SENSOR MODEL FOR AIRBORNE CCD LINEAR SCANNERS

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1 Pol 37 ENOR MOEL FOR AIRBORNE LINEAR ANNER anla Pol ETH Zurch, wtzrland KEY WOR: Pushbroom, Lnar cannrs, Gorfrncng, Orntaton modlng, lfcalbraton, Intgraton, Bundl. ABTRAT Ths papr dscrbs a snsor modl for th gorfrncng of magry acqurd by lnar array snsors wth along-track stro vwng carrd on arplan. Th modl conssts of two man algorthms for drct and ndrct gorfrncng. If postonng nstrumnts (GP,IN) provd th snsor trnal orntaton (poston, atttud) durng th mag acquston, th ground coordnats of ponts masurd n or mor stro mags can b stmatd through a forward ntrscton, takng nto account th dsplacmnts btwn th rfrnc systms nvolvd (drct gorfrncng). On th othr hand th ndrct gorfrncng algorthm s usd whn th snsor trnal orntaton s not accurat nough. In ths cas th snsor trnal orntaton s modld wth nd or st ordr polynomals dpndng on tm, wth constrants on th contnuty of th functons and thr frst and scond drvatvs btwn adjacnt sgmnts. Morovr th ntrnal orntaton and systmatc rrors du to lns dstortons and lns rotatons ar stmatd wth slf-calbraton. Usng wll dstrbutd Ground ontrol Ponts (GPs) and, addtonally, T Ponts (s), th trnal orntaton and slf-calbraton paramtrs, togthr wth th s ground coordnats, ar stmatd n a last-squar adjustmnt. Th drct and ndrct gorfrncng algorthms ar dsgnd for sngl- and mult-lns optcal systms. In cas of mult-lns snsors, addtonal paramtrs dscrbng th rlatv orntaton of ach lns wth rspct to th cntral on ar ntroducd. INTROUTION Arborn lnar scannrs ar nowadays wdly usd for hgh rsoluton mappng as altrnatv to dgtal fram camras and flm camras. Thy hav appard n th photogrammtrc communty n th lat 8's n concdnc wth th dvlopmnt and tstng of lnar scannrs for spac applcatons. In 986 th frst flght wth a lnar scannr was carrd out: th Monocular Elctro-Optcal tro cannr (M) was tstd on arborn bfor th (unsuccssful) launch n 993. A group of lnar snsors for arborn photogrammtry has bn frst usd as trrstral smulators of th corrspondng spac vrsons and thn adaptd for mor spcfc arborn mappng applcatons. Ths s th cas of a) HR-A (Hgh Rsoluton tro amra Arborn) and b) WAA (Wd Angl Arborn amra), whch wr th arborn vrson of HR and WAO (Wd Angl Optcal atllt nsor) usd for MAR96 spac msson and c) Ar-MIR, that was th smulaton of MIR (Mult Imagng pctro Radomtr), dvlopd by NAA and carrd on board of -AM platform. lnar snsors consst of a sngl-lns or mult-lns optcal systm n combnaton to a varabl numbr of arrays (lns) of lmnts placd n th focal plan. Imags ar acqurd n form of strps n a pushbroom mod: whl th arcraft movs, ach array scans a ln on th ground n a vry short tm (ntgraton tm). Thn th nstantanous scannd ln s stord closd to th othr, rsultng n a contnuos strp for ach array of. If mor arrays ar usd paralll to ach othr, prpndcular to th flght drcton and nclnd wth dffrnt vwng angls along th trajctory, th rsultng strps hav stro covrag along th flght drcton. A vry common gomtry usd both on arborn (A4, TL, PA, WAA) and satllt (MOM-, WAO) s basd on thr lns lookng forward, nadr and backward th flght drcton; dgtal mag trplt

2 38 Postr sson strps ar capturd smultanously (Fgur ). Th numbr of pls n ach array s varabl (from 5 to ). Th most rcnt snsors us a larg numbr of pls n ordr to covr a wdr swath. As th spctral charactrstcs concrns, th channls' rang usually nclud th vsbl and nar nfrard, provdng panchromatc and multspctral mags. In th most cass, th panchromatc lns ar usd for stro vwng and th mult-spctral channls ar placd n nadr drcton. In ordr to mantan hgh mag qualty n bad flght condtons, ths magrs scan th ground wth vry short ntgraton tm (- msc for scan ln) and ar usually mountd on gyro-stablzd platforms that absorb mchancal vbraton and nsur nar-nadr vwng. GP and IN nstrumnts (for ampl Applan PO for A4 and HR-A, Trmbl M75 for TL) ar carrd on board wth th magrs and gv Fgur. Gomtry of lnar array snsors wth thr vwng drctons. th snsor poston and atttud at rgular tms. Th drct masurmnt of th snsor poston and atttud durng th flght s ndspnsabl, bcaus ach mag ln s acqurd at a dffrnt tm wth a dffrnt trnal orntaton, thrfor durng th scns trangulaton th numbr of unknown paramtrs dscrbng th snsor trnal orntaton (s for ach mag ln) would b too larg and a bundl adjustmnt would not b ralstc. In cas of lnar scannrs carrd on satllt ths problms can b solvd by dscrbng th satllt movmnt wth wll-known orbtal functons; n cas of pushbroom snsors carrd on arplan th trajctory s not prdctabl and th ffcts of th flght turbulnc ar not nglgbl. For ths rason GP and IN nstrumnts masur th snsor poston and atttud at rgular nstants of tm and post-flght procssng tchnqus gv th valus at ach snsor posur. A lst of arborn lnar snsors for photogrammtrc applcatons s rportd n Tabl. HR-A was dvlopd by th Insttut of Plantary Eplotaton of LR and frst tstd n 997. nc thn, ovrag of mor than cts and larg aras has bn acqurd and procssd n Europ and UA, provdng dgtal ortho-mags and dgtal surfac modls wth an accuracy of to cm (Lhmann, ). Basd on HR-A, th narrow-angl HR-AX (Fgur c) and wdangl HR-AXW vrsons of th magr hav bn dvlopd, wth -bt radomtrc rsoluton, nstad of 8 bt, and pls pr ln nstad of 5 of HR-A. WAA and PA ar two arborn snsors dvlopd by LR and ARA rspctvly. Both snsors provd stro capablty n 3 vwng drctons along th arcraft trajctors. Thy carry GP and IN nstrumnts on board for poston and atttud masurmnts. Th pushbroom gomtry dmonstratd to b a good altrnatv to dgtal fram camras and was adoptd for th most rsnt arborn lnar snsors too: A4 and TL. A4 (Arborn gtal nsor, Fgur a) s a commrcal pushbroom snsor of LH ystms, bult by LR. Th snsor, that was frst prsntd n, conssts of on optcal systm (6.5mm focal lngth) wth 3 groups of 4 lns (rd, blu, grn, nar-nfrard) scannng backward, nadr and forward th arcraft trajctory n panchromatc and multspctral mod. Th vwng angls ar not always symmtrc wth rspct to th vrtcal and can vary n ordr to fulfll any customrs' spcal rqurmnts for optmal stro gomtr. Th snsor s carrd on arborn wth Applan GP/IMU systms to rcord th moton of th snsor durng th flght. Among th mntond arborn pushbroom snsors, TL (Thr-Ln cannr, Fgur b), or TARIMAGER, dvlopd n by tarlabo orporaton (Japan), s th only on that s not carrd on arcraft, but on hlcoptr. u to th hgh trajctory vbratons, a vry good stablzr s mountd wth th camra systm, togthr wth a Trmbl M75.

Pol 39 In th nt paragraph th snsor modl dvlopd at our nsttut for th gorfrncng of a wd class of arborn lnar array scannrs wll b prsntd.

65mm varabl 8 n vwng 3 3 5 3 3 drctons n lns 7 9 9 3 36 7 #pls/ln 584 54 6 Mod Pan (stro) M (nadr) Pan Pan(stro) M (nadr) Pan M Pan(stro) M(nadr) #pls/ln 584 54 6 Focal lngth 6.5mm 65mm 75mm.

Arborn lnar snsors: (a) A4 [ourc Lh-ystms wb], (b) TL (TARIMAGER) [ourc: tarlabo wb], (c) HR-AX [ourc:lr wb] GREFERENING OF AIRBORNE LINEAR ENOR Th am of gorfrncng s to stablsh a rlatonshp btwn mag

3 Pol 39 In th nt paragraph th snsor modl dvlopd at our nsttut for th gorfrncng of a wd class of arborn lnar array scannrs wll b prsntd. nsor A4 TL HR-AX WAA ArMIR PA ompany LH-ystms and LR tarlabo LR LR NAA AA n lnss 9 doubl Focal lngth 6.5mm 65mm 75mm.65mm varabl 8 n vwng drctons n lns #pls/ln Mod Pan (stro) M (nadr) Pan Pan(stro) M (nadr) Pan M Pan(stro) M(nadr) #pls/ln Focal lngth 6.5mm 65mm 75mm.65mm varabl 8 Tabl. Ovrvw of arborn lnar snsors. For ach snsor, th man charactrstcs ar gvn (LH-ystms wb, tarlabo wb, LR wb, andau t al., 996, NAA wb, Hanh t al., 996). (a) (b) (c) Fgur. Arborn lnar snsors: (a) A4 [ourc Lh-ystms wb], (b) TL (TARIMAGER) [ourc: tarlabo wb], (c) HR-AX [ourc:lr wb] GREFERENING OF AIRBORNE LINEAR ENOR Th am of gorfrncng s to stablsh a rlatonshp btwn mag and ground rfrnc systms, accordng to th snsor gomtry and th avalabl data. Th mag systm s cntrd n th lns prspctv cntr (P), wth -as tangnt to th flght trajctory, z-as paralll to th optcal as and pontng upwards and y-as along th ln, compltng a rght-hand coordnat systm (Fgur ). Each mag ln s th rsult of a narly paralll projcton n th flght drcton and a prspctv projcton n th ln drcton. For ach obsrvd pont, th y f X Y Z coordnats s dscrbd by: rlatonshp btwn mag [ ] and ground [ ] X Y Z X Y Z + kr( ω, ϕ, κ ) y f () X s th P poston n th ground systm, f s th focal lngth, k a scal factor and R(ω,ϕ,κ ) th rotaton matr from mag to ground systm, accordng to atttud angls whr [ Y Z ]

4 3 Postr sson [ω,ϕ,κ ]. For snsors whos optcal systms consst of mor lnss, addtonal gomtrc paramtrs dscrbng th rlatv poston and atttud of ach lns wth rspct to th nadr on ar ntroducd n th collnarty quatons (Ebnr, 99). onsdrng lns j and callng f j th focal lngth, Pj, y Pj th prncpal pont poston, j, y j, z j th rlatv poston and α j, β j, γ j th rlatv atttud, Equaton () s tndd to: X X j Pj Y Y + ( ω, ϕ, κ ) y j + kr( ω, ϕ, κ, α, β, γ ) y j () Z Z z j f j R s th complt rotaton from th mag systm cntrd n th off-nadr lns j to th ground fram and s obtand as th combnaton of two rotatons: R ( ω, ϕ, κ ) M ( α j, β j, γ j ). M(α j, β j, γ j ) transforms th mag systm cntrd n th off-nadr lns j nto th mag systm wth orgn n th cntral lns, whl (ω,ϕ,κ ) rotats th mag systm cntrd n th cntral lns nto th ground fram. It s mmdat to chck that f th optcal systm conssts of on lns, by sttng n Equaton () j and th rlatv orntaton paramtrs qual to zro, th rlaton () for on-lns snsors s obtand. olvng Equaton () wth rspct to k, w gt th collnarty quatons for mult-lns snsors: Pj y j r f j r r f j r ( X X c ) + r ( Y Yc ) + r3 ( Z Z c ) ( m + m y + m3 z) N f j ( X X c ) + r ( Y Yc ) + r3 ( Z Z c ) ( m + m y + m3 z) ( X X c ) + r ( Y Yc ) + r3 ( Z Z c ) ( m + m y + m3 z) N f j ( X X ) + r ( Y Y ) + r ( Z Z ) ( m + m y + m z) c c 3 c 3 y (3). IRET GREFERENING GP/IN nstrumnts carrd on board, togthr wth accurat fltrng tchnqus, provd th snsor trnal orntaton of ach mag ln wth accuracs n th ordr of cntmtrs for poston and arcsconds of dgr for atttud and allow th stmaton of th ground coordnats wth forward ntrscton, basd on Equatons () or (). Anyway th obsrvatons provdd by GP/IN do not rfr to th camra P: th poston data rfr to a local systm wth orgn n th GP antnna and th atttud rfr to a local fram cntrd n th IN nstrumnt (Fgur 3). Ths two systms ar shftd and rotatd wth rspct to th mag on, thrfor th rlatonshp btwn mag and ground coordnats must b tndd n ordr to nclud th offst vctors and th msalgnmnt angls btwn th mag and th GP and IN frams (Pol, ). Fgur 3. Rfrnc systms n drct gorfrncng algorthm. Thn wth forward ntrscton of homologous rays th ground coordnats of any masurd pont n or mor stro scns can b stmatd.

5 Pol 3 3 INIRET GREFERENING On of th man problms wth drct gorfrncng s that th valus of th rfrnc systms dsplacmnts and rotatons ar usually not avalabl and addtonal systmatc rrors occur n th masurmnts. Morovr vn f th offsts ar stmatd wth post-flght calbraton, th GP/IN masurmnt accuracy s not suffcnt for vry hgh rsoluton mappng, thrfor a mor prcs corrcton of th postonng data s rqurd. ffrnt approachs hav bn proposd n ordr to stmat th corrct snsor trnal orntaton for ach mag ln from th GP and IN obsrvatons. For TL snsor (hn, ) proposs to modl th trnal orntaton wth thrdordr Lagrang polynomals, whl (Grun t al., ) us both th Lagrang and pcws polynomals, accordng to th spcfc cas study. Th pcws polynomnal approach was also chosn by (L t al., ) for HYIE (Hyprspctral gtal Imagry ollcton Eprmnt) snsor dvlopd by th Naval Rsarch Laboratory at tnns pac ntr. A dffrnt approach s usd by LH-ystms for th trangulaton of A4 scns (Hnskn t al., ). Th orntaton paramtrs at ach posur ar computd from th nghborng ons plus a corrcton trm drvd from th GP/IMU obsrvatons. Th ntrpolaton coffcnts dscrbng th corrcton ar a functon of th tm dffrncs btwn th currnt posur and th posurs of th nghborng rfrnc data. In our approach w modl th snsor trnal orntaton wth pcws polynomals, bcaus ths knd of functons s vry flbl and can b adaptd to dffrnt flght condtons and to satllt snsors too (Pol, ). Apart of th trnal orntaton dtrmnaton, th snsor modl ncluds a slf-calbraton, whch s rqurd for th corrcton of th systmatc rrors du to th lns (or lnss) dstortons and th lns rotatons n th focal plan. 3.. Etrnal orntaton modlng. Th snsor trnal orntaton s modld wth pcws polynomal functons dpndng on tm (t) or on mag ln numbr (l) f th ntgraton tm s constant. Th platform trajctory s dvdd nto sgmnts accordng to th numbr and dstrbuton of avalabl GPs and s. For ach sgmnt, callng l fn th frst and last lns, th varabl t s dfnd as: tn and t fn th tm trms and l n and t tn l l t [, ] or t n [, ] t t (4) fn n l fn ln bng t th snsor posur tm and l th ln numbr. In ach sgmnt th snsor trnal orntaton (X, Y, Z, ω, ϕ, κ ) s modld wth nd ordr polynomals dpndng on t : X Y Z ω ϕ( t ) κ Xnstr + X + Xt + X t Ynstr + Y + Y t + Yt Znstr + Z + Zt + Zt ωnstr + ω + ωt + ωt ϕnstr + ϕ + ϕt + ϕt κnstr + κ + κt + κ t (5) [ X Y Z ] s th P poston obsrvd by GP, [ ϕ κ ] nstr nstr nstr ω nstr nstr nstr th P atttud obsrvd by IN and [ X XX... κκκ ] ar th unknown paramtrs for sgmnt. Equaton (5) mans that th snsor atttud and poston of ach mag ln l blongng to sgmnt ar

6 3 Postr sson modld as th sum of th GP and IN obsrvatons for that ln plus th nd ordr polynomal functon n t dscrbng th offsts and systmatc rrors contand n th nstrumnts masurmnts. At th ponts of conjuncton btwn adjacnt sgmnts, constrants on th zro, frst and scond ordr contnuty ar mposd on th trajctory functons: w forc that th valus of th functons and thr frst and scond drvatvs computd n two nghborng sgmnts ar qual at th sgmnts boundars. As th pont on th bordr btwn sgmnt and + has t n sgmnt and t n sgmnt +, mposng th contnuty constrants for X functon, w obtan: contnuty: st drvatv: nd drvatv: X + X + X t t nstr + X + X + X X nstr + X (6) + dx dx + dt dt X + X X (7) t t + d X d X + X dt dt X (8) t t In th sam way, Equatons (6), (7) and (8) ar wrttn for Y, Z, ω, ϕ and κ functons and tratd as wghtd constrants. Addtonal psudo-obsrvatons can f som or all nd ordr paramtrs and rduc th polynomal dgr to st (lnar functons). Ths opton allows th modlng of th snsor poston and atttud n ach sgmnt wth nd or st ordr polynomals, accordng to th charactrstcs of th trajctory of th currnt cas study. 3.. lf-calbraton. Th am of slf-calbraton s to stmat addtonal paramtrs modlng th systmatc rrors du to changs n th ntror orntaton (prncpal pont poston and focal lngth), lns dstortons and ln rotaton n th focal plan. Usng th sam notaton adoptd for Equaton (3), th collnarty quatons ar modfd for slf-calbraton, rsultng n: N pj f j + (9) N y y y pj f j + y and y contan th wll-known addtonal paramtrs modlng th prncpal pont dsplacmnt ( P, y P ), th focal lngth varaton ( f), th symmtrc (k,k ) and dcntrng (p,p ) lns dstorton and th scal factor n y drcton (s y ), as dscrbd n (Brown, 97), togthr wth th ffct n drcton du to th ln rotaton θ n th focal plan (Fgur 4), accordng to: bng, f P f y f f 4 P + kr P + kr P + p ( r + ) 4 + kr + kr + pp + p P P, y and r P + y P. P + p P + P sn θ ( r + ) + s y () Fgur 4. array rotaton n focal plan (,y).

7 Pol Bundl adjustmnt. GPs and s ar rqurd n ordr to solv th systm and stmat th unknown trnal orntaton paramtrs and s ground coordnats. Th functons modlng th trnal (Paragraph 3..) and th ntrnal (Paragraph 3..) orntaton ar ntgratd nto th collnarty quatons, rsultng n an ndrct gorfrncng modl. u to thr non-lnarty, th complt quatons ar lnarzd accordng to th frst-ordr Taylor dcomposton wth rspct to th unknown paramtrs. For ths opraton ntal appromatons for th unknown paramtrs ar ndd. As th trnal orntaton paramtrs concrns, s put qual to zro; ths mans that at th frst traton th snsor poston and atttud ar thos obsrvd by th GP and IN nstrumnts (X nstr, Y nstr, Z nstr, ω nstr, ϕ nstr, κ nstr ). Th ntal valus for th ground coordnats of th s ar stmatd wth forward ntrscton, usng th rough trnal orntaton obsrvd by th GP and IN nstrumnts (X nstr, Y nstr, Z nstr, ω nstr, ϕ nstr, κ nstr ). Th ntal paramtrs modlng th slf-calbraton ar st qual to zro. Th ovrall stmaton modl rsults n: GP E A GP A + B + E GP GP GP E GP E () Th vctor of unknowns contans (ncrmnts to ), (ncrmnts to ), (slfcalbraton paramtrs) and GP (ncrmnts to ground coordnats of GPs). A GP and A and ar th dsgn matrcs wth th drvatvs for for GPs and s obsrvatons; B s th dsgn matr for for s obsrvatons. GP and and ar th dsgn matrcs for for GPs and s obsrvatons and,, dscrb th constrants on zro, frst and scond ordr contnuty (Equatons (6), (7) and (8)). Th trnal orntaton and slf-calbraton paramtrs and th ground coordnats of GPs ar ntroducd as psudo-obsrvatons wth dsgn matrcs, and E rspctvly; ar th obsrvaton rrors, l th dscrpancy vctors and P th wght matrcs for ach group of obsrvatons. Th systm () s solvd wth tratv last-squars adjustmnt, usng a sutabl numbr of wll dstrbutd GPs and s. onsdrng a snsor wth lnar arrays, N GP GPs, N s and n s trajctory sgmnts, th complt systm contans (N GP +N ) collnarty quatons, togthr wth 6(n s -) quatons for ach group of constrants, plus th psudo-obsrvatons for unknown paramtrs. Th unknowns ar: 8n s trnal orntaton paramtrs, 3N ground coordnats, 8 slf-calbraton paramtrs for ach lns and paramtr for ach ln. Th procss stops whn stmatd unknown vctors ar smallr than sutabl thrsholds. 4 ONLUION AN FUTURE WORK W hav prsntd a snsor modl for th gorfrncng of lnar snsors carrd on arborn, whch ar wdly usd for photogrammtrc applcatons and rprsnt a good altrnatv to dgtal fram camras and flm camras. Th snsor modl s basd on th classcal collnarty quatons, tndd n ordr to tak nto account th partcular acquston gomtry (prspctv projcton n ach ln and paralll projcton btwn lns). Th snsor trnal orntaton, whch s dffrnt for ach mag ln, s modld wth nd ordr pcws polynomals, whl th ntrnal

8 34 Postr sson orntaton s stmatd wth a slf-calbraton adaptd for lnar arrays. Th ovrall quaton systms s solvd wth tratv last-squars mthods. Th modl nd som furthr mprovmnts, frst of all an automatc blundr dtcton and nw soluton mthods for th rducton of computng tm. REFERENE.. Brown, 97. "los-rang camra calbraton," Photogrammtrc Engnrng. Vol.37 (8), pp hn, T.,. Hgh prcson gorfrnc for arborn Thr-Ln cannr (TL) magry. Procdngs of 3 rd Intrnatonal Imag nsng mnar on Nw vlopmnt n gtal Photogrammtry, Gfu, Japan, pp LR wb: Ebnr, H., Kornus, W., Ohlhof, T., 99. A smulaton study on pont dtrmnaton for th MOM- / spac projct usng an tndd functonal modl. Intrnatonal Archvs of Photogrammtry and Rmot nsng, Vol. 9, Part B4, Washngton.., pp Grun, A., Zhang L.,. nsor modlng for aral mobl mappng wth Thr-Ln-cannr (TL) magry. Intrnatonal Archvs of Photogrammtry and Rmot nsng, Vol. 34, Part, pp Hahn, M., tallmann,., tattr,., Mullr, F., 996. Th PA-snsor systm for topographc and thmatc mappng. Intrnatonal Archvs of Photogrammtry and Rmot nsng, Vol. 3, Part B, Vnna 996, pp Hnskn, L., Mllr,., Tmplmann, U., Ubbng, R., Walkr,.,. Trangulaton of LH- ystms'a4 magry usng ORIMA GP/IMU. Intrnatonal Archvs of Photogrammtry and Rmot nsng, Vol.34, Part 3A, Graz, pp L,., Thss, H.J., Bthl, J.., Mkhal, E.M.,. Rgorous mathmatcal modlng of arborn pushbroom magng systms. Photogrammtrc Engnrng & Rmot nsng, Vol. 66, No. 4, pp Lhmann, F.,. Th HR gtal Arborn Imagr. A Rvw of Two Yars of Opratonal Producton. GoInformatcs, Vol 4, Aprl, pp. -5. Nasa wb: Pol,.,. rct gorfrncng of mult-ln mags wth a gnral snsor modl. IPR Workshop Hgh rsoluton mappng from spac, 8- ptmbr, Hannovr. Procdngs on. Pol,.,. Gnral modl for arborn and spacborn lnar array snsors. Intrnatonal Archvs of Photogrammtry and Rmot nsng, Vol. 34, Part B, nvr, pp andau, R., Edkardt, A., 996. Th stro camra famly WAO/WAA for spacborn/arborn applcatons. Intrnatonal Archvs of Photogrammtry and Rmot nsng, Vol. 3, Part B, Vnna, pp tarlabo wb: Tmplmann, U., Bornr, A., hapln, B., Hnskn, L., Mykhalvych, B., Mllr,., Rck, U., Rulk, R., Ubbng, R.,. Photogrammtrc softwar for th LH systms A4. Intrnatonal Archvs of Photogrammtry and Rmot nsng, Vol. 33, Part B, Amstrdam, pp

The Hyperelastic material is examined in this section.

The Hyperelastic material is examined in this section. 4. Hyprlastcty h Hyprlastc matral s xad n ths scton. 4..1 Consttutv Equatons h rat of chang of ntrnal nrgy W pr unt rfrnc volum s gvn by th strss powr, whch can b xprssd n a numbr of dffrnt ways (s 3.7.6):