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Advanced Remote Sensing Measurements Methods

by: Ms. Alene Howell

Advanced Remote Sensing Measurements Methods FOR 570

Marketplace > University of Idaho > Natural Resource Ecology And Mgmt > FOR 570 > Advanced Remote Sensing Measurements Methods
Ms. Alene Howell
GPA 3.99


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This 45 page Class Notes was uploaded by Ms. Alene Howell on Friday October 23, 2015. The Class Notes belongs to FOR 570 at University of Idaho taught by Staff in Fall. Since its upload, it has received 30 views. For similar materials see /class/227836/for-570-university-of-idaho in Natural Resource Ecology And Mgmt at University of Idaho.

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Date Created: 10/23/15
Introduction to LiDAR What is it How does it work LiDAR Jargon and Terms Natural Resource Applications Data Acquisition Standards Readings Western Forester April 2008 a Lidar What is it 1 Qght Detection and Ranging Essentially a laser rangefinder that has been strapped to the belly of an airplane L g The time for the light to travel to W W m m l and from thetarget is used to determine distance Distance Speed x time This distance and the position of the airplane is used to get tion and location 339 Lidar What is it Lidar Light Detection and Ranging The Basic Lidar Concept Speed distancetime d C39t d Distance meters t time seconds c speed of light lea What Is I Creauon of me Lwdar pu se L da What SIeps m the Lwdar Process Creauon of me Lwdar pu se Lwdar pu se trave s to the target SIeps m the Lwdar Process Creauon of me Lwdar pu se Lwdar pu se trave s to the target mteracnon wxm me target Lidar What is it Steps in the Lidar Process Creation of the Lidar pulse Lidar pulse travels to the target Interaction With the target Lidar pulse travels back to sensor Lidar What is it Steps in the Lidar Process Creation of the Lidar pulse Lidar pulse travels to the target Interaction With the target Lidar pulse travels back to sensor Sensor processes return signal Lidar What is it Lidar uses LASER light The light emitted by a laser occupies a very small The intensity ofa laser can exceed the sun 1mm perm Lidar What is it How does a LASER produce light h ttg39 lregairfag ecedrexel edusamlCORDleothourse01 mo d01mod01 ml How do we produce the Intense LASER light onlaser Available Lidar Light Wavelengths Source Lefsky 2005 Lidar The Pulse As the lidar pulse travels to the target the light fans out as the distance 39om the target can be several kilometers Lidar footprint Height x divergence The footprint is the effective area that the laser iight encornpasses Divergence is the degree ov Which the iight fans out frorn a straight iine measured in radians i rad degrees Tvpicai divergence 0 2574 rnradians per mourn Source Lefsky 2005 Lidar The Pulse Targai Inlemclioii mm ii W Source Lefsky 2005 Lidar The Pulse Low Divergence Canopy penetration and some pulses will reach the ground High Divergence Reduced canopy penetration and low percentage of pulses hitting and RETURNING from the ground Source Lefsky 2005 Lidar The Pulse Low Spacing Canopy penetration and some pulses will reach the ground High spacing Less pulses hitting and RETURNING from the ground Source Lefsky 2005 lLidar The Main Kinds lnvl Iiimmmim gm rimtermum 1 ii cum LN Rtmrn Lli mm Two main types Waveform Sampling Discrete Return Source Lefskx 2005quot quot1 Ai s it 1 gtll Each pulse of laser light contains a large number of photons Afew of these photons return to the sensor The 1St return might be a tree top While the last return could be from the ground It is important to note that The 1St could also be the last return The Last return might not be the ground Lidar The Main Kinds Nurmullml Z unmlutnc Rcturn lzn 39 U 02 04 LO 08 n I lt Unumu I m cl llmc 1n nunuwmnth Kclurnllncigy inlglllzvl uunm Blair and Harding NASA GSFC Lidar the Main Kinds quotrefill Sometimes the last return can be a false summit in the r signal This results in noise lanair in the resultant height data seen lt lil asdips and peaks within the lidar image quot5 1 7 trv vlquotv1uin litr nv lain Source David Harding NASA GSFC Lidar the Main Kinds Advantages of Waveform Lidar No signal processing errors 39 Enhanced ability to characterize canopy information over large areas 39 Global satellite datasets available 39 Compatible with other RS global datasets Advantages of Discrete Return Lidar 39 High spatial resolution 005200 m 39 Small diameter footprint 39 Flexibility in available data processing methods 39 Highly available Lidar What the Data Looks Like 5 J a Jt Lidar Geo morphologic Applications Volume change Utilities map power lines for signs ofdamage ms ngnghes M4 have i ToeJam Hill faultscarp Restoration Puinl lear Underwater DEMs for Coastal Mapping M mage source HEAnderson H and quotl LiDAR Data Acquisition Standards 1 it i Although the use of LiDAR is widespread in forestry people are inconsistent on how they collect the data If we want to compare measurements between different areas we need the data to be collected using standard properties 4 V j 1 One day you may be asked to get a LiDAR acquisition for your forest so its important that you know what to ask for 24 Source Evans et al PEampRS in review LiDAR Data Acquisition Standards for Forestry Pulse Repetition Frequency number of pulses per second This should be high enough so that the pulses are well distributed vertically throughout the canopy Number of Return When using Discrete Return LiDAR E for at least 3 returns per laser pulse Source Evans et al PERS in review LiDAR Data Acquisition Standards for Forestry PostSpacing Average horizontal spacing between pulses may be multiple returns per pulse Ask for a 39 post spacing of 70 cm If after shrubs or seedlings ask for a post spacing closer to 15cm Source Evans et al PERS in review 1O LiDAR Data Acquisition Standards for Forestry ScanAngle The maximum o nadir angle the sensor head swings to High scan angles carj distortthe LiDAR footprint worse on slope Ask for a 39 scanangle of 12 The total view angle is then 24 Source Evans et al PEampRS in review LiDAR Data Acquisition Standards for Forestry Flight Line Overlap This ensures fe t39tlreksm39are well sampled Ask for a flight lihe overlapIo i BOPQH When to Collect Data Avoid bad weather or snow unless you are snow modeling Do you want leaf on or leafoff data Source Evans et al PEampRS in review LiDAR Data Acquisition Standards for Forestry Accuracy Standards Vertical Root Mean Square Error lt 15cm Horizontal Root Mean Square Error lt 55cm 21 Source Evans et al PEampRS in review 11 LiDAR Data Acquisition Standards for Forestry Typical Lidar Products to ask for Ground Surface Model Digital Elevation Model Digital Surface Model surface of all non ground returns Intensity aerial Photograph Point Heights DSM DEM Canopy Height amp Density Suurce Evans et al PESRS lri review Advanced Me hods in Remote Sensing Lectures 912 Wavelet Analysis Part 1 Background to Integration and Convolution Recommended Reading FrWin vi i9 Loin ii ii gt Avery good and not too advanced generai rnatnernatios textbook Summary This lecture will introduce you to the concept of wavelet analysis and how it might be useful to remote sensing and ecological problems Essential Mathematics Wavelet analysis relies on a solid foundation in mathematics Therefore we will take the time to refreshintroduce you to integration Convolution if vod are interested in pursuing tnis sdoieot past tnat described in tnis lecture ei can point vod in tne direction of books and papers Remember Mathematics is all about rules and tricks lmaammu What n m m 835m mm mnvmmn XquotVn mu m an m mm the Rue a Imqmm pm m n Wu mm Emmaquot 5 mm m m mm mm m be mmmuywma 25m MW mm gltxltx hltx Luwww am WW unzswwsmwm mmmmummmmam mam mumWM WWW Mum go 2mm WNWmnmmmmwmmx m u m p I nu mu What About Correlation Correlation is similar to convolution EXCEPT you do not llip or reverse the function in space got fx hx fmfshsixds Where hx is the complex conjugate of hx But WHAT DOES THIS MEAN rrelation again rneasures tne AREA of overlap between one function iv and anotner function gx Errectivelv A Measure of tne direct urtransformed sirnilaritv of tne two functions Convolution Example Picture Math 1 l I I ie1 01 00 1 ie tne convolution of ie11112 two toprhat functions is 1 a big Splky triangle ie1 01 00 39 V I 1 Z l I Convolution Example 2 Picture 9 i e tne convolution ofatopnhat0110mnctlon and a triangle 0 42 l 42 o l 0 is a oroaoeneo ano sligntlv larger latteneo triangle cl 1 1 2 In general Convolution oftwo identical objects big spike J Edgenaajlm Wamvmual My Dram 4 n n n n 1 1 1 1 mm I AEwmmu mum m H mm Q mo Step1 212 2 Step 3 mm m mess by mmmv m mm W H mm mm he We EdgeDefechan Dampe NAun Vs Hunzn Edges EdgeDefechan Dampe EdgeDefechan Dampe mm vs Hmnn Edges EdgeDefechan Dampe Advanced Me hods in Remote Sensing Lectures 912 Wavelet Analysis Part 2 Background to Fourier Analysis Periodic Functions A function fx is periodic over all values of x if for a given c n ant C X CfX c is caiied tne period or x Common Exampies or periodic runctions are sine and cosine Waves 7 2p are periodic iunctions Both COS Wand Sii i mg Where The Basics of Fourier SynthesisAnalysis Mathematical discipline began is 1807 with Joseph Fourier The iust or Fourier Synthesis is tnat any 2p periodic based iunction can be or cosine Waves oryarying rreouency broken up into a set of sine i e in Matn any 2p periodic based iurictioi i can be expressed in tne form a 2mm 2mm x n cos 17 cos f 2 ZR y X y X Where 2 X as L mm X Tne series is caiied a 2 X 271m FounerSenes a I fxcos dx X n X Tne pararneters are caiied Fourier Coefficients Zmdx X fmin Freq uencyScale Analysis Fourier Synthesis was the start of functional multiscale athematics ie the analysis of functions fx that vary in 39 m scale slze Fourler analysls allowed functlol39ls to be analyzed over a senes of scales Tnls lr ollen called frecluel39lcy analysls gt or Scale Analysls The Steps 7 Create a lunctlon rm 7 Compare rm to another rdnctlone say gm 7 Use tnls cornoanson to approxlmate the snaoe ofgx e cnange the slze of rm and compare ll agaln to gx 7 Repeat the procedure overa senes of SlZeS of m Using Fourier Series to Approximate a Function Conslderthe furlotlol39l 7k when 77 ltx lt0 k and fltx27rgtfxgt when 0 ltx lt7 fXgt Thls l5 5 SQUARE WAVE and has the folloWll39lg Shape Apply Fourier Analysis to the Square Wave AS the Area ul ldel the wlve l5 Zero then an 0 Calculatlrlg aquot uSll lg the trlgorlometry relatlorl sln nx 0 1 r d aW i xkosnxx n r 1I kCOSVleXIkCOSnXdX n I 1 sin nx D sin nx 7 k k 0 7 n 77 n D Wymnmmsmm Square mm 3mm wwwmwmu a mmm IXsmmdx 1 as w wvm m Kaufmanhm Lmk LII n mmmm n aunEnNmeIg wumm u mg a mmxme My 1 mm m Annmm 1 sum necnl39WIImu Nmse mm v mum ma auamnmnxhgenuhunmn u mmaurraumwumwmnmngmmm mumsva Advanced Me hods in Remote Sensing Lectures 912 Wavelet Analysis Part 3 Introduction to Wavelet Analysis From Fourier Series to Wavelet Analysis In Fourier Series you decompose a periodic function into a linear combination of sine and cosine basis functions Hovvever VWv does your so called basis function have to be sinusoidal7 7 They don ti Thevlust need to be orthogonal This is the idea behind wavelet analysis ln l909 Haar in his PhD thesis rst hinted that discrete orthogonal basis functions or Wavelets could be used to decompose a periodic fundion ln the l930s Levy used the Haar Wavelet basis to investigate Brownian Motion These days wavelets are used in a wide variety of scienti c elds including medicine astronomy and remote sensing Wavelet Analysis The Basics ln wavelet analysis you use convolution to decompose your signal or image using discrete operators of increasing sizes called the wavelet basis functions t The Wavelet basis rnust have i Finite Energy Erlwtl2dtltoo 2 The Wavelet Basis rnust have a rnean orzero W0 0 3 The Fl must be real and Zero fornegative frequencies mm a Flumm m mm m pamcmar wam u has s mmquot was an m appncmn m mm A 5mm be chasequot m m n Pvmnmmesme me am abyensahmnrei 8mm u an um mome m mm w m wmrwwm u mmxmmmxm momma m avmmumm m mm mme 7 x u 1 mm um mlgmmnmmval u u mumm aw Vw niltum Ind tumum M u mxlnlnuamilglummlx 3911 a mum mm mm b mmm mmnmmmmm n n m quot31 mu m mm 1 mm mm 0 quotmy m mu m u m mm mmm mmm mm u m m mum mum I A afnaQy mm a Fund ummuw queuwvl mm 1 mum mummn uwwumwg Wavelet Basis are Band pass Filters When the wavelet basis meets condition 2 it also meets the criteria for a BAN DPASS lter Resuit gt The basis at a certaih siZe ohiy iets through the sighai corhoohehts Withiri a certaih rah e of frequericies i e features that are ofa sirhiiar siZe Which are oehheo by the ehergy spectrum eriergy spectrum ergy at each frequency yaiue The resuit is that Wheri you corNoiye the Wayeiet basis With a sighai ohiy those features of similar size to the Wayeiet basis are processed in the corNoiution Remember frorh conyoiution Convolution oftwo idmtical objects blg spike Convolution oftwo nonridmtical objects more attened object A Question of Scale In all forms ofwavelet analysis you convolve your wavelets of increasing size with your signal or image Ahy features that are of the same siZe aho shape to your Wayeiet basis furi tiOri vyiii oe highiighted by a big spike Therefore Wavelets highlight features of a similar scale There exist rhahy differerit shapes ofwayeiet basis Mathematically Speaking The equivalent of 2D Analysis Tomb IWXryVyaydxdy 2D CWT 20 W as Tmymummy w m Advanced Me hods in Remote Sensing Lectures 912 Wavelet Analysis Part 4 Wavelet Analysis Methods The Different Types of Wavelet Analysis There exist several different ways to use wavelets The most common include Wavelet Decomposltlol l The Wavelet Decomposltlol l Slgl lalul e 7 Texture Ahalvsls USll lg the Wavelet lhtehsltv to match feature SlZeS The Wavelet Vanal lce Ful39lCllol39l Wavelet Decomposition By convolving a signal or image a series ofwavelet bases of increasing size and therefore decreasing 39equency you p roduce a wavelet decomposition El 9 Highlights Small Features E Medium Features D Q Highlights Large Features Highlights 7 V Small Features are in general noise while large features are broadscale effec s Wavelet Decomposition The Method the smallest wavelet size convolution or zero level highlights the high 39equency features noise you can repeat the denoising technique we looked at in Fourier Analysis le vou can remove or lsolate the slghals produced bv the Wavelet decompOSlllOl l Some soltware packages l e DL s Wavelet Toolkltachleve thls result bv lettlhg vou select hoW rhahv ehergv coef clel lls vou waht to retah Th hurhberot ehergv coettlclehts ls glveh vaN Where N ls the number ofcol lvolullol39l levels new nmmnmnn sunI mmwaymmm Wm gamma 5 WWW m m dassmuuans E ui i3931 ll h 39 Em 39 755353 may mm mm humming Mammy mm 9 Wm p u an quotan H hgual mm queuxlgnmv a W w m n u mmmmwm an n mum my a c mmpnmmmmmwmxmmmm y 9mm x ummmmmmmm wuaaznunwlmmmy m m mm mem mm mm Wm H aquot We km 5 W m 5 M mm mm m Wm M WW 3 mm W Mummy hymn WWMWmummuwu autumn1ang ummwm val1m mm magnum mum u my mm nee Resumhers m usedme mm Mme mm mm m vawM vmemauvwvamm smele ana vle m u mmmmpm m a u an n EnemaIguzmm D n u wwwmmmm uavlm mum m wumw mmmm Group Exercise 1 In this exercise you as a group will consider how you could possibly use wavelet analysis to investigate Temporal Trends in Land or Sea Surface Temperatures SST LST The group is to consider 1 VWat sort of data could be used to asses SST or LST7 2 VWat are posslble factors that varythe short arld lorlg term LST7 3 How could Wavelets be used to assess tne periodicity of tnese ditrerent cycles7 4 VWat sort of results would you expect ie are tne trends hourly dally Weekly monthly yearly decadal etc 7 Explain your reasorllrlg Alter your discussion i would like you to nominate a person who hasn t preyiously spoken in a class exercise to comment on eacn question Group Exercise 2 In this exercise you as a group will consider how you could possibly use wavelet analysis to determine the endmember proportions within a hyperspectral pixel The group is to consider l Wnatmixing assumptions Will you make 2 VWat aspect ofwavelet analysis could you useVr explain your possiole method 3 Would you expect Wavelets to perform better tnan mixture modeling 7 it sorlot Why Alter your discussion l would like you to nominate a person who hasn t preyiously spoken in a class exercise to comment on eacn question Advanced Me hods in Remote Sensing Lectures 912 Wavelet Analysis Part 5 Ecological Applications ofWavelets Synthesis of research conducted by E Strand S Garrity L erling M Falkowski AMS Smith et al mm Fanquot nau mn quotIn Wm WWW Nmse areamreswnmn 5w Hwersvmm m mswv mm m nebmae Can waveth he usedm amammma wdete We mm and wavm mm ameew me Melmd 3mm cm a We mm mm mm Wm M mmquot 9 w 1 i 9 g 9 9 M m mm 2 me Melmd 212 2 In mnnwvexu a mm Inav ummvman me Melmd 212 3 H mm We snipe m Havenammmmuwxmuu sn anulylxwp mm The Method Step 4 Increase the size ofthe Hat and repeat the process Shape Image Score File v 39 7 39 l it increase x and Y Tg 1 Size The relative goodness of t is recorded The Method Step 5 Keep increasing the Hat SiZe and redo the process mm you reach the Maximum iikeiytree SiZe 39 Local Maximums givetree locations Max score gives objectwidth The Final Output Marked Point Pattern XY center location of individual Objects Estmv mum Diameter 45mm 47mm 5 4 45mm 476515 a 4 45259 4755234 2 3 45136 4755ng m 2 451567 47mg 4 4 Output table in MATLAB You can use a GIS to project the tree widths as circles around each point Wmquot M mm m m vhmnv vh mm mm m we upmnsgmm mm mam lt PM mumquot mm m Envy me mm wvx aquot 39 in Emmy m awwvx m m theve mmw ommmum NW accumw s Wquot mmmwam NW mm mm mm mhermethads mm m WWW vim 1 areth mm m wmsm em om Apphcmmns mm mgmmn Cmp ems m Landsat Em magew Where mm omens What sthe cww Owen and Ne ghbarnaad SpanaLSpema Ana vss I V nameum m unnn mum um AangmeMzmnn mm mm m mm Mam Wave ets mmpavedm Fwe d measured wavm may Detectame sue Wax2M nahS s M ubym mmquot mvzvxmmmmeuammay maan mm reanaaunananumau mm awn mum in awnWNW u mm m muaws quot 321933meIhm3mvwgmw t mm W mp u menwmmmmm V mm mmmu hlgaxmamk mm mum a it H mm mwgmm Supumllmumeuuclmluun butmequ n7 Ixugnevnvzmxmne nemwxnm Sun a mmmmmu mmmw a mum hm mum sum m Wanna Mu mm Wu m m mums n my mm mm mgmmas ssmmsm ca ed Svenm Mmure Wm SMA Wessm 113 m7 Mmure mm W Dike a 3 mm m m s msua u m Inwuguayuw m an nxemumxnmum mlngemxmnngam m m Avxs magw m1 mm a w 275 mm mm cnmk Ign m1 mum amymugvnux u mm m gxwmmm anvil Ewhguvvvhumnx 12 we mums m um Sauna mg mw mg a39yusl m wmvanem w r Common Subpixel Analysis Techniques Linear Spectral Unmixing SMA or MM Fuzzy Classi cation Additional Hyperspectral SubPier Techniques Linearly Constrained Minimum Variance Automatic Subpixel Detection Orthogonal Subspace Projection OSP Linear Spectral Unmixing Eacn surrace component Witnin a pixei is sufficiently iarge enough sucn tnat no muitipie scattering exists between tne components singer and McCord 1979 The iinear scattering approximation is valid Wnen tne size or tne pixei is smaiier tnan tne typical patch orcomponert oeing senseo e i e iinearmixing occurs at tne macroscopic scaie The Horwitz 1971 Linear Mixture Model For an image Witn N bands 0 oirrerent coxertypes x txai xzi xNy observed image values in tne i t bang and a m r mi proportion or eacn pixei Witnin eacn covertype c The Linear Mixture Mooei is oerineo as x Mr e Wnere M is an ri x 0 matrix The columns ofthis matrix represent the Spectra ofthe different eridrnernbers Source Drake et al 1999 If M and N are known then the fraction term f can be estimated using a least squares solution This least squares approach seeks to estirnate r by rhiriirhiZirig the following wow wilt Subject to the following constraints 0 lt l CONSTRAlNTl t1 t2 tquot CONSTRAiNT 2 Researchers have hoted that 2 is easy to implement butl i5 riot As a result rnost studies only implement the 2nd Constraint in the model and then applythe ist constraint to the results Assumptions Using linear spectral unmixing relies on four assumptions Settle and Drake 1993 which are There is no signiricant occurrence or rnuitipie scattering between the oirrerent surrace components E cornponent Withiri the image has surricient spectrai contrast to allow theirseparation in each pixel the total land coveris uriity Each surrace cornponent endrnernber is known Ste ps The steps involved in using linear spectral unmixing i Convert the imageryto retiectance preferably ground 2 Eridrherhber Collection 3 Chose your Constraints 4 Normalize the Fraction maps 5 interpret yourResuits mum use hnearmmure mamth 5 a need an meme m 5mm releuanee m we Pure a members dea vvmund mm mm mm be mm mm mm endmembers m endmem W M w 5v hers uken m w mav wmm mump e suf ce wrvv mm wuve Hawemr m mm m be mmmxed mmuvh mm m wave a members m Wm m mm m mann New mm ms mamas mm was WWW acmansmsmddunnv mm lmwimmcnllajlm A1ng mummy m mummmpm Human ummrmwna Iaumammemmn mmth um mm Ylwammannmm In xmmmumwmmmmwmpm v III9h Wm vmumamhwnu mm Cm nllls cam m m m lfmlv gram3 m 1 mm at am mmw usuhese MIPS w needm mmmm m canslmm m ummmmmemam Rdnlvefndmnlhvs mummy mm D wuemmu vames mm H H mm 3 a sumquot Hnmxlm unearsvem mm 5 3 MN m Wham mm mm m my m quotMM mm m mm mm an a 5m smace cmmm menus w scams wnh mump e wmvanems heme bemv mm mm mm mm 1m mmmm Nun mummmg g anezwsquot g gull11mm nun rsmmixl u mummy InnnleMnmns mm Slmnwu mce m mm mmmm mam 5 My mm m mum ms m m m mm mm mm m a nu 55mm W umwm mmm 99W ckmmcm mumsmm HI BE WEE Kulnrllnlgi mm WeermMm mme Mnure massc Nan unearM xmv 5mm NW 5 m madden Nmr ursvmnl nmxm mummy AFew Aswmm e g mm m lmmbmx n5m mums m mm Scmrers Wm m mixture uuned m e swung Warm mm mm manly u m a lab we venemvaeasAm mm A m mmm mmm H mm Sam and Senst Wman mm vaaneam e gtVllhzlwemeasnemahb a mama mmm m AM m be Mua Me My Releanee r 25 N an an varaswahekrence pm an r Hapke Theory lll Hapke 1993 describes the parameter ru as called the relative re ectance Tnis is deined as tne rellectance relative to tnat of a standad sunace consisting of an innnitelv thlcllt particulate medium of nonrabsorblngi i e A Spectralon Panel The reason We like rquot is that it alloWs us to make use of a very useful theoretical measure called the single scattenng albedo W w17y Hapke Theory lll Where The parameter ofthe combined mixture can then be obtained using Hapke 1993 w rz lwuru r39 grwl v1quotn AMI The single scattering albedo is tne probabilitv tnat given an interaction between tne photon and an individual particle tnat tne particle Will be scattered ratherthan absorbed Hapke Theory IV The reason we use w is because the nonlinear mixing effects of the re ectance can be linearly modeled with w using equations developed by Hapke 1981 1993 Hapke developed several equations for different mixing scenarios These include intimate mixtures horizontal and vertical layers etc For our intimate mixture example We have a BlNARY mixture and can use the following equations to model the combined mixture re ectance W1 W2 M1p2 Dz w 18 M2 Pi D1 M bulk densitv 7 solid densitv Assumed bv Hapke i993 to be tvoicallv equal D diameter ofthe tWo particles Hapke Theory V The bidirectional re ectance Gwhat we measure in the lab can be measured via the following equation 2 1 F g lt1294uux1294ugt VWere p 005 e and u 005 a w and e denote the ang es o ncwdence and re ecuon respectwe y


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