Azimuthal Anisotropy Adaptive Mapping Functions - Ensg

Refractivity of the Troposphere Propagation & Ray-Tracing Adaptive Mapping Functions Conclusions

Azimuthal Anisotropy Adaptive Mapping Functions Pascal Gegout1 , Richard Biancale2 , Laurent Soudarin3 Groupe de Recherche de G´ eod´ esie Spatiale - Toulouse 1 CNRS/GS, 2 CNES/GS, 3 CNES/CLS Analysis Center

Session 1 : Technical and Physical Details of Ray-Tracing Session 2 : Models derived from Ray-Traced Delays Workshop on Ray-Tracing for Space Geodetic Techniques 29-30 April 2010 - Institute of Geodesy and Geophysics - Vienna

Pascal Gegout

Azimuthal Anisotropy Adaptive Mapping Functions


1

Vertical Hybrid Coordinates Equations of State Orography and Geopotential Model Levels Heights Refractivity Refractivity Functions

Refractivity of the Troposphere Vertical Hybrid Coordinates Equations of State Orography and Geopotential Model Levels Heights Refractivity Refractivity Functions

Pascal Gegout




Time sampling The Integrated Forecasting System (IFS) implemented at ECMWF, the European Center for Medium-range Weather Forecasts, contains a four-dimensional variational (4DVAR) analysis system. The 4DVAR analysis is closely equivalent to a Kalman filter analysis. The influence of an observation in space and time is controlled by the model dynamics which increases its realism of the spreading out of the information. The atmospheric analysis procedure is a four-dimensional variational data assimilation over a time window from 9 hours before to 3 hours after the nominal analysis time (00, 06, 12 and 18 UTC). Delayed cutoff data assimilation steps are provided 3-hourly over 12 hours windows at 09 and 21 UTC.

Pascal Gegout




Horizontal Discretisation The horizontal representation of the atmospheric model is defined by the spherical harmonics truncation. at a T511L60 resolution from January 2001 to january 2006, at a T799L91 resolution from february 2006 to january 2010, and has increased to a T1279L91 resolution in january 2010.

A truncation at degree T799 (N400 gaussian grid) corresponds to a 25 km resolution which may be interpolated at a 0.225 by 0.225 regular latitude-longitude grid. A truncation at degree T1279 (N640 gaussian grid) corresponds to a 16 km resolution which may be interpolated at a 0.125 by 0.125 regular latitude-longitude grid.

Pascal Gegout




Vertical Discretisation The vertical hybrid coordinates are defined by the pressures at half-levels which are functions of the surface pressure ps (φ, λ; t) at a specific location at a given time. pk+1/2 (φ, λ; t) = Ak+1/2 + Bk+1/2 ps (φ, λ; t)

(1)

The set of constants Ak+1/2 and Bk+1/2 define the vertical hybrid coordinates system varying in time with surface pressure ps (φ, λ; t). The prognostic variables Φk , Tk , qk are represented by their values at the full-level pressures pk coherently described within the model’s vertical finite-difference scheme by ln pk =

pk+1/2 ln pk+1/2 − pk−1/2 ln pk−1/2 −1 ∆pk ∆pk = pk+1/2 − pk−1/2 . Pascal Gegout


(2) (3)



Equations of State The atmosphere is made off moist air, a mix of dry air and water vapor. The equation of state for dry air and water vapor are given by pdry = ρdry Rdry T

pvap = ρvap Rvap T

(4)

Specific humidity q is the mass of water vapor per unit mass of dry air. q = ρvap /ρdry

(5)

The virtual temperature is the temperature that dry air need to have the same density as moist air at a given pressure. Tv = T (1 + q (Rvap /Rdry − 1))

(6)

The pressure p, the density ρ and the equation of state for moist air are : p = pdry + pvap

ρ = ρdry + ρvap Pascal Gegout

p = ρRdry Tv


(7)



Area Interpolation

The complete atmospheric structure is recovered from the archived model levels fields: the surface pressure ps , the surface geopotential Φs , the temperature Tk and the specific humidity qk . For a given tracking site only the atmosphere above the local horizon is needed since radio waves do not travel through the solid Earth. Area Interpolation: Using the EMOSLIB library provided by ECMWF, the model level fields are read, decoded and interpolated on regular 0.125 x 0.125 latitude-longitude subgrids centered on the site’s nearest point with a 12.5 degrees extent (100 samples) from the tracking site.

Pascal Gegout




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Figure: Area Orography of the Tsukuba VLBI 7345 site 08/12/2008, 0Z.

Pascal Gegout




Model Levels Heights as defined in Meteorology The surface pressure ps and the surface geopotential Φs are defined at the (surface) orography. The orography is derived by averaging (and small-scales filtering) the GTOPO30 terrain elevation data set. The resulting mean topography is realistic over most land areas but is insufficient in high mountain areas where the standard deviation of the topography is added to properly trigger gravity waves. The orthometric height of the orography at the geodetic latitude φ and longitude λ is defined implicitely using the conventions of the GTOPO30 elevation model. The elevations of GTOPO30 are referenced vertically to the EGM96 geoid N(φ, λ) of WGS84. The conclusion reached is that the surface geopotential Φs (φ, λ) is defined in meteorology as the orthometric height of the orography multiplied by the gravity constant of g = 9.80665 ms −2 (WMO). Pascal Gegout




The vertical discretization of the model rely on the surface geopotential and the discrete formulation of the hydrostatic equation for the moist air : pk+1/2 Φk+1/2 − Φk−1/2 = −Rdry (Tv )k ln (8) pk−1/2 and provide the definition of the half-level geopotential using the surface geopotential as reference: Φk+1/2 = Φs +

NLEV X

Rdry (Tv )j ln

j=k+1

pj+1/2 pj−1/2

(9)

Full-level values of the geopotential are given by Φk = Φk+1/2 + αk Rdry (Tv )k with αk = 1 −

pk−1/2 pk+1/2 ln ∀k > 1 and α1 = ln 2 ∆pk pk−1/2 Pascal Gegout


(10) (11)



One should notice that in numerical weather modeling no assumption is made on the geometry of the Atmosphere, except the invariant boundary condition provided by the surface geopotential. The pressure levels are partly homothetically defined from the time variable surface pressure on the orography and geopotential levels are defined from these vertical hybrid coordinates at each time step. The parametrisation is only made off physical and dynamical parameters defined in the frame of the vertical hybrid coordinates and the influence of gravity on the atmospheric dynamic derives from the geopotential levels. /!\ We have to rebuild the geometry of the atmosphere from the dynamical geopotential levels at each time step. In order to build precisely the geometric structure of the atmosphere using a realistic geodetic model, we need to take into account the ellipsoidal shape of the Earth, the shape of the geoid and the gravity acceleration above the ellipsoid. Pascal Gegout




Model Levels Heights designed for Geodesy

geodetic height h = height relative to the ellipsoid

orthometric height H height relative to the geoid

+ geoidal undulation N

geodetic height h =

normal height H ∗

+ height anomaly ζ

A geopotential difference is related to the orthometric height or to the normal height by Z

H

Φ0 − Φk =

Z g dH =

0

Pascal Gegout

H∗

γ dH ∗

0


(12)



Model Levels Heights designed for Geodesy The normal gravity γ is given on the surface of the ellipsoid at the geodetic latitude φ by the formula of Somigliana. bγp 1 + ksin2 φ where k = −1 γ = γe p 2 2 aγe 1 − e sin φ

(13)

The normal gravity above the ellipsoid is a function of the geodetic height h : h h2 2 γh = γ 1 − 2 1 + f + m − 2f sin φ + 3 2 (14) a a The geodetic height hk (φ, λ) of the geopotential Φk (φ, λ) satisfy: hk2 hk 2 Φ0 − Φk = γhk 1 − 1 + f + m − 2f sin φ + 2 a a Pascal Gegout


(15)



Refractivity of the moist air The refractivity of the moist air is the key physical parameter which drive the propagation of GNSS signals through the troposphere. The total refractivity of moist air is expressed in terms of the (total moist air) pressure and the wet (water vapor) partial pressure. pvap k pvap pk N k = k1 + k20 + k3 2 k (16) Tvk Tk Tk This decomposition is preferred because its first term stricly follows the hydrostatic equilibrium of the model and the total refractivity does not depart two much from an hydrostatic equilibrium. The refractivity constants are empirically determined coefficients ! k1 = 77.6 10−2 K .Pa−1 , k2 = 70.4 10−2 K .Pa−1 , k3 = 3.739 103 K 2 .Pa−1 , k20 = 22.1 10−2 K .Pa−1 (Bevis et al. 1994). Pascal Gegout




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Figure: Surface refractivity N91 of the Tsukuba VLBI 7345 site 08/12/2008, 0Z.

Pascal Gegout




Refractivity Functions: Vertical interpolation Vertical refractivity functions are introduced in order to properly represent the refractivity between the model levels along the normal to the ellipsoid which points to the zenith of the local horizon. Since the hydrostatic term follows the hydrostatic equilibrium and since the vertical (total) refractivity profile is close to an hydrostatic profile, exponential functions are chosen. N(φi , λj , z) = exp(Nkc + Nkz z) ∀z ∈ [zk , zk−1 ]

(17)

Nkc (φi , λj ) = (zk−1 ln Nk − zk ln Nk−1 )/(zk−1 − zk )

(18)

Nkz (φi , λj ) = (ln Nk−1 − ln Nk )/(zk−1 − zk )

(19)

Pascal Gegout




Refractivity Functions: Horizontal interpolation For a point P(φ, λ, z) along the propagation path, the coefficients Nkc (φi , λj ) and Nkz (φi , λj ) of the four neighbouring refractivity profiles are interpolated by a bilinear interpolation to define the vertical refractivity profile by Npc (φ, λ) and Npz (φ, λ) at point P. N(φ, λ, z) = exp(Npc (φ, λ) + Npz (φ, λ)z)

(20)

The refractive index n(z) is given along the propagation path. n(φ, λ, z) = 10−6 N(φ, λ, z) + 1

(21)

The refractivity functional N(φ, λ, z) provide a continuous formulation of the vertical gradient of the refractive index. dn = (n − 1)Npz with z ∈ [zk , zk−1 ] dz Pascal Gegout


(22)



Refractivity Functions: Vertical extrapolation

If the tracking site is slightly below the bottom level (k=NLEV) or if the propagation ends above the top level (k=1) the extrapolation is made from the continuous formulation of the nearest level. With the purpose to estimate separately the respective impacts of the hydrostatic and non-hydrostatic terms, a similar continuous ¯ formulation is introduced only for the hydrostatic refractivity N. ¯ i , λj , z) = exp(N ¯ kc + N ¯ kz z) ∀z ∈ [zk , zk−1 ] N(φ

Pascal Gegout


(23)


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Figure: Surface refractivity N91 of the Tsukuba VLBI site (08/12/2008 0Z) and Latitudinal vertical refractivity profile from the surface to a height of 25 km computed every 125 m using vertical refractivity functions.

Pascal Gegout




Conclusion: Heights & Refractivity Functions

The ellipsoidal shape of the Earth and the Atmosphere, the geoid and gravity acceleration above the ellipsoid are taken into account. The model geometry is provided in ellipsoidal geodetic coordinates. The model levels heights are given in terms of orthometric heights. The model refractivities are provided by a continuous formulation by refractivity functions of the orthometric height.

Pascal Gegout



2

Eikonal Equation Ray-Tracing Tropospheric Delays Ray Bending Parallax Azimuthal Anisotropy

Propagation & Ray-Tracing Eikonal Equation Ray-Tracing Tropospheric Delays Ray Bending Parallax Azimuthal Anisotropy

Pascal Gegout




The eikonal equation in spherical coordinates is H(r , θ, λ, pr , pθ , pλ ) = 0. p2 1 p2 pr2 + 2θ + 2 λ 2 − n2 (r , θ, λ) = 0 (24) H= 2 r r sin θ Hamilton’s equations for the Hamiltonian are : dr ∂H = = pr dσ ∂pr dθ ∂H pθ = = 2 dσ ∂pθ r ∂H pλ dλ = = 2 2 dσ ∂pλ r sin θ dpr ∂H ∂n 1 pθ2 pλ2 =− =n + + 2 2 dσ ∂r ∂r r r2 r sin θ 2 dpθ ∂H ∂n pλ cot θ =− =n + dσ ∂θ ∂θ r 2 sin2 θ dpλ ∂H ∂n =− =n dσ ∂λ ∂λ Pascal Gegout


(25) (26) (27) (28) (29) (30)



The independent variable σ is the so-called generating parameter, which is related to the arclength s by ds = ndσ

(31)

and to the radio path length L of the electromagnetic wave by dT = nds = n2 dσ.

(32)

Introducing the elevation angle ε above the local horizon and the azimuth α taken counterclockwise from due south at every point P(r , θ, λ) along the ray, the conjugate momentum variables are: pr = n sin ε

(33)

pθ = n r cos ε cos α

(34)

pλ = n r cos ε sin α sin θ

(35)

Pascal Gegout




Using the following partial derivative of λ in term of the angles ε and α, n cos ε sin α dλ = dσ r sin θ

(36)

the six first-order eikonal differential equations are further reduced, by taking the longitude λ as the independent variable : r sin θ tan ε dr = dλ sin α

dθ = sin θ cot α dλ

dλ =1 dλ

The partial derivatives of the elevation ε and the azimuth α are : dε sin θ 1 dn sin θ tan ε 1 dn 1 dn = +1 + + tan ε dλ sin α n dr tan α n dθ n dλ dα sin θ 1 dn cot α 1 dn = − cos θ + − dλ cos2 ε n dθ cos2 ε n dλ Pascal Gegout


(37)

(38)

(39)



Partial derivatives of the arclength of the ray s and radio path length L of the wave through the dense atmosphere complete the eikonal differential system to provide the required solution. r sin θ dLs ds = = (40) dλ cos ε sin α dλ dL r sin θ =n dλ cos ε sin α

(41)

¯h An additional partial derivative relative to the hydrostatic delay D is required to separate the hydrostatic and non-hydrostatic terms. ¯h r sin θ dD = n¯h dλ cos ε sin α

(42)

This set of eight first-order eikonal differential equations describe the partials of the position and the direction at the current point P(r , θ, λ) and the partials of the lengths, solutions of the propagation problem. Pascal Gegout




Ray-Tracing: Integration of the differential system

Raytracing is performed by the integration of this set of eight differential equations made off the five differential equations describing the ray’s position in spherical coordinates, the ray’s direction in the tangent local frame by the elevation ε and the azimuth α, the three differential equations of the ray’s lengths.

Integration is achieved by a fourth order Runge-Kutta scheme. Integration is performed is spherical coordinates when the atmospheric model in described in ellipsoidal coordinates.

Pascal Gegout




Ray-Tracing: Initial Conditions

Initial conditions are provided by the reference site position in spherical coordinates Psite (rsite , θsite , λsite ). Initial ray direction is given by the elevation εsite and azimuth αsite . The elevation εspherical and azimuth αspherical are relative to the normal to the sphere and related to the elevation εellipsoidal and azimuth αellipsoidal relative to the ellipsoid normal by εellipsoidal = εspherical + (π/2 − φ − θ) cos α

(43)

αellipsoidal = αspherical + (π/2 − φ − θ) tan ε

(44)

Pascal Gegout




Ray-Tracing: Refractivity at Ray During raytracing, the current point position along the ray path expressed in spherical coordinates is converted in ellipsoidal coordinates P(φ, λ, z) in order to retrieve the refractivity. For a position in spherical coordinates, the height z in ellipsoidal geodetic coordinates is retrieved. The model level k is determined and hence the coefficients of refractivity functions of the four neibourghring columns. The coefficients characterizing the total and hydrostatic vertical ¯ pc (φ, λ), N ¯ pz (φ, λ) are refractivity profiles Npc (φ, λ), Npz (φ, λ), N retrieved by bilinear interpolation. Estimated at the height z, the refractive indices n(φ, λ, z) and n¯(φ, λ, z) are provided to the eikonal differential system.

Pascal Gegout




Ray-Tracing: Refractivity Gradients at Ray

The vertical gradient is straightfully derived from the vertical refractivity profile coefficient Npz . The vertical gradient expressed along the normal to the ellipsoid is converted in the radial gradient along the normal to the sphere as required during the numerical intergration. Horizontal gradients are estimated from the refractivity values at the same height z. Horizontal gradients are hundred times smaller than vertical gradient and can be neglected. The ray trajectory is inside a plane.

Pascal Gegout




Propagation Lengths The arclength LS is simply the geometrical length of the ray. Z Z Z Z Z LS = ds L= n ds = ds + (¯ nh − 1) ds + (˜ nnh − 1) ds S

S

S

S

S

(45) The radio path length L account for : an increase of length due to the wave propagating at a speed v = c/n, in the medium of refractive index n > 1 (slower than the speed of light c it would have in vacuum), an increase of the arclength LS of the ray path S, due to the bending of the ray path induced by the gradients of the refractive medium (lengthier than the straigth line path SL it would follow in vacuum).

Pascal Gegout




Tropospheric Delays The geometric delay DG is defined as the difference in length of the paths S and SL , the arclength LS minus the straight line LG . Z Z DG = LS − LG = ds − ds (46) S

SL

The chosen decomposition of the refractivity lead to the total delay Dt , sum of the hydrostatic Dh and the non-hydrostatic Dnh delays. Z Z Z Dt = (n−1) ds Dh = (¯ nh −1) ds Dnh = (˜ nnh −1) ds (47) S

S

S

The radio delay D is the difference between the radio length L and the geometric distance LG . Z Z D = L − LG = n ds − ds (48) S Pascal Gegout

SL Azimuthal Anisotropy Adaptive Mapping Functions



Refractive Indices & Tropospheric Delays

Refractive Index (/10) & Delay (m)

40

30

20

N NH NN T H N

10

0 0

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80000

Altitude (m)

Figure: Variation of refractive indices & delays with altitude during ray-tracing for a specific ray ε = 5o and α = 90o , for Tsukuba on August 12, 2008, 00Z Pascal Gegout




Elevation (degree)

Ray Bending E 5,2

5,1

5,0

Azimuth (degree)

90 0

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80000

A

89 88 87 86 0

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60000

80000

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Figure: Variation of Elevation and Azimuth with altitude during ray-tracing, for this specific ray, for the Tsukuba VLBI 7345 site on August 12, 2008, 00Z Pascal Gegout




Psat

Lout

Ψtop

Ptop ray ε'top L Ray

εtop

out ε'top Ψtop

Lg

εsite Psite rsite rtop Ψtop

Ψsat

rsat

Figure: The Parallax Problem Pascal Gegout




Parallax The ray-tracing is on purpose stopped on a reference sphere of radius rtop = 6450 km at the top of the troposphere. Propagation above the troposphere is here assumed to be propagation in vacuum. To determine the ray direction which reach the satellite position Psat (rsat , θsat , λsat ; t) the outgoing ray direction have to satisfy the ray out continuity condition ε0top = ε0top . The direction of the satellite from the top sphere is provided by a out family of lines which encounter the sphere with an elevation ε0top . out ε0top = arctan

rtop cos Ψtop − rsat cos Ψsat rtop sin Ψtop − rsat sin Ψsat

(49)

The solution to this boundary condition is unique if the elevations of both family of rays are assumed to be monotonic functions. Finding the outgoing elevation angle ε0top is solved iteratively. Pascal Gegout




Initial condition is provided by εsite = εsat (straight line problem). The iterative process to obtain a new estimate is the following : AMF-D provide the top outgoing point, the center angle Ψtop , and the outgoing direction from the satellite : ε0topout = arctan

rtop cos Ψtop − rsat cos Ψsat rtop sin Ψtop − rsat sin Ψsat

(50)

AMF-E provide the ray outgoing direction ε0ray top . ε0topray = εtop − Ψtop

(51)

A new estimate of the solution εsite is obtained by old 0 out 0 ray εnew site = εsite + εtop − εtop

(52)

Trigonometric considerations avoid the use of a shooting method. The required angles to solve the parallax problem provide the outgoing ray position and direction on top of a reference sphere. Pascal Gegout




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−1 2

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−90˚ 0˚

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−6 −3 0

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AMF−R−F3A4C5G2−7345−21730S007−21408.0000AMF−R−F3A4C5G2−7345−21730S007−21408.0000

Figure: Radio Delay (left) Radio Delay with Azimuthal Mean Removed (right) for the Tsukuba VLBI 7345 site on August 12, 2008, 00Z Pascal Gegout




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60 ˚

−1

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degree 0.0

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0.000

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0.004

AMF−E−F3A4C5G2−7345−21730S007−21408.0000AMF−E−F3A4C5G2−7345−21730S007−21408.0000

Figure: Outgoing Ray Elevation (left) with Azimuthal Mean Removed (right) for the Tsukuba VLBI 7345 site on August 12, 2008, 00Z Pascal Gegout




Conclusion: Ray-Tracing

Pascal Gegout



3

Adaptive Mapping Functions Partial Derivatives Fitting Mapping Functions Performance of Adaptive Mapping Functions Mapping Elevations and Delays Time Discretisation

Adaptive Mapping Functions Adaptive Mapping Functions Partial Derivatives Fitting Mapping Functions Performance of Adaptive Mapping Functions Mapping Elevations and Delays Time Discretisation

Pascal Gegout




The basic underlying idea is to insert a dependency to the azimuth α inside each fraction coefficient aif by introducing a Fourier serie in α : aif = aif ,0 +

iαX =nα

aicf ,iα cos iα α + aisf ,iα sin iα α

(53)

iα =1

The numerators and denominators at each level index if are Nnf = 1 + anf Nif = 1 + aif /Nif +1

Dnf = sin ε + anf

(54)

Dif = sin ε + aif /Dif +1

(55)

These recursive definitions lead to the numerator Nf and denominator Df of the mapping function f . f is scaled by a scaling factor Sf . Nf = 1 +

a1 a2 1 + 1+...

Df = sin ε +

a1 a2 sin ε + sin ε+...

f = Sf

Nf (56) Df

The elevation mapping functions are specific cases f ε = f cos ε. Pascal Gegout




The usual mapping function f3,0 (ε) truncated at the third fraction nf = 3 does not depend on α (nα = 0), 1+ f = Sf

sin ε +

a1 a 1+ 1+a2

3

a1 a2 sin ε+ sin ε+a

(57) 3

The adaptive mapping function f3,1 (ε, α) truncated at the third fraction nf = 3 with α-series truncated at the first term nα = 1 is Nf3,1 = 1 +

Df3,1 = sin ε +

c s a1,0 + a1,1 cos α + a1,1 sin α

1+

c cos α+as sin α a2,0 +a2,1 2,1 c cos α+as sin α 1+a3,0 +a3,1 3,1

c s a1,0 + a1,1 cos α + a1,1 sin α

sin ε +

Pascal Gegout

c cos α+as sin α a2,0 +a2,1 2,1 c cos α+as sin α sin ε+a3,0 +a3,1 3,1


(58)

(59)



Gradients are introduced in two alternative forms: a formulation where the gradient is embedded in the fraction form by addition of four terms in the fraction coefficient a1 : a1g = a1 + (Ecc cos α + Ecs sin α) cos ε

(60)

+ (Etc cos α + Ets sin α) tan ε

(61)

OR the usual formulation : f = Sf

Nf (1 + (D c cos α + D s sin α) cot ε) Df

Pascal Gegout


(62)



Partial Derivatives Partial derivative relative to each coefficient of the mapping function are needed to fit the mapping function on the traced rays. Theses partial derivatives are computed recursively and derived from the derivative of the fraction coefficient aif . 1 dN1 = da1 N2 dNif ai −1 dNif −1 =− f daif Nif Nif +1 daif −1 dNnf an −1 dNnf −1 =− f danf Nnf daif −1

Pascal Gegout

dD1 1 = da1 D2

(63)

dDif aif −1 dDif −1 =− daif Dif Dif +1 daif −1

(64)

dDnf an −1 dDnf −1 =− f danf Dnf daif −1

(65)




Partial derivatives relative to the coefficient aicf ,0 which does not depend on the azimuth α are df dNif dDif = Sf Df − Nf Df−2 (66) daif ,0 daif daif and in the specific case of the elevation mapping function df df ε = cos ε daif ,0 daif ,0

(67)

Partial derivatives relative to the α-series coefficients are df df = cos iα α daicf ,iα daif ,0

df df = sin iα α daisf ,iα daif ,0

(68)

The partial derivative relative to the scaling factor is f df = dSf Sf Pascal Gegout


(69)



Partial Derivatives : Gradients The partial derivative relative to the embedded gradient is df df cos α cos ε = c dEc da1,0

df df sin α cos ε = s dEc da1,0

(70)

df df = cos α tan ε dEtc da1,0

df df = sin α tan ε dEts da1,0

(71)

The partial derivative relative to the classical gradient is df = f cos α cot ε dD c

Pascal Gegout

df = f sin α cot ε dD s


(72)



Levenberg-Marquardt : Non-Linear Least-Squares Method

Determining the mapping function coefficients at a given site and a given epoch is an overdetermined and non-linear problem and require to fit iteratively all the coefficients of the mapping functions including the scale factor, the azimuthal coefficients and the gradient coefficients for the raytraced delays or the bending elevations. The non-linear fit is achieved by the Levenberg-Marquardt method, the standard of nonlinear least-squares method.

Pascal Gegout




Levenberg-Marquardt : The Merit Function The set of nr ray-traced delays (or bending angle) r (εi , αi ) with i = 1...nr , the εi elevation refer to the normal to the ellipsoid, is now modeled by the mapping function f (εi , αi ; p1 ...pnp ) which nonlinearly depends on the set of np unknown parameters pj with j = 1...np of the function f. {p1 , ..., pnp } = {Sf , aif ,0 , aicf ,iα , aisf ,iα , D c , D s },

(73)

The merit function include a realistic statistical model of the elevation dependent variability of the delays. 2

χ =

i=n Xr i=1

r (εi , αi ) − f (εi , αi ; p1 ...pnp ) σi

Pascal Gegout

2


(74)



The gradient and a second partial derivation of the χ2 merit function are used to take a step down (the gradient) and to define the Hessian matrix and the second member, leading to the set of linear equations: Hk,l =

1 ∂ 2 χ2 2 ∂pk ∂pl

Sk = −

1 ∂χ2 2 ∂pk

l=np

X

Hk,l δpl = Sk ∀k ∈ {1...np } (75)

l=1

Some issues lead to remove second derivatives from H. Hk,l =

i=n X 1 ∂f (εi , αi ; p1 ...pnp ) ∂f (εi , αi ; p1 ...pnp ) ∂pk ∂pl σi2

(76)

i=1

The idea of the Levenberg-Marquardt method is to force the modified Hessian matrix H’ to be diagonally dominant by multiplying the diagonal terms of H by a factor (1 + κ). 0 0 Hl,l = Hl,l (1 + κ) and Hk,l = Hk,l ∀ k 6= l

Pascal Gegout


(77)



Levenberg-Marquardt : Iterative Scheme

The linear system H 0 δp = S is solved iteratively. Given an initial guess p and an initial κ = 0.001, the merit function χ2 (p) is computed. The iterative scheme takes two steps. The linear system is solved for δp and χ2 (p + δp) is evaluated. A convergence test is performed on the merit function: If χ2 (p + δp) ≥ χ2 (p), κ is increased by a factor ten, else if χ2 (p + δp) < χ2 (p), κ is decreased by a factor ten and the trial solution is updated p = p + δp.

Iterations continue until the merit function decrease less than 10−6 .

Pascal Gegout




Performance of Adaptive Mapping Functions Several fraction truncations and α-series truncations are now tested, including classical mapping functions. A convention is useful to name specific mapping functions: F3 is the fraction truncation number nf = 3, A4 is the truncation of the serie at nα = 4, C5 is the elevation cutoff at 5 degrees, G2 is the two terms classical gradient and G4 the embedded gradient form.

Fitted Mapping Functions : AMF-D : Site Elevation minus Outgoing Point Elevation AMF-E : Site Elevation minus Outgoing Ray Elevation AMF-H : Hydrostatic Delay AMF-N : Non-hydrostatic Delay AMF-T : Total Delay = Hydrostatic + Non-Hydrostatic Delays AMF-R : Radio Delay = Total + Geometric Delays Pascal Gegout




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AMF−R−F3A0C5G2−7345−21730S007−21408.0000AMF−R−F3A0C5G2−7345−21730S007−21408.0000AMF−H−F3A0C5G2−7345−21730S007−21408.0000AMF−H−F3A0C5G2−7345−21730S007−21408.0000

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AMF−T−F3A0C5G2−7345−21730S007−21408.0000AMF−T−F3A0C5G2−7345−21730S007−21408.0000AMF−N−F3A0C5G2−7345−21730S007−21408.0000AMF−N−F3A0C5G2−7345−21730S007−21408.0000

Figure: (up) Radio AMF-R-F3A0C5G2 (left) Hydro AMF-H-F3A0C5G2 (right) (down) Total AMF-T-F3A0C5G2 (left) Non-Hydro AMF-N-F3A0C5G2 (right) for the Tsukuba VLBI 7345 site on August 12, 2008, 00Z Pascal Gegout




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AMF−D−F3A4C5G2−7345−21730S007−21408.0000 AMF−D−F3A4C5G2−7345−21730S007−21408.0000 AMF−E−F3A4C5G2−7345−21730S007−21408.0000 AMF−E−F3A4C5G2−7345−21730S007−21408.0000 AMF−R−F3A4C5G2−7345−21730S007−21408.0000 AMF−R−F3A4C5G2−7345−21730S007−21408.0000

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AMF−D−F3A4C5G2−7345−21730S007−21408.0000 AMF−D−F3A4C5G2−7345−21730S007−21408.0000 AMF−E−F3A4C5G2−7345−21730S007−21408.0000 AMF−E−F3A4C5G2−7345−21730S007−21408.0000 AMF−R−F3A4C5G2−7345−21730S007−21408.0000 AMF−R−F3A4C5G2−7345−21730S007−21408.0000

Figure: AMF-D & AMF-E & AMF-R (F3A4C5G2) for the PARALLAX problem (up) mapping function fit & residuals; (down) gradient & azimuthal anisotropy; for the Tsukuba VLBI 7345 site on August 12, 2008, 00Z

Pascal Gegout




0˚

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log10 -1.00

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Moyenne : -.3285E-03

0.000

Ecart-type : 0.3788E-01

0˚

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0˚ 0 180˚ Nombres de points : 1547 011_001_coupe_CR9FEH2_2008_223_stations_libre.amfx.100420_1328.0 : TSKB (140.09/36.11)

Pascal Gegout


Adaptive Mapping Functions Partial Derivatives Fitting Mapping Functions Performance of Adaptive Mapping Functions Mapping Elevations and Delays Time Discretisation ˚ 20

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180˚ 180˚ 180˚ 180˚ 180˚ 180˚ AMF−R−F3A4C5G2−7345−21730S007−21407.8750 AMF−R−F3A4C5G2−7345−21730S007−21407.9167 AMF−R−F3A4C5G2−7345−21730S007−21407.9583 AMF−R−F3A4C5G2−7345−21730S007−21408.0000 AMF−R−F3A4C5G2−7345−21730S007−21408.0417 AMF−R−F3A4C5G2−7345−21730S007−21408.0833

0.0

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0˚

−0.1

50

−1

12

−0.2

0˚

30

0˚

m

0.2

−3

˚

0˚

0˚

0.1

0˚ −6

0.0

0˚

−0.1

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50

−1

12

−0.2

−3

˚ 30

0˚

m

0.2

0˚ −6

0.1

15

˚

0˚

50

−1

0˚

0˚ −6

0˚ −6

0.0

−3

˚

30

0˚

m −0.1

15

˚

0˚

12

0˚

0˚

12

12

−3 0˚

˚ 30

0˚

−0.2

15

˚

50

−1

˚

50

−1

0˚ −6 −3 0˚

15 0˚

˚

0˚

−1 20

15

˚ 50

−1

m

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−0.1

0.0

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m

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˚ −1 20

90˚ ˚ 60

−90˚ ˚

90˚

˚ −1 20

60

−90˚ ˚

90˚

˚ −1 20

60

−90˚ ˚ 60

−90˚ ˚

˚

˚ −1 20

60

−90˚ ˚ 60

90˚

˚ −1 20

−90˚

90˚

0.0

0˚

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−3

˚ 30

0˚

30

0˚

0.1

˚

0˚

m

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−6

0.1

15

˚

12

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−6

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30

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m

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50

−1

12

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−6

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0˚

m

0.2

0˚

50

−1

12

0˚

0˚

0˚

0.1

15

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50

−1

−6

−6

0.0

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30

0˚

m −0.1

15

˚

0˚

12

0˚

0˚

12

12

0˚

−3 0˚

˚ 30

0˚

−0.2

15

˚

50

−1

−1 20

50

−1

−6 −3 0˚

15 0˚

˚

0˚

90˚

15

˚ 50

−1

m

0.2

−0.2

−0.1

0.0

0.1

m

0.2

−0.2

−0.1

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˚ −1 20

90˚ ˚ 60

−90˚ ˚

90˚

˚ −1 20

60

−90˚ ˚

90˚

˚ −1 20

60

−90˚ ˚ 60

−90˚ ˚

˚

−1 20

60

−90˚ ˚ 60

90˚

−1 20

−90˚

90˚

0.0

0˚

−0.1

−3

˚ 30

0˚

30

0˚

0.1

0.2

m −0.2

−0.1

0.0

˚

0˚

m −0.2

0˚

0.2

−6

0.1

0˚

12

0˚

0.0

12

−0.1

0˚

30

0˚

m −0.2

15

˚

−1

0˚

0.2

−3

˚

0˚

0˚

0.1

50

−6

0.0

0˚

−0.1

−3

˚ 30

0˚

m −0.2

0˚

0.2

−1

−6

0.1

0˚

50

12

0˚

0˚

0˚

−3

˚

30

0˚

m 0.0

−6

−6

0˚

−3 0˚

˚ 30

0˚

−0.1

−1

15

˚

0˚

50

12

0˚

0˚

12

12

−6 −3 0˚

−0.2

15

˚

0˚

−1

˚

−1

15

˚

50

−1 20

50

˚

−1

15 0˚

˚

0˚

90˚

15

˚ 50

0.1

0.2

m −0.2

−0.1

0.0

0.1

0.2

AMF−R−F3A4C5G2−7345−21730S007−21408.6250 AMF−R−F3A4C5G2−7345−21730S007−21408.6667 AMF−R−F3A4C5G2−7345−21730S007−21408.7083 AMF−R−F3A4C5G2−7345−21730S007−21408.7500 AMF−R−F3A4C5G2−7345−21730S007−21408.7917 AMF−R−F3A4C5G2−7345−21730S007−21408.8333

Figure: Azimuthal anisotropy of the tropospheric radio delay for Tsukuba site for 24 hourly steps beginning on August 11, 2008, 21Z until August 12, 20Z. Pascal Gegout



Conclusions

This study provide a detailed discussion of some physical and mathematical formulations needed to maintain as far as possible a millimetric accuracy in handling the atmospheric model, ray-tracing, bending, fitting mapping functions and solving the parallax problem. As the tropospheric delays are very sensitive to elevation, especially at low elevation, the parallax problem has to be properly solved. Determining the proper ray elevation at the site is as critical as properly fitting the azimuthal anisotropy.

Pascal Gegout



Conclusions

Although the millimetric accuracy is a concern which motivates the introduction of some formulations, the time discretization may not guarantee such precision and further invertigations are required. Undergoing investigations of a large number of meteorological situations depending on the site and the season and practical orbitographic studies should provide realistic performances of Azimuthal Anisotropy Adaptive Mapping Functions.

Pascal Gegout
