trd(AC')m?xwhereoBcuis determinedtfromothelalignmentnandax is the metraxsociate UPith. th maxgest eievlue~ skewA-A Il symtInulrvlctImti=hs eeet r Bu,n prisularposiivedfinit re,iaoal smmercmatrx, somit ca copued directly from thc i outniputeso theayost quto bedaoaiedubnerhooalsmlaiy rnsomain (7)= is sove by' an numrialtceewihreut nA h Hence, ma matrx BI7 1 is rtognaizD (nraie)bIn fmn Now, without loss of generolvingdirectonlcosies=iniwogimprtant2rspects First andh (4) contains only four unknowns, thesacomponentsroflq,eas of three-dimensional vectors thcompare to (7) wich cont ain nIney unnwns the ele- 1 3 trx (AC') max ma IIUDUII , mensxo B.Seond noe o te othoonaiztio prce {4,}il i111=1 dure sugeste for otining) C fromD) th soDluino(7ar whee 0is ny nitvecor nd , 0, 0 1 s a orhonrm knoswns to be poptmlv hresteothgnlztinpoe ma -JD max = IllDU)4~I a IIDRefultiplcatos,treadiosanoedvso. whn s henrmlie egevctrcorepndngt X. IAC Thoreically, Thore 3toudh e usedotatrnon find siuatin opti where,fagai, {0 = {U,00'~ and} isaotherse nofs orhonrale mals C even-od forthen, B.obtAine From(7).rInhpractice,ehow-en similarity truansfrmtion seo si(D) the areP arhgnd, sinc P isemterixe B.ItisgharactteristABOicof theceuler fou9pramte positiv-definie, the igenvaues of must b positie,Cmatix cx thatlgrtmfo tape-on nria ai Theore 2atisn"nowEaTrdirectroconsequencecronofSyLemma v1. fomain [6] is th theoretical base fory the norma izaion 1 []PR ams iieDmnioa etrSae.Pictn .. scheme. ~ ~ ~ ly anNosthrawl-nowd mthdinole9te5ueranl Theorem : If A i an arbirary rea invertile13IXYrepresetzation althug this regep"resegntliation gnrtesonlyu marxwtehere exsnyuists anctorthovigonal maretryixn whihert o the unkntowns,osthe diffreniaIE equatins. thosate must blec cmionizest Iand 3 -CI aFrthrmhoore,lCet A(A 'A ectors srolved arsemnnina, vo.AES5 copare to8 (4)n (7),mbe whic6ar havig al fist cmponntsequa to eroAgan,uingLe line]Ar, Raltnd, in adirtiCon,thrse iNuexistsal mAthmaalysse sinulr-: Prof i Fromllema2otwssficett maximizeIty (gimbal lok. Methods base onl th1 Ele6nge5ep Chrl(AC'). Usaring There 2,r wnte dronsoxb showin that seentation ar4e knowesntolb lessoefciaent winh ghenea thange 3n optrSene i uretrsac neetr Stcattitudecdterionationabysi adiretionhalcsticntes;a hEnce,theys tRic(aC') =D. Bro(P) faonnNwYorkCCitA(An'A)g"2. are also les reffcivent thante optimMS.,ald meho.degivens in Fro Thornem 1h fandlt forthi matrlixh Dc,inwe ha iverst,TAnc,N..wereiCPetyhima,Dprteto Mahene, adCmue cene ei h uhro MTI EHD Aaei Pes n OENITO DUCOR DFFEENIA EUATON (cham' Othe Sris,McGawHil3 tr( C)= r(P ' = - X.(PC) - X.(P. -. - .. .1 ,I I. s 326 Egyptian Pottery Color plate 5.3 An incomplete pilgrim flask from el-Ahaiwah, no. 6-18525 Color plate 5.4 A spherical jug recovered from Tomb A542 and made in one piece, no. 6-18519 ?t + (WI. + co? ) x vtt - g, = f t (14) Yt ie it t t Yp However because of errors in the navigation system, the following variables appear in the above equation: vt + 5 vt- (15) t 2 XP (9)i 9t + 6gt (16) t 2 oz + 5cot (17) x t i t i t Fig. 3. Relation between platform frame and true frame. (wt )i ='Wt + 6wt (18) ie ie ie Substituting (5),(6), and (9) into (7) and subtracting (3) where subscript 2 refers to a perturbation analysis error from the result yields variable. Also, instead of flt , only ftp + 5fp are available from the accelerometers. Thus, the equations tllat the navi- 5 ?C + (COC + WC ) x 5 VC - 6gc, x ff + 5jfc. (10) gation system solves are 1 ie ic 1 1 B. Platform Frame to Computer Frame Error Equations + (cot )i] x Mi - (9)i = fp + bfp - (19) ( V)i + KU it I ie Since the navigation system assumes that the platform An angular relation between the platform frame and the axes are coincident with the computer axes, the gyros are true frame is defined by the angles shown in Fig. 3. For torqued with wq . However, the gyros control the platform IC small angles, the direction cosine matrix relating the two along platform axes (except for nonorthogonalities) and, frames is further, the gyros have a drift rate eP. Thus, the platform rate is I oz ___OY W? = co' + EP. O 1) CP I = I - 0 X (20) lp ic t _11z OX An alternative explanation of (I 1) is that the platform rate OY ___Ox I wP is equal to the torquing rates applied to the gyros plus __j ip the gyro drift rate eP. The torquing rate numbers are the and the components of vectors transform as desired angular velocity of the platform. Since the naviga- tion system assumes the platform frame is always coincident fp = Cpfl = ft - 0 X f (21) with the computer frame, this angular velocity is wq . t IC Substituting (8) into (I 1) yields, to first order in w and Substituting ( 17), (I 8), and (2 1 ) into ( 19) yields, after ep subtracting ( 14), wp =OXWP +6P. (12) 5 + (W. + wt ) x 5 vt + (5cot + 5cit) x V" CP ip ?2' te it 2 ie t wP is the angular velocity of the platform with respect to -5 t =-OXft+6ft- (22) cp 0 92 the computer frame and, for smafl angles, is equal to ViP (see Fig. 2),' consequently giving the famfliar 0 equation ?p B. Platform Frame to True Frame Error Equations -(,)p. x P + ep. (13) ip M. Perturbation Error Analysis The navigation system assumes that the platform axes are coincident with the true frame. However, Wt is not avail- it able for gryo torquing, but only wt + Uot. The gyros con- A. Velocity Error Equations it trol the platfon-n along platform axes and, further, they have In the perturbation analysis, the frame in which the navi- a drift rate eP. gation equations are assumed to be solved is the true t frame Consequently, the platform rate is and in this frame the ideal solution is Color Plate Section 329 Color plates 9.7-8 Painted pattems on Early Ptolemaic vessels made in Egypt, possibly in Athribis; 7 = Illrd century B.C., 8 = Late Illrd- early llnd century B.C. Color plate 9.9 Fragments of a vessel with painted figural pattems llnd century B.C. 330 Egyptian Pottery Color plate 9.10 Products of mid- Ptolemaic pottery workshops in Athribis: globular vessels with splashed decoration. Color plates 9.11 -12 Imported Hellenistic wares with starnped, incised and painted decoration, found in Tell Atrib; 1 1 = Sherds of "Gnathic"-type pottery, the plate probably is from Teano (Campania); 12 = small bowls. 332 Egyptian Pottery d I el Ce = CeCn = Ce [I+ 50 X]. c n c n Thus, Ze 61 sin L 0 61 cos L (58) (59) (60) MERIDI? Ye Also from Fig. 6, 5R N=(R +h)6L 6R -61(R+h)cosL. E Substituting (59) and (60) into (58) yields 0 6RE tan LI(R + h) 60 x - - 6R tan L l(R + h) 0 E L- 6RNI(R + h) 6REI(R + h) Xe Fig. 6. Relation of local4evel north-pointing frame to the Earth. Fig. 6, plumb-bob gravity wif be modeled as _?g gn (56) t g + U _j where g - goR 2 l(R + h)2 (57) and where go = plumb-bob gravity at the Earth's surface ?, 77 = vertical deflections Ag = gravity anomaly h = altitude R = Earth's radius L = latitude 1 = longitude. From Fig. 6, (61) mmoma Color Plate Section 337 b c 338 Egyptian Pottery a b C Color plate 10. 6 Nile silt fabric from a tdgen bowl made by the Abu Raguan potter (W-3 1; figure 10. 12.4): a) color section of fabric; b) and c) SEM views of the fabric groundmass at 400X and 1000X respectively. Paste consists of granular silt with reticulated clay texture containing well-rounded to subangular silt grains, abundant tabular clays, biocarbonate (biosparite) sand-sized fragments and sand-sized ash. bk? i Color Plate Section 339- a b Color plate 10. 7 Nile silt fabrics: a) coarse Nile silt fabric belonging to a zTr from the Fayum (W-64; figure 10.2.1); b) straw or chaff tempered Nile silt from a sahfa bowl from the Fayum (W-71; figure 10.7.2). I 1, i? j I r iI I i If i i I i r Color Plate Section 339 340 Egyptian Pottery a b Color plate 10.8 Nile silt fabrics: a) zir water jar from Minouf (W-52; figure 10.3.2); b) 'olla from Minouf (W-61; figure 10.8.2). im I c Color Plate Section 34] a b Color plate 10.9 Nile silt fabrics: a) zTr water jar from Abu Raguan (14.9; figure 10.3.3); b) zTr water jar from Badrashein (16.1; figure 10.3.1). j ok- kd Color Plate Section 341 342 Egyptian Pottery a b Color plate 10.10 Sinai silt fabrics: a) color section of black bowl rim (13.115; figure 10.13.1); b) color section of tab n fragment of anomalous fabric (13.75; not drawn). r 6= I Color Plate Section 343 a b c Color plate 10.11 Black fabric of Nile silt belonging to a b sa cookpot from Sharqiya province (W-47; figure 10.9.2): a) color section of fabric; b) and c) SEM shots of fabric groundmass at 400X and lOOOX, respec- tively. Paste consists of granular silt with reticulated clay texture with a few subangular to angular sand- sized quartz and feldspar grains. Very minor vitrification. A l I U K- ii I Color Plate Section 343 I 344 Egyptian Pottery a b Color Plate 10.12 Mixed Nile silt and marl clay fabrics: a) color section of fabric belonging to an abrr' from Cairo (W-50; figure 10.8.5); b) color section of fabric from a Cairo 'olla (W-5 1; figure 10.8.4). Color Plate Section 345 a 4. i - tI b c Color Plate 10.13 Mixed Nile silt and marl clay 'olla from Cairo (W-39; figure 10.8.6): a) color section of fabric; b) and c) SEM views of fabric groundmass at 400X and lOOOX, respectively. Paste exhibits a rounded silt and granular micrite matrix with common sand-sized pores and minor silt-sized pores. Angular to rounded sand-sized mudstone fragments are abundant. Also present are a few silt-sized grains of magne- tite, common silt-sized rounded quartz and very well-formed calcium oxide coated pores. Vitrification is very minor. - U I I Color Plate Section 345 346 Egyptian Pottery a Color Plate 10.14 Sinai silt fabric ("orange-brown sandy" ware) belonging to a flowerpot (13.68; figure 10.17.8): a) color section of fabric; b) and c) SEM views of fabric groundmass at 400X and 1OGOX, respec- tively; d) SEM backscatter view (1OOOX) and EDAX energy spectrum showing elemental distribution for very fine silt-sized barite (barium sulfate) grain. Paste exhibits subrounded to subangular silt and tabular clays with calcium oxide coated pores. Sand-sized pores are common; silt-sized pores are rounded and uncommon. Minor fragments of grog are present. The barite, which almost certainly comes from an old sedimentary environment, may serve as a useful marker mineral for the source area. im M- Color Plate Section 347 C C_~~~~~~~~~~ C~~~~~~~~~~~~4 , _ d - U 348 Egyptian Pottery a b Color Plate 10.15 Flowerpot fabrics from El Qanatar workshop: a) color section of Nile silt fabric (bottom; 15.4; figure 10. 16.8) and mixed Nile silt and Tebbine clay fabric (top; 15.2; not drawn); b) and c) SEM views of Nile silt (15.4) at 400X and 1OOOX respectively; d) and e) SEM views of mixed fabric (15.2) at 400X and lOOOX respectively. SEM views of sample 15.4 show a paste composed of well-rounded to subangular silt with tabular clays that appear to be partially vitrified and sand-sized angular pores which are dominantly tensile in nature. Some sand-sized ash frag- ments are present, along with rare organics. SEM views of sample 15.2 illustrate a groundmass with a granulated silt texture containing well-rounded to subangular silt-sized mineral grains in a highly porous structure with abundant calcium carbonate silt. The pores are dominantly rounded and silt-sized. Angular sand-sized mudstone fragments are present. Welding appears to be minor. Color Plate Section 349 C d e JULY 1975 VOLUME AES-1 1 NUMBER 4 Published Bimonthly A PUBLICATION OF THE IEEE AEROSPACE AND ELECTRONIC SYSTEMS SOCIETY (Contents Continued on Back Cover) 352 Egyptian Pottery a b Color Plate 10.18 SEM photographs of two different Nile silt fabric pastes: a) and b) 400X and IOOOX views, respectively, of the groundmass of sample W-21, a small magur bowl from Minya (figure 10.12.2); c) and d) 400X and IOOOX views, respectively, of the groundmass of W-69, a handb from the Fayum (figure 10.7.3). W-21 has a paste consisting of well rounded to subrounded silt grains with tabular clays and some calcium oxide coated pores. Pores range from sand to silt-sized and are rounded to angula; some of the angular pores are in tensile configurations. Vitrification is very minor. The silt is composed dominantly of quartz and feldspar with minor heavy minerals. Some of the sand-sized mineral grains are angular. Ash is also present. The groundmass of W-69 consists of granular silt with an organic cast texture. The organic casts are dominated by carbon and phytolithic debris; many contain original cellular structure. Casts vary in size from silt to sand and are dominantly elongated. Individual phytoliths are present in the silt matrix and consist of grass short cells and non-segmented hair and hair-based forms. The tabular clay texture is overshadowed by the plant cast texture. Vitrification is very minor. Some of the silt appears to be carbonate grains. L by the vector differential equation x(t) = A (t) x (t) + B(t) w(t) (1) where x is an n-dimensional plant state vector, w is an rn-dimensional disturbance vector,, and A and B are n X n n X m system matrixes, respectively. Let w(t) be a vector of white noise processes with mean gwL(t), and let the covari- ance matrix associated with w(t) be defined by E{ [w(t) - g(t)] [w(Tr) - jW(r)] T}=-Qw(t)6 (t-,r) (2) where 6(.) is the Dirac delta function. Let P(t) represent the state covariance matrix, i.e., 1(t) = E{ [x(t) - ,lx(t)] [X(t) - 1(t)]I T} (3) where yxt(t), the mean of x(t), may be determined from (1) by replacing w(t) by g,4,(t) and x(t) by gx(t)'. It has been shown that 1(t) satisfies the matrix differential equation [1], [21, [5], [7] given by This result is sometimes referred to as the direct covariance algorithm [111 .1 Suppose Qw(t) in (2) and (4) is not known exactly but lies somewhere on the bounded range between Qw, (t) and Qw 2 (t). The corresponding values of1P(t) from (45 may be calculated to yield PI(t) and P2(t). Such an error analysis based on deterministic variations from some nominal con- ditions, such as Qw(t) = QwN/t) and 1(t) = PA(t), may be simplified when bounded variations bQw(t) occur above and below QwN$t), i.e., (Q N- QW