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<p>Collocation theory is a “smoothing” approach to the prediction of a signal or of some signal functionals from a set of observations. Namely, the predicted signal/functional and the actual signal do not belong to the same space, the first being more regular than the actual one. In Physical Geodesy, collocation theory is applied to the determination of the anomalous potential of the gravity field <i>T</i>. The predicted <i>T</i>, in this case, belongs to a space harmonic down to a Bjerhammar sphere internal to the Earth surface. A space more regular than that of the actual <i>T</i>, which is harmonic only outside of the Earth. However, it was proved that the predicted <i>T</i> can approximate the actual field as well as we like. In addition, it was proved that the collocation solution for the Geodetic Boundary Value Problem converges to the actual <i>T</i>, when the number of observations tends to infinity. In the present paper, the same convergence problem is tackled and solved under quite general conditions when assuming that the observations are noiseless. Moreover, the approximation to the solution of geodetic BVP’s from noisy observations at the boundary is proved to be true in the mean-square sense.</p>

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The convergence problem of collocation theory: a return

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年6月25日

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  • Fernando Sansó,
  • Giovanna Venuti

摘要

Collocation theory is a “smoothing” approach to the prediction of a signal or of some signal functionals from a set of observations. Namely, the predicted signal/functional and the actual signal do not belong to the same space, the first being more regular than the actual one. In Physical Geodesy, collocation theory is applied to the determination of the anomalous potential of the gravity field T. The predicted T, in this case, belongs to a space harmonic down to a Bjerhammar sphere internal to the Earth surface. A space more regular than that of the actual T, which is harmonic only outside of the Earth. However, it was proved that the predicted T can approximate the actual field as well as we like. In addition, it was proved that the collocation solution for the Geodetic Boundary Value Problem converges to the actual T, when the number of observations tends to infinity. In the present paper, the same convergence problem is tackled and solved under quite general conditions when assuming that the observations are noiseless. Moreover, the approximation to the solution of geodetic BVP’s from noisy observations at the boundary is proved to be true in the mean-square sense.