Test of Math Mode

Actually, both math and LaTeX2html. Taken from RefBendDelayCalc but with the figures left out.

Another test: E = mc^2 and the A_n = 1 old way.

The ALMA Delay Server

The ALMA delay server is a software module which is responsible for distributing the delay correction for each antenna to the various bits of hardware in the antenna and to the correlator. The delay correction is defined as:

T_{delay}=T_{cause}-T_{geometric}-T_{cable}-T_{atmosphere}

where T_{cause} is some arbitrary offset to make sure all delays are positive, T_{cable} is a per antenna constant representing the electronic delay, T_{geometric} is the geometric delay derived by making a call to CALC, and T_{atmosphere} is the atmospheric delay correction (which can be gotten from CALC but is (as of 2008/03/18) not used and set to zero. In the long term the atmospheric delay will be derived using ATM to calculate the zenith delay coupled to an appropriate mapping function.

Overview

The delay experienced by an incoming signal due to its propagation through the Earth's atmosphere is given by:

\tau_{atm} = \int_s\left(n-1\right)ds

where s is the path through and n is the refractive index of the atmosphere. Since n is very nearly unity for the Earth's atmosphere one normally uses the "refractivity" (N) instead of the index of refraction. Refractivity and refractive index are related as follows:

N = 10^6 \left(n-1\right)

Furthermore, the atmospheric delay can be separated into contributions due to the dry and wet atmosphere:

\tau_{atm} = \tau_d+\tau_w

where \tau_d is the contribution due to dry air while \tau_w is the contribution due to wet air. In general \tau_d and \tau_w are parameterized in terms of a zenith contribution to the delay which is dependent upon local atmospheric conditions (Z) and a "mapping function" (M) which relate delays at an arbitrary elevation angle E to that at the zenith:

\tau_{atm}=Z M=Z_d M_d+Z_w M_w

Since the elevation angle E is the unrefracted source elevation, refraction effects are included in the mapping functions M. In the following I describe calculations of Z and M.

Zenith Delay

The contribution to the atmospheric delay at the zenith (Z) is a measure of the integrated refractivity of the atmosphere at the zenith. In general, the refractivity of moist air at microwave frequencies depends upon the permanent and induced dipole moments of the molecular species that make up the atmosphere. The primary species that make up the dry atmosphere, nitrogen and oxygen, do not have permanent dipole moments, so contribute to the refractivity via their induced dipole moments. Water vapour does have a permanent dipole moment. Permanent dipole moments contribute to the refractivity as N\propto\frac{p}{T^2} , while induced dipole moments contribute as N\propto\frac{p}{T} , where p is the pressure and T is the temperature of the species.

A simple parameterization of the refractivity at the zenith is given by the Smith-Weintraub equation (Smith & Weintraub 1953):

N=k_1\frac{p_d}{T}+k_2\frac{p_w}{T}+k_3\frac{p_w}{T^2}

where p_d and p_d are the partial pressures due to dry and wet air, T is the temperature of the atmosphere, and k_1, k_2, and k_3 are constants. The pressures and temperature are usually taken to be equivalent to the local ambient values at the antenna station under study. The dry and wet air refractivities are then given by:

Z_d=k_1\frac{p_d}{T} Z_w=k_2\frac{p_w}{T}+k_3\frac{p_w}{T^2}

The dry air contribution to this refractivity (Z_d) is primarily due to oxygen and nitrogen, and is nearly in hydrostatic equilibrium. Therefore, Z_d does not depend upon the detailed behaviour of dry air pressure and temperature along the path through the atmosphere, and can be derived based on local atmospheric temperature and pressure measurements. The wet air refractivity (Z_w) can be inferred from local water vapour radiometry measurements. Also, one can use an atmospheric model to derive the total atmospheric refractivity using an atmospheric model (such as ATM) with local atmospheric conditions as input.

Mapping Function

The simplest form for the mapping function (M), which relates the delay at an arbitrary elevation angle E to that at the zenith, is given by the plane-parallel approximation for the Earth's atmosphere:

M=\frac{1}{\sin(E)}

This simple form is in fact inadequate, which led Marini (1972) to consider corrections to this simple functional form which accounted for the Earth's curvature. Assuming an exponential atmospheric profile where the atmospheric refractivity varies exponentially with height above the antenna, Marini (1972) developed a continued fraction form for the mapping function:

M=\frac{1}{\sin(E)+\frac{a}{\sin(E)+\frac{b}{\sin(E)+c}}}

where I include only the first three terms in the continued fraction. A slight modification to the Marini (1972) continued fraction functional form which forces M=1 at the zenith replaces the even numbered \sin(E) terms (i.e. the second, fourth, sixth, etc.) with \tan(E)

M=\frac{1}{\sin(E)+\frac{a}{\tan(E)+\frac{b}{\sin(E)+c}}}

Chao (1974) introduced this modification by truncating the Marini (1972) form to include only two terms.

A more generalized continued-fractional form for the mapping function was developed by Yan and Ping (1995):

M=\frac{1}{\sin(E)+\frac{a}{I^2\tan(E)+\frac{b}{\sin(E)+\frac{c}{I^2\tan(E)+d}}}}

...where...

l=\sqrt{\frac{r_0\sec(E)}{2H}}

...is the "normalized effective zenith argument" of function which includes the "normalized effective height" of the atmosphere (H) defined as:

H=\frac{1}{N_0}\int^\infty_{h_0}N(h)dh

For an exponentially-distributed atmosphere:

N(h)=N_0\exp\left(\frac{-(h-h_0)}{H}\right)

Normally, h_0=0 (i.e. start the integration from the ground).

The constants a, b, c, d, etc. in the continued fraction forms above are generally derived from analytic fits to ray-tracing results either for standard atmospheres or for observed atmospheric profiles based on radiosonde measurements. The mapping functions derived in Niell (1996) and Davis (1985) are derived in this way.

A physically more correct mapping function has been derived by Lanyi (1984). Unlike previous mapping functions, Lanyi does not fully separate the dry and wet contributions to the delay, which is a more physically correct approximation. It is based on an ideal model atmosphere whose temperature is constant from the surface to the inversion layer h_1, then decreases linearly with height at rate W (the lapse rate) from h_1 to the tropopause height h_2, then is assumed to be constant above h_2. This mapping function is designed then to be a semi-analytic approximation to the atmospheric delay integral that retains an explicit temperature profile that can be determined using meteorological measurements. The mapping function is expanded as a second-order polynomial in Z_d and Z_w, plus the largest third-order term). It is nonlinear in Z_d and Z_w. It also contains terms which couple Z_d and Z_w, thus including terms which arise from the bending of the signal path through the atmosphere. The functional form for the atmospheric delay in this Lanyi (1984) model is given by:

\tau_{atm}=\frac{F(E)}{\sin(E)}

where

F(E)=F_d(E) Z_d+F_w(E) Z_w+\frac{F_{b1}(E) Z^2_d+2 F_{b2}(E) Z_d Z_w+ F_{b3}(E) Z^2_w}{\Delta}+\frac{F_{b4}(E) Z^3_d}{\Delta^2}

where

Z_d= \textrm{Dry atmospheric zenith delay}, \\ 
Z_w=\textrm{Wet atmospheric zenith delay}, \\
F_bn=\textrm{n-th bending contributions to the delay}, \\
\Delta=\textrm{Dry~atmospheric~scale~height}=\frac{kT_0}{mg_c}, \\
k= \textrm{Boltzmann's constant}, \\
T_0= \textrm{Daily average surface temperature}, \\
m= \textrm{Mean molecular mass of dry air}, \\
g_c=\textrm{air column center of gravity gravitational acceleration} \\

With standard values of k, m, T_0=292 K (appropriate for mid-latitudes), and g_c=978.37 cm/s^2, \Delta=8.6 km.

The dry, wet, and bending contributions are expressed in terms of moments of the refractivity. The bending terms are evaluated for the ideal model atmosphere and thus give the dependence of the delay on the four parameters T_0, W, h_1, and h_2. Therefore, the Lanyi (1984) model relies upon accurate surface meteorological measurements at the time of the observations to which the delay model is applied.

Antenna Height Correction to Total Atmospheric Delay

In the calculation of the zenith atmospheric delay at an antenna it is assumed that the atmospheric properties (P, T, RH) are the values measured at the focal plane of the antenna. For example, in VLBI each station has a set of associated weather measurements which are used to calculate Z. For a clustered array like the VLA or ALMA, the affect of the differences in antenna focal plane height above some reference point need to be accounted for.

For the VLA (not EVLA), CALC was not used to calculate the atmospheric delay. The antenna height correction was incorporated with a simple atmospheric delay correction by correcting for the path difference between each VLA antenna and a reference point at the center of the array. For the VLA case, the extra atmospheric path due to a difference in antenna height above the center-of-the-array reference point (\Delta H, in ns) is given by:

\Delta T=\frac{10^{-6} N_0 \Delta H+T_z * \frac{w}{\sin(E)}}{\sin(E)}

where N_0 is the atmospheric refractivity, T_z is the atmospheric zenith delay calculated using the VLA weather station (which is located near the center of the array), w is the geometric w of the antenna (in ns). The first term is the antenna height correction to the zenith delay, while the second term is a simple atmospheric delay correction. For EVLA, CALC will be used to calculate both geometric and atmospheric delay. We believe (though have not confirmed) that CALC also calculates the antenna height correction (first term in the equation above) given antenna heights relative to the reference point at the center of the array. ALMA will need to include this antenna height correction term.

A simple estimate of the magnitude of the antenna height difference correction at the zenith can be gotten by assuming that the pressure P changes linearly with height. Then 52 cm of additional antenna height (the current difference in height between the two ATF antennas) out of a total atmospheric height of 8 km would correspond to:

\frac{52 cm}{8 km} P =0.068 mb

where I have assumed P = 1053 mb. The dry term zenith atmospheric delay changes approximately like 2.3 mm/mb of pressure change. A pressure change of 0.068 mb corresponds to approximately 156 micron of path difference. This is consistent with alternate back-of-the-envelope calculations of this quantity.

Differential Excess Atmospheric Delay Between Two Antennas

NOTE: The following is just an aside. Since CALC or any other analysis of the atmospheric delay at an antenna calculates the total integrated delay along the path of observation, the differential delay between two antennas is accounted for in any differencing calculations done during baseline determination.

The differential delay induced in an interferometer by a horizontally stratified troposphere results from the difference in zenith angle of the source at the antennas. Thompson, Moran, and Swensen (2001), pp. 516-518 discuss the atmospheric delay induced along an interferometer baseline. The excess path length is given by:

L=10^{-6} N_0 \int_0^\infty\exp\left(-\frac{h}{h_0}\right) dy

where N_0 is the refractivity at the Earth's surface, h is the height above the Earth's surface, h_0 is the atmospheric scale height, y is the length coordinate along the direction to the source, and E is the antenna elevation while observing the source. Note that refraction is neglected. One can relate y, h, h_0, and E as follows (see Figure 13.4 in TMS, page 517; reproduced below) using the cosine rule on the triangle formed by r_0, y, and r_0+h:

\left(r_0+h\right)^2=r^2_0+y^2_0-2 r_0y\cos(180-z)

Solving for h and using elevation rather than zenith angle yields:

h=y\sin(E)+\frac{y^2-h^2}{2r_0}

For the triangle which is formed by sides y, h, and the side which is equal to y\sin(z_i), we can write:

y^2-h^2\simeq\left(y\sin(z_i)\right)^2

Since r_0\simeq 6370 km and h\simeq 8 km (the height of the troposphere), r_0\gg h. Since z_i\simeq z +\frac{h}{r_0}, z_i \simeq z. The equation for h in terms of y, E, and r_0 then becomes:

h\simeq y \sin(E)+\frac{y^2}{2r_0}\cos^2(E)

(Thanks to Dick Thompson for filling-in some of the details of this calculation).

We can now write the expression for L as follows:

L \simeq10^{-6} N_0 \int_0^\infty\exp\left(-\frac{y}{h_0}\sin(E)\right)\exp\left(-\frac{y^2}{2r_0h_0}\cos^2(E)\right)dy

Since \frac{y}{r_0h_0} \ll 1, the second term in the equation above can be expanded with a Taylor series so that:

L\simeq10^{-6} N_0\int_0^\infty\exp\left(-\frac{y}{h_0}\sin(E)\right)\left(1-\frac{y^2}{2r_0h_0}\cos^2(E)+\frac{y^4}{8r^2_0h^2_0}\cos^4(E)+...\right)dy

Integration yields:

L\simeq10^{-6}N_0h_0\csc(E)\left(1-\frac{h_0}{r_0}\cot^2(E)+\frac{3h^2_0}{r^2_0}\cot^4()+\right)

Writing this equation in terms involving \csc(E), the excess path length L becomes:

L \simeq 10^{-6}N_0h_0 \\ 
\left[\left(1+\frac{h_0}{r_0}+\frac{3h^2_0}{r^2_0}\right)\csc(E)-\left(\frac{h_0}{r_0}+\frac{6h^2_0}{r^2_0}\right)\csc^3(E)+\frac{3h^2_0}{r^2_0}\csc^5(E)+...\right]

Taking the derivative of L with respect to E and multiplying this derivative by the baseline length D divided by r_0 yields the atmospheric differential delay between two antennas separated by baseline D:

\frac{dL}{dE} \simeq \frac{-DN_0h_0\cot(E)}{10^{3}r_0} \\
\left[\left(1+\frac{h_0}{r_0}+\frac{3h^2_0}{r^2_0}\right)\csc(E)-3\left(\frac{h_0}{r_0}+\frac{6h^2_0}{r^2_0}\right)\csc^3(E)+\frac{15h^2_0}{r^2_0}\csc^5(E)+...\right]

...where D is in m, h_0 is in km, and r_0 is in km. Note that once must calculate N_0 using a suitable atmospheric model which uses measurements of the local atmospheric pressure, temperature and relative humidity to derive the resultant differential residual delay.

-- PatrickMurphy - 2009-01-21
Topic revision: r2 - 2009-12-07, PatrickMurphy
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