Sect_6.2.8

6.2.8 Measurement Biases and Errors

TROPOSPHERIC DELAY

Caused by the signal refraction in the electrically neutral (or non-ionised) atmospheric layer called the troposphere, extending from the earth's surface to about 8km (though approximately twice as thick at the equator). Another component of the neutral atmosphere is the stratosphere, extending up to an altitude of about 50km, to the base of the ionospheric layer. For the purposes of discussing neutral atmospheric delay, under the term "tropospheric delay" will be included both the components due to the troposphere and the stratosphere because the troposphere, although being relatively thin, contains most of the mass of the neutral atmosphere and practically all of the water vapour.

Tropospheric delay is a function of the satellite elevation angle and the altitude of the receiver, and is dependent on the atmospheric pressure, temperature, and water vapour pressure (BRUNNER & WELSCH, 1993). However, a good starting point is to define it in terms of the refractive index, integrated along the signal ray path:

(6.2-6a)

or in terms of the refractivity of the troposphere N_trop = 10⁶(n – 1):

(6.2-6b)

The tropospheric refractivity can be partitioned into the two components, one for the dry part of the atmosphere and the other for the wet part:

N_trop = N_wet + N_dry (6.2-7a)

and the total tropospheric delay is:

d_trop = d_dry + d_wet (6.2-7b)

The d_trop can then be estimated by separately considering its two constituents d_dry and d_wet. About 90% of the magnitude of the tropospheric delay arises from the dry component, and the remaining 10% from the wet component. There are several models of the wet and dry refractivities, and a number of models for the tropospheric delay based on the numerical or analytical integration of eqn (6.2-6) -- see HOFMANN-WELLENHOF et al (1998) for details. The following treatment is mostly taken from this reference.

A model for the dry and wet refractivity at the earth's surface is:

(6.2-8a)

(6.2-8b)

where p is the atmospheric pressure in millibars (mb), e is the partial pressure of water vapour in millibars and T is the temperature in degrees Kelvin. Note that these coefficients are empirically determined and cannot fully describe the local situation even when locally measured met parameters are available.

The variation of N_dry,h with height must also be empirically estimated, perhaps from radiosonde data. For example, Hopfield (HOFMANN-WELLENHOF et al, 1998) derived a representation of the dry refractivity as a function of the height h (in metres) above the surface as:

(6.2-9)

where h_d = 40136 + 148.72(T - 273.16) (in metres). Hence the dry part of the tropospheric delay (sometimes referred to as the hydrostatic delay) can be obtained by applying eqn (6.2-6b), solving the integral along the zenith using the variation of N_dry along the signal path defined by eqn (6.2-9) and neglecting the curvature of the signal path, leading to an expression for the dry zenith delay:

(6.2-10)

This is scaled by an appropriate mapping function to any arbitrary elevation angle:

(6.2-11)

where E is the satellite elevation angle in degrees.

The wet component is much more difficult to model because of the strong variations in the distribution of atmospheric water vapour in space and time. Hence, due to a lack of an appropriate alternative, the Hopfield model assumes the same functional model for the wet component as for the dry component (eqn (6.2-9)):

(6.2-12)

where hw is set to a value of between 11000 to 12000m. The wet zenith delay is:

(6.2-13)

This is also scaled by a mapping function, to give the delay at any arbitrary elevation angle:

(6.2-14)

Alternative functions for the wet and dry refractivities at the earth's surface (eqn (6.2-8)), and for the upward continuation of these refractivities (eqns (6.2-9) and (6.2-12)) result in new zenith tropospheric delay models. The mapping functions may also be changed. For example, modified Hopfield models can be produced by varying several of these functions (HOFMANN-WELLENHOF et al, 1998). Another popular model for GPS reductions is that due to Saastamoinen (SEEBER, 1993):

(6.2-15)

where the parameters p, e , T and E have been defined earlier. The B term is dependent upon the altitude of the station (and is interpolated from a table of values), and the R correction term is dependent upon the altitude of the station and the elevation angle of the satellite (and is also interpolated from a table of values) -- see HOFMANN-WELLENHOF et al (1998).

Note that in the models described here (and other commonly used models such as the Lanyi model, the Chao model, Marini and Murray model, etc.), measured values of p, e and T are assumed to be available at the GPS receiver site. Alternatively, it is possible to use profile functions to express pressure, water vapour pressure and temperature as a function of the height H above mean sea level (in kms). Examples of such profile functions are:

(6.2-16)

where p_o,e_o and T_o are "standard" values at sea level (say, 1013.25mb, 15mb, and 291.2°K respectively), r is relative humidity (in %), t is temperature (in degress Celsius) and e_sat is the saturation water vapour pressure (in mbs)

The choice of zenith tropospheric delay model is somewhat arbitrary. All are very good at modelling the zenith dry tropospheric delay, but can only reasonably model the wet delay when it is very small (that is, when the atmosphere is very dry). For example, the zenith dry tropospheric delay at sea level is of the order of 2.3m. The zenith wet tropospheric delay, however, may vary from a few millimetres to as much as 40cm. The variability of the dry component is relatively low and can be estimated with a precision approaching 1% when pressure is known (to mm accuracy). On the other hand, the wet component of the delay is notoriously difficult to estimate and errors of 10-20% are common.

There are several mapping functions to chose from and all give line-of-sight tropospheric delay values which are reasonably close to each other (less than 5mm difference) for satellite elevation angles greater than about 30°. Some models are superior to others when elevation angles below 15° are considered. However, even the best may produce errors of several decimetres at elevation angles below about 5°, in a total line-of-sight tropospheric delay of the order of 20-30m! The tracking of low elevation angle satellites is therefore to be avoided because the uncertainties in modelling both the wet and dry tropospheric delay are amplified at low elevation angles.

There are a number of further comments that can be made regarding the tropospheric delay and its impact on GPS positioning:

With respect to microwaves up to frequencies of 15GHz the neutral atmosphere is a non-dispersive medium, and hence the tropospheric delay is not frequency-dependent. It cannot therefore be eliminated through linear combinations of L1 and L2 observations as in the case of the ionospheric delay.

The magnitude of the tropospheric delay is the same for both L1 and L2 observations, and for pseudo-range and carrier phase measurements.

There is significantly less tropospheric delay at high altitude than at sea level. For example, if two receivers are deployed, one at sealevel and the other at 3000m, and assuming a satellite elevation angle of 20°, the tropospheric refraction error for the receiver at sealevel will be approximately 7.8m, while for the receiver positioned at high altitude the tropospheric refraction error is of the order of 4.8m.

The tropospheric delay can be predicted using values of temperature, pressure, and humidity, input into one of a number of atmospheric refraction models. Such models can account for approximately 90% of the delay (corresponding mainly to the dry troposphere component), however the remaining 10% (largely due to the wet troposphere component) will still seriously bias high accuracy positioning.

For surveys of less than a few tens of kilometres in extent, the tropospheric delay will tend to be the same at both ends of a baseline and thus assuming: (a) the same tropospheric model is applied uniformly for the campaign processing, or (b) no model is applied; the delay effect will largely cancel. The error introduced in baseline components under a wide range of conditions has been estimated as being of the order of 1ppm.

Neglecting to apply tropospheric refraction results in an absolute scale error. For example, a 1metre tropospheric delay in the zenith direction causes a scale effect of 0.4ppm -- that is, the baselines appear longer (BEUTLER et al, 1989). (As the total zenith value is of the order of 2.3m, neglecting tropospheric refraction introduces 1ppm relative error, as stated above.)

In practice, surface meteorological observations are rarely made as it appears that the baseline results are worse when they are used! For small scale surveys (less than a few tens of kilometres in extent) it is often recommended that modelled values of met parameters be used (upward continuation of "standard" sea level values using eqn (6.2-16)), instead of observed meteorological values because small systematic errors in the meteorological values (such as due to calibration errors of the measuring instruments and observer error in the field) can often introduce larger biases than the use of a "standard atmosphere". Furthermore, most commercial GPS software will not accept input of field measured met values.

GPS surveyors should always be mindful of the influences of local meteorological conditions that may exist at individual stations (for example, fog, temperature inversions, precipitation, etc.), which could lead to significant differential (or residual) tropospheric refraction biases.

Any uncertainty in modelling the differential tropospheric refraction bias results mostly in a degradation of the height component in the solution. BEUTLER et al (1989) have suggested the "rule-of-thumb": 1mm differential tropospheric bias causes a height error of about 3mm. There is only a minimal effect on latitude and longitude.

TROPOSPHERIC DELAY BIAS

MAGNITUDE:

Zenith value at sea level 2.3m.
Near horizon at sea level 20 - 30m.

90% due to dry atmosphere which can be modelled very well. 10% due to wet component of atmosphere is difficult to account for.

OPTIONS:

IGNORE the bias -- but avoid tracking low elevation satellites, generally no observations taken below 20°.

CORRECT data using a STANDARD TROPOSPHERIC REFRACTION MODEL (Saastamoinen, Hopfield, etc.) -- with or without surface meteorological readings.

ESTIMATE residual tropospheric effect as an additional parameter -- only for very high precision work.

DIFFERENCE data between sites -- effect of error is minimised due to high correlation over short to medium baselines.

MEASURE wet path component directly using a Water Vapour Radiometer -- too expensive, no longer considered a viable option.

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