Summary
A new method for cubature calculations is presented. Spline approximations are constructed for the discharge variations in the different rivers of a tidal region. Derived quantities as ebb- and flood-volumes can be evaluated in a simple way. A computer program based on this method is described. Some results for the basin of the Schelde are shown.
Zusammenfassung
Eine neue Methode für die Ausführung einer Kubizierung wird vorgestellt. Spline Approximationen werden für die Variationen des Abflusses in den Flüssen eines Gezeitengebietes konstruiert. Abgeleitete Größen wie Ebb- und Flutstromvolumen können damit in einer einfachen Weise berechnet werden. Ein auf dieser Methode basierendes Computer-Programm wird beschrieben. Es werden einige Ergebnisse aus der Anwendung für das Scheldebecken gezeigt.
Résumé
Une nouvelle méthode pour les calculs d'une cubature est présentée. Des approximations splines sont construites pour les variations du débit dans les diverses rivières d'un bassin soumis à la marée. Des grandeurs dérivées comme les volumes de jusant et de flot peuvent être évaluées d'une manière simple. Un programme d'ordinateur basé sur cette méthode est décrit. Quelques résultats d'une application au bassin de l'Escaut sont montrés.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Abbreviations
- b (x, z) :
-
storage width
- b v, w :
-
measured storage width
- g, h :
-
number of knots of the two-dimensional spline function
- H :
-
tidal range
- HW:
-
high water
- HWS:
-
high water slack
- k, l :
-
degrees of the two-dimensional spline function
- LW:
-
low water
- LWS:
-
low water slack
- m :
-
number ofx q-values
- M :
-
number of\(\hat x_v \)-values
- M i, k+1 (x) :
-
normalized B-spline of degreek, with respect to the knotsλ i,λ i+1,...,λ i+k+1
- n :
-
number oft r-values
- N :
-
number ofz w-values
- N j, l+1 (t) :
-
normalized B-spline of degreel, with respect to the knotsμ j,μ j+1,...μ j+l+1
- Q (x,t) :
-
discharge of the river
- \(\tilde Q(x,t)\) :
-
discharge of the river, taking into account the tributaries
- Q o :
-
constant discharge at the place where the tidal flow is retained
- s(x), S(t) \(\tilde S(t)\) :
-
one-dimensional spline functions
- s(x, t) :
-
two-dimensional spline function
- t HW :
-
point of time of highest water level
- t HWS :
-
point of time of high water slack
- t LW :
-
point of time of lowest water level
- t LWS :
-
point of time of low water slack
- t r :
-
point of time of water level measurements
- V E :
-
ebb volume or ebb power
- V F :
-
flood volume or flood power
- \(\tilde x\) :
-
abscissa of the mouth of a tributary
- x * :
-
abscissa indicating the place in the river where the tidal flow is retained
- x q :
-
abscissa indicating a place in the river where water levels are measured
- \(\hat x_v \) :
-
abscissa indicating a place in the river where the storage width is measured
- z (x,t) :
-
altitude of the water level with respect to a plane of reference
- z HW :
-
highest water level
- z LW :
-
lowest water level
- z, q, r :
-
measured altitude of water level
- z w :
-
water level for measuring the storage width
- λ i :
-
knot of the two-dimensional spline, in the first dimension
- μ j :
-
knot of the two-dimensional spline, in the second dimension
- Φ:
-
lateral water supply
References
Bonnet, L. and J. Blockmans, 1936: Etude du régime des revières du bassin de l'Escaut maritime par cubature de la marée moyenne décennale 1921–1930, Extrait Annal. Trav. Publ. Belgique.89, Fasc. 3.
Boor, C. de, 1972: On calculating with B-splines. J. Approx. Theory.6, 50–62.
Chow, Ven Te, 1959. Open-channel hydraulics. New York: McGraw-Hill Book Comp. XVIII, 680 S.
Cox, M. G., 1972: The numerical evaluation of B-splines. J. Inst. Math. Appl.10, 134–144.
Dierckx, P., 1977: An algorithm for leastsquares fitting of cubic spline surfaces to functions on a rectilinear mesh over a rectangle. J. Comput. Appl. Math.3, 113–129.
Gaffney, P. W., 1976: The calculation of indefinite integrals of B-splines. J. Inst. Math. Appl.17, 37–41.
Greville, T. N. E., 1969: Theory and applications of spline functions. New York: Academic Press, XI, 212 S. m. Tab.
Knuth, D. E., 1968: The art of computer programming; fundamental algorithms, London: Addison-Wesley Publ. Comp. S. 347–361.
McDowell, D. M. and B. A. O'Connor, 1977: Hydraulic behaviour of estuaries. London [usw.]: Macmillan Press Ltd. 292 S. m. Abb. u. Tab.
Pierrot, J. A. and L. Van Brabandt, 1907: Etudes sur le régime des rivières du bassin de l'Escaut maritime. In: Recueil de documents relatifs à l'Escaut maritime. Ed. by the Ministry of Public Works, Belgium.
Sterling, A. and P. Roovers, 1967: Modelstudies aangaande de verbetering van de bevaarbaarheid van de Westerschelde. Ingenieur. No 19, 20.
Temmerman, M., 1980: Debietberekeningen in de Schelde door middel van splinefunkties. Katholieke Univ. Leuven, Appl. Math. Progr. Div. (in Dutch).
Valcke, E., H. Vandervelden, A. Sterling et al., 1966: Stormvloeden op de Schelde. 5 Vol. Bestuur der Waterwegen, Ministerie van Openbare Werken, Brussel.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Dierckx, P., Smets, E., Piessens, R. et al. A new method of cubature using spline functions. Deutsche Hydrographische Zeitschrift 34, 61–79 (1981). https://doi.org/10.1007/BF02226587
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02226587