A GENERIC METHOD TO MODEL FREQUENCY-DIRECTION WAVE SPECTRA FOR FPSO MOTIONS

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1 Proceedgs o the ASME 7th Iteratoal Coerece o Oshore Mechacs ad Arctc Egeerg OMAE8 Jue 5-, 8, Estorl, Portugal OMAE A GENERIC METHOD TO MODEL FREQUENCY-DIRECTION WAVE SPECTRA FOR FPSO MOTIONS Hermoe J. va Zutphe Shell Iteratoal Explorato ad Producto Rjswjk ZH, The Netherlads Hermoe.vaZutphe@SHELL.com Phlp Joatha Shell Global Solutos Shell Techology Cetre Thorto, Chester UK Kev C. Ewas Shell Iteratoal Explorato ad Producto Rjswjk ZH, The Netherlads ABSTRACT We report a ew approach to model the requecy-drecto spectrum, whch the requecy-drecto spectra rom measuremets or hdcast studes are tted smultaeously two dmesos, requecy ad drecto. Depedg o the amout o wd orcg o the partto, ether a umodal (swell) or bmodal (wd-sea) wave spreadg ucto s adopted together wth the spectral orm whch best ts the requecy spectrum. Ths paper descrbes the ew method ad presets the results o a measured dataset. Key words: requecy-drecto spectra, waves, wave spreadg, vessel motos INTRODUCTION Waves are a mportat drver the desg o oshore structures ad loatg systems. A better uderstadg o the wave clmate reduces the ucertates the desg o oshore systems. A recet developmet drectoal wave spectral aalyss s the sotware package XWaves or spectral parttog. Based o a approach, orgally developed by the Appled Physcs Lab o Joh Hopks Uversty (Haso & Phllps, ), the sotware extracts the wd-sea ad swell compoets the wave spectrum ad the amout o wd orcg o each compoet, or large datasets. Addtoally, t tracks the developmet o the swell systems, whch ca be geerated by storms thousads o klometres away. Presetly, the resultg D requecy-drecto compoets o the spectrum (parttos) are reduced to requecy spectra ad tted to a partcular spectral orm, such as a JONSWAP spectrum. The resultg parameters are the used log term respose aalyses o vessel motos. Whle the mea drecto o each compoet s retaed, geeral orms or the drectoal spreadg uctos are used; the specc drectoal spreadg detals are thereore lost. Prevous studes (or example, HSE ) have show that (turret) moored vessels respod deretly to short crested seas tha to log crested seas, ad thereore, addg drectoal ormato drectly rom the wave spectrum to the parametrzed spectrum should mprove the accuracy o the calculated resposes. Studes rom or example Ewas, 998 ad Hwag et al,, have show that wd seas are bmodal at requeces greater tha the peak requecy, whle swell compoets have bee show to be umodal (Ewas, ). I ttg the spectra to drectoal dstrbuto models or both wd-sea ad swell, the b-modalty or wd-seas s retaed. NOMENCLATURE (, ) (, ) S requecy-drecto spectrum A spreadg ucto p requecy (rage) peak requecy drecto (rage) Copyrght xx by ASME

2 agular separato H L o the drecto peaks, β, ttg parameters or σ ad error term ε LOCATION OF FREQUENCY-DIRECTION SPECTRA For the valdato o the model, observatos o requecydrecto spectra made ear the ste o the Mau-A platorm o the West Coast o New Zealad are used. Parameter values or the ts obtaed durg studes to derve drectoal dstrbutos or wd-seas (Ewas, 998) ad swell (Ewas, ) are used ths study as startg guess or the ttg procedure. The Mau locato has bee prove to produce well-deed etch-lmted seas due to south-east wds (Ewas ad Kbblewhte, 99) whch coorm closely to the JONSWAP spectrum. Also, at ths locato, a more or less costat swell compoet orgatg rom the Souther Ocea s observed. THEORETICAL BACKGROUND The -dmesoal requecy-drecto spectrum S (, ) o a sea state ca be descrbed as the sum o ts wd sea (dex ) ad to swell parttos: (, ) ( ) (, ) + ( ) (, ) S S A S A where ( ) ad A (, ) π (.) S deotes the requecy spectrum or partto the spreadg ucto whch yelds: ( ) A, d (.) π Whle case o buoy measuremets, the drectoal spreadg ucto ca be estmated usg ether a model-depedet estmate (lke the maxmum etropy method (MEM) or the maxmum lkelhood method (MLM)) rom the Fourer coecets, ths case, we eed a parametrc descrpto o the drectoal spreadg ucto. Wd seas have bee show by or example Ewas, 998 ad Hwag et al., to be bmodal at requeces larger tha the peak requecy. Ewas (998) proposed a double-peaked orm or the drectoal dstrbuto, based o a double Gaussa uctos deed as: A, exp ( ) ( ) ( H ( ) π k ) σ ( ) ( L ( ) π k ) σ ( ) 8πσ k + exp (.3) where σ s the agular wdth ad a measure or the spreadg o each compoet (crcular rms spreadg) ad ad L H are the locatos o the peaks cetered at equal agles o each sde o the mea wave drecto. The crcular rms spreadg σ ca be parametrzed usg the ollowg orm: 4 + or < σ 5 temp + or p p 3 p p (.4) Ewas (998) proposed the ollowg values or the rms spreadg or wd-seas: temp The locato o the peaks o the spectrum s gve by: H L + ( ) ( ) ( ) ( ) ( ) ( ) (.5) (.6) wth the agular separato, H L,o the peaks. The agular separato ca be wrtte the orm (Ewas, 998): β or < exp βtemp β or > wth p p p (.7) Copyrght xx by ASME

3 β β β temp (.8) Ocea swell s less spread ad less lkely to have a bmodal drectoal dstrbuto (Ewas, ), ad thereore, the drectoal dstrbuto ucto reduces to: ( ( ) π k ) σ ( ) κ A (, ) exp πσ ( ) k κ,,..., (.9) where the crcular rms spreadg s parametrzed usg a smlar orm as (.4): 4 + or < σ 5 + or p p temp 3 p p wth the ollowg values obtaed rom Ewas t: temp or each compoet. (.) (.) GENERALIZED PARAMETRIC DESCRIPTION OF FREQUENCY-DIRECTION SPECTRUM The model ca be geeralzed by assumg a commo orm or both wd-sea ad swell compoets, by otg the act that the drectoal dstrbuto or swell compoets has the same orm as a wd-sea compoet wth a agular separato (umodal). Ths geeralzato wll oer the advatage that the parameters ca be modeled smoothly tme. The parametrc descrpto o the spreadg dstrbuto uctos ca the be geeralzed as ollows: S (, ) S ( ) N (, ),,... (.) wth dcatg the wd-sea compoet ad > or the swell compoets. N (, ) whch ow take the commo orm: are the ormalzed D spectra N κ, exp 8πσ ( ) k κ ( ) ( H ( ) π k ) σ ( ) ( L ( ) π k ) σ ( ) + exp (.) For swell compoets, L H ad (.) wll reduce to (.9). Usg (.4) ad (.), the commo orm or the agular rms spreadg ca ow be wrtte as: 4 + or < p p σ ( ) (.3) 5 temp + 3 or p p At p we have a equalty costrat: + + whch ca be used to elmate temp 3 temp to yeld a sgle expresso: 4 σ ( ) + ( I ) + I... p + p p where I x 3 p p 5 I s a dcator ucto equal to or (.4) x ad otherwse. Further, or the agular separato, we ca ow wrte: ( ) H ( ) ( ) + (.5) ( ) ( ) ( ) L ad takes the orm: β or < ( ) exp β β or > temp p p At p we have a equalty costrat: p (.6) 3 Copyrght xx by ASME

4 ( ) β exp β β whch ca be used to elmate temp β to yeld a sgle expresso: ( ) β exp β p + where ( x ) + x ad otherwse. (.7) The models ca ow be solved usg a parameter ttg procedure, ths case a Maxmum Lkelhood Method. The requecy spectra are assumed to be kow, wth or example a JONSWAP or Gaussa spectrum. For each wave compoet,,...,, we eed to t the parameters o the geeralzed model or the requecy-drecto spectra: {,, 3, 4, 5,, } amout o work or the solver, the expoets {, } β β. However, the decrease the ca be 4 5 set to xed values lke or example to values speced the Mau drectoal dstrbutos. MAXIMUM LIKELIHOOD ESTIMATION We assume that the measured spectra are take rom the model: yk Sk + εk εk ~ N (, ν ) (3.) where the dex k reers to the grd o values x (, ) k j at whch wth (,,..., m, j,,..., m, m m m ) the spectral data s avalable. The error terms ε are detcally ad depedetly ormally-dstrbuted wth costat varace ν. The lkelhood o the observed spectral data { k} by: L ( y ( ; )) k S xk λ y k m exp (3.) k πν ν ad the egatve log-lkelhood by: l * log ( π ) + m log ( ν ) + k ( y ( ; )) k S xk λ m m ν Sce m ad ν are costats, ths s eectvely l : m k ( ( ; )) k k l y S x λ s gve (3.3) (3.4) whch s smply least squares. We ote however that maxmum lkelhood provdes a atural ramework or corporato o (e.g.) measuremet error ν whch vares wth requecy ad drecto. Maxmum lkelhood estmates are obtaed by settg the partal dervatves o l wth respect to each compoet o ad β to zero ad solvg: l η,,, β, β { } 3 η Sce S S ( x ; λ ) (3.5), we ca solve usg the cha rule: k l l λ j η λ η where λ j { σ, } j j I terms o S, we have: (3.6) ( ) m l S x λ k j ( S ( xk ) yk ) η j k λ j η S x (3.7) To solve, we eed to evaluate the dervatves λ j η ( ) λ or all possble {,,,, } {, } λ σ j. η β β 3 j k ad ad IMPLEMENTATION The maxmum lkelhood equatos are solved usg MATLAB (wth Optmzato Toolbox). The ttg algorthm s tally ed wth a startg guess, or whch the spectrum s calculated. By perturbg the startg guess, the algorthm tres to d the mmum o the egatve log-lkelhood. The best guess o ths rst set o perturbatos s the used to start a secod set o perturbatos. The umber o perturbatos s mportat to make sure that the algorthm s dg the soluto at the lowest (global) mmum o the egatve log-lkelhood uctos, stead o a local mmum, whch wll retur a less optmal soluto.. Establsh qualty o least squares t at startg guess {,,,, } g β β o 3. Uorm radom perturbato go to d ew startg guesses or the least squares problem. Optmzato wth smaller tolerace. 3. Iterato o step 4. Optmal soluto s retured. The algorthm was tested wth model spectra obtaed or the theoretcal dstrbutos rom the Mau data or 3 cases (see Table ): bmodal sea state cosstg o proouced wd sea ad swell compoet. umodal wd-sea wth a very small swell compoet umodal swell wth a very small wd-sea compoet 4 Copyrght xx by ASME

5 Table Sgcat wave heght Hs ad peak perod Tp or the 3 theoretcal model spectra. Spectrum Wd-sea Swell Hs (m) Tp (s) Hs (m) Tp (s) The model data s geerated to coorm to the parameter values gve (.5) ad (.8) or wd-seas ad (.) or swells. To test the covergece o the algorthm, both a good guess (the orgal parameters) ad a bad guess are used to t the data: Table Good ad bad startg guesses used to test the method. Wd-sea ( ) 3 β β Good Bad Swell ( > ) Good Bad 55 The value or 4 s xed 7.9 or wd sea ad 5 or swell compoets, ad s xed to respectvely ad.3. 5 Fgures, ad 3 gve the resultg log-spectra o the model spectra. I Table 3 the mea, stadard devato ad the RMS error o the startg guess ad the optmum t are gve relatve to the true spectrum. The ttg algorthm s well able to t the eergy peaks ad the rght varato the spectrum. The goodess o t results show that the ttg procedure has sgcatly mproved the startg guess o the spectrum ad that both the mea value ad the stadard devato are retaed the resultg optmum. However, the tal o speccally the wd-sea compoet s much wder spread tha the true spectrum ad whe the eergy cotet o the wd-sea s low, the bmodalty seems to dsappear. A possble mprovemet to the method would be to preeretally pealse poor ttg o spectral tals by troducg o-costat ν, or alteratvely to reormulate the maxmum lkelhood orm to t to the log-spectrum stead o the ormal spectrum. Table 3 Goodess o t results or the optmum ts or the 3 theoretcal model spectra wth a bad startg guess. Spectrum Mea Std RMS error True spectrum Startg guess e-6 Optmum Ft e- Spectrum Mea Std RMS error True spectrum Startg guess e-4 Optmum Ft e- Spectrum 3 Mea Std RMS error True spectrum Startg guess e-5 Optmum Ft e- APPLICATION TO MAUI DATA Addtoally, the ttg method has bee appled to observed wave spectra a Mau usg the good startg guess as descrbed Table. The results or oe selected case are show Fgure 6, Fgure 7 ad Fgure 8. Table 4 Goodess o t results or the optmum ts to a observed wave spectrum at Mau-A. Observed Spectrum Mea Std RMS error True spectrum Optmum Ft e-6 For a gve requecy, the drecto dstrbuto shows a much wder spectrum tha the orgal data. Ths s also the reaso why the eergy at the peak s lower the optmum t tha the orgal. The eergy cotet o the spectrum s coserved, ad thereore whe the t retur a wder spectrum at a certa requecy, the peak eergy level decreases at ths requecy. The log spectra Fgure 8 show that the trasto area betwee the wd-sea ad swell compoet s ot well captured by the ttg process. However, ths mght mprove whe ttg the log-spectrum stead o the ormal spectrum, or puttg a weghtg o the low eergy parts o the spectrum. DISCUSSION & CONCLUSION The test cases have show that the peaks o the deret compoets are well captured. However, the hgh requecy tals o the partto ad the bmodalty o the wd-sea spectrum s sometmes ot well captured, especally whe the 5 Copyrght xx by ASME

6 eergy cotet s low. Ths s also observed or the measured spectra rom Mau-A. A possble mprovemet would be to gve a weghtg to the low eergy cotet o the spectrum or to t to the log-spectrum stead o the ormal spectrum. I addto here, we have assumed xed values or ad or all compoets. By 3 also ttg these terms, we would make our model orm cosderably more lexble. Coversely, oe should be aware o the lmtatos o the data. The measured wave spectra are usually obtaed rom 3- compoet drectoal wave buoys returg 3 orthogoal traslatos, whch are the trasormed to the rst 4 compoets o a Fourer expaso. Thereore, the resoluto o the drectoal dstrbuto s lmted. Smlarly, hdcast models are lkely to be lmted to the accuracy o the physcs the model; e.g. dscrete teracto approxmato or the 3 rd order resoace wave-wave teractos ad umercal accuraces the propagato o swell rom the org to the cosdered locato. Ultmately, the D ttg method ca be used to eed back to the parttog process, assstg the detcato o compoets ad also ther cotuty through tme. 3 ACKNOWLEDGMENTS The authors would lke to thak Shell, Todd Ol Servces Ltd or the use o Mau wave data. REFERENCES [] Ewas K.C., 998, Observatos o the Drectoal Spectrum o Fetch Lmted Waves, Joural o Physcal Oceaography, Vol. 8, No 3., March 998 [] Ewas K.C.,, Drectoal Spreadg Ocea Swell, Proceedgs o Ocea Wave Measuremet ad Aalyss (), [3] Forrstall G.Z ad Ewas K.C., 998, Worldwde Measuremets o Drectoal Wave Spreadg, Joural o Atmospherc ad Oceac Techology, Vol. 5, Aprl 998 [4] Haso, J. L ad Phllps, O. M,. Automated Aalyss o Ocea Surace Drectoal Wave Spectra. Joural o Atmospherc ad Oceac Techology, 8, [5] Hwag P.A., Wag D.W., Walsh E.J., Krabll W.B. ad Swt R.N.,, Arbore Measuremets o the Waveumber Spectra o Ocea Surace Waves. Part II: Drectoal Dstrbuto, Joural o Physcal Oceaography, Vol.3. 6 Copyrght xx by ASME

7 Fgure Log-spectra o case, proouced wd-sea. Although the eergy peaks s well tted, the tal o the wd-sea partto s much wder r the optmal soluto tha the true spectrum. Fgure Log-spectra o case, proouced swell partto. The wdth ad eergy peak o the swell s well tted. The small wd- sea compoet s much more spread the t tha the true spectrum ad the bmodalty the tal has ot bee properly captured. 7 Copyrght xx by ASME

8 Fgure 3 Log-spectra o case 3: proouced sea ad swell partto. For both spectra, the peak s well tted, but aga, the tal o the wd d-sea s clearly much more spread tha the true spectrum. 8 Copyrght xx by ASME

9 Fgure 4 Spectrum or case 3: proouced wd-sea ad swell spectrum. Fgure 5 Frequecy dstrbuto ad drecto dstrbuto or case 3. The lower plot gves the dereces or each grd pot (, j ) o the put spectrum wth the startg guess (black) ad the optmum soluto (red). 9 Copyrght xx by ASME

10 Fgure 6 Optmum soluto compared to a measured spectrum at Mau-A platorm. The t s much wder tha the measuremet at a gve requecy, ad thereore, the peak eergy level at that requecy s lower. Fgure 7 Optmum soluto compared to the observed spectrum at Mau-A platorm. The drecto dstrbuto captures the two peaks, but s less peaked tha the measuremet. The lower plot shows the spectral dereces at each combato o requecy ( ) ad drecto, j Copyrght xx by ASME

11 Fgure 8 Log-spectra o a measured spectrum at Mau-A. Copyrght xx by ASME