NMF2 mapping function 

Memo:   NMF2 mapping function
Author: L. Petrov
Date:   2004.03.11

Introduction

Attempts to use numerical data for computation of the neutral atmosphere path delay were made recently by P.-H. Andersen, J. Boehm, A. Niell and L. Petrov. One of the approaches was to find a better regression model for the mapping function (dependence of path delay with the geometric, refraction-free elevation above the horizon). The motivation of the efforts to find a good regression model was to reduce computation time and to make using advanced mapping function easier.

Isobaric mapping function (IMF)

Isobaric mapping function was built be the following way:
  • Raytracing integrals has been computed using 730 radiosonde profiles collected for 26 sites during 1992.01.01 1992.12.31 for every 12 hours. These integrals were computed for nine elevation from 3 t o90 degrees.

  • Mapping function was parameterized by a continuous fraction
    m(e) = [1 + a/(1+b/(1+c)))]/[(sin(e) + a/(sin(e) + b/((sin(e) + c))]
    where e is the refraction-free elevation angle of the source above the horizon.

  • Linear regression was found between the coefficients a,b,c for hydrostatic path delay and the geopotential height at the 200 mbar level, geocentric latitude and height above the reference ellipsoid.

  • Linear regression was found between the coefficients a,b,c for wet path and the ratio of the humidly integral at the elevation 90 degrees and 3.3 degrees, and the height above the reference ellipsoid.

    • Assuming that this regression derived for radiosounde profiles is valid of numerical weather models, one can compute the IMF for the sites where no radiosounde profiles are available with use of numerical weather models interpolated in space and time.

    Niell mapping function second revision (NMF2)

    Computation of isobaric mapping function requires availability of the numerical weather models, which even for moderate resolution 200x250 km are very voluminous: about 1 Gb per year. For circumventing efforts for providing an interface between ephemeric numerical weather models and sophisitcated space geodesy software, the following approach is proposed:
    1. For each station of interest a long time series of the coefficients a,b,c for both hydrostatic and wet mapping function are computed.

    2. Using this time series coefficients of a 14 parameters model are computed for the coefficients a,b,c as well as two tilt angles of normals to the surface of geopotential height for each station separately using least squares. The model includes:
    Thus, for each station, the set of 112 regression time-independent coefficients is computed.

    Results of baseline length repeatability test with different mapping functions

    Baseline lenght statistics is computed among the baselines with 16 or more sessions. Only experiments after 1993.01.01 were taken into account. Weighted root mean square with respect to the result of fit of linear model of baseline lenghts (with possible jumps dure to episodic motion) for a time series of baseline lengths for a given baseline is called repeatabiity. The model sqrt (A2 + (B*L)2 ) was computed.

    Three solutins were made:

    Results of estimation of harmonic site position variations with different mapping functions

    Three solution were made for estimation of harmonic site position variations for 40 VLBI stations, for 32 frequencies. Detailed description of the technique can be found here .

    In solution olc_22b IMF was used.
    In solution olc_22c NMF was used.
    In solution olc_22d NMF2 was used.

    The ratio of the weighted sum of residual harmonic site position variations to its mathematical expectation. If no signal presents, the ratio should be less than 1.22.
    # Name NMF IMF NMF2
        vert. horz. vert. horz. vert. horz.
    1 s4 0.81 0.96 0.82 1.01 0.79 0.94
    2 m4 0.71 0.88 0.72 0.90 0.69 0.85
    3 s3 1.02 1.51 1.01 1.56 1.02 1.50
    4 m3 0.63 1.10 0.63 1.15 0.62 1.09
    5 empt-2 1.00 1.00 1.00 1.00 1.00 1.00
    6 k2 1.75 1.53 1.72 1.65 1.70 1.52
    7 s2 2.63 1.87 2.71 1.95 2.56 1.86
    8 t2 1.06 1.18 1.07 1.24 1.04 1.17
    9 l2 1.39 0.75 1.44 0.78 1.36 0.74
    10 m2 1.19 1.89 1.27 1.99 1.17 1.87
    11 ni2 0.90 0.89 0.94 0.93 0.88 0.88
    12 n2 1.04 0.97 1.10 1.00 1.00 0.97
    13 mi2 1.17 1.05 1.18 1.08 1.15 1.03
    14 2n2 1.95 1.19 1.98 1.18 1.91 1.17
    15 oo1 0.88 1.49 0.87 1.41 0.90 1.46
    16 empt-1 1.00 1.00 1.00 1.00 1.00 1.00
    17 j1 0.89 1.40 0.91 1.37 0.92 1.37
    18 fi1 0.76 1.03 0.83 1.00 0.79 1.01
    19 psi1 1.09 1.34 1.14 1.30 1.13 1.30
    20 k1 1.77 3.50 1.64 3.45 1.65 3.51
    21 s1 4.03 2.29 3.74 2.30 3.82 2.25
    22 p1 1.05 0.97 1.11 0.99 1.10 0.96
    23 m1 0.95 1.16 0.96 1.09 0.96 1.14
    24 o1 0.89 1.40 0.90 1.35 0.91 1.36
    25 ro1 1.16 0.90 1.20 0.90 1.17 0.87
    26 q1 0.52 1.19 0.51 1.13 0.53 1.15
    27 empt-0 1.00 1.00 1.00 1.00 1.00 1.00
    28 mtm 1.71 0.94 1.67 0.99 1.73 0.94
    29 mf 1.66 1.04 1.06 1.08 1.63 1.05
    30 mm 1.71 0.91 1.52 0.97 1.68 0.91
    31 ssa 2.84 1.23 1.96 1.20 1.89 1.21
    32 sa 7.77 2.74 6.66 2.60 6.60 2.42

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    Last update: 11-MAR-2004 14:20:04