عنوان مقاله [English]
Components of verticaldeflection, i.e., North-South component and East-West component ,are used for accurate determination of geoid or quasigeoid. Moreover, vertical deflection components area useful source for determination of variations in subsurface density and geophysical interpretations. Generally, there are two definitions for verticaldeflection. According to Helmert definition, vertical deflection at any given pointis the angle between the actualgravity vector (actual plumb line) and a line that is normal to the reference ellipsoid(a straight line perpendicular to the surface of reference ellipsoid). Another definition of vertical deflection is proposed by Molodensky. According this definition, vertical deflection at any given point is the angle between actualgravity vector and normal gravity vector (normal plumb line). Some relations have been introduced to convert Molodensky vertical deflection to Helmert vertical deflection. Helmert vertical deflection is estimated using astrogeodetic observations (combination of astronomical and geodetic observations).
Presently, global geopotential models (GGMs) have been expanded to the degree of2190, which is equivalenttoabout 5-min spatial resolution. Vertical deflectionat any point on the Earth can be calculated using the GGM. The resulting vertical deflection is consistent with Molodensky definition.Unfortunately, accuracy of GGMs is not sufficient for estimation of verticaldeflection.In other words, since GGMs are expanded up to a limited degree due to their resolution, omission error(or truncation error) occurs in computation of the earth’s various gravity field functionals, such as the geoidal height and verticaldeflection. Combining GGM with a digital terrain model (DTM) is a method used to reduce omission error.It should be noted that DTM has a higher spatial resolution as compared to GGM.In this method, the omitted signals of GGM can be modeled using residual terrain model (RTM) derived from subtracting high resolution DTM from a reference smooth surface. The reference smooth surface is obtained from eitherapplying average operator to DTM or expanding global topography into spherical harmonics. Fortunately, DTMs with spatial resolution of 3seconds or more,and reference smooth surface based on 2190 degree spherical harmonics are publicly available.
The present study seeks to assess vertical deflectionderived from a combination of GGM and DTM in Iran. Previously, Jekeli(1999) has studied EGM96 geopotential model with the aim of computingvertical deflection in the USA. Hirt(2010) and Hirt et al. (2010a) have assessed vertical deflection in Europe and the Alps using a combination of EGM2008 and RTM models.In Iran, GO_CONS_GCF_2_TIM_R4, a GOCE-only model, and EGM2008 geopotential model have been used toobtain vertical deflection and the results have been evaluated byKiamehr and Chavoshi-Nezhad(2014).
Materials & Methods
To implement the present study,a EGM2008 model with a spatial resolution of about 5-min is selected asGGM and a SRTM model with 3-sec spatial resolution is considered as DTM. To obtain RTM, DTM2006 model based on2190 degree spherical harmonicsis selected as the reference smooth surface.To compute the residual topography effect, prism method was used in an ellipsoidalmulti-cylindrical equal-area map projection system. First, we compute vertical deflectionusing EGM2008 model. It is also calculated using a combination of EGM2008 model and RTM(EGM2008/RTM method). In the next step, vertical deflection derived from the first method (EGM2008 model) and the second one (combination of EGM2008 model and RTM) are compared with vertical deflectionderived from astrogeodetic observations in 10 available Laplace stations in Iran.
Results & Discussion
Results indicate that there is a 1.2sec difference between North-South component of vertical deflection (i.e.) obtained from EGM2008 model and astrogeodetic observations.With RTM, this will reach 1 sec, which shows a 15% improvement. Moreover, there is a5.7secdifference between East-West component of vertical deflection () obtained from EGM2008 model and astrogeodetic observations, while this value will reach 5.6sec using RTM. Improvement in East-West component () is1.4%, which is smaller than the improvement of North-South component (). Based on the computations, we found that values of and in the Laplace stations canreach 17sec (RMS=7sec) and 15sec (RMS=8sec), respectively. Therefore, it is concluded that the relative error ofNorth-South component ()computation using EGM2008/RTM method is about 6% and the relative error ofEast-West component ()computation is about 37%.
The present research has studied the RTM effect on the improvement of GGM used for the determination of vertical deflectionin Iran. To performthe study, EGM2008 model with around 5-min spatial resolution was selected as GGM. RTM is also derived from subtracting the DTM2006 model (based on2190 degree spherical harmonics)from the 3-sec spatial resolutionSRTM model. Numerical findings indicate that a combination of RTM and GGM can improve the results of vertical deflectioncomputation, as compared to the results obtained from GGM-only approach. The improvement in North-South component of vertical deflection () is about15%and East-West component of the vertical deflection () undergoes about 1.4% improvement. In general, EGM2008 model and its combination with RTM have been more successful in the computation of component as compared to computationin the geographical region of Iran. There is no clear explanation for this difference, but it can be due to errors in theastronomical or geodetic observations oflongitude in Laplace stations.