Mapping of Niell

Fundamentals
Title	Mapping of Niell
Author(s)	J. Sanz Subirana, JM. Juan Zornoza and M. Hernandez-Pajares, University of Catalunia, Spain.
Level	Advanced
Year of Publication	2011

A mapping function that does not require surface meteorology measurements, but provides comparable accuracy and precision to others that requires such data is presented in [Niell, 1996]. This mapping only use the receiver geographic location and measurement time as inputs.

The Niell's mapping involves two different obliquity factors, for the dry and wet components, which are computed from the equations (1) to (4), where [math]\displaystyle{ (E) }[/math] is the elevation of ray and [math]\displaystyle{ (H) }[/math] is the receiver height, in kilometres:

Hydrostatic mapping function:

[math]\displaystyle{ \begin{array}{l} M_{dry}(E,H)= m(E,a_d,b_d,c_d)+ \Delta m(E,H)\\ \qquad\mbox{with}\\ \Delta m(E,H)= \left[ \frac{1}{sin E} - m(E,a_{ht},b_{ht},c_{ht}) \right ] \cdot H \end{array} \qquad\mbox{(1)} }[/math]

Wet Mapping Function:

[math]\displaystyle{ M_{wet}(E)= m(E,a_w,b_w,c_w) \qquad\mbox{(2)} }[/math]

where, [math]\displaystyle{ m(E,a,b,c) }[/math] is the [Marini, 1972] mapping normalised to unity at zenith:

[math]\displaystyle{ m(E,a,b,c)= \frac{1+\frac{a}{1+\frac{b}{1+c}}}{\sin E+\frac{a}{\sin E+\frac{b}{\sin E+c}}} \qquad\mbox{(3)} }[/math]

• The Hydrostatic parameters [math]\displaystyle{ a_d }[/math], [math]\displaystyle{ b_d }[/math], [math]\displaystyle{ c_d }[/math] are time [math]\displaystyle{ (t) }[/math] and latitude ([math]\displaystyle{ \phi) }[/math] dependent parameters given by:

[math]\displaystyle{ \xi(\phi,t)=\xi_{_{avg}}(\phi)-\xi_{_{amp}}(\phi) \cos \left ( 2\pi \frac{t-T_0}{365.25} \right ) \qquad\mbox{(4)} }[/math]

where, [math]\displaystyle{ t }[/math] is the time from January [math]\displaystyle{ 0.0 }[/math], in days, and [math]\displaystyle{ T_0 }[/math] is taken as DoY [math]\displaystyle{ 28 }[/math] (i.e., [math]\displaystyle{ T_0=28 }[/math]). The parameters [math]\displaystyle{ \xi_{_{avg}}(\phi_i) }[/math] and [math]\displaystyle{ \xi_{_{amp}}(\phi_i) }[/math] are linearly interpolated from table 1 between the nearest [math]\displaystyle{ \xi(\phi_i) }[/math]. The [math]\displaystyle{ a_{ht} }[/math], [math]\displaystyle{ b_{ht} }[/math], [math]\displaystyle{ c_{ht} }[/math] are taken directly from the same table 1.

• The wet parameters [math]\displaystyle{ a_w }[/math], [math]\displaystyle{ b_w }[/math], [math]\displaystyle{ c_w }[/math] are latitude dependent and are linearly interpolated from table 2 between the nearest [math]\displaystyle{ \xi(\phi_i) }[/math].

Table 1: Coefficients of the hydrostatic mapping function.

Table 2: Coefficients of the wet mapping function.