RECOMMENDATION ITU-R P Propagation by diffraction

Transcription

1 Rec. ITU-R P.- 1 RECOMMENDATION ITU-R P.- Propagatio by diffractio (Questio ITU-R / The ITU Radiocommuicatio Assembly, ( cosiderig a that there is a eed to provide egieerig iformatio for the calculatio of field stregths over diffractio paths, recommeds 1 that the methods described i Ae 1 be used for the calculatio of field stregths over diffractio paths, which may iclude a spherical earth surface, or irregular terrai with differet kids of obstacles. Ae 1 1 Itroductio Although diffractio is produced oly by the surface of the groud or other obstacles, accout must be take of the mea atmospheric refractio o the trasmissio path to evaluate the geometrical parameters situated i the vertical plae of the path (agle of diffractio, radius of curvature, height of obstacle. For this purpose, the path profile has to be traced with the appropriate equivalet Earth radius (Recommedatio ITU-R P.. If o other iformatio is available, a equivalet Earth radius of km may be take as a basis. Fresel ellipsoids ad Fresel zoes I studyig radiowave propagatio betwee two poits A ad B, the iterveig space ca be subdivided by a family of ellipsoids, kow as Fresel ellipsoids, all havig their focal poits at A ad B such that ay poit M o oe ellipsoid satisfies the relatio: λ AM + MB AB + (1 where is a whole umber characterizig the ellipsoid ad 1 correspods to the first Fresel ellipsoid, etc., ad λ is the wavelegth. As a practical rule, propagatio is assumed to occur i lie-of-sight, i.e. with egligible diffractio pheomea if there is o obstacle withi the first Fresel ellipsoid. The radius of a ellipsoid at a poit betwee the trasmitter ad the receiver is give by the followig formula: 1/ λ d1 d R ( d1 + d

2 Rec. ITU-R P.- or, i practical uits: 1/ d1 d ( 1 d + d f R ( where f is the frequecy (MHz ad d 1 ad d are the distaces (km betwee trasmitter ad receiver at the poit where the ellipsoid radius (m is calculated. Some problems require cosideratio of Fresel zoes which are the zoes obtaied by takig the itersectio of a family of ellipsoids by a plae. The zoe of order is the part betwee the curves obtaied from ellipsoids ad 1, respectively. Diffractio over a spherical earth The additioal trasmissio loss due to diffractio over a spherical earth ca be computed by the classical residue series formula. A computer program GRWAVE, available from the ITU, provides the complete method. A subset of the outputs from this program (for ateas close to the groud ad at lower frequecies is preseted i Recommedatio ITU-R P.. At log distaces over the horizo, oly the first term of this series is importat. This first term ca be writte as the product of a distace term, F, ad two height gai terms, G T ad G R. Sectios.1 ad. describe how these terms ca be obtaied either from simple formulae or from omograms. It is importat to ote that: the methods described i.1 ad. are limited i validity to trashorizo paths; results are more reliable i the deep shadow area well beyod the horizo; atteuatio i the deep shadow area will, i practice, be limited by the troposcatter mechaism..1 Numerical calculatio.1.1 Ifluece of the electrical characteristics of the surface of the Earth The etet to which the electrical characteristics of the surface of the Earth ifluece the diffractio loss ca be determied by calculatig a ormalized factor for surface admittace, K, give by the formulae: i self-cosistet uits: 1/ 1/ π ae K H [( ε 1 + ( λ σ ] for horizotal polarizatio ( λ ad [ ε + ( λ σ ] 1/ K V K H for vertical polarizatio ( or, i practical uits: KH [( ε 1 + (1 σ / ] 1/ 1/. ( ae f f (a KV [ ε + ( 1 σ / f ] 1/ K H (a

3 Rec. ITU-R P.- a e : effective radius of the Earth (km ε : effective relative permittivity σ : effective coductivity (S/m f : frequecy (MHz. Typical values of K are show i Fig. 1. FIGURE 1 Calculatio of K 1 Normalized factor for surface admittace, K 1 Vertical ε σ ε σ ε 1 σ ε σ Horizotal ε σ ε 1 σ ε σ khz khz 1 MHz MHz MHz 1 GHz GHz Frequecy -1

4 Rec. ITU-R P.- If K is less tha.1, the electrical characteristics of the Earth are ot importat. For values of K greater tha.1, the appropriate formulae give below should be used..1. Diffractio field stregth formulae The diffractio field stregth, E, relative to the free-space field stregth, E, is give by the formula: E log F( X + G( Y1 + G( Y db ( E where X is the ormalized legth of the path betwee the ateas at ormalized heights Y 1 ad Y E (ad where log is geerally egative. E I self-cosistet uits: X 1/ β π d ( a λ e Y 1/ β π h ( a λ e or, i practical uits: d : a e : h : f : Y path legth (km 1/ / e X. β f a d (a / 1/ e. β f a h (a equivalet Earth s radius (km atea height (m frequecy (MHz. β is a parameter allowig for the type of groud ad for polarizatio. It is related to K by the followig semi-empirical formula: K +. K β ( 1 +. K + 1. K For horizotal polarizatio at all frequecies, ad for vertical polarizatio above MHz over lad or MHz over sea, β may be take as equal to 1. For vertical polarizatio below MHz over lad or MHz over sea, β must be calculated as a fuctio of K. However, it is the possible to disregard ε ad write: K σ. (a / / k f where σ is epressed i S/m, f (MHz ad k is the multiplyig factor of the Earth s radius.

5 Rec. ITU-R P.- The distace term is give by the formula: F( X 11 + log ( X 1. X ( The height gai term, G(Y is give by the followig formulae: 1/ G( Y 1. ( Y 1.1 log ( Y 1.1 for Y > (11 For Y < the value of G(Y is a fuctio of the value of K computed i.1.1: G ( Y log ( Y +.1Y for K < Y < (11a [ log ( Y / K 1] G ( Y + log K + log ( Y / K + for K / < Y < K (11b G( Y + log K for Y < K / (11c. Calculatio by omograms Uder the same approimatio coditio (the first term of the residue series is domiat, the calculatio may also be made usig the followig formula: E log F( d + H( h1 + H( h db (1 E E : E : d : h 1 ad h : received field stregth field stregth i free space at the same distace distace betwee the etremities of the path heights of the ateas above the spherical earth. The fuctio F (ifluece of the distace ad H (height-gai are give by the omograms i Figs.,, ad. These omograms (Figs. to give directly the received level relative to free space, for k 1 ad k /, ad for frequecies greater tha approimately MHz. k is the effective Earth radius factor, defied i Recommedatio ITU-R P.. However, the received level for other values of k may be calculated by usig the frequecy scale for k 1, but replacig the frequecy i questio by a hypothetical frequecy equal to f / k for Figs. ad ad f / k, for Figs. ad. Very close to the groud the field stregth is practically idepedet of the height. This pheomeo is particularly importat for vertical polarizatio over the sea. For this reaso Fig. icludes a heavy black vertical lie AB. If the straight lie should itersect this heavy lie AB, the real height should be replaced by a larger value, so that the straight lie just touches the top of the limit lie at A. NOTE 1 Atteuatio relative to free space is give by the egative of the values give by equatio (1. If equatio (1 gives a value above the free-space field, the method is ivalid.

6 Rec. ITU-R P.- FIGURE Diffractio by a spherical earth effect of distace GHz 1 1 GHz Frequecy for k 1 1. GHz GHz Frequecy for k / Distace (km 1 Level (db i relatio to free space MHz MHz 1 1 Horizotal polarizatio over lad ad sea Vertical polarizatio over lad (The scales joied by arrows should be used together -

7 Rec. ITU-R P.- FIGURE Diffractio by a spherical earth height-gai Frequecy for k 1 k / 1 1 Height of atea above groud (m 1 GHz 1. GHz 1. Height-gai (db H(h 1 1 GHz 1 1 GHz 1 MHz MHz Horizotal polarizatio lad ad sea Vertical polarizatio lad -

8 Rec. ITU-R P.- FIGURE Diffractio by a spherical earth effect of distace GHz 1 1 GHz Frequecy for k 1 1. GHz GHz Frequecy for k / Distace (km 1 Level (db relative to free space MHz MHz 1 1 Vertical polarizatio over sea (The scales joied by arrows should be used together -

9 Rec. ITU-R P.- FIGURE Diffractio by a spherical earth height-gai Frequecy for k 1 k / 1 1 Height of atea above groud (m 1 GHz 1. GHz 1. Height-gai (db H(h 1 1 GHz 1 1 GHz 1 MHz MHz A B Vertical polarizatio sea -

10 Rec. ITU-R P.- Diffractio over obstacles ad irregular terrai May propagatio paths ecouter oe obstacle or several separate obstacles ad it is useful to estimate the losses caused by such obstacles. To make such calculatios it is ecessary to idealize the form of the obstacles, either assumig a kife-edge of egligible thickess or a thick smooth obstacle with a well-defied radius of curvature at the top. Real obstacles have, of course, more comple forms, so that the idicatios provided i this Recommedatio should be regarded oly as a approimatio. I those cases where the direct path betwee the termials is much shorter tha the diffractio path, it is ecessary to calculate the additioal trasmissio loss due to the loger path. The data give below apply whe the wavelegth is fairly small i relatio to the size of the obstacles, i.e., maily to VHF ad shorter waves ( f > MHz..1 Sigle kife-edge obstacle I this etremely idealized case (Figs. a ad b, all the geometrical parameters are combied together i a sigle dimesioless parameter ormally deoted by ν which may assume a variety of equivalet forms accordig to the geometrical parameters selected: ν h λ d1 d (1 ν θ 1 λ d d (1 h θ ν ( ν has the sig of h y θ (1 λ ν d λ α1 α ( ν has the sig of α1 ad α (1 h : d 1 ad d : d : height of the top of the obstacle above the straight lie joiig the two eds of the path. If the height is below this lie, h is egative distaces of the two eds of the path from the top of the obstacle legth of the path θ : agle of diffractio (rad; its sig is the same as that of h. The agle θ is assumed to be less tha about. rad, or roughly 1 α 1 ad α : agles betwee the top of the obstacle ad oe ed as see from the other ed. α 1 ad α are of the sig of h i the above equatios. NOTE 1 I equatios (1 to (1 iclusive h, d, d 1, d ad λ should be i self-cosistet uits.

11 Rec. ITU-R P.- 11 FIGURE Geometrical elemets (For defiitios of θ, α 1, α, d, d 1, d ad R, see.1 ad. θ > d 1 α 1 h > d α a α 1 d 1 h < θ < α d b d 1 h d R α 1 α d c -

12 1 Rec. ITU-R P.- Figure gives, as a fuctio of ν, the loss (db caused by the presece of the obstacle. For ν greater tha. a approimate value ca be obtaied from the epressio: J ( ν. + log ( ν + + ν db (1 FIGURE Kife-edge diffractio loss J(ν (db ν -. Fiite-width scree Iterferece suppressio for a receivig site (e.g. a small earth statio may be obtaied by a artificial scree of fiite width trasverse to the directio of propagatio. For this case the field i the shadow of the scree may be calculated by cosiderig three kife-edges, i.e. the top ad the two sides of the scree. Costructive ad destructive iterferece of the three idepedet cotributios will result i rapid fluctuatios of the field stregth over distaces of the order of a wavelegth. The followig simplified model provides estimates for the average ad miimum

13 Rec. ITU-R P.- 1 diffractio loss as a fuctio of locatio. It cosists of addig the amplitudes of the idividual cotributios for a estimate of the miimum diffractio loss ad a power additio to obtai a estimate of the average diffractio loss. The model has bee tested agaist accurate calculatios usig the uiform theory of diffractio (UTD ad high-precisio measuremets. Step 1: Calculate the geometrical parameter ν for each of the three kife-edges (top, left side ad right side usig ay of equatios (1 to (1. Step : Calculate the loss factor j(ν J(ν/ associated with each edge from equatio (1. Step : Calculate miimum diffractio loss J mi from: or, alteratively, J mi ( ν log + + j1 ( ν j( ν j( ν db (1 Step : Calculate average diffractio loss J av from: ( log db 1 ( ( ( J a ν ν + + (1 j ν j ν j ν. Sigle rouded obstacle The geometry of a rouded obstacle of radius R is illustrated i Fig. c. Note that the distaces d 1 ad d, ad the height h above the baselie, are all measured to the verte where the projected rays itersect above the obstacle. The diffractio loss for this geometry may be calculated as: A J( ν + T ( m, db ( a J(ν is the Fresel-Kirchoff loss due to a equivalet kife-edge placed with its peak at the verte poit. The dimesioless parameter ν may be evaluated from ay of equatios (1 to (1 iclusive. For eample, i practical uits equatio (1 may be writte: ν 1/ ( d 1 + d.1 λ d1 d h (1 where h ad λ are i metres, ad d 1 ad d are i kilometres. J(ν may be obtaied from Fig. or from equatio (1. Note that for a obstructio to lieof-sight propagatio, ν is positive ad equatio (1 is valid. b T(m, is the additioal atteuatio due to the curvature of the obstacle: T(m, k m b k b. +. [1 ep ( 1. ] (a (b (c

14 1 Rec. ITU-R P.- ad 1/ d1 + d π R m R ( d1 d λ / π R h R ( λ ad R, d 1, d, h ad λ are i self-cosistet uits. T(m, ca also be derived from Fig.. Note that as R teds to zero, m, ad hece T(m,, also ted to zero. Thus equatio ( reduces to kife-edge diffractio for a cylider of zero radius. It should be oted that the cylider model is iteded for typical terrai obstructios. It is ot suitable for tras-horizo paths over water, or over very flat terrai, whe the method of should be used. FIGURE The value of T(m, (db as a fuctio of m ad T(m, (db m -

15 Rec. ITU-R P.- 1. Double isolated edges This method cosists of applyig sigle kife-edge diffractio theory successively to the two obstacles, with the top of the first obstacle actig as a source for diffractio over the secod obstacle (see Fig.. The first diffractio path, defied by the distaces a ad b ad the height h 1, gives a loss L 1 (db. The secod diffractio path, defied by the distaces b ad c ad the height h, gives a loss L (db. L 1 ad L are calculated usig formulae of.1. A correctio term L c (db must be added to take ito accout the separatio b betwee the edges. L c may be estimated by the followig formula: ( a + b ( b + c L c log ( b ( a + b + c which is valid whe each of L 1 ad L eceeds about 1 db. The total diffractio loss is the give by: L L 1 + L + L c ( The above method is particularly useful whe the two edges give similar losses. FIGURE Method for double isolated edges h' 1 h' a b c - If oe edge is predomiat (see Fig., the first diffractio path is defied by the distaces a ad b + c ad the height h 1. The secod diffractio path is defied by the distaces b ad c ad the height h. FIGURE Figure showig the mai ad the secod obstacle M h' h 1 h T a b c - R

16 1 Rec. ITU-R P.- The method cosists of applyig sigle kife-edge diffractio theory successively to the two obstacles. First, the higher h/r ratio determies the mai obstacle, M, where h is the edge height from the direct path TR as show i Fig., ad r is the first Fresel ellipsoid radius give by equatio (. The h, the height of the secodary obstacle from the sub-path MR, is used to calculate the loss caused by this secodary obstacle. A correctio term T c (db must be subtracted, i order to take ito accout the separatio betwee the two edges as well as their height. T c (db may be estimated by the followig formula: T c p q 1 log a p ( 1 π with: p λ ( a + b + c ( b + c a 1/ h 1 q λ ( a + b + c ( a + b c 1/ h b( a + b + c ta α ac 1/ ( h 1 ad h are the edge heights from the direct path trasmitter-receiver. The total diffractio loss is give by: L L L ( The same method may be applied to the case of rouded obstacles usig I cases where the diffractig obstacle may be clearly idetified as a flat-roofed buildig a sigle kife-edge approimatio is ot sufficiet. It is ecessary to calculate the phasor sum of two compoets: oe udergoig a double kife-edge diffractio ad the other subject to a additioal reflectio from the roof surface. It has bee show that, where the reflectivity of the roof surface ad ay differece i height betwee the roof surface ad the side walls are ot accurately kow, the a double kife-edge model produces a good predictio of the diffracted field stregth, igorig the reflected compoet. T c. Geeral method for oe or more obstacles The followig method is recommeded for the diffractio loss over irregular terrai which forms oe or more obstacles to lie-of-sight propagatio. The calculatio takes Earth curvature ito accout via the cocept of a effective Earth radius (see Recommedatio ITU-R P.,.. This method is suitable i cases where a sigle geeral procedure is required for terrestrial paths over lad or sea ad for both lie-of-sight ad trashorizo. A profile of the radio path should be available cosistig of a set of samples of groud height above sea level ordered at itervals alog the path, the first ad last beig the heights of the trasmitter ad receiver above sea level, ad a correspodig set of horizotal distaces from the trasmitter. Each height ad distace pair are referred to as a profile poit ad give a ide, with idices icremetig from oe ed of the path to the other. Although it is ot essetial to the method, i the followig descriptio it is assumed that idices icremet from the trasmitter to the receiver. It is preferable but ot essetial for the profile samples to be equally spaced horizotally.

17 Rec. ITU-R P.- 1 The method is based o a procedure which is used from 1 to times depedig o the path profile. The procedure cosists of fidig the poit withi a give sectio of the profile with the highest value of the geometrical parameter ν as described i.1. The sectio of the profile to be cosidered is defied from poit ide a to poit ide b (a < b. If a + 1 b there is o itermediate poit ad the diffractio loss for the sectio of the path beig cosidered is zero. Otherwise the costructio is applied by evaluatig ν (a < < b ad selectig the poit with the highest value of ν. The value of ν for the -th profile poit is give by: ν h d / λd d ( h h + [d a d b / r e ] [(h a d b + h b d a / d ab ] h a, h b, h : vertical heights as show i Fig. 11 d a, d b, d ab : horizotal distaces as show i Fig. 11 r e : λ : effective Earth radius wavelegth ad all h, d, r e ad λ are i self-cosistet uits. ab a b (a The diffractio loss is the give as the kife-edge loss J(ν accordig to equatio (1 for ν >., ad is otherwise zero. Note that equatio ( is derived directly from equatio (1. The geometry of equatio (a is illustrated i Fig. 11. The secod term i equatio (a is a good approimatio to the additioal height at poit due to Earth curvature. FIGURE 11 Geometry for a sigle edge Poit Poit b h Poit a h h b h a Sea level Earth bulge r e d a d b d ab -11

18 1 Rec. ITU-R P.- The above procedure is first applied to the etire profile from trasmitter to receiver. The poit with the highest value of ν is termed the pricipal edge, p, ad the correspodig loss is J(ν p. If ν p >. the procedure is applied twice more: from the trasmitter to poit p to obtai ν t ad hece J(ν t ; from poit p to the receiver to obtai ν r ad hece J(ν r. The ecess diffractio loss for the path is the give by: L J(ν p + T [ J(ν t + J(ν r + C ] for ν p >. (1a L for ν p. (1b C : empirical correctio D : total path legth (km C. +.D ( ad T 1. ep [ J(ν p /. ] ( Note that the above procedure, for trashorizo paths, is based o the Deygout method limited to a maimum of edges. For lie-of-sight paths it differs from the Deygout costructio i that two secodary edges are still used i cases where the pricipal edge results i a o-zero diffractio loss. Where this method is used to predict diffractio loss for differet values of effective Earth radius over the same path profile, it is recommeded that the pricipal edge, ad if they eist the auiliary edges o either side, are first foud for media effective Earth radius. These edges should the be used whe calculatig diffractio losses for other values of effective Earth radius, without repeatig the procedure for locatig these poits. This avoids the possibility, which may occur i a few cases, of a discotiuity i predicted diffractio loss as a fuctio of effective Earth radius due to differet edges beig selected.. Fiitely coductig wedge obstacle The method described below ca be used to predict the diffractio loss due to a fiitely coductig wedge. Suitable applicatios are for diffractio aroud the corer of a buildig or over the ridge of a roof, or where terrai ca be characterized by a wedge-shaped hill. The method requires the coductivity ad relative dielectric costat of the obstructig wedge, ad assumes that o trasmissio occurs through the wedge material. The method is based o the Uiform Theory of Diffractio (UTD. It takes accout of diffractio i both the shadow ad lie-of-sight regio, ad a method is provided for a smooth trasitio betwee these regios. The geometry of a fiitely coductig wedge-shaped obstacle is illustrated i Fig. 1.

19 Rec. ITU-R P.- 1 FIGURE 1 Geometry for applicatio of UTD wedge diffractio Φ π s 1 s Φ 1 Source Field poit face face π -1 The UTD formulatio for the electric field at the field poit, specializig to two dimesios, is: e UTD ep( jks1 s e 1 D ep( jks ( s1 s ( s1 + s e UTD : e : s 1 : s : electric field at the field poit relative source amplitude distace from source poit to diffractig edge distace from diffractig edge to field poit k : wave umber π/λ D : diffractio coefficiet depedig o the polarizatio (parallel or perpedicular to the plae of icidece of the icidet field o the edge ad s 1, s ad λ are i self-cosistet uits. The diffractio coefficiet for a fiitely coductig wedge is give as: D ep ( jπ/ πk π + ( Φ Φ + cot 1 F( kla ( Φ Φ1 π ( Φ Φ + cot 1 F( kla ( Φ Φ1 π ( Φ + Φ + R cot 1 ( ( Φ + F kla π + ( Φ + Φ + + R cot 1 F( kla ( Φ + Φ 1 Φ1 (

20 Rec. ITU-R P.- Φ 1 : icidece agle, measured from icidece face ( face Φ : diffractio agle, measured from icidece face ( face : eteral wedge agle as a multiple of π radias (actual agle π (rad j 1 ad where F( is a Fresel itegral: t t F d ep( j ep(j j ( ( π t t t t d ep( j j (1 d j ep( ( The itegral may be calculated by umerical itegratio. Alteratively a useful approimatio is give by: ( π d j ep( A t t ( + < otherwise j ( j ep( if j ( j ep( j 1 ( d c b a A ( ad the coefficiets a, b, c, d have the values: a +1. b -. c +. d +.11 a b 1 +. c 1 -. d 1 +. a -. b -. c +. d -.11 a -.1 b -. c +. d +. a +.1 b -. c +. d +.1 a -.1 b +.11 c -.1 d +.11 a -. b -.11 c +.11 d a -.1 b -1.1 c -. d +. a +.1 b -. c +. d -. a -.1 b +.1 c +. d +.1 a -. b -.11 c -.11 d -.1 a b c d s s s s L + ( β π β ± ± cos ( N a (1 1 β Φ ± Φ (

21 Rec. ITU-R P.- 1 I equatio (1, ± N are the itegers which most early satisfy the equatio. N ± β ± π π ( R R, are the reflectio coefficiets for either perpedicular or parallel polarizatio give by: si( Φ η cos( Φ R ( si( Φ + η cos( Φ b si( Φ η cos( Φ R ( b si( Φ + η cos( Φ Φ Φ 1 for R ad Φ ( π Φ for R η ε r j 1 σ / f ε r : σ : f : relative dielectric costat of the wedge material coductivity of the wedge material (S/m frequecy (Hz. Note that if ecessary the two faces of the wedge may have differet electrical properties. At shadow ad reflectio boudaries oe of the cotaget fuctios i equatio ( becomes sigular. However D remais fiite, ad ca be readily evaluated. The term cotaiig the sigular cotaget fuctio is give for small ε as: π ± β ± cot F( kla (β [ π kl sig(ε klε ep(jπ/ ] ep(jπ/ ( with ε defied by: ε π + β + πn for β Φ + Φ1 ( ε π β + πn for β Φ Φ1 ( The resultig diffractio coefficiet will be cotiuous at shadow ad reflectio boudaries, provided that the same reflectio coefficiet is used whe calculatig reflected rays.

22 Rec. ITU-R P.- The field e LD due to the diffracted ray, plus the lie-of-sight ray for ( Φ Φ1 π, is give by: < ep( jks e + for Φ < Φ + π e UTD 1 LD s ( eutd for Φ Φ1 + π s : straight-lie distace betwee the source ad field poits. Note that at ( Φ Φ1 π the d cotaget term i equatio ( will become sigular, ad that the alterative approimatio give by equatio ( must be used. The field stregth at the field poit (db relative to the field which would eist at the field poit i the absece of the wedge-shaped obstructio (i.e., db relative to free space is give by settig e to uity i equatio ( ad calculatig: s : s e E log UTD UTD ( ep( jks straight-lie distace betwee the source ad field poits. Note that, for ad zero reflectio coefficiets, this should give the same results as the kife edge diffractio loss curve show i Fig.. A MathCAD versio of the UTD formulatio is available from the Radiocommuicatio Bureau. Obstacle surface smoothess criterio If the surface of the obstacle has irregularities ot eceedig h, where R : λ : h. Rλ m (1 obstacle curvature radius (m wavelegth (m the the obstacle may be cosidered smooth ad the methods described i ad. may be used to calculate the atteuatio.