Estymacja metodą najmniejszych kwadratów (LS) jest Jednym z najważniejszych narzędzi w analizowaniu danych geodezyjnych. Jednakże powszechne korzystanie z rej metody nie zawsze idzie w parze z pełnym uświadomieniem sobie jej podstaw. W standardowym formalizmie teorii estymacji LS w rzeczywistości istnieje kilka paradoksalnych i osobliwych zagadnień rzadko formułowanych wprost. Celem niniejszej pracy jest przedstawienie niektórych z tych zugadnień i przedyskutowanie ich konsekwencji w analizie danych gcodezyjnvch oraz problematyce estymacji parametrów. W pierwszej części pracy przedstawiony Jest alternatywny pogląd na podstawy statystyczne, które są tradycyjnie łączone z estymacją LS. W SZC7.ególności pokazano. że właściwość nieobciąźoności dla zwykłych estymatorów LS może być zastąpiona przez inne. równowazne JeJ uwarunkowanie, które powoduje, że zakres numeryczny nieznanych parametrów jest nieograniczonv. \V drugiej części pracy przedstawiono wady meiodv LS 7. czysto algebraicznego punktu widzenia. bez uwzględnienia pojęć z zakresu prot abilisryczncgo/sratysrycznego teorii estymacji. W szczególności ,, yjaśnione zostało. cło czego odnosi się 'najmnicjsz, · (least) w metodzie najmniejszych kwadratów. Z pewnością nie odnosi się cło błędów· wyznaczanych parametrów modelu. Ponadto stwierdzono, że w bielej inwersji modelu liniowego opartej na metodzie LS istnieje krytyczna zamiana pomiędzy normami euklidesowymi błędów wyznaczanych parametrów i wyrównanych residuów.

The main theme of this paper is to study two important aspects of precise geoid determination using Helrnerts second method of condensation. This work illustrates via numerical investigations the importance of using actual density information of topographical bulk and the effects that different gravimetric reductions have on gravity interpolation in Helmert geoid computational process, in addition to the commonly used Bouguer scheme. A rugged area in the Canadian Rockies bounded by latitude between 49°N and 54°N and longitude between 236°E and 246°E is selected to carry out numerical investigations. The lateral density information is used in all steps of the Helmert geoid computational process. The Bouguer and residual terrain modelling (RTM) topographic reductions, the Rudzki inversion scheme, and the topographic-isostatic reductions of Pratt-Hayford (PH) and Airy-Heiskanen (AH) are used for gravity interpolation. Results show that the density information should be applied in all steps of the Helmert geoid computational process and that the topographic-isostatic gravimetric reduction schemes like the PH or AH models or the RTM reduction, should be applied for smooth gravity interpolation instead of the commonly used Bouguer reduction scheme for precise Helmert geoid determination.

This paper investigates the terrain-aliasing effects on geoid determination using different gravimetric reduction schemes. The high resolution of digital terrain model (DTM), if available, should be used for every gravimetric reduction scheme since it can precisely map the details of the terrain. The reduction methods used in this study are the Rudzki inversion method, Helmert's second method of condensation, the residual terrain model (RTM) method, and the Pratt-Hayford (PH) topographic-isostatic reduction technique. The effect of using different DTM grid resolutions of 6", 15", 30", 45", I' and 2' on gravity anomalies and absolute geoid undulations is studied for each of these reduction schemes. A rugged area in the Canadian Rockies bounded by latitude between 49°N and 54°N and longitude between 236°E and 246°E is selected to conduct numerical tests. Our results suggest that a DTM grid resolution of 6" or higher is required for precise geoid determination with an accuracy of a decimetre or higher for any gravimetric reduction method chosen to treat the topographical masses above the geoid in rugged areas. The most precise geoid models obtained in this test are the ones obtained using Rudzki, Helmert, and RTM methods with 6" DTM resolution.

A number of new satellite-only Global Gravity Models (GGMs) become progressively available based on the CHAMP and GRACE satellite mission data. These models promise higher (compared to older GGMs) accuracy in the determination of the low and medium harmonics of the Earth's gravity field. In the present study, the latest GGMs generated from CHAMP and GRACE data (namely EIGEN2, EIGEN3p, GGM0IC, GGM0IS and GRACED IS) have been studied with respect ro their accuracy and performance when used in gravity field approximation. A spectral analysis of the new models has been carried out, employing their degree and error-degree variances. In this way, their performance against each other and with respect to EGM96 was assessed, and the parts of the gravity field spectrum that each model describes more accurately have been identified. The results of the analysis led to the development of a combined geopotential model, complete to degree and order 360, whose coefficients were those of CHAMP until degree 5, then GRACE until degree 116, and EGM96 for the rest of the spectrum. Finally, a validation of all models (the combined included) has been performed by comparing their estimates against GPS/levelling data in land areas and TOPEX/Poseidon sea surface heights in marine regions. All rests have taken place over Greece and the eastern part of the Mediterranean Sea. From the results obtained it was concluded that the combined GGM developed provides more accurate results (compared to EGM96), in terms of the differences with the control datasets, at the level of 1-2 cm geoid and 1-2 mGal for gravity (ICT). Furthermore, the absolute geoid accuracy that the combined GGM offers is 12.9 cm (ICT) for 11 = 120, 25 cm for 11 = 200 and 33 cm for n = 360, compared to 29 cm, 36 cm and 42 cm for EGM96, respectively.

Marine geoid modelling in the Atlantic coastal region of Argentina is problematic. Firstly, because of the insufficient amount of available shipborne gravity data, which renders a purely gravimetric solution not feasible. Secondly, because of the very strong ocean currents, that affect the quality of satellite altimetry data, so that a purely altimetrie model is too noisy, even after low-pass filtering the Sea Surface Heights (SSHs) to remove (part of) the influence of the oceanographic signals. Thus, the recommended solution is to employ a combination method and the use of all the available gravity and altimetry data together. This is a suitable solution since (i) combination methods such as least squares collocation and Input Output System Theory (!OST) inherently low-pass filter and weigh the data, and (ii) will make use of the altimetrie heights to fill the gaps of the shipborne gravity data. Following this idea, purely altimetrie, gravimetric and combined (using the !OST method) marine geoid models have been estimated for Argentina, employing all available shipborne gravity data, satellite altimetry SSHs and the latest Earth Gravity Models (EGMs) developed from CHAMP and GRACE missions. The new EGMs are especially useful to assess the quality of the new geoid models, especially against EGM96, which was used in an older ERSl-only solution for the same area. From the comparison of the estimated geoid models with respect to stacked TOPEX/Poseidon SSHs, the authors found that the altimetrie model provides the best agreement while the combined one improves the accuracy (I a) of the gravimetric solution.