the 1 0 E l r h uick model The standard deviation of the residuals << ±0 1 TECu confirms a good quality of fit g the 300º The standard deviation of the residuals, v << ±0.1 TECu, confirms a good quality of fit. g the 300 dian from 100 The second experiment consists in to fit the expansion (6.a) to the NeQuick sTEC using the mapping dian, from 100 000 k f The second experiment consists in to fit the expansion (6.a) to the NeQuick sTEC using the mapping function: 000 km of function: ht, at 14 LT, for 2 12 E r h dr ch (MAR) 0 sin cos , L l l L L eNQ dr vTEC x P z h N r v . (6.c) ch (MAR), ember (SEP) 1 ,0 0 cos E L l l L L eNQ l r h z ember (SEP), 1 E r h Th t d d d i ti f th id l i ( t th i i t) t 5 3 TEC f h e (JUN) and The standard deviation of the residuals increases (w.r.t. the previous experiment) to v = ±5.3 TECu for L h = ember (DEC) v L 350 km ±3 2 TECu for 400 km ±3 7 TECu for 450 km and ±5 7 TECu for 500 km ember (DEC), for low (LSA) 350 km, ±3.2 TECu for 400 km, ±3.7 TECu for 450 km, and ±5.7 TECu for 500 km. The results of this experiment are summarized in Fig 6 for low (LSA) hi h (HSA) The results of this experiment are summarized in Fig. 6. high (HSA) 190 3 r activity, 190 3 ulated with the 2.5 ulated with the uick electron 160 The last experiment consists in to add the IFB to the Eq (6c) and to estimate their values together with uick electron 2 The last experiment consists in to add the IFB, , to the Eq. (6.c) and to estimate their values together with th i ffi i t sity model (the 7 the expansion coefficients: line at 6720 km 130 10.7 1.5 2 12 E r h d line at 6720 km esents the F 2 12 sin ( ) cos E dr vTEC x P h z h N r v (6d) esents the h i l t 1 ,0 0 sin ( ) cos , cos L l l L L L eNQ l h vTEC x P h z h N r v z . (6.d) spheric layer at 100 1 0 cos E l r h z km). 0.5 The inclusion of additional unknowns reduces (w r t the previous experiment) the standard deviations of the The inclusion of additional unknowns reduces (w.r.t. the previous experiment) the standard deviations of the residuals: = ±3 5 TECu for h = 350 km ±2 6 TECu for 400 km ±3 0 TECu for 450 km and ±4 2 TECu 1 2 3 4 5 6 7 8 9 10 11 12 70 0 residuals: v ,= ±3.5 TECu for L h = 350 km, ±2.6 TECu for 400 km, ±3.0 TECu for 450 km, and ±4.2 TECu month for 500 km. Nevertheless, this fact does not imply any improvement in the estimation of the L vTEC : in fact, L h results significantly greater than in the previous experiment (Fig 7) Fig 6 Uppe figurer: TEC estimated from the Eq (6c) L h results significantly greater than in the previous experiment (Fig. 7). Fig. 6. Uppe figurer: L vTEC estimated from the Eq. (6.c) for L h = 350, 400, 450 and 500 km (curves of different (5 a) for L h 350, 400, 450 and 500 km (curves of different l ) d NQik TEC (bl k ) i TEC f Since the NeQuick sTEC is not affected by DCB, any deviation from zero of the estimated ( ) L h (5.a) colours) and NeQuick vTEC (black curve) in TECu, for L unknowns of Eq (6 d) must be interpreted as an error ( ) h due to the propagation of the mapping high solar activity and March; middle: variation with the unknowns of Eq. (6.d) must be interpreted as an error, ( ) L h , due to the propagation of the mapping OS ( E z from 0º height of the ionospheric layer h (km) of h function errors to the estimated unknowns. OS ( E z from 0 height of the ionospheric layer, L h (km), of L h The systematic component of this error is characterized by the average of all the observed LOS ( z from 0º (points) and h (bars) in TECu for high solar The systematic component of this error is characterized by the average of all the observed LOS ( E z from 0º (points) and L h (bars), in TECu, for high solar ii d M h b i i i h h l to ±90º) from the 7 GNSS receivers: (5.b) activity and March; bottom: variation with the solar 20º 1 R (5.b) activity and month of h in TECu 20 1 ( ) ( ) R h h (7 a) activity and month of ,0 L h in TECu. 40º ( ) ( ) 7 L L h h , (7.a) 40º 7 R and the varying component is characterized by the average of the (absolute value) deviations: (5 c) and the varying component is characterized by the average of the (absolute value) deviations: 20º 1 (5.c) 20 1 R h h h (7 b) 7 L L L h h h . (7.b) ediate low (100 40º 7 R ediate-low (100 2 th f th Both estimates are shown in Fig. 8. 2 months of the Both estimates are shown in Fig. 8. 190 3 2.5 160 2 7 130 F10.7 1.5 F 1 100 0.5 70 0 1 2 3 4 5 6 7 8 9 10 11 12 70 th 0 month 550 Fig 8 Left: ( ) h (points) and ( ) h (bars) estimated from the Eq (6 d) for h = 350 400 Fig. 8. Left: ( ) L h (points) and ( ) L h (bars) estimated from the Eq. (6.d) for L h = 350, 400, 450 and 500 km, for high solar activity and March; right: variation with the solar activity and month of h (TECu) ,0 L h (TECu). It should be noted that: i) the ionospheric layer height h that cancels the systematic bias in the It should be noted that: i) the ionospheric layer height, ,0 L h , that cancels the systematic bias in the vTEC estimation is, in general, different from the ionospheric layer height, 0 ' L h , that cancels the 550 ,0 L systematic bias in the IFB estimation (Fig 9 left); and ii) the simultaneous estimation of vTEC and systematic bias in the IFB estimation (Fig. 9, left); and ii) the simultaneous estimation of vTEC and DCB k th t ti TEC bi iti ( t th ti ti f TEC ith t DCB) t L h (bars), in DCB makes the systematic vTEC bias more sensitive (w.r.t. the estimation of vTEC without DCB) to L and June (06) Fig. 7. Upper figure: L vTEC estimated from the Eq. (6.d) ionospheric layer height changes (Fig. 9.right). and June (06). L for h = 350 400 450 and 500 km (curves of different ionospheric layer height changes (Fig. 9.right). ayer height h for L h 350, 400, 450 and 500 km (curves of different l ) d NQik TEC (bl k ) i TEC f ayer height ,0 L h colours) and NeQuick vTEC (black curve) in TECu, for high solar activity and March; bottom: variation with the 15 190 height of the ionospheric layer, L h (km), of L h 180 190 height of the ionospheric layer, L h (km), of L h ( it) d h (b ) i TEC f hi h l ti it 10 (points) and L h (bars), in TECu, for high solar activity 160 160 3 and March 5 0.1 3 and March. 140 .7 7 25 F10. 0 130 10.7 2.5 120 5 F 2 100 -5 0.05 2 100 10 100 1.5 6 C l i 80 -10 6. Conclusions 80 15 1 Th h i ht f i h i l i k t f th 1 2 3 4 5 6 7 8 9 10 11 12 month -15 1 2 3 4 5 6 7 8 9 10 11 12 70 0 The height of ionospheric layer is a key parameter of the month month 0.5 thin layer ionospheric model. There is not any fixed height Fi 9 L ft diff ' h h h (k ) b t th i h i l h i ht ' h th t f lfil that reduces to zero the effects of the mapping function Fig. 9.Left: difference ,0 ,0 ,0 ' L L L h h h (km) between the ionospheric layer heights, ,0 ' L h that fulfils 9 10 11 12 0 that reduces to zero the effects of the mapping function errors on the vTEC and DCB estimation A fixed height for the condition ' h = 0 and h that fulfils the condition h = 0; right: sensitivity 9 10 11 12 errors on the vTEC and DCB estimation. A fixed height for the condition ,0 ' L h = 0 and ,0 L h that fulfils the condition ,0 L h = 0; right: sensitivity a given solar activity and month can be selected to reduce (TECu/km) of the systematic bias ( ) h to the ionospheric layer height h ; to zero the average of the vTEC bias for a given attitudinal (TECu/km) of the systematic bias ( ) L h to the ionospheric layer height, L h ; to zero the average of the vTEC bias for a given attitudinal range but still remains a latitudinal varying residual error range, but still remains a latitudinal varying residual error.
Transcript of 5.5 Assessment of the in the vTEC and Issess e t o t e t e ... · f ll i fi titi i i id dfollowing...
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Claudio Brunini Emilio Camilion Francisco Azpilicueta Claudio Brunini, Emilio Camilion, Francisco Azpilicueta Facultad de Ciencias Astronómicas y Geofísicas Facultad de Ciencias Astronómicas y Geofísicas Universidad Nacional de La Plata Universidad Nacional de La Plata
and CONICET Argentina and CONICET, Argentina claudiobrunini@yahoo com [email protected]
Fi Fig. cuts dens
1 Introduction denscomp1. Introduction comp
QAs responsible of the International Terrestrial Reference evaluates the errors that are produced when NeQAs responsible of the International Terrestrial Reference Frame (ITRF) densification in Latin America SIRGAS
pthe model is used to reproduce those alongFrame (ITRF) densification in Latin America, SIRGAS
(Si t d R f i G é t i l A é i ) the model is used to reproduce those ionospheric scenarios
alongmerid(Sistema de Referencia Geocéntrico para las Américas) ionospheric scenarios. meridt 10manages a continuously operational GNSS network with to 10manages a continuously operational GNSS network with
more than 200 receivers SIRGAS uses this network to
heighmore than 200 receivers. SIRGAS uses this network to t i l f ti l T t l El t C t t
gMarccompute regional maps of vertical Total Electron Content MarcSepte(TEC)(Fif. 1) that are released to the community through Septe( )( ) y g
the SIRGAS web page (www sirgas org) Junethe SIRGAS web page (www.sirgas.org). Dece
As other similar products (e.g.: Global Ionospheric Maps Deceand fp ( g p p
computed by the International GNSS Service) SIRGAS and f
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simuNeQlayer of infinitesimal thickness at a fixed height and a
geometrical mapping function that relates the vertical to NeQ
geometrical mapping function that relates the vertical to th l t TEC
densthe slant TEC. solid This contribution aims to characterize the errors
solid repreThis contribution aims to characterize the errors
i t d t th thi l i h i d Th k repreiassociated to the thin layer ionospheric mode. The work ionos
relies upon a numerical simulation performed with an 350 krelies upon a numerical simulation performed with an empirical model of the Earth’s ionosphere which allows
350empirical model of the Earth s ionosphere, which allows
ti li ti b t t ll d i h i i
Fig 1 SIRGAS vTEC for day 205 2005 0 UT creating a realistic but controlled ionospheric scenario, Fig.1. SIRGAS vTEC for day 205, 2005, 0 UT. and then 4 Assessment of the mapping function error 4. Assessment of the mapping function error 2 Thin layer ionospheric model The vTEC error due to the use of the mapping function is: 2. Thin layer ionospheric model e C e o due to t e use o t e app g u ct o s
r h r h The whole ionosphere is represented by one spherical layer of infinitesimal thickness at a fixed height, Lh . 2 2
( )E Er h r hdrh TEC TEC h N N d
p p y p y g , L
Th l t TEC ( C ) l i t llit t i li f i ht (LOS) i l t d t th ti l TEC cos , ( , )
cosL L L L Lh vTEC vTEC z h N r N r drz
. The slant TEC (sTEC ) along a given satellite- to-receiver line-of-sight (LOS) is related to the vertical TEC 1 1
cosE Er h r hz
( LvTEC ) at the ionospheric penetration point (IPP) by: The systematic component of this error is characterized by the average of all the observed L( LvTEC ) at the ionospheric penetration point (IPP) by: The systematic component of this error is characterized by the average of all the observed Lto ±90º) from the 7 GNSS receivers: sec L L LsTEC z h vTEC , (1) to ±90º) from the 7 GNSS receivers: sec L L LsTEC z h vTEC , (1)
90º1 1 Ez
1/ 222
1 1L L Eh h dz , where ( )L Lz h is the LOS Ez zenith angle at the IPP and 2sec 1 sinL L E E E Lz h r z r h
90º7 180ºL L Eh h dz
, ( )L L E g L L E E E L 90ºR Ez
( Ez is the zenith angle at the observing point and Er the Earth’s radius) is the geometrical mapping function). and the varying component is characterized by the average of the (absolute value) deviations: ( E g g p E ) g pp g ) and the varying component is characterized by the average of the (absolute value) deviations: 90ºz The LvTEC is parameterized as a function of time, t , and the geographic latitude, L , and longitude, L , of 901 1 Ez
h h h d
L p , , g g p , L , g , L ,the IPP:
7 180ºL L L Eh h h dz , the IPP: 90º7 180
R Ez
0, , ; ,...,L L L LvTEC f t x x . (2) The error assessment is performed for 5 levels of solar activity: low (F10 7 = 70 SFU) interme 0, , ; ,...,L L L L nvTEC f t x x . (2) The error assessment is performed for 5 levels of solar activity: low (F10.7 = 70 SFU), intermeSFU) i t di t (130 SFU) i t di t hi h (160 SFU) d hi h (190 SFU) f th 12The parameters, ix , 0,...,i n , and the satellite + receiver inter-frequency biases (IFB), R S , are SFU), intermediate (130 SFU), intermediate-high (160 SFU), and high (190 SFU); for the 12The parameters, ix , 0,...,i n , and the satellite receiver inter frequency biases (IFB), R S , are
ti t d f th GNSS b ti b d th f ll i ti f b ti year; and for 11 heights of the ionospheric layer, from 350 km to 550 km, with steps of 25 km. estimated from the GNSS observations based on the following equation of observation: year; and for 11 heights of the ionospheric layer, from 350 km to 550 km, with steps of 25 km. Th lt f thi t i d i Fi 4 d 5 sec ;L z h f t x x (3) The results of this assessment are summarized in Figs. 4 and 5. 0sec , , ; ,...,I I L L L L L nL z h f t x x , (3)
where L is the dual-frequency GNSS ionospheric observable and v is the associated observational error where IL is the dual-frequency GNSS ionospheric observable and Iv is the associated observational error.
5070 / 03
5070 / 06
3 N i l i l ti Fi 2 G t f th i l i l ti 3. Numerical simulation Fig. 2. Geometry for the numerical simulation. The vTEC distribution in the SIRGAS region is The vTEC distribution in the SIRGAS region is
0h L)>
0h L)>
characterized by large horizontal gradients 0
< (h 0
< (hy g g
caused by the Appleton Anomaly (Fig 3) caused by the Appleton Anomaly (Fig. 3). These gradients are particularly noticeable during g p y gdaytime and particularly in the North-South 300 350 400 450 500 550
-5
h300 350 400 450 500-5
hdaytime and particularly in the North South direction
hL hL
190 / 03 190 / 06direction. 5190 / 03
5190 / 06
In order to study the errors caused by these In order to study the errors caused by these gradients in the thin layer ionospheric model the gradients in the thin layer ionospheric model, the f ll i fi titi i i id d (Fi 2) following fictitious scenario is considered (Fig.2): 0(h
L)>
0(hL)>
14 LT;
< (
< (
14 LT; A GNSS satellite moving along the 300º A GNSS satellite moving along the 300
meridian; Pl f th fi 300º idi 5 5meridian; Plane of the figure ≡ 300º meridian; 300 350 400 450 500 550-5
hL
300 350 400 450 500-5
hL
An array of 7 GNSS receivers along the same g
G ≡ geocentre; hL hL
An array of 7 GNSS receivers along the same meridian one every 10º of latitude from 40º to
G ≡ geocentre; Fig. 4. Variation with the height of the ionospheric layer, Lh (km), of Lh (points) and meridian, one every 10º of latitude from –40º to º GQ ≡ projection of the Equator on the meridian plane;
g g p y , L ( ), L (p )TECu for low (F10 7 = 70 SFu) and high (F10 7 = 190 SFu) solar activity and for March (03) a20º; GQ ≡ projection of the Equator on the meridian plane; TECu, for low (F10.7 = 70 SFu) and high (F10.7 = 190 SFu) solar activity, and for March (03) a
LOS zenith angles from every receiver R ≡ fictitious GNSS receive at geography latitude R ; The systematic effect of the mapping function error i e : h cancels at an ionospheric la LOS zenith angles from every receiver R fictitious GNSS receive at geography latitude R ; The systematic effect of the mapping function error, i.e.: Lh , cancels at an ionospheric lacovering all zenith angles, from 0º to ±90º S ≡ fictitious GNSS satellite on the meridian plane; that varies with the solar activity and month. g g ,(North and South of the observing point)
p ;h ≡ position of a point P along the LOS
y (North and South of the observing point).
Er r h , ≡ position of a point, P , along the LOS. 190 450 190
The NeQuick electron density model is used to simulate the electron density distribution 190 450 190
The NeQuick electron density model is used to simulate the electron density distribution. For this particular geometry NeQuick reduces to a two-dimensional function, ,eNQN r , so that: 160 160p g y Q , ,eNQ ,
h
160
400
160
2Er h
( )400
( , )eNQ LvTEC N r dr , (4.a) 13010.7
13010.7
1
( , )E
eNQ Lr h
( )
F1 F1
1Er h
h350
2Er h drTEC N
d (4 b) 100 100
,eNQsTEC N r , and (4.b) 1
cosE
Qr h z
1E
TEC h TEC (4 ) 1 2 3 4 5 6 7 8 9 10 11 1270 300
1 2 3 4 5 6 7 870
cosL L LvTEC z h sTEC , (4.c) 1 2 3 4 5 6 7 8 9 10 11 12month
1 2 3 4 5 6 7 8month L L L
h h 100 k d h 1000 k th i l b d f th i h Fig 5 Variation with the solar activity and month of h (km) (left) and h (TECu) (right)where 1h ~100 km and 2h ~1000 km are the nominal bounds of the ionosphere. Fig. 5. Variation with the solar activity and month of ,0Lh (km) (left) and ,0Lh (TECu) (right).
European Geophysical Union, 2010 General Assembly, Vienna, May 2-7, 2010.p p y y y
5. Assessment of the in the vTEC and IFB estimation errors 5 ssess e t o t e t e C a d est at o e o sTh ( )h di d i S ti 4 t t th k f E (3) h th i l The ( )Lh error discussed in Section 4 propagates to the unknowns of Eq. (3) when their values are estimated from the GNSS observations and causes an incorrect estimation of both the x x parameters estimated from the GNSS observations and causes an incorrect estimation of both, the 0x ,…, nx parameters f th TEC f ti d th IFB of the LvTEC function and the IFB, .
In order to study the error propagation a Legendre’s functions expansion is used to represent the latitudinal In order to study the error propagation, a Legendre s functions expansion is used to represent the latitudinal i ti f th TEC l th 300º idi f 50º t 30º f hi l tit d t 14 LT variation of the vTEC along the 300º meridian, from –50º to +30º of geographic latitude, at 14 LT: g g g
12
12
0 0; sinl lvTEC f x x x P (6 a) 0 ,00
; ,..., sinn l ll
vTEC f x x x P . (6.a)
3 L tit di l 0l
Th fi t i t i t i t fit b L t S th i (6 ) t th N Q i k TEC 3. Latitudinal The first experiment consists in to fit by Least Squares the expansion (6.a) to the NeQuick vTEC: of the electron
y ( )212 Er h
sity distribution 212
sin ( )E
vTEC x P N r dr v (6 b) sity distribution puted with the
,00
sin ( , )l l Ll
vTEC x P N r dr v . (6.b) puted with the 1
0E
l r h
uick model The standard deviation of the residuals << ±0 1 TECu confirms a good quality of fit g the 300º
The standard deviation of the residuals, v << ±0.1 TECu, confirms a good quality of fit. g the 300 dian from 100 The second experiment consists in to fit the expansion (6.a) to the NeQuick sTEC using the mapping dian, from 100
000 k f The second experiment consists in to fit the expansion (6.a) to the NeQuick sTEC using the mapping function: 000 km of function:
ht, at 14 LT, for
212 Er h dr
, ,
ch (MAR) 0 sin cos ,L l l L L eNQdrvTEC x P z h N r v . (6.c) ch (MAR),
ember (SEP)
1
,00
,cos
E
L l l L L eNQl r h z
( )
ember (SEP), 1Er h
Th t d d d i ti f th id l i ( t th i i t) t 5 3 TEC f h e (JUN) and The standard deviation of the residuals increases (w.r.t. the previous experiment) to v = ±5.3 TECu for Lh = ( )ember (DEC)
( ) v L
350 km ±3 2 TECu for 400 km ±3 7 TECu for 450 km and ±5 7 TECu for 500 km ember (DEC), for low (LSA)
350 km, ±3.2 TECu for 400 km, ±3.7 TECu for 450 km, and ±5.7 TECu for 500 km. The results of this experiment are summarized in Fig 6 for low (LSA)
hi h (HSA) The results of this experiment are summarized in Fig. 6.
high (HSA) 190 3
r activity, 190 3
y,ulated with the 2.5ulated with the uick electron 160The last experiment consists in to add the IFB to the Eq (6 c) and to estimate their values together with uick electron 2
The last experiment consists in to add the IFB, , to the Eq. (6.c) and to estimate their values together with th i ffi i t sity model (the
7
the expansion coefficients: y ( line at 6720 km 13010
.7
1.5212 Er h d line at 6720 km esents the
F
212
sin ( ) cosE drvTEC x P h z h N r v (6 d) esents the
h i l t 1 ,0
0sin ( ) cos ,
cosL l l L L L eNQl h
vTEC x P h z h N r vz
. (6.d) spheric layer at 1001
0 cosE
l r h z
km). 0.5The inclusion of additional unknowns reduces (w r t the previous experiment) the standard deviations of the ) The inclusion of additional unknowns reduces (w.r.t. the previous experiment) the standard deviations of the residuals: = ±3 5 TECu for h = 350 km ±2 6 TECu for 400 km ±3 0 TECu for 450 km and ±4 2 TECu
1 2 3 4 5 6 7 8 9 10 11 1270 0residuals: v ,= ±3.5 TECu for Lh = 350 km, ±2.6 TECu for 400 km, ±3.0 TECu for 450 km, and ±4.2 TECu
monthfor 500 km. Nevertheless, this fact does not imply any improvement in the estimation of the LvTEC : in fact, , p y y p L , h results significantly greater than in the previous experiment (Fig 7)
Fig 6 Uppe figurer: TEC estimated from the Eq (6 c) Lh results significantly greater than in the previous experiment (Fig. 7).
Fig. 6. Uppe figurer: LvTEC estimated from the Eq. (6.c) for Lh = 350, 400, 450 and 500 km (curves of different
(5 a) for Lh 350, 400, 450 and 500 km (curves of different
l ) d N Q i k TEC (bl k ) i TEC f Since the NeQuick sTEC is not affected by DCB, any deviation from zero of the estimated ( )Lh (5.a) colours) and NeQuick vTEC (black curve) in TECu, for y , y ( )Lunknowns of Eq (6 d) must be interpreted as an error ( )h due to the propagation of the mapping high solar activity and March; middle: variation with the unknowns of Eq. (6.d) must be interpreted as an error, ( )Lh , due to the propagation of the mapping
OS ( Ez from 0º g y ;
height of the ionospheric layer h (km) of h function errors to the estimated unknowns. OS ( Ez from 0 height of the ionospheric layer, Lh (km), of Lh The systematic component of this error is characterized by the average of all the observed LOS ( z from 0º (points) and h (bars) in TECu for high solar The systematic component of this error is characterized by the average of all the observed LOS ( Ez from 0º (points) and Lh (bars), in TECu, for high solar
i i d M h b i i i h h l to ±90º) from the 7 GNSS receivers:
(5.b) activity and March; bottom: variation with the solar )
20º1 R (5.b) yactivity and month of h in TECu
201( ) ( )R
h h
(7 a) activity and month of ,0Lh in TECu. 40º
( ) ( )7L Lh h , (7.a)
40º7R
and the varying component is characterized by the average of the (absolute value) deviations: (5 c)
and the varying component is characterized by the average of the (absolute value) deviations: 20º1 (5.c)
201 R
h h h
(7 b) 7L L Lh h h . (7.b)
ediate low (100 40º7
R
ediate-low (100 2 th f th Both estimates are shown in Fig. 8. 2 months of the Both estimates are shown in Fig. 8.
190 3
2.5
160
2
7
130
F10.
7
1.5
F
1
100
0.5
70 01 2 3 4 5 6 7 8 9 10 11 12
70
th
0
month
550
Fig 8 Left: ( )h (points) and ( )h (bars) estimated from the Eq (6 d) for h = 350 400 Fig. 8. Left: ( )Lh (points) and ( )Lh (bars) estimated from the Eq. (6.d) for Lh = 350, 400, 450 and 500 km, for high solar activity and March; right: variation with the solar activity and month of , g y ; g y
h (TECu) ,0Lh (TECu). It should be noted that: i) the ionospheric layer height h that cancels the systematic bias in the It should be noted that: i) the ionospheric layer height, ,0Lh , that cancels the systematic bias in the vTEC estimation is, in general, different from the ionospheric layer height, 0'Lh , that cancels the
550
, g , p y g , ,0L ,systematic bias in the IFB estimation (Fig 9 left); and ii) the simultaneous estimation of vTEC and
systematic bias in the IFB estimation (Fig. 9, left); and ii) the simultaneous estimation of vTEC and DCB k th t ti TEC bi iti ( t th ti ti f TEC ith t DCB) t Lh (bars), in DCB makes the systematic vTEC bias more sensitive (w.r.t. the estimation of vTEC without DCB) to L ( ),
and June (06) Fig. 7. Upper figure: LvTEC estimated from the Eq. (6.d) ionospheric layer height changes (Fig. 9.right). and June (06). g pp g L q ( )for h = 350 400 450 and 500 km (curves of different
ionospheric layer height changes (Fig. 9.right).
ayer height h for Lh 350, 400, 450 and 500 km (curves of different
l ) d N Q i k TEC (bl k ) i TEC f ayer height ,0Lh colours) and NeQuick vTEC (black curve) in TECu, for high solar activity and March; bottom: variation with the 15 190
height of the ionospheric layer, Lh (km), of Lh 180190
height of the ionospheric layer, Lh (km), of Lh ( i t ) d h (b ) i TEC f hi h l ti it
10
(points) and Lh (bars), in TECu, for high solar activity 160 1603
and March 50.1
3
and March. 140
.7 72 5
F10. 0
13010.72.5
120
5
F
2
100
-50.05
2
100
101001.5 6 C l i 80
-10 6. Conclusions 80
151 Th h i ht f i h i l i k t f th 1 2 3 4 5 6 7 8 9 10 11 12month
-151 2 3 4 5 6 7 8 9 10 11 12
70 0 The height of ionospheric layer is a key parameter of the month
month0.5 thin layer ionospheric model. There is not any fixed height Fi 9 L ft diff 'h h h (k ) b t th i h i l h i ht 'h th t f lfil
y p y gthat reduces to zero the effects of the mapping function Fig. 9.Left: difference ,0 ,0 ,0'L L Lh h h (km) between the ionospheric layer heights, ,0'Lh that fulfils
9 10 11 120
that reduces to zero the effects of the mapping function errors on the vTEC and DCB estimation A fixed height for , , , ,
the condition 'h = 0 and h that fulfils the condition h = 0; right: sensitivity 9 10 11 12 errors on the vTEC and DCB estimation. A fixed height for
the condition ,0'Lh = 0 and ,0Lh that fulfils the condition ,0Lh = 0; right: sensitivity a given solar activity and month can be selected to reduce (TECu/km) of the systematic bias ( )h to the ionospheric layer height h ;
g yto zero the average of the vTEC bias for a given attitudinal (TECu/km) of the systematic bias ( )Lh to the ionospheric layer height, Lh ; to zero the average of the vTEC bias for a given attitudinal range but still remains a latitudinal varying residual error range, but still remains a latitudinal varying residual error.
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