Changeset 4207

Timestamp:

Jun 9, 2005, 5:25:46 PM (21 years ago)

Author:

eugene

Message:

added refraction stuff from ken

File:

• trunk/doc/pslib/psLibADD.tex (modified) (5 diffs)

trunk/doc/pslib/psLibADD.tex

@@ -1 @@
-%%% $Id: psLibADD.tex,v 1.75 2005-06-09 05:16:23 eugene Exp $
+%%% $Id: psLibADD.tex,v 1.76 2005-06-10 03:25:46 eugene Exp $
 \documentclass[panstarrs]{panstarrs}
 
@@ -1087 +1087 @@
 For ease of conversion, UTC should be represented as the number of
 seconds since the UNIX epoch of ``1970-01-01T00:00:00Z'',
-non-inclusive of leap-seconds.  \tbd{what does this statement
-actually mean? what is the source of time (gettimeofday?  let's be
-explicit here}
-
-\tbd{Times will always be expressed in the 'UTC timezone'.  Use of
-the local timezone is forbidden.  -- this makes no sense given that we
+non-inclusive of leap-seconds.  \tbd{what does this statement actually
+mean? what is the source of time (gettimeofday?  let's be explicit
+here}
+
+\tbd{Times will always be expressed in the 'UTC timezone'.  Use of the
+local timezone is forbidden.  -- this makes no sense given that we
 define LST.  In any case, this statement, or somethign equivalent,
 belongs in the SDRS not the ADD}
@@ -1151 +1151 @@
 
 \subsubsection{Universal Time (UT1)}
-\label{sec:ut1}
 
 UT1 is directly tied to the rotation of the Earth.  Historically, time
@@ -2092 +2091 @@
 
 \subsubsection{Ray Tidal Model : {\tt psEOC\_PolarTideCorr}}
+\label{Raymodel}
 
 The Ray Model tidal corrections to X, Y, and dT are given by the the
@@ -2134 +2134 @@
 \paragraph{Atmospheric Refraction}
 
-\tbd{add in summary of Ken's paper}
+{\em The following discussion is adapted from an article by Ken Chambers}
+\newcommand\citep{\em}
+\newcommand\citet{\em}
+
+The hypsometric structure and index of refraction of the Earth's
+atmosphere produces:
+\begin{itemize}
+\item atmospheric refraction, resulting in an apparent positional
+displacement of astronomical objects towards smaller topocentric
+zenith distances,
+\item chromatic dispersion, along great circles intersecting the
+topocentric zenith with shorter wavelengths having smaller zenith
+distances; and
+\item extinction, from scattering and absorption of light by
+atmospheric gases (including water vapor) and aerosols.
+\end{itemize}
+
+Atmospheric refraction $R(\lambda) = z_{vac} - z_1$ is the difference
+between the topocentric zenith angle {\it in vacuo}, where the index
+of refraction is $n \equiv 1$, and the observed refracted zenith angle
+at the observatory $z_1$.  (All subscripts ``1" in this discussion
+indicate the local values of quantities at a given observatory site,
+and all units are SI.)  There are various ways to express the equation
+for zenith angle refraction; common ones include the ``general
+equation", e.g. \citep{s1936,s1962}, the Auer \& Standish
+transformation \citep{as1979,as2000}, and the Saastamoinen
+approximation \citep{saas73a,saas73b,saas73c}.  All are derived from
+the ``refractive invariant'', the fact that along the refracted light
+path described by the locus of points $r(z,n)$, where $r$ is the
+radial distance from the center of the Earth and $z$ and $n$ are the
+local values of the refracted zenith angle and the index of refraction
+of air respectively, the product $ n r {\rm sin} z = \rm {constant} $
+remains invariant.
+
+We have revisted the Saastamoinen approximation in search of an
+improved analytical expression accurate to $\sim 1$ mas up to zenith
+angles of 75 degrees.  Although different approximations are made at
+different stages, both the general equation and the Saastamoinen
+approximation lead to an expression for the refraction $R(\lambda)$ in
+the form of a series in odd powers of tan$z_1$:
+\begin{equation}
+ R(\lambda) = R_1 {\rm tan} z_1 + R_3 {\rm tan}^3 z_1 + 
+              R_5 {\rm tan}^5 z_1 + R_7 {\rm tan}^7 z_1 + 
+              R_9 {\rm tan}^9 z_1 + \dots                    
+\end{equation}
+where we have chosen to label the coefficients with a subscript reflecting
+the exponent of the tan$z_1$ terms.
+We find (c.f. Stone 1996)
+%
+\footnote{ The $R_1$ coefficient and the first two terms in the $R_2$
+          coefficient in equation 2 above give results similar to that
+          of Stone's equation (4), if it is modified such that the
+          $\kappa$ factor multiplies only his $\beta$, rather than the
+          whole coefficient as is done for both coefficients.  We
+          suspect this is a typographical error; his equation equation
+          as written gives much greater residuals when compared to the
+          Pulkovo Refraction Tables than he reports in his Table 2,
+          whereas the agreement is comparable, if his equation is
+          modified as described, and computed for the Pulkovo baseline
+          model of $\lambda = 590$nm, $15\deg C$, 101325 Pa, zero
+          water vapor, and mean sea level at latitude 45 degrees.  An
+          exact comparison is not possible because the residuals in
+          his Table 2 are averages computed from a wide but
+          unspecified range of meteorological conditions.  }
+%
+\begin{equation}
+ R(\lambda) = \gamma_1(1 - h_1/r_1){\rm tan} z_1 + 
+                [\gamma_1(\gamma_1 / 2  - h_1/r_1) + \delta_1]
+                {\rm tan}^3 z_1                                  
+\end{equation}
+in radians, where we define for convenience the variable 
+$$ \gamma_1 = n_1 -1 $$ where $n_1(\lambda)$ is the index of
+refraction of air at the observatory, which is dependent on the
+observatory's altitude and the meteorological conditions at the time
+of the observation (see Section 3 below).  Each of the other variables
+Eq.(2) take detailed discussions which, for the sake of clarity, are
+divided into separate subsections.
+
+\subsection{Observatory height}
+
+The height of the observatory from the geometric center 
+of the Earth is 
+\begin{equation}
+\begin{array}{ll}
+ r_1 = r_e + r_h, \qquad  r_h \approx r_n + r_o,                    
+\end{array}
+\end{equation}
+where 
+$r_e$ is the local radius of the reference ellipsoid,
+$r_h$ is the local height above the reference ellipsoid,
+$r_n$ is the local geoid height, normal to the reference ellipsoid,  
+$r_o$ is the orthometric height, which is the height above the geoid
+      (or Mean Sea Level) in the direction of normal gravity
+      (i.e. a local plumb line). 
+
+\subsection{The magnitude of normal gravity at the observatory }
+                                                                                
+The local magnitude of normal gravity
+%
+\footnote{ Stone uses a common
+  \citep[e.g.][]{allen1973,seid1992,allen2001} expression for the
+  normal gravity as a function of lattitude and altitude, which
+  apparently first appeared in \citet{lamb1949}.  However its
+  derivation is unclear -- cited as {\it ``Some notes on the
+  calculation of the geopotential", unpublished manuscript},
+  \citet{lamb1949}.  }
+%
+$g_1$
+is the acceleration due to the combination
+of the gravity of the ellipsoid, the local mass distribution,
+and the centripetal acceleration from the Earth's rotation,
+the vector sum being directed opposite to the local zenith by definition,
+being close but not identical to the normal to the ellipsoidal, but
+not intersecting the the geometric center of the Earth.
+i.e. the atmospheric topocentric zenith differs from the geocentric
+astronomical zenith, the impact of this difference on calculating
+the atmospheric refraction is minimal, see \citep{seid1992}. 
+The acceleration at the observatory is 
+\begin{equation}
+ g_1(r_h,\phi) = g(\phi)\left[ 1 - 2( 1 + f + m_r - 2 f {\rm sin}^2\phi)
+                   \left(r_h \over a \right) +
+                   3 \left(r_h \over a \right)^2 \right]
+\end{equation}
+where
+\begin{equation}
+ g(\phi) = g_e \left( { {1 + k_s {\rm sin}^2 \phi}\over
+            \sqrt{1 - \epsilon^2 {\rm sin}^2 \phi}} \right)
+\end{equation}
+and $\phi$ is the latitude of the observatory and the other constants
+are given in Table 2.
+\begin{table}[!ht]
+\caption{WGS-84 World Geodetic System Reference Ellipsoid for GPS}
+\smallskip
+\begin{center}
+{\small
+\begin{tabular}{lr}
+%\tableline
+\noalign{\smallskip}
+Semi-major axis $a$                              & 6356752.3142 $m$\\
+Flattening $f =(a-b)/a$                          & 0.003352811 \\
+Eccentricity $\epsilon = \sqrt{a^2 +b^2}/a$      & 0.081819\\
+Polar gravity $g_p$                              & 9.8321849378 $m s^{-2}$ \\
+Equatorial gravity $g_e$                         & 9.7803253359 $m s^{-2}$ \\
+Somigliana's Constant $k_s = ((b/a)(g_p/g_e)-1)$ & 0.001931853 \\
+Angular velocity of the Earth $\omega$           & 0.00007292115 $rad/s$\\
+$GM_{earth}$ including atmosphere         & 3986004.418$\times 10^8 m^3
+s^{-2}$\\
+Gravity ratio $m_r = \omega^2 a^2 b /(GM)$       & 0.003449787\\
+\noalign{\smallskip}
+%\tableline
+\end{tabular}
+}
+\end{center}
+\end{table}
+
+
+\subsection{The scale height above the observatory} 
+
+The scale height of the atmosphere above the observatory is
+\begin{equation}
+ h_1 = \mathcal{Z}_1\mathcal{R}T_1/g_1 M_a[1 - x_w(1-M_w/M_a)],  
+\end{equation}
+where
+$T_1$ is the local air temperature at the observatory in degrees Kelvin,
+$\mathcal{R} = 8.314472 {\rm J \ mol^{-1} K^{-1}}$ 
+is the gas constant,
+\begin{equation}
+\begin{array}{ll} \mathcal{Z}_1 = 1 - (P_1/T_1)\left[a_0 + a_1t_1 + a_2t_1^2 + 
+                  (b_0 + b_1)t_1 x_w + (c_0+c_1t_1)x_w^2 \right] \\
+                  \qquad \qquad +(P_1/T_1)^2(d+ex_w^2)  
+                \end{array}
+\end{equation}
+is the compressibility of moist air at local air temperature 
+$t_1 = T_1 - 273.15$  
+in degrees Celsius. 
+The values of the other constants in $\mathcal{Z}_1$ are given in
+Appendix A. 
+The quantity  
+$M_a = 0.0289635 + 1.2001 \times 10^{-8}(x_{c1} - 400)$ 
+kg/mole
+is the molar mass of dry air with a $CO_2$ concentration $x_{c1}$ 
+in $\mu$mol/mol,
+$M_w = 0.018015 $ kg/mole is the molar mass of water vapor, 
+and
+$x_w$ is the molar fraction of water vapor in moist air, which 
+depends on the local humidity.   
+
+To calculate both the scale height $h_1$ and 
+the index of refraction of moist air at the observatory $n_1$
+we need to calculate $x_{w1}$, the molar fraction of water vapor
+from local measurements of the relative humidity or, much better, the
+dew point. Simple formulas such as \citet{davis1992}
+are only suitable above $0\deg C$, whereas at Mauna Kea observatories the
+temperature is often below $0\deg C$, and occasionally the surrounding 
+ground is covered in snow or ice, which also alters saturation vapor pressure.
+Thus, we adopt the best available equation for the 
+saturation water vapor pressure $p_{sv}$, 
+the IAPWS equations \citep{huang1998}. 
+Haung's equations are:
+
+\begin{equation} 
+\begin{array}{ll}
+   \Omega = T + K_9/(T - K_{10}) \qquad A = \Omega^2 +K_1\Omega + K_2 \\ 
+   B = K_3\Omega^2 + K_4\Omega + K_5 \qquad C = K_6\Omega^2 + K_7\Omega + K_8 \\
+   X = - B + \sqrt{B^2 - 4AC} \qquad p_{sv}(t) = 10^6(2C/X)^4  \\
+\end{array}
+\end{equation} 
+For saturation vapor pressure over ice or snow, use 
+\begin{equation} 
+\begin{array}{ll}
+   \Theta = T/273.16  \qquad Y=A_1(1-\Theta^{-1.5})+A_2(1-\Theta^{-1.25}) \\ 
+   p_{sv}(t) = 611.657 e^Y  \qquad  \\
+\end{array}
+\end{equation} 
+The constants are given in Table 1.  
+\begin{table}[!ht]
+\caption{Constants for Compressibility and Humidity Equations} 
+\smallskip
+\begin{center}
+{\small
+\begin{tabular}{lcl}
+%\tableline
+\noalign{\smallskip}
+ $a_0=1.58123\times 10^{-6}$KPa$^{-1}$     &  &  $K_1= 1.16705214528E+03$ \\  
+ $a_1=-2.9331\times 10^{-8}$Pa$^{-1}$      &  &  $K_2= -7.24213167032E+05$ \\  
+ $a_2=1.1043\times 10^{-8}$K$^-1$Pa$^{-1}$ &  &  $K_3= -1.70738469401e+01$ \\  
+ $b_0=5.707\times 10^{-6}$KPa$^{-1}$       &  &  $K_4= 1.20208247025E+04$ \\  
+ $b_1=-2.051\times 10^{-8}$Pa$^{-1}$       &  &  $K_5= -3.23255503223E+06$ \\  
+ $c_0=1.9898\times 10^{-4}$KPa$^{-1}$      &  &  $K_6= 1.49151086135E+01$ \\  
+ $c_1=-2.376\times 10^{-6}$Pa$^{-1}$       &  &  $K_7= -4.82326573616E+03$ \\  
+ $d  =1.83\times 10^{-11}$K$^2$Pa$^{-2}$   &  &  $K_8= 4.05113405421E+05 $ \\  
+ $e  =-0.765\times 10^{-8}$$^2$KPa$^{-2}$  &  &  $K_9= -2.38555575678E-01$ \\  
+                                           &  & $K_{10}=6.50175348448E+02 $ \\  
+                                           &  & $A_1=-13.928169$           \\  
+                                           &  & $A_2=34.7078238$           \\  
+\noalign{\smallskip}
+%\tableline
+\end{tabular}
+}
+\end{center}
+\end{table}
+
+\noindent Now, to calculate the mole fraction of water vapor $x_w$, we need the
+so called ``enhancement factor"
+\begin{equation}
+   f(p,t)= a' + b' p + c' t^2 
+\end{equation}
+where $a' = 1.00062$,$b'= 3.14 \times 10^{-8}$, and 
+$c' = 5.60\times 10^{-7}$, and where $p$ and $t$ are the air pressure in
+Pascals and air temperature. 
+
+If you have the more precise measurement of the dew point $t_d$ (or frost point)then
+\begin{equation} 
+x_w = f(p,t) \times p_{sv}(t_d)/p.
+\end{equation}
+On the other hand if you only have the relative humidity $RH$, a less accurate
+expression is 
+\begin{equation} 
+x_w = (RH/100) \times f(p,t) \times p_{sv}(t_d)/p.
+\end{equation}
+
+\subsection{The index of refraction of moist air at the observatory}
+
+The Ciddor equation for the index of refraction of moist air 
+\citep{ciddor1996} has been adopted by the International Association
+of Geodesy (IAG) as the standard as it is believed to provide the most
+accurate results under the largest
+range of wavelength, temperature, and humidity conditions (300 to 1690 nm;
+-40 to 100 C; 0-80\% RH). Note for astronomy, the air temperature at the
+tropopause in the 1976 standard atmosphere is 216.6 K, below the stated range ofvalidity. Astronomical observers often observe up to 90\% RH, where
+water droplets can form and change the effective index of refraction.
+Note that developments in the equation for the index of refraction of moist
+air have been poorly tracked in the astronomical literature and it is critical to examine in
+every application what equation is actually being used. \footnote{
+Edlen's (1953) original fit to the available data covered the wavelength
+range 2752 to 6440 /AA with reasonable residuals. His 1953 constants still
+survive in the astronomical literature in the equations of \citet{allen1973};
+%Allen (1973);
+\citet{stone1996}; \citet{allen2001}; 
+%Stone(1996); Allen(2001), 
+Roe (2002 - who extrapolates the 1953 equation to
+K band to correct for dichoric adaptive optics) and in the computer codes
+SLALIB and ZEEMAX. None discuss the range of validity. However, Elden himself
+revised them \citep{elden1966},
+%(Elden 1966), 
+and these were further discussed and updated by
+\citet{pr1972, bd1993, bd1994, ciddor1996, bp1998}. 
+\citet{rueg1998} and \citet{sz2004} make convincing arguments 
+for the Ciddor equation, and the latter's approach is followed here.}  
+
+The index of refraction of air at standard temperature and pressure
+is \citep{ciddor1996} 
+\begin{equation}
+\gamma_{as} = 10^{-8} \left( { \left[ { k_1 \over k_0 - \sigma^2} \right] 
+                    + \left[ { k_3\over k_2 - \sigma^2} \right] } \right),
+\end{equation} 
+where $\sigma = 1/\lambda$ is the wavenumber of wavelength of light in microns.
+Adjusting for the (annually varying and secularly increasing) value of 
+atmospheric $CO_2$ concentration $x_{CO2}$ in units of $\mu$mole/mole, 
+the expression for dry air becomes  
+\begin{equation}
+\gamma_{axs} = \gamma_{as} 
+               \left[ 1 + 5.34\times10^{-7} 
+               (x_{CO2} - 450\ \mu{\rm mole/mole} )\right]. 
+\end{equation} 
+For water vapor under standard conditions, Ciddor finds 
+\begin{equation}
+\gamma_{ws} = 1.022 \times 10^{-8}
+              \left[\omega_0 + \omega_1 \sigma^2 + 
+              \omega_2 \sigma^4 + \omega_3 \sigma^6 \right]. 
+\end{equation} 
+Following \citep{owens1967}, 
+the indicies can be combined in proportion to their densities,
+thus the index of refraction of moist air at the observatory 
+$n_1 = \gamma_1 +1 $ is given by     
+\begin{equation}
+\gamma_1 = (\rho_a/ \rho_{axs})\gamma_{axs} + 
+           (\rho_w/ \rho_{ws})\gamma_{ws},  
+\end{equation} 
+where
+\begin{equation} 
+\begin{array}{ll}
+   \rho_{a}   = (1 - x_w) P_1 M_a/({\cal Z}_m {\cal R} T_1) \\ 
+   \rho_{w}   = x_w P_1 M_w/( {\cal Z}_m {\cal R} T_1) \\ 
+   \rho_{axs} = P_{STP} M_a/( {\cal Z}_a {\cal R} T_{STP}) \\ 
+\end{array}
+\end{equation} 
+and $\rho_{ws}$ is given in Table 3.  
+
+\begin{table}[!ht]
+\caption{Constants in the Ciddor Eq. for index of refraction of moist air}  
+\smallskip
+\begin{center}
+{\small
+\begin{tabular}{lll}
+%\tableline
+\noalign{\smallskip}
+$k_0 = 23.0185\ \mu {\rm m}^{-2}$       &
+$\omega_0 = 295.235\  \mu {\rm m}^{-2}$ & 
+$P_{STP} = 101325$ Pa                   \\ 
+
+$k_1 = 5792105\ \mu{\rm m}^{-2} $       &
+$\omega_1 = 2.6422\   \mu {\rm m}^{-2}$ & 
+$T_{STP} = 288.15 $ K                   \\
+
+$k_2 = 57.362\ \mu{\rm m}^{-2}  $       & 
+$\omega_2 = -0.03238\ \mu {\rm m}^{-4}$ & 
+${\cal Z}_{a} = 0.9995922115$           \\
+
+$k_3 = 167917\ \mu {\rm m}^{-2} $       & 
+$\omega_3 = 0.004028\ \mu {\rm m}^{-6}$ &
+$\rho_{ws} = 0.00985938\ {\rm kg\ m}^3$ \\
+\noalign{\smallskip}
+%\tableline
+\end{tabular}
+}
+\end{center}
+\end{table}
+
+
+\subsection{The tropopause term in the equation of refraction}
+
+The final term in the Refraction Equation (2) 
+is $\delta_1$, which comes from our re-derivation of the 
+Saastamoinen approximation: 
+\begin{equation}
+ \delta_1 = 5 \left( {h_1 \over {r_1 T_1}}\right)^2
+              \left[ {{{\gamma_{ft}T_{ft}^2 - \gamma_t T_t^2} \over
+                                {1 - (h_1\beta/T_1)}}} 
+                                 + \gamma_tT_t^2 \right]   
+\end{equation}
+where $\beta$ is the lapse rate of the troposphere, 
+and the index of refraction of the free troposphere is given by 
+\begin{equation}
+ \gamma_{ft} = \gamma_t(T_{ft}/T_t) ^{-(T_1/h_1 \beta)-1}       
+\end{equation}
+and the index of refraction of the tropopause is given by  
+\begin{equation}
+ \gamma_t = \gamma_1 
+           {\rm exp} \left[ {T_1 (r_t - r_1) \over T_t h_1 }\right]    
+\end{equation}
+where $\beta$ is the lapse rate in K/m,  
+the temperature of the tropopause $T_t$, and height of the tropopause $r_t$ 
+are all determined from contemporaneous meterological data 
+(radiosonde or modern forecast models). Then the temperature of the 
+free troposphere is given by 
+\begin{equation}
+ T_{ft} = T_t - \beta(r_t - r_1).    
+\end{equation}
+
+\subsection{Calculating the atmospheric refraction from both 
+            the observed and true zenith angle} 
+
+The monochromatic refraction can now be calculated for any given wavelength
+$\lambda_{air}$
+(formally only within the range of validity - 300 to 1670 nm) given
+the altitude of the observatory
+$h_1$; 
+contemporaneous meteorological measurements at the observatory of   
+air temperaure $T_1$ (K); 
+atmospheric pressure $P_1$ (Pa);
+percent relative humidity $RH$, or preferably dew point temperature $t_d$ ($\deg$C);
+as well as a small set of additional meteorological data:   
+the lapse rate of the troposphere $\beta$ (K/m);
+the radius of the tropopause $r_t = r_e+h_t$(m), where
+$h_t$ is the height above MSL of the tropopause; 
+the temperature of the tropopause $T_t$(K);
+and the atmospheric concentration of CO$_2$ 
+$x_{CO2}$ ($\mu$mole/mole). 
+The characterization of the troposphere and tropopause can 
+be determined either by interpolation between radiosonde measurements 
+or from hourly updated meteorological models available at many observatory 
+sites, or, worst case, simply adopting the 1976 Standard Atmosphere 
+values:\footnote{
+\citet{seid1992}
+%Seidelmann 1992 
+contains a typographical error quoting this value in units of K/km} 
+$\beta = -0.0065$ K/m, 
+$h_t = 11000$m,
+$T_t = 216.6$ K and assuming 
+$x_{C02} = 375 \mu$mole/mole.\footnote{
+Closed rooms have higher CO$_2$ concentration, thus the STP laboratory measurements 
+have concentrations near 450 $\mu$mole/mole (the prefered unit to parts per million per volume). The secular
+increase of atmospheric CO$_2$ in the industrial age is well documented, with 
+an annual cycle superimposed due to terrestrial biomass and ocean exchange. 
+}
+
+
+If the number of electrons being created from the illumination from the source in
+the interval of wavelength $d\lambda$ is $N_{\lambda}d\lambda$, 
+then the mean refraction is  
+\begin{equation}
+\bar R = {{\int R(\lambda) N_{\lambda} d \lambda} \over 
+         {\int N_{\lambda} d\lambda}}  
+\end{equation}
+
+\section{Atmospheric Dispersion}    
+
+The atmospheric dispersion is then
+\begin{equation}
+{ \overline{(R- \bar R)^2}}  = {{\int (R - \bar R)^2 N_{\lambda} d \lambda} \over 
+                           {\int N_{\lambda} d\lambda}}  
+\end{equation}
+
+\section{Air Mass and Extinction}    
+
+By Laplace's theorem, the monochromatic airmass (mass per unit area
+along the refracted path) is 
+\begin{equation}
+M(z_1) = (P_1/g_1) R(\lambda) / {\rm sin} z_1
+\end{equation}
+in kg/m$^2$. Thus
+\begin{equation}
+ M(z_1) = (P_1/g_1) \left( \gamma_1(1 - h_1/r_1){\rm sec} z_1 + 
+                [\gamma_1(\gamma_1 / 2  - h_1/r_1) + \delta_1]
+                {\rm sec}^3 z_1 \right).                                  
+\end{equation}
+This is generally normalized to $P_1/g_1$, i.e. in spite of the
+daily changes in barometric pressure, and thus daily changes in the 
+true mass of air over the observatory, the resulting change in extinction is
+generally treated as a drift in photometric zeropoint.  
+For a survey program like Pan-STARRS, one could instead normalize to a 
+standard barometric pressure (i.e. the altitute pressure), and thus
+during a low in atmospheric pressure, the airmass would be less than
+1 at zenith, and during periods of high pressure the airmass would
+be greater than 1 at zenith. From the variation with zeropoint and
+temperature and barometric pressure, this would remove most of the
+observed variation in zeropoint in the CFHT legacy program. 
+(Magnier, private comunication). 
+
+The mean airmass is then
+\begin{equation}
+\bar M(z_1) = \left( P_1 \over g_1 \right) \int 
+               \left( \gamma_1(1 - h_1/r_1){\rm sec} z_1 + 
+                [\gamma_1(\gamma_1 / 2  - h_1/r_1) + \delta_1]
+                {\rm sec}^3 z_1 \right) N_{\lambda} d \lambda.                                   
+\end{equation}
+and depends weakly on the filter bandpass. Use of this more
+accurate expression for airmass should lead to improved extinction
+corrections at high airmass.  
+
+\section{Limits to ground based relative and absolute astrometry}   
+
+The limits to ground based astrometry may well be our abilitiy to 
+measure the atmospheric profile along the line of sight of a given 
+observation, and the systematic limit of the telescope axes encoders
+(and sophistication of the telescope mount model.) 
+The refratction model above requires only the additional data of
+the temperature, height, and presure of the tropopause, but much more
+detailed atmospheric information will be available for PS1 from our sky
+probes which measure atmospheric absorption for each field and even,
+phase drifts of GPS clocks from Rubidium or Cesium standard clocks. 
+These can be converted directly into a nearby line of sight index of refraction
+at optical wavelengths. Thus we encourage wide field survey telescopes to
+err on the side of over instrumenting the accuracy and repeatibility 
+of the axes encoders.  
+
+
 
 \subsection{Projections}