Modeling of gravity field has many application in geosciences such as geodesy، geophysics، and oceanography and tectonic. Marine Gravity observations due to factor such as oscillation and accelerations of ship، also mechanical errors that are greater in water have low precision. Satellite altimetry provide detailed information about geoid changes and its slope changes over the oceans. in geodesy we can convert geoid and its slope (deflection of vertical component)، to gravity anomaly using spectral analysis methods. There are two integral equation to estimate gravity anomaly from geoid and deflection of vertical، stokes integral and vening meinesz integral. In this paper method of marine gravity anomaly deriving from satellite altimetry are introduced and then using satellite altimetry data estimate marine gravity anomaly over Oman Sea that is a Deep region. Then for control precision of methods، use marine gravity anomaly points that have corrected and collected by national geophysical data center (NGDC). our result show that in gravity anomaly estimation from satellite altimetry using Deflections of vertical as observation and spectral kind of vening meinesz integral as method have a RMS of 4. 2 mgal in comparison with shipborne data، while the using geoid as observation and spectral kind of stokes integral as method، have 4. 38 mgal in comparison with shipborne data. This indicate spectral kind of vening meinesz integral have advantage in comparison with spectral kind of stokes integral.

To solve the geodetic boundary-value problem the gravity anomalies have to be continued from the Earth's surface down onto the geoid surface. Helmert’s gravity anomalies (multiplied by the geocentric radius) are harmonic above the geoid. It is thus possible to use them in the Poisson solution of the inverse Dirichlet boundary value problem to get their values on the geoid. The downward continuation (DWC) of Helmert gravity anomalies has raised some criticism, however, originating from the suspicion that the anomalies, even though they are smoother than free-air and Faye anomalies, are still too rough to yield a reasonably good solution particularly for denser grid of gravity anomalies. No topography (NT) space is the gravity Earth's field after the removal of the gravitational effect of the all masses above geoid. Vaníček et al., 2003 proposed the Spherical Complete Bouguer Anomaly (SCBA) as a 3D solid gravity anomaly that is suitable for DWC. The SCBA is a harmonic and smooth function that conforms to an accurate and stable downward continuation. The topography above the geoid contributes toward the high frequency portion of the free-air gravity anomaly on the Earth surface. By contrast, the SCBA is a smooth function because it is already corrected for the topographical effect. Hence, the Bouguer anomaly, especially the SCBA, is much smoother with attenuated high frequency content, compared to the Helmert anomaly. By the nature of the Downward Continuation (DWC) process, which amplifies high frequencies, the SCBA is better suited (Heck, 2003). However, in contrast to the Helmert anomaly (a free-air type), the SCBA involves the disturbed isostasic equilibrium of the crust on the mantle because of the removal of the topographical effect. This phenomenon results in a powerful low frequency content in the SCBA and a large indirect effect (PITE), ultimately amounting to a few hundred meters which is detrimental to the accuracy of the geoid solution. To reduce the large effect to a few decimeters, the SCBA on the geoid is transformed back to the Helmert space. In this paper we formulate the Stokes-Helmert BVP using No topographical space and SCBA in a three space scenario. 1. transforming the observed gravity anomaly on the terrain to the NT space on the same position, 2. downward continuation of the SCBA anomalies from the observation positions to the geoid level, 3. transforming the SCBA on the geoid to the Helmert space, 4. evaluation of the co-geoid using the generalized Stokes integral, 5. transforming the co-geoid back to the geoid. i.e., from Helmert’s space to the real space by applying the PITE. In theory, all different reductions yield the same result in geoid determination. We compared the geoid by NT deduced Stokes-Helmert method and geoid by Stokes-Helmert method in Iran. The long wavelengths part of geoid up to degree and order 180/180 is determined using the EGM08 model. As external evidence, the two gravimetric geoids were compared to the geometric GPS-levelling solution at 213 points. The RMS of differences between two geoids and GPS-levelling data are 46cm without any applying correction surface. However, the RMS of difference between two geoid models is about 4cm and it can be reached up to 1.67meter in mountainous area.

Due to wide spread usage of the satellite positioning techniques especially GPS, we need to precisely determine geoid model in order to use GPS measurements for height determination, as an alternative of traditional leveling techniques in geodetic applications. Precise local geoid modelling using GPS/Leveling data, apart from the existing models such as geopotential models and gravimetric geoid models could be an interesting investigation topic. An important question is, ‘What accuracy level can be achieved using this approach?’ However precession of this modelling could be influenced by some issues such as data quality or modelling techniques. In this paper, we attempt to assess the implementation of modern learning-based computing techniques including artificial neural networks and adaptive network-based fuzzy inference systems compared with multivariate polynomial regression equations in GPS/Leveling Geoid modeling. This assessment carried out in a small and dense network of GPS/Leveling benchmarks in contrast with previous studies, located in shahin-shahr, Isfahan. And these high quality data make it possible to achieve an accuracy of better than 1 cm. The results show a few millimeter superiority of ANN and ANFIS derived geoid models in terms of root mean square error, as well as in terms of coefficient of determination. And RMSE=8cm, R2=0.9949 and RMSE=7cm, R2=0.9964 achieved for this models respectively. Therefore ANFIS derived geoid model provide the most accurate geoid heights in the study area.

By increasing the accuracy and resolution of gravity data derived from terrestrial, airborne and satellite methods, the accuracy and resolution of global geopotential models have significantly been improved. For example, EGM2008 and EIGEN-6C4 are among the most accurate global geopotential models which have been expanded up to degree 2190. Nevertheless, global geopotential models do not have an adequate accuracy everywhere. Therefore, the local gravity field modeling based on the geodetic boundary value problem approach and the local gravity data has always been an interesting subject. In Iran, first-, second- and third-order gravity networks with the spatial resolutions 30ʹ, 15ʹ, and 5ʹ have been designed for geodetic applications. Now, these important questions arise: (1) How important is the spatial resolution of the ground gravity data in the local modeling? (2) Can local gravity data with any resolution improve the global models? To answer these questions, the effect of the spatial resolution of the Iranian ground gravity data to determine the local geoid based on the geodetic boundary value problem solution by the remove-compute-restore technique is studied. In this line, four regions over Iran with different spatial resolutions are selected as test regions. Region 1 consists of 1738 gravity data with spatial resolution 5.7ʹ, region 2 consists of 165 gravity data with spatial resolution 21.6ʹ, region 3 consists of 234 gravity data with spatial resolution 18ʹ and region 4 consists of 1728 gravity data with spatial resolution 5.7ʹ. Then, the geodetic boundary value problem is separately solved for each region, where the EGM2008 global model up to degrees 360, 720, 1080 and 2160 is used as reference model. Finally, the computed local geoids and the global geoids are compared with the GPS/Leveling geoid. From results we found that the local geoid in the regions 1 and 4 has an accuracy of about 23 cm in terms of the root mean square error (RMSE), while the local geoid in the regions 2 and 3 has an accuracy of about 32 cm. This means that the local geoid in the regions 1 and 4, where the spatial resolution of gravity data is higher, is more accurate than the local geoid in the regions 2 and 3. Moreover, we found that the local geoid of region 1 is more accurate than the global geoids up to degrees 360, 720 and 1080, while the accuracy of the local geoid is consistent with the global geoid up to degree 2160. Such a result is obtained for the region 4. For the regions 2 and 3, the local geoid is more accurate than the global geoid only up to degree 360, while the accuracy of the local geoid is consistent with the global geoids up to degrees 720, 1080 and 2160. This is due to the fact that the spatial resolution of gravity data in the regions 1 and 4 is 5.7ʹ which is equivalent to degree about 2160, while the spatial resolution of gravity data in the regions 2 and 3 are 21.6ʹ and 18ʹ, respectively, which are equivalent to degrees about 500 and 600. Therefore, it is concluded that when the spatial resolution of ground gravity data is lower than the corresponding degree of the reference model, the local geoid does not outperform the corresponding global geoid.

The main topic of this research is to investigate different gravimetric reduction in the context of precise geoid determination. Gravimetric reduction perform an essential role on precise geoid determination, particularly in rugged areas. A numerical investigation was performed in the rugged area of the Northwest Iran within the geographical boundaries 35.5> φ >39.5 and 44.5>>λ >49.5 to study gravimetric geoid solutions based on the Rudzki inversion scheme, Hеlmert’s second method of condensation, RTM, and the topographic-isostatic reduction methods of Airy-Heiskanen (AH) and Pratt-Hаyford (PH). The results shows Rudzki gеoid performs as well as the Hеlmert and RTM geoids (in terms of standard deviation and range of minimum and maximum values) when comparing to comparison with the GPS-levelling geoid of the test area. Rudzki inversion the sole gravimetric reduction scheme which doesnt change the equipotential surface and thus doesnt need the calculation of the indirect effect.

In geodesy, three levels are considered: the physical surface of the earth on which mapping measurements are made, the ellipsoidal reference surface (geometric datum) which is the basis of mathematical calculations, the geoid physical surface (physical datum) which is the basis for measuring heights. Satellite positioning systems measure the height of points relative to the ellipsoid surface. The geoid is one of the equipotential surfaces of the earth's gravity field, which approximates the mean sea level (MSL) by least squares. Geoid is very important in geodesy as a representative of the physical space or the space of observations made on the earth and also as the base level of elevations. The separation between the geoid and the geocentric reference ellipse is called geoid height (N). Although there is only one equipotential surface called geoid, various methods are used to determine it. These methods include: geometric method, geoid determination by satellite method, Gravimetric methods and geoid determination using GPS/leveling.

In this paper, the aim is to estimate the height of the local geoid using machine learning models. To do this, artificial neural network (ANN), adaptive neuro-fuzzy inference model (ANFIS), support vector regression (SVR) and general regression neural network (GRNN) models are used. The geodetic coordinates of 26 GPS stations in the north-west of Iran along with their orthometric height (H0) and normal height (h) were obtained from the national cartographic center of Iran. In all stations, the difference of orthometric height and normal height is considered as geoid height (N). Therefore, the geodetic longitude and latitude of the GPS stations are considered as the input of the machine learning models, and the corresponding geoid height was considered as the output. In order to test the results of machine learning models, two modes of 4 and 7 test stations are considered. Also, the output of the models is compared with the local geoid model IRG2016 presented by Saadat et al. for the Iranian region and also the global geoid model EGM2008.

Due to the availability of a complete set of observations of GPS stations along with orthometric height obtained from leveling in the north-west region of Iran, the study and evaluation of the models proposed in the paper has been carried out in this region. Observations of 26 GPS stations of North-west of Iran were prepared from the national cartographic center (https://www.ncc.gov.ir/). Two modes are considered for training and testing of ANN, ANFIS, SVR and GRNN models. In the first case, the number of training stations is 22 and the number of test stations is 4. But in the second case, by increasing the number of test stations to 7 stations, the error evaluation of the models has been done. It should be noted that the distribution of training and test stations is completely random. After the training step of machine learning models and choosing the optimal structure, the test step is performed in two different modes (4 and 7 stations). At this step, the value of the geoid height in the test stations is estimated and compared with the value obtained from the difference of orthometric height and normal height as a basis. Two statistical indices of relative error in percentage and RMSE in centimeters were calculated for all models and presented in Table (1) for the first case.Table 1. Relative error (%) of ANN, ANFIS, SVR, GRNN and IRG2016 models in the test stations considered for the first case According to the results of Table (1) and comparing the relative error values of all models in the test stations, it shows that the ANFIS model was more accurate than other models. After ANFIS model, IRG2016 model has higher accuracy than ANN, SVR and GRNN models. It should be noted that the IRG2016 local model uses the observations of all Iranian plateau stations to model the local geoid, and therefore it is expected that this model will be more accurate in the study area than other models.

The evaluations show that in the case of 22 training stations and 4 test stations, the RMSE of ANN, ANFIS, SVR, GRNN and IRG2016 models in the test step are 37.32, 19.83, 49.34, 53.82 and 29.65 cm, respectively. However, in the case of 19 training stations and 7 test stations, the error values of the models are 36.63, 58.31, 39.64, 41.29 and 24.68 cm, respectively. Comparison of RMSE shows that ANN model with less number of training stations provides higher accuracy than ANFIS, SVR and GRNN models. The results of this paper show that by using ANN and ANFIS models, geoid height can be estimated and used with high accuracy locally in civil and surveying applications.