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early versions of the IEEE preprint included a derivation of the elevation angle correction $\delta e^*$, which accounted for both elevation angle bending $\delta e$ and the linear delay. it might be worthwhile to revisit that in a separate paper.

only the first term below is currently implemented

it's already implemented in drafts but not released yet.

Originally posted by @fgnievinski in #1 (comment)

could you compare this nominal case to ray-tracing (for a horizontal surface and atmosphere), please?

you can reuse the scripts previously used here:
https://github.com/ufrgs-gnss-lab/atm-interf-rtr-dev/issues/3

here is the updated m library:
https://drive.google.com/file/d/1Gf6lLfnzvEMppD0XAAW_e_-UeeqDK1OW/view?usp=sharing
https://drive.google.com/file/d/1E6UQ14sHqTEVRH0KZBTNsS7plK6x5DYZ/view?usp=sharing

PS: refractivity for the nominal case should be taken from the RTR results.

eq.27-30:

early versions of the IEEE preprint included a derivation of the propagation of uncertainty, from input $N$ and $\delta e$ ($\sigma_N$ and $\sigma_{\delta e}$), into output, $d$ and $\Delta H$ ($\sigma_d$ and $\sigma_{\Delta H}$). it might be worthwhile to revisit that in a separate paper.

after #2

m=500; e = linspace(10, 90, m)';
[dt0, ~, ~, Ht0] = get_atm_interf_met (e, H);
[dt1, ~, ~, Ht1] = get_atm_interf_met (e, H, [], [], [], [], struct('der_aided',true));

%fpk e [dt1, dt0, dt1-dt0]
fpk e [Ht1, Ht0, Ht1-Ht0]
fpk e [Ht1-Ht0]

the impact is sub-mm for H=10m and 500 points between 0<e<90deg; it should be worse for larger heights and fewer points.

the discrepancy here is mainly due to the differences in elevation bending and its rate of change:

Bennet:

        P = PaToMbar(P);
        T = KelvinToCelsius(T);
        num = (P - 80)./930;
        den = 1 + 8e-5.*(Rm + 39).*(T - 10);
        k = num./den;
        R = k.*Rm;  % Bennet (1982), p.258
        dkr = (num./den.^2).*(-1).*(0+8e-5.*(1+0).*(T-10));  % dk/de
        dRr = dkr.*Rm + k.*Rmr;  % dR/de
        de = R./60;
        der = dRr./60;

• archive trunk into branch "prelim"
• start new branch for v5preprint with #3 #6
• start new branch for v6preprint with #1
• check dependencies
• post code on public repo https://github.com/ufrgs-gnss-lab/atm-interf
• release new versions, incrementing version number as per semantic versioning
• reply to Sajad and to Kristine, who are asking for it

it used to be:

    if opt.H_approximate
        tmp = deg2rad(de).*tand(e);
        Hg = -H.*der + H.*tmp;
    end

that's before #3 and #5.

nominal:

    tmp = sqrt((1+N).^2 - cose.^2);
    dt = 2*H.*(tmp - sine);
    Ht = -der.*H;

general:

    dt = (1+N).*Dip - Di;
    Ht = H.*(1-(1+N).*cosep_cose.*(1+der));

using the same nominal elevation bending de:

    ep = acosd(cose./n);  % e': "refracted or apparent elevation", "e-prime"
    de = ep - e;  % de=e'-e: "elevation bending"

possibly also the same bending rate der:

    tmp = sqrt(n.^2 - cose.^2);
    epr = sine./tmp;  % de'/de: "e-prime-rate"
    der = epr - 1;  % dde/de=de'/de-de/de: "delta-e-rate"

either nominal analytical der as above or nominal numerical der, $\partial \delta e / \partial e$).

provide components only optionally:

• geometric + along-path (or curvilinear)
• linear + angular + mixed

given in eq.(20):

not to be confused with geometric + along-path, eq.(21).