Calculation by: Dr. Seyed Abdoreza Saadat
1 Introduction of the Iranian height system and datum
One of the important issues in the surveying and civil projects is to determine the elevation of points from a reference surface in a specified height system. In Iran, due to the installation of tidegauge stations on the southern coast of the country, the elevation of the DNG1001 station in BandarAbbas has been considered as the height datum of the Iranian precise levelling networks. The height of this point was determined 3.777 m above an estimated mean sea level based on continuous fiveyear observations of the tidegauge, located at ShahidRajaee Port in 1995.
In 1998, by expanding the precise levelling networks in Iran, the height of the levelling benchmarks has been provided to the users without applying the gravityfield and related corrections as a purely geometric height, these calculations called the Iranian Height System 1998 (IRHS1998).
Then, by equipping the National Cartographic Center (NCC) of Iran with digital levels, based on the decisions taken, the precise levelling measurements were repeated from the second half of 2001 to the end of 2009. Subsequently, by performing the necessary processing and completing the acquisition of additional data, the elevation of the stations of the precise levelling network was adjusted in the Helmert's orthometric height system under the name of the Iranian Height System 2014 (IRHS2014). In this calculation, in addition to determining the orthometric correction, the additional corrections such as refraction, thermal expansion coefficient and calibration of the levelling rods have been applied. Here, similar to the old height system IRHS1998, the elevation of the DNG1001 station has been considered as the height datum.
Therefore, it is obvious that apart from applying the additional corrections and changes in the earth's crust due to the subsidence and tectonic movements over recent years, the new height system IRHS2014 has a minimum height difference corresponding to the orthometric correction (OC) compared with the old height system IRHS1998, where the difference in some parts of the country reaches about 1.5 m.
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2 Height reference surface and regional geoid model in Iran
The geoid, as a geopotential surface of the Earth's gravity field which approximates mean sea level, has been an important issue for geodesists and geoscientists for many years. Gauss was the first one who describe the above approximation for the geoid in 1828 as the mathematical figure of the earth (Heiskanen and Moritz 1967 p. 49, Torge 1991 p. 2, Gauss 1828). Later, Listing applied the term "geoid" for the surface defined by Gauss (Torge 1991 p. 2, Listing 1873). The practical application of the definition of this reference surface is the conversion of geodetic heights measured by GNSS receivers to orthographic height using Eq. 2 and significant reduction of the levelling campaign as a costly and timeconsuming operations (Fig. 1).
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Figure 1: Geoid height conception.
In order to determine the geoid model, it is necessary to collect the gravity data with uniform distribution in the whole of the Earth. In practice, however, by combining satellite and terrestrial data, the long wavelengths part of the geoid height, which have a major contribution of the computation, can be determined by using global geopotential models, while for calculating short wavelengths, an appropriate local modeling using groundbased gravity data is applied.
Precise geoid determination is not a new subject in Iran. Various efforts have been made by different working groups using different techniques for this purpose. The first efforts in this field date back to 30 years. Weber and Zomorrodian (1988) employed a method that is based on an integral formula in order to improve the GPM2 geopotential model (Wenzel, 1985) tailored with terrestrial gravity anomalies for regional geoid determination in Iran. With respect to the Doppler and levelling data in the area of interest, the first accuracy was approximately± 1.4m. Hamesh and Zomorrodian (1992) applied extracted Digital Terrain Model (DTM) from 1:250,000 maps to calculate the terrain effect of terrestrial gravity data supplied by the Bureau Gravimetrique International (BGI) based on the removerestore technique for geoid computation. Ardalan (1999) and Safari et al. (2005) determined a regional geoid for Iran using the ellipsoidal Bruns formula based on the inversion of the AbelPoisson integral. Najafi (2004), and Nahavandchi and Soltanpour (2005) employed a gravimetric method based on the StokesHelmert scheme method for precise geoid determination in Iran. Kiamehr (2006) computed the gravimetric geoid model of Iran using the leastsquares modification of the Stokes's formula. Sedighi et al. (2008) calculated the effect of mass density variation in precise geoid determination. Then, Hatam (2010) determined the Stokesbased geoid model (IRGeoid10) using the removerestore technique and Helmert’s second condensation method. He reached an accuracy of about 0.26 m at 819 independent GNSS/levelling control points over Iran. Finally, Saadat et al. (2018) calculated the novel regional geoid model of Iran entitled IRG2016 combining terrestrial and satellite gradiometry data based on radial basis functions (Fig. 2). The calculations were carried out in the area of interest using 21525 latestrefined gravity data supplied by the NCC of Iran (Fig. 3), which is contrary to most of the previous mentioned studies that have used part of the gravity data supplied by the BGI data centre, in which their positioning accuracy is not completely clear in Iran.
The statistical results of the geoidal height differences at 1288 GNSS/Levelling control points based on global geopotential model EGM2008, the RBFbased geoid model IRG016 and gridded RBFbased geoid model IRG2016, before and after 3 and 6parameter fitting have been presented in Table1. Figure 4 shows the distribution of 1288 GNSS/Levelling control points over Iran.
Figure 2: Calculated RBFbased geoid model IRG2016 at 2.5’×2.5’ grid points over Iran.
Figure 3: Distribution of the latestrefined input gravity data over Iran.
Table 1. Statistical results of the geoid height difference at 1288 independent control points before and after 3 and 6parameter fitting for the EGM2008, the RBFbased geoid model IRG2016 and the 2.5’×2.5’ gridded RBFbased geoid model IRG2016 (in m)
Geoid model 
Calculation 
Min 
Max 
Mean 
STD 
RMS 
2.5’×2.5’ EGM2008 
Alltrans EGM2008 
1.261 
1.624 
0.024 
0.297 
0.297 
2.5’×2.5’ gridded RBFbased geoid model IRG2016 
Before fitting 
0.871 
1.008 
0.078 
0.254 
0.265 
After 3parameter fitting 
0.864 
0.962 
0.000 
0.236 
0.236 

After 6parameter fitting 
0.877 
0.935 
0.000 
0.231 
0.231 

RBFbased geoid model IRG2016 
Before fitting 
0.866 
0.988 
0.072 
0.248 
0.258 
After 3parameter fitting 
0.851 
0.968 
0.000 
0.231 
0.231 

After 6parameter fitting 
0.861 
0.943 
0.000 
0.227 
0.227 
Figure 4: Distribution of 1288 GNSS/Levelling control points over Iran
The distributions of the geoidal height difference at all GNSS/Levelling control points over Iran using the RBFbased geoid model IRG2016 after 6parameter fitting has been shown in Figure 5. The largest differences are in the region of the country that lacks appropriate data and it reveals the need for dense data collection in these areas.
Figure 5: Distributions of the geoidal height difference at all GNSS/Levelling control points over Iran using the RBFbased geoid model IRG2016 after 6parameter fitting.
Also, in addition to calculating the geoidal height, it is also possible to convert the height of the points from the Iranian new height system IRHS2014 to the old height system IRHS1998.
 Of course, as already mentioned, the more precise computation of the geoid model requires the data collection with appropriate distribution and accuracy over the country, which should be considered more attention. At present, the accuracy of the regional geoid model IRG2016 is only suitable for the preparation of the maps with a scale smaller than 1:2000 and the contour line spacing of more than two meters.
3 Iranian regional geoid model IRG2016
The IRG2016 is an RBFbased regional geoid model for Iran, presented by the National Cartographic Center (NCC) of Iran and under a PhD thesis at University of Tehran (Saadat et al. 2018). It is bounded between 25°E and 40°E in latitude and between 44°N and 63.5°N in longitude. The input observations were considered according to 21525 latestrefined gravity data supplied by the NCC of Iran.
There are two options for geoidal height calculation based on IRG2016 online calculator, one is the pointwise and other is the fromfile calculation. Here, "Pointwise" or "From File" options can be selected from "Input format" section. If the "Pointwise" option is selected, the Input coordinates must be entered based on desired format, while for the "From File" option; the input file must be uploaded after login. If you don't have username and password, please register first.
The IRG2016 program can calculate the geoidal height in two ways. If the "2.5'*2.5' grid interpolation" is chosen from the "Calculation type" section, the geoid undulation is computed using only the latitude and longitude of point (the ellipsoidal height is not necessary) based on an appropriate interpolation method on 2.5'x2.5' arc minutes gridded points. The appropriate interpolation method (Nearest, Bilinear, Spline, and Bicubic) can be selected by the user from the relevant section. (The default options are good.) But if the "Single calculation" option is selected, the geoidal undulation will be calculated using the spherical harmonics coefficients and radial basis function (RBF), which is slightly more time consuming than interpolation method. In this case, in addition to latitude and longitude, the ellipsoidal height of the point on the surface of the Earth should also be taken into account in the input parameters.
The gravimetric geoid model IRG2016 can be fitted to 1288 GNSS/Levelling control points over Iran to determine the regional height reference surface (Hybrid geoid) by applying a corrector surface, which can be applied by choosing "Fitted to the GNSS/Levelling control points" option.
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Also, the program provides the ability to convert height from the Iranian new height system IRHS2014 to the old height system IRHS1998.
31 Pointwise calculation:
In this case, the geoid height and the correction value from the new height system of Iran to the old height system are computed by specifying the coordinates of the input point and the desired parameters in two ways, based on the interpolation on 2.5'×2.5' arc minutes gridded points or using single calculation:
 If the "2.5'*2.5' grid interpolation" is chosen, the geoid undulation and the correction value are calculated using only the latitude and longitude of point. Here, the ellipsoidal height is not necessary.
 If the ellipsoidal height is entered, the orthometric height of the point will also be calculated.
The desired interpolation method such as Nearest, Bilinear, Spline, and Bicubic can be selected by from the relevant section
 if the "Single calculation" option is selected, the geoidal undulation will be calculated using the spherical harmonics coefficients and radial basis function. Here, in addition to latitude and longitude, the ellipsoidal height of the point on the surface of the Earth should also be entered as an input parameters.
Also, the fitting method to the GNSS/Levelling control points based on 3 or 6parameter corrector surface can be specified from "Fitted to the GNSS/Levelling control points" option.
The output of the program is the regional geoid height and the correction value from the Iranian new height system to the old one. In the case of the introduction of ellipsoidal elevation, the orthographic height and height value in the old height system IRHS1998 is also calculated.
 If the geoid height and the correction value are equal to 9999, it means that the coordinates of the input point are outside the defined region.
 If the geoid height and the correction value are 8888, it means that the input parameters are not correct and should be contacted to the Admin.
32 Introducing input file
If the "From file" option is selected, the desired input file can be introduced by entering the appropriate username and password.
 The input file is a "*.txt" file separated by a comma whose first line is a header. For the "Single calculation", in addition to the station name, latitudes and longitudes (in decimal degrees), the ellipsoidal height of the point on the surface of the Earth should be entered in the input file. In this case, the input file format is as follow:
Inputfile.txt
St, Latitude, Longitude, h A1, 31.2165, 51.3268, 1200.326 A2, 35.2169, 52.3126, 1187.326 ... .. . 
 For the "2.5'*2.5' grid interpolation", the input file format without ellipsoidal height is as follow:
Inputfile.txt
St, Latitude, Longitude A1, 31.2165, 51.3268 A2, 35.2169, 52.3126 ... .. 
 Note 1: If the ellipsoidal height is inserted in the "2.5'*2.5' grid interpolation" case, the orthometric height will be computed in the output file.
Ho = h – N
 Note 2: The column "IRHS1998" in the output file is related to the Iranian old height system 1998.
Finally, browse your input file, then press "Upload Input File" button.
The output file will be sent to the registered email address.
Reference
Ardalan A.A., 1999. High resolution regional geoid computation in the World Geodetic Datum 2000 based upon collocation of linearized observational functionals of the type gravity potential and gravity intensity. Phd dissertation, Stuttgart university. 250 pp.
Ardalan A.A., Hatam C.Y., Sharifi M.A., Safari A., Gazavi K., Motagh M., 2002. Determination of Precise geoid for Iran based on Potential approach. Technical report, National Cartographic Center of Iran (NCC), Tehran, Iran.
Gauss C.F., 1828. Bestimmung des Breitenunterscchiedes zwischen den Sternwarten von Gottingen und Altona, Gottingen.
Hamesh M., Zomorrodian H., 1992. Iranian gravimetric geoid determination, second step. NCC J Surveying 6: 1724, 5263.
Hatam C.Y., 2010. Etablissement des nouveaux reseaux multiobservations geodesiques et gravimetriques et determination du geoide en Iran, Geophysics, Universite Montpellier 2, French.
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Safari A., Ardalan A. A. and Grafarend, 2005. A new ellipsoidal gravimetric, satellite altimetry and astronomic boundary value problem, a case study: The geoid of Iran. J. Geodyn., 39: 545568.
Sedighi M., Najafialamdari M., Tabatabaie S.H., 2008. Gravity Field Implied Density Modeling, for Precise Determination of the Geoid, Journal of Applied Science 8 (19): 33713379, ISNN 18125654.
Torge W., 1991. Geodesy, Walter de Gruyter, Berlin, 264 pp.
Weber G., Zomorrodian, H., 1988. Regional geopotential model improvement for the Iranian geoid determination. Bull. Géod., 62: 125141.
Wenzel H.G., 1985. Hochauflösende Kugelfunktionsmodelle für das Gravitationspotential der Erde. Wissenschaftliche Arbetien der Fachrichtung Vermessungswesen der Universität Hannover, Nr.137, Hannover.