Numerical Experiment on Atmospheric Delay in Very Long Baseline Interferometric Observation*

Transcription

1 Journal of the Geodetic Society of Japan Vol. 29, No. 4, (1983), pp Numerical Experiment on Atmospheric Delay in Very Long Baseline Interferometric Observation* Masaru KAIDZU Geographical Survey Institute (Received February 23, 1984) ABSTRACT The neutral atmosphere is one of the most influential noise sources in Very Long Baseline Interferometry. To improve the accuracy of the atmospheric correction, it is important to figure out where exactly the problem lies. For this purpose, a series of numerical experiments was carried out. Main results are as follows. The accuracy of the atmospheric correction depends on the accuracy of the available information on water vapor distribution. Marini-Murray model has an unnegligible bias in the low elevation angle under hot and humid circumstances. Solving for zenith path delay as one of the estimated parameters reduces the residual significantly. 1. Introduction Very Long Baseline Interferometry (VLBI) provides a very powerful tool to astrometry and geodesy. In geodetic application, it is quite important that the baseline length observed with VLBI is principally determined by only geometry of the radio sources and the baseline and that there is no chance of getting scale factor through the data processing such as in satellite ranging which employs the orbital determination procedure. This means the baseline length determined through VLBI observations must be a good standard * Read at 60th meeting of the Geodetic Society of Japan in October 1983.

2 Numerical Experiment on Atmospheric Delay in Very Long Baseline 263 of scale. The accuracy of the VLBI observations depends on system noise, clock drift, delay caused by propagation media and modeling of geophysical and relativistic effects. Among these, the delay by the neutral atmosphere is very influential and not yet fully modeled. In the routine analysis of POLARIS data, Marini-Murray formula (Marini and Murray [1]) is currently used. The proposers of this formula claim that the formula is accurate to 1.3% and they tested it against a ray trace model. Rogers (Rogers [2]) compared the Marini-Murray formula with a simple ray trace model and suggested that the formula has a bias in low elevation angle. As Rogers' ray trace model assumed an improper water vapor distribution and the test was done in a standard atmosphere only, we need further investigation on Marini-Murray formula (Appendix 2). In this paper, the effect of deviation of lapse rate and water vapor distribution from standard atmosphere on the amount of atmospheric correction is investigated using a ray trace technique and then the accuracy of Marini-Murray formula is tested against the ray trace model. 2. Ray trace model In the ray trace model used for this work, the atmosphere is assumed to be a hydrostatically stratified ideal gas with molecular weight of g/mol. The earth is treated as a sphere which has a radius equal to the mean redius of curvature of the reference ellipsoid at a given latitude. The gravitational attraction at the surface of the earth is given by the standard gravity formula As the reference ellipsoid, Geodetic Reference System 1980 (equatorial radius= m, flattening=1/ ) is used. To calculate refractivity, the water vapor partial pressure at the surface of the earth e0 (mb) is calculated through the atmospheric temperature T (K), barometric pressure PO (mb) and the relative humidity Rh(%) by using the following formulae. e0=e8.rh/(1-(1- (Rh/100))e,/P0).100) where e8=6.105 exp (25.22((T )/T-5.31 log (T/273.15))). The water vapor partial pressure e(mb) above the surface of the earth is calculated with Saastamoinen's formula (Saastamoinen, [3], Appendix 3) as e=e0 (P/P0) 4, where P is the atmospheric pressure at given height, The schema of the model used in ray trace is shown in Fig. 1. When we represent gas constant as Rg, atmospheric molecular weight as m0, refractive index as n, angle of insidence to the atmospheric layer at the calculated height as i and the gravitational attraction at the surface of the earth as g0, we can compose the system of differential equations to be solved through Snell's law and hydrostatic equilibrium as follows by using the notations in Fig. 1.

3 264 Masaru KAIDZU Fig. 1. Schema of the model and notations in this study. dp/dr= - ((mogo)/rg) (PIT) (Re/r)2 dz/ds = - ((sin i) /n) (dn/dr) i=z-ľ dt/dr= -6.5 K/km (in Troposphere) dt/dr=0.0 (above Tropopause) d (correction for optical depth)/ds=n=n-1 geometric correction = ) In above expression, s represents the path length along the light path and refractivity N is given by the formula given by Bean and Dutton (Bean and Dutton [4]) as N= (77.6P/ T x 105e/T2) x 10-g. The scheme of integration is Adams' scheme. To generate four initial values, modified Euler's scheme is used. Intervals of integration are 500m for Adams' scheme and 5m for modified Euler's scheme. 3. Results of the experiments In the delay caused by the neutral atmosphere, about 90% is due to dry component. Therefore, the effect of the deviation of lapse rate from the standard value -6.5 K/km on the atmospheric correction is examined. The calculation is carried out for a typical meteorological condition at Tokyo. Fig. 2 shows the difference between the atmospheric correction for the lapse rate -6.0 K/km and the correction for the standard lapse rate

4 Numerical Experiment on Atmospheric Delay in Very Long Baseline 265 Fig. 2. Effect of change in lapse rate used for ray trace. and the difference between the atmospheric correction for the lapse rate of -7.0 K/km and the correction for the standard lapse rate. Differences are less than 1 cm at the elevation angle larger than 10, degrees, and the difference is proportional to total atmos pheric correction in the elevation over 20 degrees. This figure shows that the variation of lapse rate in the realistic range does not affect the atmospheric correction much and we can neglect the effect in present accuracy of the geodetic VLBI observations. As the standard model of the water vapor distribution, we used Saastamoinen model in the ray trace model. The accuracy of the Saastamoinen model is not mentioned in the paper in which the model was proposed (Saastamoinen, [3]). To evaluate the accuracy of this model, we modeled the water vapor distribution as a slightly generalized form as e = e0(p/p0)ƒ, and calculated a using the radio sonde date taken in Japan (Tokyo Astronomical Obser vatry, [5]). Calculated mean a is 4.1 which is almost identical with Saastamoinen model. Thus the Saastamoinen model is correct as a mean, but the value of a shows a large scatter. Fig. 3 shows the relation between a and the water vapor partial pressure at the surface. a scatters in 4±1 and there is no meaningful relations between them. Fig. 4 shows the relation between a and the temperature at 700 mb height. There seems a slight correla tion, but the correlation is not good enough to be modeled. Therefore, we must examine how much the scatter of a affects the atmospheric correction. In order to assess the effect of the scatter of a, the atmospheric carrections for a=3 and a=5 were calculated and compared with the standard atmospheric correction. The meteorological condition at the surface is again taken to be the typical condition at Tokyo. Fig. 5 shows the result of calculation. This figure shows the scatter of a affects the atmospheric correction

5 266 Masaru KAIDZU Fig. 3. Relationship between water vapor pertial pressure at 1000 mb height and alpha. Fig. 4. Relationship between temperature at 700 mb height and alpha. Fig. 5. Effect of variation of alpha on atmospheric correction calculated through ray trace.

6 Numerical Experiment on Atmospheric Delay in Very Long Baseline 267 quite much. To achieve a cm accuracy in geodetic VLBI observations, it is definitely necessary to improve the model of the water vapor distribution. The best way is the direct measurement. When the direct measurement is not available, there is a possibility that we may bypass the problem of poor model of the water vapor distribution using the fact that the amount of effect of a on the atmospheric correction is almost proportional to the total amount of correction. The significance of the scale factor for the atmospheric correction as one of the parameters estimated in VLBI data processing will be discussed later. So far the effects of the deviation of the atmosphere from the standard model on the atmospheric correction has been discussed. Here, we will test the accuracy of the Marini- Murray model, which is currently used in routine analysis of POLARIS data, against the ray trace model. a. Effect of the daily variation of the temperature field in the planetary boundary layers on the accuracy of the Marini-Murray model It is well known that in the planetary boundary layers, the profile of the temperature field shows a large daily variation. Fig. 6 shows an example of such a daily variation of the temperature field observed with a CO2 radiometer. As the Marini-Murray model uses only the surface values of temperature, barometric pressure and humidity to calculate the atmospheric correction, the correction calculated by ray trace using the observed temperature distribution may be different from the correction calculated by the Marini- Fig, 6. An example of temperature distribution in planetary boundary layer, (Observed at Kawasaki in Feb. 1975)

7 268 Masaru KAIDZU Fig. 7. Effect of the daily variation of the temperature distribution in the planetary bound ary layer on the accuracy of the Marini-Murray model. Murray model. In order to see the effect of temperature distribution, the surface values of the barometric pressure and the relative humidity are held fixed. The calculated differences between the correction through the ray trace and the correction calculated by the Marini-Murray model are shown in Fig. 7. The daily variation of the temperature field does not affect the accuracy of the Marini-Murray model. However, the difference between the correction through ray trace and the correction calculated by Marini-Murray model is as large as 4 cm at elevation of 10 degrees. Whether this difference depends on the meteorological condition is the next problem to be investigated. b. Accuracy of the Marini-Murray model in different meteorological conditions As was suggested by Rogers, the Marini-Murray model has some bias in low elevation in a certain meteorological condition. In order to see whether the accuracy of the Marini- Murray model differs according to the meteorological condition, the atmospheric correction calculated by the Marini-Murray model is compared with the correction through the ray trace in various meteorological condition. The result of the comparison shows that the accuracy of the Marini-Murray model depends mostly on the surface temperature. Fig. 8 shows the difference between the atmospheric correction through the ray trace and the correction calculated by the Marini-Murray model in low temperature. When the surface temperature is low, the Marini-Murray model is rather accurate. Fig. 9 shows the result of a similar comparison in high temperature. When the surface temperature is high, the Marini-Murray model has a large bias in low elevation angle. Similar comparison was carried out in different conditions and it was shown that the other factors such as the barometric pressure, the relative humidity and the latitude of the station do not affect the accuracy of the Marini-Murray model as the temperature does. In summer season, the climate at Tokyo is quite hot and humid. This means the Marini-Murray model is

8 Numerical Experiment on Atmospheric Delay in Very Long Baseline 269 Fig. 8. Difference of the atmospheric correction calculated through ray trace from Marini- Murray model at Oslo in January. Fig. 9. Difference of the atmospheric correction calculated through ray trace from Marini- Murray model at Tokyo in August. not accurate in low elevation at Tokyo in summer time. In scheduling the observations in Japan, we should avoid the observation of the source whose elevation angle is less than 20 degrees. c. Significance of the scale factor of the atmospheric correction as one of the parameters estimated in VLBI data analysis As the effects of the deviation of watervapor distribution and the lapse rate from the standard model are almost proportional to the total amount of atmospheric correction, there is a possibility that we may bypass the problem of the poor model of the water vapor distribution by estimating the scale factor of the atmospheric correction with other

9 274 Masaru KAIDZU Fig. 14. Coherency of the baseline length calculated from the data set of MERIT short campaign. The solid line shows the scatter of base line lengths calculated without adjusting the scale factor of atmo spheric correction. The dashed line shows the scatter of baseline lengths calculated with the scale factor of atmospheric correction adjusted. parameters such as baseline components or clock parameters. In order to examine whether the estimation of such a scale factor as one of the parameters is statisticaly gignificant or not, the data sets of MERIT short campaign were processed with and without the scale factors and the residuals were tested using AIC (Akaike's Information Criterion (Akaike, [6]). AIC is given as AIC= -21og (Maximum likelihood) +2 (number of independently adjusted parameters). In processing the data, coordinates of the three stations and the clock parameters of the three clocks are estimated. Earth orientation parameters have been pre-adjusted and held fixed in present processing. When the scale factors of the atmospheric correction were estimated as a part of parameters, AIC decreased for 13 experiments out of 14. This means that we should estimate the scale facters with other parameters. However, it does not always mean that estimating the scale factors improves the accuracy of the baseline components. To test the significance of estimating the scale factors of the atmospheric correction, the scatter of the baseline lengths calculated through one day observation around the alithmetic mean of 14 such observations was calculated with and without adjusting the scale factor of the atmospheric correction. Fig. 10 shows the scatter of the baseline length as a function of the baseline length. This figure shows that adjusting the scale factors of the atmospheric correction does not improve the coherence of the baseline length. This will be partly because of the azimuth dependent atmospheric delay. That is because such an azimuth dependent error in atmospheric correction can not be eliminated by adjusting the scale factor.

10 Numerical Experiment on Atmospheric Delay in Very Lontr Baseline Conclusions Through the numerical experiments discussed above, we have reached the following conclusions. a. The accuracy of the atmospheric correction depends greatly on the accuracy of the available information on water vapor distribution. The best way to achieve a high accuracy is a direct measurement of the water vapor using a reliable equipment. When the direct measurement of water vapor is not available, we should adjust the scale factor of the atmospheric correction with other parameters in data processing. b. Marini-Murray model of the atmospheric delay has a large bias in low elevation angle. The bias is larger in hot and humid circumstances. As the Japanese climate is hot and humid especially in summer, in the scheduling of the observations, we should avoid the observation of the sources whose elevation is less than 20 degrees. c. Adjusting the scale factors for the atmospheric correction as a part of the estimated parameters has been shown to be significant. However, it does not improve the coherence of the baseline length. This may be partly because of the azimuth dependence of the atmospheric delay. 5. Acknowledgement The author wishes to extend his sincere thanks to National Geodetic Survey, NOS, NOAA for providing the opportunity to carry out this work, especially to Dr. ROBERTSON and Dr. FALLON for their fruitful discussion and friendly assistance. The stay of the author in the United States was financially supported by Japanese government through Science and Technology Agency. That is also appreciated. References [1] MARINI, J. W. and C. W. MURRAY, Jr.: CORRECTION OF RADIO RANGF TRACKING DATA FOR ATMOSPHERIC REFRACTION AT ELEVATIONS ABOVE 10 DEGREES, unpublished memo (1974). [2] ROGERS, A. E. E.: Elevation dependence of the atmospheric delay- a preliminary investigation, unpubli shed memo (1982). [3] SAASTAMOINEN, J.: Atmospheric Correction for the Troposphere and Stratosphere in Radio Ranging of Satellites. Geophysical Monogr. 15, AGU, Washington D. C. (1982). [4] BEAN, B. R. and E. J. DUTTON: Radiometeorology. NBS Monogr. 92, (1966). [5] Tokyo Astronomical Observatory (ed): Rika-Nenpyo (Science Almanac) 1981, Maruzen, Tokyo (1980). (in Japanese) [6] AKAIKE, H.: Information theory and an extension of the maximum likelihood principle. 2nd International Symposium on Information Theory. B. N. Petrov and F. Csaki (eds.), Akademiai Kiado, Budapest, (1973). Appendix 1. Rogers tested the accuracy of Chao model and the Marini-Murray model of the atmospheric delay against a ray trace model. In the ray trace model, the differential equations are integrated with Euler's scheme. The scheme is robust against instability

11 272 Masaru KAIDZU but is not very accurate. The model of water vapor distribution in the ray trace is that the relative humidity of the atmosphere at any point of the troposphere is the same as the relative humidity at the surface of the earth. The radio sonde data taken in Japan show that this model over-estimates the content of water vapor in the atmosphere. Appendix 2. Marini-Murray model is the model of the atmospheric correction for a radio ranging. The formula approximates the atmospheric correction by continued fraction. In the ap proximation, the water vapor partial pressure at the surface of the earth and latitude dependence of the gravitational attraction are taken into account. The correction R is given as R= (1/f(ƒÓ,H)) ((A+H)/(sin E+((P/(A+B))/(sin E+0.015))) where A= (P0+(1255/T0+0.05)e0) B= ~ 10-3 exp ( H) f (ƒó,h) = (1-2 sin2ƒó) H E: elevation angle ƒó: latitude of the station H: height of the station. Appendix 3. Saastamoinen modeled the distribution of the water vapor in the troposphere as e=e0(t/t0)-4g/(rg*dt/dr) In the same article in which Saastamoinen proposed above formula, pressure is modeled as P=P0(T/T0)-g(Rg*dT/dr) Therefore, we can rewrite the expression of the water vapor as e=e0(p/p0)4. The last model is preferable in ray trace. That is because in the ray trace procedure, we calculate the pressure step by step to a good accuracy, where as the original model assumes as simplified distribution of the pressure.