Proceedings of the 7 th International Conference on Offshore echanics and Arctic Engineering OAE June 5-, 8, Estoril, Portugal DRAFT OAE8-5784 REAL TIE WAVE FORECASTING FOR REAL TIE SHIP OTION PREDICTIONS Ir. Peter Naaijen (Delft University of Technology) Prof. Dr. Ir. Rene Huijsmans (Delft University of Technology) ABSTRACT This paper presents results of a validation study into a linear short term wave and ship motion prediction model for long crested waves. odel experiments have been carried out during which wave elevations were measured at various distances down stream of the wave maker simultaneously. Comparison between predicted and measured wave elevation are presented for 6 different wave conditions. The theoretical relation between spectral content of an irregular long crested wave system and optimal prediction distance for a desired prediction time is explained and validated. It appears that predictions can be extended further into the future than expected based on this theoretical relation. INTRODUCTION Within the offshore industry there are various operations for which a motion-prediction based decision-support system can be beneficial: Top-side installations (lifting or float-over), LNG connecting and helicopter/automatic UAV landing are examples of operations for which safety and operability can be increased if a reliable prediction of the vessel motions were available. In 6 an international Joint Industry Project called Onboard Wave and otion Estimation (OWE) was launched to develop, test and demonstrate a practical system to predict quiescent periods of ship and platform motions some 6 seconds in advance. The approach of the system is based on measuring the wavefield around the vessel by means of an Xband radar having a reach of appr. km. Within the project Delft University of Technology (DUT) has been commissioned to provide a wave propagation model which uses the measured wave field to make a prediction of the vessel s motions. It was agreed, for the sake of minimum calculation time, robustness and maintainability and due to the uncertainty of the accuracy of the input wave measurement (which has a large effect on the accuracy of higher order prediction models) to use linear wave theory. This paper presents a first study on the accuracy of a linear long crested wave propagation model for various wave conditions. An optimization study for the distance between measurement and prediction location, depending on the desired forecast time has been carried out and results are presented. EXPERIENTS The experiments with scale :3 were carried out at towing tank # of the DUT Ship Hydromechanics Laboratory, having the following particulars: Length: 5 m Width: 4. m Depth:.4 m During the experiments, waves were created by the wave maker consisting of one flap hinged at the bottom of the towing tank. Alongside the towing tank, down stream of the wave maker, an array of 9 wave probes was installed. See Figure for the schematic experimental set-up. Figure, probe positions experimental set-up The position of the wave probes was chosen such that for each wave condition, at least one probe was positioned at a distance Copyright xx by ASE
from the reference probe that would theoretically just allow a s prediction within the predictable area based on theoretical spectrum of the wave condition. (For the swell condition, the limited length of the basin allowed only a 9 s) The words predictable area are explained in detail in the theory paragraph. The campaign existed of 6 measurements of about hours full scale duration with different wave conditions. 5 Jonswap spectra were teseted and one typical swell spectrum: type Significant Wave Height H sig Peak period T p Wave length / water depth λ p /h [-] [m] [s] [-] Jonswap 3.8 4.98.53 Jonswap 4.7 5..59 Jonswap 5.54 6.6.79 Jonswap 6.48 6.8.8 Jonswap 7 3.54 8.. Swell.3 3.8 4.79 Table, Properties of tested wave conditions (as observed) Before sampling, low-pass analogue filtering was applied with a cut-off frequency of Hz. The applied digital sampling rate of the data acquisition amounted to Hz (model scale) which corresponds to 8.3 Hz prototype scale. High-pass digital filtering was applied to eliminate the possible effect of seishes in the towing tank. THEORY This paragraph describes the theory that is used to predict the wave elevation at a certain downstream location by using an upstream wave elevation measurement which is based on linear wave modeling. As done by orris et al. [] and Edgar et al. [3], the linear wave propagation modeling problem can be represented schematically by a time-distance diagram, see Figure. Consider a time trace of one dimensional dispersive waves satisfying the linear wave equation measured at location A. A Fourier Transform of this irregular time trace yields the amplitude and phase angles of a limited number of regular wave components that it contains. (In order to reduce the endeffects a tanh-shaped window function with steep slopes was applied to the input trace before the Fourier transformation was carried out. This slightly improved the results.) For a sampled irregular input trace at location A of N points length yields: = N / i, n i nt A() t a, ne ε e ω n= () This trace is indicated by the thick line OT at the lower side of the leftmost triangle in Figure. With the linear dispersion relation the wave number k of each of the components can be determined and for the predicted wave at location B at a future time t+ t can be written: ( ) N / B, n= i( εn kn( xb xa) ) iωn ( t+ t) a n () t+ t = e e where the wave is supposed to propagate in positive x direction and location B has a larger x-co-ordinate than location A. This trace is indicated by the thick line CE in Figure resulting from shifting OT by t horizontally and by x B x A vertically. As the overall aim of the project is to predict quiescent periods in the vessel motions, it s not the deterministic wave or motion that we are interested in but rather it s envelope. The envelope of the prediction can simply be determined from the deterministic prediction by taking the absolute value of its Hilbert transform Predictable area However, for physical significance of the forecasted trace from equation (), t and x B x A cannot be chosen freely: The longest and shortest wave components found in the irregular measured trace OT determine the so-called predictable area in space and time where prediction is possible using trace OT. The predictable area, represented by the triangular region OTB, is bounded by the line OB of which the slope equals the phase velocity of the shortest wave components. At any location above OB, the shortest wave components contained by irregular input trace OT have not arrived yet. The other boundary of the predictable area is formed by line TB whose slope equals the phase velocity of the longest wave component. At any location below TB, the longest wave components have passed already. Of the forecasted trace CE only the part JG is useful: The leftmost part (CH) is outside the predictable area so the shorter waves from OT haven t arrived at location x B yet, while the part HI, though being within the predictable area, is useless as it represents no forecast but hindcast: obviously, the analysis of input trace OT can be started only if it has been completely acquired, which is at time T (= time I). The time required to do the analysis is represented by IJ which means that the analysed result is no sooner avaliable than at time J. The rightmost part, GH, is outside the predictable area as the longer wave components from OT have passed already at location x B. Copyright xx by ASE
3 4 5 6 7 75.6 X B 6 H Error [-] C I G E Envelope [m] X-distance [m] 45 3 5 -.6 -. -.8 envelope [m] Error [-] X A -.4 3 4 5 6 7 O T out T.6.4 prediction measurement Figure, Wave propagation time-distance diagram The analysis of the measurement is done on a moving time window of length D. The predictable area for the next time step is represented by the right triangular region in Figure. When the analysis of the first time step has been completed, which is at time J (= time F), a usefull forecast is available having a duration that equals JG. (As the analysis of the next input time trace (represented by the base of the rightmost triangle) can start at F, and as it will again take computational time to obtain its result, the minimal time window into the future during the proces of predicting will be of length JG - IJ. Duration D and distance x B x A can be optimized in order to minimize the prediction error for a certain required forecast time and wave condition. This has been examined using the model test data and will be presented in the next paragraphs. Spectral truncation A validation of the linear propagation model was carried out by comparing predicted and measured wave traces. The wave measurement at the first wave probe (represented by A in equation ()) was used as the input of the model and predictions up to 4 s ahead were made at the locations of the other probes. As in real life the wave measurement would be done by means of a radar having a significantly lower sample rate than 8.3 Hz, the input wave data was down-sampled to Hz (prototype scale). In general a long duration D appeared to be favourable in terms of prediction accuracy. 5 s was considered as an optimum as for longer durations the computation time increased significantly only resulting in marginal increase of accuracy. [m]. I -. -.4 -.6 -.8 inside predictable area G 5 54 56 58 6 6 64 Figure 3, example of realization, condition Jonswap 3, probe 3 Figure 3 shows a visualization of one simulation step for wave condition Jonswap3 at wave probe #3. In the upper most figure a time distance diagram similar to Figure is presented with the same capitals indicating beginning of forecast, end of predictable part etc. At OT the envelope of the input trace is plotted. Underneath CE, both predicted and measured envelopes are plotted and above CE the absolute difference between them is shown. The vertical axis for these plots can be found at the right hand side of the diagram. The lower figure zooms in on the most relevant part of the predicted trace representing the forecast, part IE. Both predicted and measured wave elevation and envelope are shown. The average results over 4 realizations, each of whose input and predicted trace was shifted s. in time, is shown in figure 3. E 3 Copyright xx by ASE
Figure 4, Average error deterministic wave and envelope, Jonswap 3, probe 3 The dotted and solid lines represent the normalized standard deviation of the error of the predicted envelope and deterministic wave respectively: σ E Prediction Error [-].5..5..5 () t = Average error predicted envelope and deterministic wave Envelope Distance: 6 m wave Deterministic Input Duration: 5 s Prediction:4 s m= σ E ( * Bm () t Bm () t ) H sig hindcast percentage 7 of spectral area 8 included in 9 prediction 3 4 5 6 time [s] (3) 3 spectral area [%] 4 5 6 the full predicted trace doesn t fit within the predictable area: for part GE the longest wave components have passed the prediction site already. For this wave condition the distance needs to be increased in order to make sure the whole prediction fits within the predictable area. The dashed-dotted line in Figure 4 indicates how much of the wave content in the measured input trace is present at the time and location of prediction. The phase velocity of the longest wave component that just has passed at the prediction location at time T out in Figure 3 is: ( )/( ) c = X X T T (5) out B A out Where: c out is the phase velocity, X A and X B are the input measurement and prediction location respectively. T and T out are start time of forecast part of predicted trace and considered moment in time outside predictable trace respectively. The corresponding wave frequency,, follows from the linear dispersion relation. All wave components in the measured input trace OT having frequencies lower than ω H have passed the considered prediction location already at time T out. The relative amount of wave energy that these components represent, the spectral truncation, is the spectral area below ω H divided by the total spectral area. ω out. σ E () t = m= * ( Bm () t Bm () t ) H sig (4) S [m s].5. Where: σ = the standard deviation of the error of the deterministic E wave prediction. is the number of realizations (4). Bm, and * m are the mth realization of predicted and measured wave elevation at prediction location X B respectively. H is the significant wave height. Tildes indicate the sig envelope. The dashed dotted line in Figure 4 indicates the relative amount of wave energy in the predicted trace that can originate from the input trace: In the upper graph of Figure 3, the left and right hand side sloped lines OH and TG indicate the propagation of the shortest and longest wave components found in input trace OT respectively. As explained they bound the predictable area. Apparently for the shown example, the distance between reference and prediction location is such that.5.5 ω H.5.5 ωout ω [rad/s] Figure 5, Spectral truncation of wave spectrum The relative amount of wave energy of the remaining wave components (that don t have passed yet) is plotted in Figure 4 by the dashed-dotted line (vertical axis on right hand side of Figure 4). The theoretical relation between the mentioned relative amount of wave energy and the accuracy was strongly confirmed by the experimental results for small fetch (distance between input measurement and prediction location) over wave length ratios. However, it was observed that for higher ratios spectral truncations up to 9% were allowed. Depending on the fetch, 4 Copyright xx by ASE
this can result in a significant increase of the maximum forecast time. fetch for the long wave components is smaller resulting in a smaller error for those components. SHIP OTIONS As mentioned, the final purpose of the project in which this study was carried out is to predict motion behavior of floating structures. Applying linear wave theory and linear motion wave transfer functions, the step from wave prediction to motion prediction is a simple and straight forward one. For any motion ( j ) the prediction can be written as: wave spectrum (S ) heave ( 3 ) pitch ( 5 ) wave spectrum (S ) RAO heave ( H 3 ) RAO pitch ( H 5 ) ( ) N / i( ε n kn( xb xa) ) iωn ( t+ t), H, (6) t+ t = e e jb a n j n n= where: H j is the complex transfer function of the motion for mode j. As no measurements of motions were carried out during the measurement campaign, the ideal situation is assumed now that the motion transfer is perfect. This means a motion measurement can be obtained by applying the transfer funtion to the measured wave at the prediction location. Then, analogue to equations (3) and (4), the standard deviation of the error of the motion can be written as: σ E j () t = m= * ( () ()) jbm t jbm t SDA j (7) Where: SDA j is the significant double amplitude of the motion for mode j. Depending on it s shape related to the wave spectrum, the transfer function, acting as a filter, can result in an increase or a decrease of the motion prediction error compared to the wave prediction error. Prediction simulations have been carried out for the heave and pitch motion in head waves of an offshore support vessel having the following particulars: Length: 6 m, Beam: m, Draft: 6. m For both the heave and pitch transfer functions of the vessel only the tail coincides with the frequency range where the wave energy is located. Figure 6 shows the absolute values of these transfer functions (RAO s) together with one of the measured wave spectra. The fact that the joint frequencies are located at the low frequency range of the wave spectrum results in motion prediction errors that are smaller than the errors of the wave prediction. This might be explained by the fact that the relative ω Figure 6, Wave spectrum and transfer functions of heave and pitch (axes labels omitted for confidentialily reasons) - Jonswap 4, probe 4, [m] predicted measured.77 58 6 6 64 66 68 Figure 7, typical sample of time trace of predicted and measured wave elevation Jonswap 4, probe 4...448 58 6 6 64 66 68 Figure 8, sample of time trace of predicted and measured corresponding heave motion.5 -.5.793 58 6 6 64 66 68 Figure 9, sample of time trace of predicted and measured corresponding pitch motion 5 Copyright xx by ASE
For that reason a poor wave prediction can still result in a fairly good motion prediction as is shown by the example realizations of wave, heave and pitch prediction in Figure 7, Figure 8 and Figure 9 respectively. The number in the top right corner of each plot indicates the standard deviation of the prediction error. (The thick vertical line indicates the theoretical end of the predictable part as explained in the paragrapgh predictable area. The end of the shown trace is the end of the predictable part as it was found from the average result of 4 realizations.) where k p = peak wave number of wave spectrum [rad/m] X = fetch [m] As can be seen a fairly linear relation between error (on the vertical axis) and the mentioned estimator (on the horizontal axis) is found. Of all points in Figure that are marked with an arrow,indicating the corresponding wave condition, sample time traces of prediction and measurement can be found in Annex B, Figure. It can be concluded that a 6 s accurate forecast of wave elevation is very well feasable for all considered wave conditions. otion predictions are even more accurate. RESULTS AND DISCUSSION Predictable area As mentioned already, the predictable area (as it can be expected from the explained theory) gives an indication for the maximum prediction time that can be achieved for a certain fetch. The experimental results however showed that predictions can be extended significantly long beyond this theoretical predictable area. Qualitatively it can be said that the the higher the higher the fetch over wave length ratio, the further the prediction can be extended beyond the predictable area. This is why the probe distances in the experimental set-up were actually larger than necessary for the aimed forecast times: Except for the Jonswap 7 and Swell condition a forecast time of 6 s. appeared to be feasable at probe 3. (A mismatch between the theoretical Jonswap spectra and the observed spectra during the experiments also partly explains the large difference between expected required probe distances on which the set-up was based and the found required distances from the experiments.) Annex B, Figure shows sample time traces of predicted and measured wave elevation, heave motion and pitch motion at probe 3 for all Jonswap conditions and at probe 9 for the swell condition. Again the thick vertical lines indicate the theoretical end of the predictable part as explained in the paragrapgh predictable area. The end of the shown trace is the end of the predictable part as it was found from the average result of 4 realizations. ACKNOWLEDGENTS This paper is published by courtesy of all participants and partners of the OWE-JIP for which they are greatfully acknoledged. NOENCLATURE REFERENCES [] orris E.L., Zienkewicz H.K., Belmont.R. Short term forecasting of the sea surface shape, International Shipbuilding Progress Vol. 45, No. 444, 998 [] Trulsen K., Stansberg C.T. Spatial evolution of water surface waves, Proc. Eleventh Intl Offshore & Polar Engng Conf. pp. 7 77, [3] Edgar D.R.,Horwood J..K., Thurley R., Belmont.R. The effects of parameters on the maximum prediction time possible in short term forecasting of the sea surface shape, International Shipbuilding Progress Vol. 47, No.45, Accuracy See Annex A, Figure. Following Trulsen [] the standard deviation of the wave prediction error (as defined in equation ε (4)) is plotted against being the non-dimensional steepness squared times non-dimensional fetch where: π H sig non-dimensional steepness, gt where T p k p X ε = (8) = peak period of wave spectrum [s] p 6 Copyright xx by ASE
ANNEX A ERROR OF PREDICTION AGAINST FETCH-STEEPNESS PARAETER.3 Error W ave Elevation 5.5..5 7 8 9 79. 9 34 Swell.5 4 Jonswap 7 9 6 36 marker color indicates wave condition 9 49 7 Jonswap 4 89 Jonswap 5 Jonswap 3 66 88 Jonswap 6 numbers indicate maximum forecast time 4 marker shape indicates probe: probe probe 3 probe 4 probe 5 probe 6 probe 7 probe 8 probe 9...3.4.5.6.7 ε k p X [-] Figure, standard deviation of prediction error against non-dimensional steepness squared times non-dimensional fetch 7 Copyright xx by ASE
ANNEX B SAPLE S OF PREDICTED AND EASURED TIE TRACES.5 Jonswap 3, probe 3, [m].57..7595..847 -.5 -.5 54 56 58 6 Jonswap 4, probe 3, [m].5.4 56 58 6 6 Jonswap 5, probe 3, [m] - 4 6 -.6989 Jonswap 6, probe 3, [m].57 56 58 6 Jonswap 7, probe 3, [m] - 5 5 53 54 55 Swell, probe 9, [m].7667.734 -.. -.. 54 56 58 6.366 56 58 6 6.5976 -. 4 6.5 -.5.5.75994 56 58 6 -.5 5 5 53 54 55.55.58 -..5 -.5 54 56 58 6.385 56 58 6 6.736-4 6 -.59763 56 58 6.6388-5 5 53 54 55.83466-6 6 64 66-6 6 64 66-6 6 64 66 Figure, typical samples of time traces of predicted and measured wave elevation, heave motion and pitch motion for all wave conditions 8 Copyright xx by ASE