**Abstract**

One of the main steps within the geoid computation methodology without applying the Stokes formula is downward continuation of the harmonic residual observables from the surface of the Earth down to the surface of the reference ellipsoid. This downward continuation is done via the Abel-Poisson integral and its derivatives. This integral in which the unknowns, i.e. harmonic residual potential values on the surface of the reference ellipsoid, are under integral sign, is a Fredholm integral equation of the first kind. Solution of the aforementioned integral equation is an unstable problem and like any unstable problem requires regularization. One of the most important issues of every regularization method is estimation of the regularization parameter.

The aim of this paper is the comparison of different methods for estimation of the regularization parameter of the Tikhonov regularization method when applied to the downward continuation of incremental gravity observables for the geoid computation without applying the Stokes formula. For this purpose, the following regularization parameter selection methods, which are free from the knowledge of norm of vector of observation errors, are considered: (i) Discrepancy Principle (DP), (ii) Generalized Cross-Validation (GCV), (iii) L-Curve (LC), and (iv) Flattest Slope (FS). Each regularization parameter estimation method has its own concept for identification of optimum regularization parameter and as such they can result in different regularization parameters for the same problem. For example, in the DP method, the optimum regularization parameter is selected in a way that the estimated factor variance is less sensitive to the variations of the regularization parameters. In the GCV method, the optimum regularization parameter is the one that is less sensitive to the reduction of input information. LC makes a balance between regularization of the solution and the introduced error by the regularization. In FS, the estimation of optimum regularization parameter is based on having the least changes in the solution of the problem vs. changes of the regularization parameter.

The aforementioned methods are applied to: (i) the real data for geoid computation without applying the Stokes formula in a geographical region of Iran (43.5?E< <64.5?E, 23.5?N< <40.5?N) based on a methodology, which algorithmically consists of remove, downward continuation using ellipsoidal Abel-Poisson integral, restore, and application of ellipsoidal Bruns formula, and (ii) a simulation which is designed for the same geographical area.

According to the simulation study the LC method results in (i) least relative error, (ii) Largest Effective Number of Degree of Freedom and (iii) closest regularization parameter to the actual one. Therefore, it can be concluded that LC amongst the tested methods for the estimation of regularization parameter, is the most efficient one and its application is recommended for the geoid computation methodology without applying the Stokes formula.

The aim of this paper is the comparison of different methods for estimation of the regularization parameter of the Tikhonov regularization method when applied to the downward continuation of incremental gravity observables for the geoid computation without applying the Stokes formula. For this purpose, the following regularization parameter selection methods, which are free from the knowledge of norm of vector of observation errors, are considered: (i) Discrepancy Principle (DP), (ii) Generalized Cross-Validation (GCV), (iii) L-Curve (LC), and (iv) Flattest Slope (FS). Each regularization parameter estimation method has its own concept for identification of optimum regularization parameter and as such they can result in different regularization parameters for the same problem. For example, in the DP method, the optimum regularization parameter is selected in a way that the estimated factor variance is less sensitive to the variations of the regularization parameters. In the GCV method, the optimum regularization parameter is the one that is less sensitive to the reduction of input information. LC makes a balance between regularization of the solution and the introduced error by the regularization. In FS, the estimation of optimum regularization parameter is based on having the least changes in the solution of the problem vs. changes of the regularization parameter.

The aforementioned methods are applied to: (i) the real data for geoid computation without applying the Stokes formula in a geographical region of Iran (43.5?E< <64.5?E, 23.5?N< <40.5?N) based on a methodology, which algorithmically consists of remove, downward continuation using ellipsoidal Abel-Poisson integral, restore, and application of ellipsoidal Bruns formula, and (ii) a simulation which is designed for the same geographical area.

According to the simulation study the LC method results in (i) least relative error, (ii) Largest Effective Number of Degree of Freedom and (iii) closest regularization parameter to the actual one. Therefore, it can be concluded that LC amongst the tested methods for the estimation of regularization parameter, is the most efficient one and its application is recommended for the geoid computation methodology without applying the Stokes formula.

**Keywords**