GNSS-R as a source of opportunity for remote sensing of the cryosphere

5 REMOTE SENSING OF DRY SNOW This Chap e is dedica ed o analysis o he use o GNSS-R o emo e sensing o hick d y snow masses. Mo i a ed by he p elimina y esul s shown in p e ious Sec ion 3.3, whe e he e lec ed wa e o ms o An a c ic d y snow appea o be composed by su ace and sub-su ace con ibu ions, Sec ion 5.1desc ibes he o wa d model de eloped o simula e such beha io and hen o in e in o ma ion om he da a such as he dep h om he con ibu ing laye s. The esul s ob ained owa ds his pu pose and u he wo k a e shown in Sec ion 5.2. Table 19 p o ides a lis o publica ions issued om he s udy p esen ed along his Chap e . Mos ele an no el y wi h espec o p e ious GNSS-R s udies: Empi - ical emo e sensing o deep d y snow laye s based on mul iple e lec- ions. Ti le Re e ence Moni o ing sea ice and d y snow wi h GNSS e lec ions Fab a e al. (2010) An empi ical app oach owa ds cha ac e iza ion o d y snow laye s using GNSS-R Fab a e al. (2011a) GNSS Re lec ome y o he emo e sensing o sea ice and d y snow Fab a e al. (2011d) GNSS-R o he Re ie al o In e nal Laye s’ In o ma ion om D y Snow Masses Fab a e al. (2011c) Cha ac e iza ion o D y-snow Sub-s uc u e using GNSS Re lec ed Signals Ca dellach e al. (2012) Sun e lec ions o An a c ica’s snow sub-s uc u al laye s In p epa a ion o Geophysi- cal Resea ch Le e s Table 19.: Lis o publica ions a isen om he wo k p esen ed in his Chap e . 137 emo e sensing o d y snow 5.1MODELING AND PROCESSING GNSS-R OVER DRY SNOW: A NOVEL APPROACH 5.1.1PROPERTIES OF THE REFLECTED SIGNALS The ecei ed e lec ed signals om his expe imen p esen cohe ence imes longe han 1 second. This is a e y long cohe ence ime compa ed o GNSS signals e lec ed o o he ypes o su aces. Fo ins ance, e lec ions o he ough ocean p esen se e al millisecond cohe ence only. Pa o his long cohe ence ime can be unde s ood by he ac ha he ecei e c oss-co ela es he e lec ed signals wi h a signal model ha includes eal- ime in o ma ion ob ained by he di ec adio-link. In o he wo ds, he ecei e ends o s op he e lec ed signal using he dynamics o he di ec one. The o he ac o o unde s and he long cohe ence is he ac ha he snow su ace is e y smoo h and he sca e ing is essen ially specula . The snow su ace opog aphy in he a ea is essen ially la , wi h global slope <0.2◦, and oughness cha ac e ized by e y long au o-co ela ion leng h (compa ed o GPS elec omagne ic wa eleng hs) and ∼1cm RMS e ical dispe sion (Pe i e al., 1982; Six e al., 2004). Tha is, he su ace oughness is no la ge enough o induce di use sca e ing, which would ha e in oduced luc ua ion in he phase. Specula e lec ions end o gene a e wa e o ms wi h he shape o he signal modula- ion’s au o-co ela ion unc ion, wi h no u he de o ma ions (in opposi ion o di use sca e ing). The ecei ed wa e o ms, ne e heless, do no show he expec ed iangle shape om he GPS C/A code (2ρC/A ≃600 m wid h in he space domain), bu a se ies o dis o ed iangles, wi h added ails and seconda y peaks. Mo eo e , hese shapes change g adually in ime, as displayed in Figu e 74. We can see ha he e lec ed wa e- o ms a e no cons an , bu oscilla e, and ha hese oscilla ions di e be ween dis an lags. An example o he ampli ude oscilla ion pa e n ound is gi en in Figu e 75: a sho ime se ies is chosen o pe cei e he high a e componen s o he in e e ence and hei epea abili y. I shows a sequence o 1-second in eg a ed ampli udes o wo di e en lags and ou di e en days: lag 22, which app oxima ely co esponds o he peak o he di ec wa e o m (nominal ze o delay); and lag 37, delayed by ∼225 me e (see Figu e 74 o loca e bo h lags wi hin he wa e o m). The oscilla ion pa e ns a lag 22 and lag 37 p esen simila i ies, bu do no pe ec ly ma ch wi h each o he . Fu he delayed lags end o sligh ly inc ease he a e: double peaks appea some imes whe e only one peak was de ec ed in lag 22. Fo example, i happens a ound ele a ion ∼44.65◦,∼44.9◦, o ∼45.28◦. We could hink hose a e e ec s o he noise (lowe SNR le els a he end o he ailing edge, lag 37), howe e , some o hese new peaks ha e signi ican SNR le els, and a hin o hem seemed o eme ge in lag 22. Mo eo e , all days p esen he same pa e ns, hus sugges ing ha hey canno be jus noise, bu some signal. In he GNSS geodesy communi y he e m mul ipa h designa es a pa icula ype o e lec ions, nea he ecei ing sys em (Elósegui e al., 1995; Byun e al., 2002). Mul ipa h is usually an undesi ed e ec which migh mask o de e io a e he GNSS obse ables, because i in e e es wi h he main ay gene a ing oscilla ing pa e ns in he ampli ude and he phase, as i happened in G eenland’s campaign o sea ice emo e sensing, whose da a analysis was desc ibed in Chap e 4. The equency o hese pa e ns is gi en by M=−1 λ dρM d (65) 138 5.1 modeling and p ocessing gnss- o e d y snow:a no el app oach 0 5 10 15 20 Wa e o m ampli ude (a.u.) 10 20 30 40 50 60 Delay (15−m lag) −10 0 10 20 Wa e o m I/Q (a.u.) −10 0 10 20 Wa e o m I/Q (a.u.) Figu e 74.: Ampli udes o a sequence o 1-second in eg a ed complex wa e o ms collec ed wi h he GOLD-RTR ecei e (PRN 13, Decembe 16,2009), be ween 44.5◦and 45.5◦ele a- ion (only 1ou o e e y 10 wa e o ms a e he e shown, o a oid o e loading he plo ). [Top] In-phase and Quad a u e componen s (in g ay and black espec i ely). [Bo om] To al ampli ude. Figu e om Ca dellach e al. (2012). 139 emo e sensing o d y snow 0 10 20 Ampli ude (a.u.) 44.5 45.0 45.5 Ele a ion angle (deg) 0 10 20 Ampli ude (a.u.) 44.5 45.0 45.5 Ele a ion angle (deg) 0 10 20 Ampli ude (a.u.) 44.5 45.0 45.5 Ele a ion angle (deg) 0 10 20 Ampli ude (a.u.) 44.5 45.0 45.5 Ele a ion angle (deg) 0 10 20 Ampli ude (a.u.) 44.5 45.0 45.5 Ele a ion angle (deg) 0 10 20 Ampli ude (a.u.) 44.5 45.0 45.5 Ele a ion angle (deg) 0 10 20 Ampli ude (a.u.) 44.5 45.0 45.5 Ele a ion angle (deg) 0 10 20 Ampli ude (a.u.) 44.5 45.0 45.5 Ele a ion angle (deg) 0 10 20 Ampli ude (a.u.) 0 10 20 Ampli ude (a.u.) 0 10 20 Ampli ude (a.u.) 0 10 20 Ampli ude (a.u.) 0 10 20 Ampli ude (a.u.) 0 10 20 Ampli ude (a.u.) 0 10 20 Ampli ude (a.u.) 0 10 20 Ampli ude (a.u.) Figu e 75.: Ampli ude o lag-22 ( op) and lag-37 (bo om) o he 1-second in eg a ed wa e o ms, be ween 44.5◦and 45.5◦ele a ion, o PRN 13. Days 16 o 19 Decembe 2009 ha e been plo ed in di e en hues o g ey. No e ha some o hese poin s a e explici ly con ained in he bo om panel in Figu e 74 (Decembe 16). Figu e om Ca dellach e al. (2012). 140 5.1 modeling and p ocessing gnss- o e d y snow:a no el app oach whe e ρMis he ange delay be ween he e lec ed and he di ec signals; and λis he elec omagne ic wa eleng h o he signal. Assuming plana ho izon al e lec o s, he phase wi h which he mul ipa h signal eaches he ecei e , measu ed in cycles wi h espec o he phase o he main signal, can be modeled as ΦM=ρM λ=2HM λsin(ε)(66) being ε he ele a ion angle o obse a ion, and HM he e ical dis ance a which he e- lec ing su ace is loca ed wi h espec o he ecei ing an enna. No e ha his exp ession is consis en wi h p e ious Equa ion (33) a e aken in o accoun he adian- o-cycles con- e sion. The mul ipa h ield sums cohe en ly wi h he main ield, wi h a o a ed phase wi h espec o he main signal as gi en in Equa ion (66). As he condi ions change (ei- he HMo he ele a ion angle o obse a ion ε), he mul ipa h phase in Equa ion (66) changes oo, in oducing a o a ion o he added ield wi h espec o he main one. This phenomena in oduces oscilla ions in bo h ampli ude and phase. In ou expe imen al se -up, he only dynamic pa ame e is he ele a ion angle. As will be shown la e , he a e o change o he ele a ion angle o obse a ion is oo low o a nea -by e lec o (o e en om he shel e a he base o he owe ) o in oduce he scin illa ing pa e ns obse ed in he da a. A nea -by mul ipa h phenomena, hus, canno be he sou ce o in e e ence. In he GNSS adio-occul a ion communi y, he e m mul ipa h, o oposphe ic mul- ipa h, is applied o he phenomena ha occu s unde ce ain a mosphe ic condi ions, o which he GNSS signals spli in se e al ays. A echnique called adio-holog aphy is hen used in GNSS adio-occul a ions o iden i y and sepa a e a mosphe ic mul ipa h (Iga ashi e al., 2000). Simila ly, a new adio-holog aphic obse able will be la e in o- duced in Sec ion 5.1.3 o shed some ligh on he sou ce o he pa e ns. 141 emo e sensing o d y snow 5.1.2FORWARD MODEL: MULTIPLE-RAY SINGLE-REFLECTION Rays may be e lec ed o bo h he ex e nal snow su ace and in e nal snow in e aces. Gi en he clea mul iple in e e ence pa e ns obse ed in he da a, we ha e neglec ed olume ic sca e ing (con a y o wo k done by Wiehl e al. (2003)), which would no p oduce in e e ence pa e ns, and we ha e ocused on sca e ing o in e nal laye s. We ha e aken a geome ical op ics app oach, whe e he di e en con ibu ions a e modeled as ays bouncing in di e en laye s. A gene al iew o he componen s o he model we implemen ed a e ske ched in Figu e 76. We assume locally ho izon al laye s, pa allel incidence, and p opaga ion/ e lec ion h ough he snow laye s ollowing he Snell’s law: n(i+1)sin θ(i+1)=nisin θi(67) whe e θiis he incidence angle and niis he e ac i e index o he i-laye . The pe mi i - i y p o iles, p e iously shown in Figu e 29 om Chap e 3, a e compu ed om he in-si u d y snow measu emen s desc ibed in Appendix E.3. The model conside s a se o ays con ibu ing o he o al ecei ed signal, whe e each ay migh su e single- e lec ions solely, so we call i Mul iple-Rays Single-Re lec ion (MRSR) model. The ollowing sub-sec ions will desc ibe he equa ions o he delay ρi(Sec ion 5.1.2.1) and he ampli ude Ui(Sec ion 5.1.2.2) wi h which a ield ha incises in o he snow, p opaga es down o he i-laye , ebounds, and p opaga es back o he snow-ai in e ace, inally eaches he ecei e . Wi h his in o ma ion, he complex wa e o m ecei e can be cons uc ed (Sec ion 5.1.2.3). Re lec ed GNSS signals Di ec GNSS signal D y snow laye s n2 n1 n3 n4 Figu e 76.: Basic scheme o he mul iple- ay single- e lec ion model (MRSR). 142 5.1 modeling and p ocessing gnss- o e d y snow:a no el app oach 1 n n2 n3 0 nH0 H1 H2 H3 D3 D2 D1 θ1 θ2 θ3 A1 A3 A2 ρSR ρTS ρTR θ0 εε ε ε R S Figu e 77.: Ske ch o he single- e lec ion app oach implemen ed o model he snow in e nal e lec ions. Each laye has a cons an e ac i e index ni. 5.1.2.1Delay o he i-laye con ibu ion The i s s ep o he model is o compu e, o each i-laye , he delay o he ay such ha manages o p opaga e down in o he i-laye , is e lec ed o he bo om o ha laye , and p opaga es upwa d owa ds he ecei e . These delays, ρi, a e gi en wi h espec o he di ec ecep ion o he signal ( adio-link om he ansmi e o he ecei e wi h no e lec ion). As desc ibed below, mos o he con ibu ions o he i-delay a e also common o he ays ha ha e been e lec ed om he laye s abo e i . The e o e, an i e a i e app oach can easily sol e he p oblem. In ou no a ion 0-laye is he ex e nal ai , so he 0-delay co esponds o he e lec ion o he snow’s ex e nal su ace (poin S in Figu e 77): ρ0=ρTS +ρSR −ρTR (68) whe e subsc ip s TS,SR, and TR mean T ansmi e -Specula e lec ion poin , Specula e lec ion poin -Recei e , and T ansmi e -Recei e (di ec adio-link) espec i ely. No e ha in his case, ρ0is equi alen o ρgeo in he no a ion employed du ing he analysis o sea ice desc ibed in Chap e 4, whe e only su ace e lec ions we e unde s udy. The ay which p opaga es in o he i s laye o snow, o e ac i e index n1, and ge s e lec ed o o i s bo om, is delayed wi h espec o he di ec -link by ρ1. This delay has se e al con ibu ions: (1) he one gi en by he in e nal p opaga ion h ough laye 1,ρin −1; (2) he dis ance om he specula poin o he ecei e , ρSR; (3) he dis ance om he ansmi e o he poin in which his ay en e s he snow. This dis ance is equal o he dis ance be ween he ansmi e and he specula e lec ion poin , excep o he segmen be ween A1and he specula poin S; (4) inally, in o de o e e ence he delay o he di ec adio- 143 emo e sensing o d y snow link ecep ion, we need o sub ac he di ec dis ance be ween he ansmi e and he ecei e , ρTR. These ou con ibu ions a e summa ized in he ollowing equa ion: ρ1=ρin −1+ρSR + [ρTS −A1S]−ρTR (69) whe e ρin −1=2n1H1 cos(θ1)(70) and A1S=D1sin(θ0)(71) D1being he ho izon al ex en o he p opaga ion inside he 1-laye o snow (see Fig- u e 77): D1=2H1 an(θ1)(72) A compac way o exp ess i is: ρ1=ρ0+ρin −1−D1sin(θ0)(73) Simila ly, he ay ha manages o p opaga e down o laye -2, ge e lec ed o i s bo - om, and p opaga e upwa d o each he ecei e is: ρ2=ρin −2+ρin −1+ρSR + [ρTS −A2S]−ρTR (74) whe e ρin −2is he delay-con ibu ion om he in e nal p opaga ion h ough laye -2 ρin −2=2n2H2 cos(θ2)(75) and he dis ance be ween he specula poin Sand he poin A2,A2S, is: A2S= (D1+D2)sin(θ0)(76) being D2=2H2 an(θ2). To comple e hese examples, he 3 d laye would ead: ρ3=ρin −3+ρin −2+ρin −1+ρSR + [ρTS −(D1+D2+D3)sin(θ0)] −ρTR (77) wi h ρin −3=2n3H3 cos(θ3)(78) and D3=2H3 an(θ3). The e o e, he gene al exp ession o he i-laye is: ρi=ρ0+ k=i ∑ k=1 2nk Hk cos(θk)− k=i ∑ k=1 Dk!sin(θ0)(79) being Dk=2Hk an(θk)(80) Figu e 78 shows he delay a which he signals e lec ed o each snow laye each he ecei e when conside ing he pe mi i i y p o ile displayed in Figu e 29. No e ha because he incidence angle o he obse a ion cons an ly e ol es in ime, he delay be ween he di ec and e lec ed signals also changes in ime. A ecei e acking he di ec di ec signal bu ecei ing a con ibu ion om ano he ay-pa h om a e lec- ion o he i-laye , does no lock his o he componen because he la e a i es wi h a di e en equency. This in e e ome ic equency (wi h espec o he di ec one) has a mul ipa h-like beha io and he e o e can be compu ed using Equa ion (65). 144 5.1 modeling and p ocessing gnss- o e d y snow:a no el app oach 0 200 400 600 800 1000 1200 1400 1600 Delay (m) 0 100 200 300 400 500 Snow dep h (m) 0 200 400 600 800 1000 1200 1400 1600 Delay (m) 0 100 200 300 400 500 Snow dep h (m) 0 200 400 600 800 1000 1200 1400 1600 Delay (m) 0 100 200 300 400 500 Snow dep h (m) 0 200 400 600 800 1000 1200 1400 1600 Delay (m) 0 100 200 300 400 500 Snow dep h (m) 0 200 400 600 800 1000 1200 1400 1600 Delay (m) 0 100 200 300 400 500 Snow dep h (m) 0 200 400 600 800 1000 1200 1400 1600 Delay (m) 0 100 200 300 400 500 Snow dep h (m) 0 200 400 600 800 1000 1200 1400 1600 Delay (m) 0 100 200 300 400 500 Snow dep h (m) 0 200 400 600 800 1000 1200 1400 1600 Delay (m) 0 100 200 300 400 500 Snow dep h (m) 80 deg 0 200 400 600 800 1000 1200 1400 1600 Delay (m) 0 100 200 300 400 500 Snow dep h (m) 10 deg 0 200 400 600 800 1000 1200 1400 1600 Delay (m) 0 100 200 300 400 500 Snow dep h (m) 0 30 60 90 Ele a ion (deg) 0 30 60 90 Ele a ion (deg) 0 30 60 90 Ele a ion (deg) 100 m 0 30 60 90 Ele a ion (deg) 200 m 0 30 60 90 Ele a ion (deg) 300 m 0 30 60 90 Ele a ion (deg) Figu e 78.: A ay p opaga ing in o he snow, down o he x-me e dep h laye , e lec ed a ha laye , and p opaga ed upwa d o each he ecei e , a i es wi h y-delay wi h espec o he di ec signal (acco ding o he model in Equa ion (79)). The delay depends on he ele a ion angle o obse a ion: g ey hues, om 10◦ o 80◦ele a ion, in s eps o 10◦, ligh e o da ke espec i ely. The dependency on he ele a ion angle is also shown on he igh ame, whe e he delays in he signals e lec ed by laye s 100,200, and 300 me e deep a e plo ed as unc ion o he ele a ion angle. Figu e om Ca dellach e al. (2012). 145 emo e sensing o d y snow 0.00 0.05 0.10 Wa e o m ampli ude (a.u.) 10 20 30 40 50 60 Delay (15−m lag) −0.05 0.00 0.05 0.10 Wa e o m I/Q (a.u.) −0.05 0.00 0.05 0.10 Wa e o m I/Q (a.u.) Figu e 83.: [Top] Real and imagina y pa s o he complex wa e o ms syn hesized using he model in Equa ion (99). An ampli ude o 0.05 has been assigned o he LHCP leakage o he di ec signal. The geome ic condi ions ep oduce hose in Figu e 74, ha we e ob ained wi h eal da a. [Bo om] To al ampli ude o he wa e o ms on op. Figu e om Ca dellach e al. (2012). 152 5.1 modeling and p ocessing gnss- o e d y snow:a no el app oach 5.1.3LAG-HOLOGRAPHIC ANALYSIS Radio holog aphy uses cohe en p ope ies o he signals p opaga ing h ough a medium (Iga ashi e al., 2000). These p ope ies a ise due o he high s abili y o he GNSS signal and i s high sensi i i y o laye ed s uc u es. This app oach seeks o ob ain he maximum spa ial comp ession o he main ay sepa a ely om ha o he o he ays ajec o ies. This makes i possible o e alua e he in ensi y o adio wa es a each ay ajec o y and o de e mine he co esponding equency displacemen om a e e ence ay. A e e ence wa e ield ( e e ence ay) is used o e eal he spec a om he o al ecei ed ield. We choose he di ec signal (wi h no e lec ion) as a e e ence ield, aiming o see he es o possible con ibu ing ays p esen in he da a. Once he e e ence ield has been used o coun e - o a e he phase o he e lec ed signal, a spec al analysis is pe o med. In GNSS adio-occul a ion applica ions, he holog aphy is applied a he peak o he ecei ed wa e o m solely (because his is he only da a p o ided by s anda d and adio- occul a ion ecei e s). We he e p esen a new holog aphic obse able ha uses each o he lags o he ecei ed wa e o m. The gene a ion o he lag-holog am ollows he s eps below: •Time se ies o N( ypically we will use 128) complex (I/Q) wa e o ms a 1second sampling ob ained om he on -end o he ecei e connec ed o he ho izon- looking an enna a e aken: w (τw, ). •The phases o each lag τwo hese wa e o ms a e hen coun e - o a ed by he phase o he di ec signal. The di ec signal is he e de ined as he peak o he wa e o m (lag 22) ob ained by he on -end connec ed o he zeni h-looking an enna (wd). Then: w (τw, )e−iφd(22, ). •A Fou ie analysis (by FFT) is conduc ed independen ly on he ime se ies o each lag o ob ain wha we e e as lag-holog am: W(τw, I) = F{w (τw, )e−iφd(22, )}(100) •Since he geome ic pa ame e ha changes wi h ime (and hus o ces he po en- ial in e e ence o change) is he ele a ion angle, i is mo e p ac ical o exp ess he equency in e ms o ele a ion a e (oscilla ion cycles/deg ee-ele a ion) a he han Hz (oscilla ion cycles/second). The con e sion be ween hem is gi en by Icycle deg −el = I[Hz] dε d [deg −el/s]= I[Hz] dε d [ ad/s] 2π 360 (101) •Finally, each lag τwo he lag-holog am is no malized (independen ly o he o he lags): W(τw, I) = 1 ∑kkW(τw, I,k)kW(τw, I)(102) Ano he possible no maliza ion would ha e been a single ac o o he en i e lag- holog am. We ha e chosen he lag no maliza ion o gi e mo e ela i e powe o he ea u es a he end o he wa e o m, on hose lags o he wa e o m whe e he o e all powe is weak, unmasking equency con ibu ions ha o he wise would be oo low 153 emo e sensing o d y snow −30 −20 −10 0 10 20 30 In e .F eq.(cycle/deg−ele .) 10 20 30 40 50 60 Delay (15−m lags) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 Powe (a.u.) −30 −20 −10 0 10 20 30 In e .F eq.(cycle/deg−ele .) 10 20 30 40 50 60 Delay (15−m lags) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 Powe (a.u.) Figu e 84.: [Le ] Lag-holog am o a se ies o 128 syn hesized complex wa e o m a ound ∼45◦ ele a ion. [Righ ] Lag-holog am o a ime se ies o N=128 1-second measu ed com- plex wa e o ms unde he same geome y as on he le panel. The ze o- equency co - esponds o he e e ence ay, he di ec one. Acco ding o Equa ion (104), equencies mo e nega i es han −5.8 cycle/deg-el co espond o sca e ing o e lec ing-elemen s below he snow-ai in e ace. compa ed o he alues a ound he lag-holog am’s peak powe . The disad an age o he lag no maliza ion is ha weake equencies in powe ul lags migh also become masked. An example o he e ec p oduced by bo h ypes o no maliza ion is shown in Figu e 103 om Sec ion 5.2.4.1. The lag-holog am esul ing o a se ies o 128 syn he ic wa e o ms which include he ones p esen ed in he examples abo e (Figu e 83) is shown in he le panel in Figu e 84. The equency componen s a e gi en in cycle pe ele a ion deg ee (see Equa ions (65) and (101)). The lag-holog am clea ly shows a disc e e se o in e e ence equency bands, a he han a con inuous o b oad spec a. The band-s uc u e is clea ly odd, indica ing ha he equency componen s a e phaso - o a ions a he han ampli ude modula ions (which would gene a e symme ic posi i e and nega i e bands). The ze o- equency co esponds o he di ec signal, while ∼ −5.8 cycle/deg co esponds o he heo e ical in e e ome ic equency o a e lec ion o he snow su ace (H0=46 m and ε=45◦): su I[Hz] = −1 λ dρ0 d =−2H0 λcos(ε)dε d (103) su I[cycle/deg −el] = −2H0 λcos(ε)2π 360 (104) The igh panel in Figu e 84 displays he lag-holog am gene a ed om eal obse a- ions unde he same condi ions. Any e lec ion o plana -elemen s abo e he snow su ace would co espond o equencies compu ed wi h HMsmalle han H0, hus e- quencies slowe (less nega i e) han su I. Figu e 85 compiles he i s nega i e peak o he lag-holog ams obse ed du ing Decembe 16 2009, which clea ly ollow Equa- ion (104). The e o e, i seems ha he main bulge o equency con ibu ions gene ally cap u ed by he lag-holog ams, which a e as e (mo e nega i e) han su I, migh come ei he om dep hs below he snow ex e nal su ace, o om e lec o s which do no ol- low Equa ion (66). Such e lec o s could be il ed e lec ing su aces loca ed abo e he snow, bu a some ho izon al dis ances. In p e ious Figu e 27, he en i onmen o he obse a ion owe was displayed. The Conco dia S a ion buildings a e a ∼800 m dis- 154 5.1 modeling and p ocessing gnss- o e d y snow:a no el app oach −10 −9 −8 −7 −6 −5 −4 −3 −2 In e .F eq.(cycle/deg−ele .) 10 20 30 40 50 60 Ele a ion (deg) −10 −9 −8 −7 −6 −5 −4 −3 −2 In e .F eq.(cycle/deg−ele .) 10 20 30 40 50 60 Ele a ion (deg) Figu e 85.: A e aged loca ion o he i s nega i e peak in he lag-holog ams collec ed du ing Decembe 16,2009. The solid line is he heo e ical in e e ome ic equency co e- sponding o a e lec ion o he snow-ai in e ace, su Iin Equa ion (104). The dis- pe sion (e o -ba s) a e consis en wi h he equency esolu ion o he lag-holog am, gi en by he leng h o he ime se ies (128 seconds) and Equa ion (101): wi h dε/d in he ange o alues be ween 0.003 −0.008◦/s, he equency esolu ion lies be ween 1.0-2.6 cycle/deg −el. F equencies mo e nega i es mus come om snow sub-su ace e lec ions. Figu e om Ca dellach e al. (2012). ance om he owe , in he di ec ion opposi e o he main beam o he an enna. These dis ances a e oo long o be cap u ed wi hin ou wa e o m (42 lags a e he di ec signal, which is 630 m ange delay wi h espec o he ecep ion o he di ec adio-link). A small shel e is close o ou an ennas, a ∼100 m in he back-lobe di ec ion. The ac ha i is in he blind a ea o he an enna, oge he wi h i s small size (which hinde s he gene a ion o con inuous mul ipa h, om any incidence angle), makes i di icul o belie e i migh gene a e he s ong and con inuous pa e ns obse ed in he da a. Thus, ou hypo hesis is ha he p esence o mul iple e lec ions wi hin he d y snow sub-s uc u al laye s a e esponsible o he in e e ome ic pa e ns ound in he da a. 5.1.3.1Snow dep h e ie al and spa ial esolu ion o he lag-holog am F om Sec ion 5.1.2.1we know ha because he incidence angle o he obse a ion con- s an ly e ol es in ime, he delay be ween he di ec and e lec ed signals also changes in ime, p oducing hen an in e e ome ic equency (wi h espec o he di ec signal) ha can be ob ained om Equa ions (65) and (101), whe e ρMis now he delay o he laye - e lec ed signal wi h espec o he di ec one. The in e e ome ic equencies co e- sponding o each snow laye , compu ed om he MRSR model (ha ing he snow densi y p o ile as a unique inpu ), a e displayed in Figu e 86. Wi h his con e sion me hod, we can ans o m he equency-axis o he lag-holog am in o a dep h-axis, and hus he spec al s ipes ela e o he snow laye s ha e lec signal owa ds he ecei e . The e ical esolu ion o he iden i ied laye s is mainly gi en by he leng h o he ime se ies used o gene a e he spec al analysis (FFT esolu ion). O he seconda y ac o s a e he geome y (see di e en slopes in Figu e 86 as unc ion o he ele a ion angle due o 155 emo e sensing o d y snow 0 10 20 30 40 50 F equency (cycle/deg−el) 0 100 200 300 400 500 Snow dep h (m) 0 10 20 30 40 50 F equency (cycle/deg−el) 0 100 200 300 400 500 Snow dep h (m) 0 10 20 30 40 50 F equency (cycle/deg−el) 0 100 200 300 400 500 Snow dep h (m) 0 10 20 30 40 50 F equency (cycle/deg−el) 0 100 200 300 400 500 Snow dep h (m) 0 10 20 30 40 50 F equency (cycle/deg−el) 0 100 200 300 400 500 Snow dep h (m) 0 10 20 30 40 50 F equency (cycle/deg−el) 0 100 200 300 400 500 Snow dep h (m) 0 10 20 30 40 50 F equency (cycle/deg−el) 0 100 200 300 400 500 Snow dep h (m) 0 10 20 30 40 50 F equency (cycle/deg−el) 0 100 200 300 400 500 Snow dep h (m) 80 deg 0 10 20 30 40 50 F equency (cycle/deg−el) 0 100 200 300 400 500 Snow dep h (m) 10 deg 0 10 20 30 40 50 F equency (cycle/deg−el) 0 100 200 300 400 500 Snow dep h (m) 0 30 60 90 Ele a ion (deg) 0 30 60 90 Ele a ion (deg) 0 30 60 90 Ele a ion (deg) 100 m 0 30 60 90 Ele a ion (deg) 200 m 0 30 60 90 Ele a ion (deg) 300 m 0 30 60 90 Ele a ion (deg) Figu e 86.: A ay p opaga ing in o he snow, down o he x-me e dep h laye , e lec ed a ha laye , and p opaga ed upwa d o each he ecei e , a i es wi h a y- equency wi h espec o he di ec signal (Equa ions (65) and (101)). Fo a gi en ele a ion a e o a ia ion, i depends on he ele a ion angle o obse a ion: g ay hues, om 10◦ o 80◦ele a ion, in s eps o 10◦, ligh e o da ke espec i ely. The dependency on he ele a ion angle is also shown on he igh ame, whe e he in e e ome ic equency in he signals e lec ed by laye s 100, 200, and 300 m deep a e plo ed as unc ion o he ele a ion angle. Values om he MRSR model assuming he snow densi y p o ile gi en in Figu e 134. Figu e om Ca dellach e al. (2012). 156 5.1 modeling and p ocessing gnss- o e d y snow:a no el app oach di e en ele a ion a es). Fo 128 samples o 1-sec in eg a ion, he app oxima e e ical esolu ion anges be ween 5and 15 me e . This could be imp o ed by inc easing he leng h o he ime se ies. Howe e , he impac o signi ican geome ic changes su e ed by he obse a ion along he –longe – e en migh wo sen he esul s (as la e analysis will show in Sec ion 5.2.2.3). The ho izon al esolu ion o he measu emen in he di ec ion pe pendicula o he line-o -sigh is o he o de o 5-10 me e , and i is gi en by he i s F esnel zone. The esolu ion along he line-o -sigh di ec ion is gi en by he displacemen s o he specula poin s in deep laye s ( ela ed o Dkin Figu e 77 and Equa ion (80)). Assuming ha he cap u ed e lec ions migh occu down o ∼300 m dep h, he esolu ion along he line-o -sigh is o he o de o ∼350 m. No e ha line-o -sigh esolu ion wo sens wi h dep h. 5.1.3.2Dep h sensi i i y o inaccu acies in he snow densi y p o ile We ha e seen ha each equency s ipe in he lag-holog am is assigned o a gi en dep h o he e lec ing laye by means o he MRSR model wi h a snow densi y p o ile. This sec- ion ies o unde s and how sensi i e he e ical loca ion o he laye is o inaccu acies o his p o ile. To assess his ques ion, a se o syn he ic uns ha e been pe o med. We employ a ealis ic bu smoo h analy ic exp ession ob ained by means o an exponen ial i o he g ound u h disc e e densi y p o ile: ρs=0.92 −0.6e−δs 60 +δ2 s 30000 (105) A se o pe u ba ions o his smoo h p o ile ha e been added. A pe u ba ion is he e a Gaussian unc ion added o he smoo h p o ile, o ela i ely la ge in ensi y (0.1g /cm3, ep esen ing mo e han 10 % o he highes densi y) and e ical size (∼20 me e hal - wid h), and a iable dep h-loca ion. I is illus a ed in he op panel in Figu e 87. Fo each o hese pe u ba ions, he model has been un o ind he ela ionship be ween in e e ome ic equency and snow dep h. This has been done o a geome y a ound 45◦ele a ion angle. Di e en geome ies would yield di e en esul s, bu o simila o de o magni ude. The e o in oduced by hese unce ain ies is gi en in he bo om panel in Figu e 87. I is clea ha e en la ge unce ain ies such as he Gaussian pe u ba ions added in he smoo h p o ile in oduce e o s below 2.5%. In 5ou o he 6cases he e o is lowe han 1%. 5.1.3.3Disc e iza ion e ec s in he lag-holog am The model elies on a disc e ized se o laye s, gi en by he disc e e sampling o he densi y p o ile. This may induce ake in e aces, jus because he pe mi i i y is no con inuously sampled: Acco ding o Equa ions (85) and (86), he limi in which he pe mi i i y eds is smoo h and con inuous would esul in no in e nally- e lec ed signal. The ac ha ou pe mi - i i y is gi en in a disc e e se o snow dep hs a i icially in oduce jumps in eds, which migh p oduce a i icial in e e ences w ongly in e p e ed as in e nal e lec ions. 157 emo e sensing o d y snow 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 Snow densi y [g /cm3] 0 50 100 150 200 250 300 350 Snow dep h [me e ] Re e ence 0.0 0.5 1.0 1.5 2.0 2.5 Rela i e dep h e o [%] −30 −25 −20 −15 −10 −5 In e . eq.[cycles/deg ee−ele .] Dep h = 20m Dep h = 50m Dep h = 100m Dep h = 150m Dep h = 200m Dep h = 250m Figu e 87.: [Top] A smoo h analy ical exp ession o he snow densi y is used o es ima e a gene ic ela ionship be ween in e e ome ic equency and snow dep h. The smoo h p o ile is hen pe u bed by Gaussian bulks, o la ge in ensi y (>10% o he highes densi y), ∼20 me e e ical hal -size, and loca ed a di e en dep hs. [Bo om] The link be- ween in e e ome ic equency and dep h o he snow e lec ing laye is compu ed assuming a gi en densi y p o ile. This panel shows he e o in he snow dep h loca- ion in oduced by each Gaussian pe u ba ion, wi h espec o he dep hs ob ained wi h he non-pe u bed e e ence p o ile. Figu e om Ca dellach e al. (2012). 158 5.1 modeling and p ocessing gnss- o e d y snow:a no el app oach 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Snow densi y [g /cm3] 0 50 100 150 200 250 300 350 Snow dep h [me e ] Figu e 88.: G ound u h densi y p o ile, in ed, oge he wi h he smoo hed analy ical exp ession (exponen ial i ), in g een, used o check he e ec s o discon inuous sampling o he densi y in o he MRSR model. In o de o check how sensi i e is ou app oach o he le el o disc e iza ion gi en by he densi y p o ile, we ha e un wo examples (see Figu e 88): •1-cm disc e e sampling: he model has been un using he analy ical exp ession o he snow densi y in Equa ion (105) a 1-cm disc e e laye s, a ound 50◦ele a ion angle, in 256-0.006◦s eps ( om su ace o 200 me e dep h solely). •1-m disc e e sampling: same as abo e bu using 1me e esolu ion. The ampli ude o he e lec ed signals coming om each o he 1-cm and 1-m disc e e laye s a e shown in Figu e 89, oge he wi h he delay o a e lec ion o each o hese laye s. The igu e shows ha he only signi ican con ibu ion comes om he ex e nal in- e ace (ai -snow), and he ampli udes quickly d op a e wa d (quicke in 1cm- esolu ion han 1m- esolu ion). Bo h disc e iza ion le els p oduce he same delays (delay compu a- ion no a ec ed by disc e iza ion). The esul ing lag-holog ams a e displayed in Figu e 90. The only clea e lec ion when disc e izing a 1-cm le el comes om he ex e nal in e ace, a ∼-5cycle/deg-ele a ion. The 1-me e disc e iza ion in oduce some a i icious e lec ion bands a he end o he ailing edge. The esolu ion o he g ound u h densi y p o ile gi en by IFAC s ays below o a ound 1me e du ing he i s 90 me e s, and ∼1.5me e s a e wa ds. The e o e, he le el o a i icious e lec ion bands should be weak and mos ly a ec ing he end o he ail o he log-holog ams. 159 emo e sensing o d y snow Figu e 89.: Ampli ude (le ) and delays ( igh ) o in e nal e lec ions in a smoo h medium (Equa- ion (105) and Figu e 88), sampling a 1-cm laye esolu ion ( ed) and 1-me e (g een). −20 −15 −10 −5 0 5 10 15 20 F equency (cycle/deg−ele .) 10 20 30 40 50 60 Delay−lag (15 m.−lag) −20 −15 −10 −5 0 5 10 15 20 F equency (cycle/deg−ele .) 10 20 30 40 50 60 Delay−lag (15 m.−lag) Figu e 90.: Lag-holog ams o he disc e iza ion-exe cise: [Le ] 1-cm laye esolu ion on a smoo h medium (Equa ion (105) and Figu e 88); [Righ ] same a 1me e disc e iza ion. 160 5.1 modeling and p ocessing gnss- o e d y snow:a no el app oach 5.1.4ALTERNATIVE: THREE-REFLECTION MODEL Based on he Single- e lec ion model, we ha e also implemen ed a model in which 3 e lec ions occu wi hin a laye . Tha is, a e lec ion o he bo om o he laye ; a e lec ion o he op o he laye ; and inally a las e lec ion o he bo om again. Wi h his odd- numbe o e lec ions he esul ing pola iza ion is LHCP (as in single- e lec ion). An ske ch is p o ided in Figu e 91. Because o he low powe expec ed om hese so o e lec ions, o he o de o <3 c oss, we ha e no implemen ed a ull model o build complex wa e o ms based on hese 3- e lec ion e en s, bu we jus use he model o ha e an es ima ion o he in e e ome ic equency ha such an e en would in oduce. The delay o a ay e lec ing 3 imes wi hin he i-laye , would jus be (compa e wi h Equa ion (79)): ρi=ρ0+ k=i−1 ∑ k=1 2nk Hk cos(θk)+4ni Hi cos(θi)− k=i−1 ∑ k=1 Dk!+2Di!sin(θ0)(106) The esul o playing wi h his model, assuming iple- e lec ions wi hin laye -a eas in which he eal densi y p o ile seems o acili a e hese so o e en s a e: •T iple- e lec ion wi hin a laye 3-me e hick loca ed a 74 me e dep h a i es wi h a delay o he o de o ∼500 me e (change wi h incidence angle). Delays o his o de o magni ude came om single- e lec ions a laye s in he ange o ∼100 o ∼200 me e dep h. •The in e e ome ic equency is no necessa y highe han single- e lec ion e en s (i changes wi h incidence angle), bu o he same o de o magni ude. 1 n n2 n3 0 nH0 H1 H2 H3 θ1 θ2 ρSR ρTR θ0 D3 θ3 D3 θ3 ρTS A3 ε R S ε Figu e 91.: Ske ch o he delays induced by a 3- e lec ions e en wi hin a ce ain laye . 161 emo e sensing o d y snow −20 −15 −10 −5 0 5 10 15 20 F equency (cycle/deg−ele .) 10 20 30 40 50 60 Delay−lag (15 m.−lag) −20 −15 −10 −5 0 5 10 15 20 F equency (cycle/deg−ele .) 10 20 30 40 50 60 Delay−lag (15 m.−lag) −20 −15 −10 −5 0 5 10 15 20 F equency (cycle/deg−ele .) 10 20 30 40 50 60 Delay−lag (15 m.−lag) −20 −15 −10 −5 0 5 10 15 20 F equency (cycle/deg−ele .) 10 20 30 40 50 60 Delay−lag (15 m.−lag) Figu e 96.: Repea abili y o he ele a ion/ele a ion- a e a e aged lag-holog am, co esponding o he cell (47.5±2.5◦,0.0075±0.0005◦/s), Decembe 17 o 20,2009 (le o igh , op o bo om). All igu es use he same colo -scale (a bi a y uni s). 168 5.2 expe imen al esul s 5.2.2CONSISTENCY WITH THE MODEL 5.2.2.1Compa ison wi h he ele a ion/ele a ion- a e a e ages Gi en ha he ele a ion/ele a ion- a e a e aged cells should be be e ep esen ed by he model (in which bo h ele a ion and ele a ion- a e can be uned), we i s compa e he cell-a e aged da a wi h uns o he model co esponding o he cen al pa ame e s o he ele a ion/ele a ion- a e cells. The cen al alues o he cells selec ed in Table 20 and shown in Figu e 95 a e hus simula ed wi h he model, and he esul ing lag-holog ams a e displayed in Figu e 97. As expec ed, he model un a he cen al pa ame e s o he cell do no accoun o he di e si y wi hin he cell, hus p oducing sha pe (less blu and ading) images, o highe equency esolu ion. Besides his e ec , he main ea u es o he model a e p esen in he a e aged da a lag-holog ams, including he esolu ion loss expe ienced a low alues o ele a ion and ele a ion- a e. −20 −15 −10 −5 0 5 10 15 20 In e .F eq.(cycle/deg−ele .) 10 20 30 40 50 60 Delay (15−m lags) −20 −15 −10 −5 0 5 10 15 20 In e .F eq.(cycle/deg−ele .) 10 20 30 40 50 60 Delay (15−m lags) −20 −15 −10 −5 0 5 10 15 20 In e .F eq.(cycle/deg−ele .) 10 20 30 40 50 60 Delay (15−m lags) −20 −15 −10 −5 0 5 10 15 20 In e .F eq.(cycle/deg−ele .) 10 20 30 40 50 60 Delay (15−m lags) −20 −15 −10 −5 0 5 10 15 20 In e .F eq.(cycle/deg−ele .) 10 20 30 40 50 60 Delay (15−m lags) −20 −15 −10 −5 0 5 10 15 20 In e .F eq.(cycle/deg−ele .) 10 20 30 40 50 60 Delay (15−m lags) −20 −15 −10 −5 0 5 10 15 20 In e .F eq.(cycle/deg−ele .) 10 20 30 40 50 60 Delay (15−m lags) −20 −15 −10 −5 0 5 10 15 20 In e .F eq.(cycle/deg−ele .) 10 20 30 40 50 60 Delay (15−m lags) −20 −15 −10 −5 0 5 10 15 20 In e .F eq.(cycle/deg−ele .) 10 20 30 40 50 60 Delay (15−m lags) Figu e 97.: Ele a ion/ele a ion- a e modeled lag-holog ams. Colo scale gi en in Figu e 84. Le o igh and op o bo om co espond o he lis so ing in Table 20. Compa e wi h Figu e 95. 169 emo e sensing o d y snow 5.2.2.2Compa ison wi h non-a e aged lag-holog ams The consis ency wi h he model has been checked o a wide ange o ele a ion angles o obse a ion. We will conside again PRN13 collec ed du ing Decembe 16,2009 (as he p elimina y esul s shown in Sec ion 5.1.3). 128 1-second samples a e used o compu e he FFTs. This ope a ion is epea ed a ound ele a ion angles 15◦,25◦,35◦,45◦,55◦and 65◦(along he g ound ack). Figu es 98 o 99 display he lag-holog ams ob ained wi h eal da a, as well as he ou pu o he model. No ice ha only wo cases achie e he minimum ele a ion and ele a ion- a e le els (a ound 30◦and 0.006◦/s espec i ely) o ge he bes pe o mance. F om he esul s ob ained, we can obse e ha each pai o model/da a lag-holog ams ag ee on he app oxima e loca ion o he ew i s equency-bands. A signi ican di e - ence elies on he impac o he i s e lec ion band (close o he su ace a a ound -6cycle/deg-e, depending on he geome y) on he whole lag-holog am. While he eal da a ou pu shows ela i ely s ong bands be ween -8and -12 cycle/deg-e in he lag- delay ange 12-44, he model is appa en ly masking hese laye ’s con ibu ions a e he no maliza ion. Tha would mean ha e lec ions coming om he i s 50 me e deep in snow a e s onge han expec ed (compa able o he –ai /snow– su ace e lec ion). Mo eo e , some high nega i e equency componen s seem o pe sis in lag-delay a - eas which a e in-consis en wi h he model. Fo ins ance, he bands a ∼-25 and ∼- 28 cycle/deg-e isible be ween lags 30 and 40 would co espond o e lec ions o e y deep laye s, eaching he ecei e a e y long delays. These e y long delays should no con ibu e in o hese lags, because o he code-delay il e ing o he GPS signals. As −30 −25 −20 −15 −10 −5 0 5 In e .F eq.(cycle/deg−ele .) 10 20 30 40 50 60 Delay (15−m lags) −30 −25 −20 −15 −10 −5 0 5 In e .F eq.(cycle/deg−ele .) 10 20 30 40 50 60 Delay (15−m lags) −30 −25 −20 −15 −10 −5 0 5 In e .F eq.(cycle/deg−ele .) 10 20 30 40 50 60 Delay (15−m lags) −30 −25 −20 −15 −10 −5 0 5 In e .F eq.(cycle/deg−ele .) 10 20 30 40 50 60 Delay (15−m lags) −30 −25 −20 −15 −10 −5 0 5 In e .F eq.(cycle/deg−ele .) 10 20 30 40 50 60 Delay (15−m lags) −30 −25 −20 −15 −10 −5 0 5 In e .F eq.(cycle/deg−ele .) 10 20 30 40 50 60 Delay (15−m lags) Figu e 98.: Indi idual lag-holog ams (Up-model, Down- eal da a) o PRN13, in FFT windows o 128 1-sec samples, om Decembe 16 h. Di e en ele a ion angles and ele a ion- a es a e conside ed: [Le ] 15◦and 0.0073◦/s, [Cen e ] 25◦and 0.0075◦/s, [Righ ] 35◦and 0.0076◦/s. Model and eal-da a colo scales, gi en in Figu e 84. 170 5.2 expe imen al esul s −30 −25 −20 −15 −10 −5 0 5 In e .F eq.(cycle/deg−ele .) 10 20 30 40 50 60 Delay (15−m lags) −30 −25 −20 −15 −10 −5 0 5 In e .F eq.(cycle/deg−ele .) 10 20 30 40 50 60 Delay (15−m lags) −30 −25 −20 −15 −10 −5 0 5 In e .F eq.(cycle/deg−ele .) 10 20 30 40 50 60 Delay (15−m lags) −30 −25 −20 −15 −10 −5 0 5 In e .F eq.(cycle/deg−ele .) 10 20 30 40 50 60 Delay (15−m lags) −30 −25 −20 −15 −10 −5 0 5 In e .F eq.(cycle/deg−ele .) 10 20 30 40 50 60 Delay (15−m lags) −30 −25 −20 −15 −10 −5 0 5 In e .F eq.(cycle/deg−ele .) 10 20 30 40 50 60 Delay (15−m lags) Figu e 99.: Indi idual lag-holog ams (Up-model, Down- eal da a) o PRN13, in FFT windows o 128 1-sec samples, om Decembe 16 h. Di e en ele a ion angles and ele a ion- a es a e conside ed: [Le ] 45◦and 0.0074◦/s, [Cen e ] 55◦and 0.0066◦/s, [Righ ] 65◦and 0.003◦/s. Model and eal-da a colo scales, gi en in Figu e 84. explained in Sec ion 5.1.4, he iple- e lec ion model nei he explains hese bands ( ipe- e lec ion model does no p edic so much high in e e ome ic equencies, and i does p edic e y long delays, inconsis en wi h he lag-loca ion o hese bands). The e o e, o he easons should be in es iga ed. Howe e , no e ha hese e ec s dec ease a e a e aging he esul s om se e al obse a ions (as shown in Figu es 95 and 96). 5.2.2.3Lag-holog ams wi h highe esolu ion In addi ion o he conside ed FFT-leng hs o 128 samples om he p e ious lag-holog aphic analysis, longe windows o da a-se ies ha e been also es ed. Due o basic Fou ie - ans o m p ope ies, o he same geome ic condi ions and sampling a e, he numbe o samples is di ec ly p opo ional o he equency esolu ion (and hus dep h esolu- ion). The esul s ob ained unde he same geome ic condi ions as he example gi en in Figu e 84 a e displayed in Figu e 100. We can obse e how, in spi e o he esolu ion imp o emen , he equency bands ound in he da a show poo ag eemen wi h he models. The impac o signi ican geome ic changes su e ed by he obse a ion along he –longe – e en migh wo sen he esul s, hus inc easing he inconsis encies ound in p e ious Sec ion 5.2.2.2. 171 emo e sensing o d y snow −30 −25 −20 −15 −10 −5 0 5 In e .F eq.(cycle/deg−ele .) 10 20 30 40 50 60 Delay (15−m lags) −30 −25 −20 −15 −10 −5 0 5 In e .F eq.(cycle/deg−ele .) 10 20 30 40 50 60 Delay (15−m lags) −30 −25 −20 −15 −10 −5 0 5 In e .F eq.(cycle/deg−ele .) 10 20 30 40 50 60 Delay (15−m lags) −30 −25 −20 −15 −10 −5 0 5 In e .F eq.(cycle/deg−ele .) 10 20 30 40 50 60 Delay (15−m lags) −30 −25 −20 −15 −10 −5 0 5 In e .F eq.(cycle/deg−ele .) 10 20 30 40 50 60 Delay (15−m lags) −30 −25 −20 −15 −10 −5 0 5 In e .F eq.(cycle/deg−ele .) 10 20 30 40 50 60 Delay (15−m lags) Figu e 100.: Lag-holog ams (Up-model, Down- eal da a) o PRN13 a a ound 45◦o ele a ion (0.0074◦/s o ele a ion- a e) om Decembe 16 h. Di e en FFT windows o 1-sec samples a e conside ed: [Le ] 256 samples, [Cen e ] 512 samples and [Righ ] 1024 samples. Colo -scale in a bi a y uni s. 172 5.2 expe imen al esul s 5.2.3APPLICATION: DEPTH OF THE CONTRIBUTING LAYERS As men ioned in he in oduc o y sec ion, he main scien i ic ques ion we seek o answe is whe he he GNSS-R echniques ha e po en ial o iden i y he dep h o he snow laye s om which he signal is mos ly e lec ed. To do so and as an in e media e s ep, we i s in eg a e he lag-holog ams along he lag-axis, o ob ain he o al spec al powe as a unc ion o he snow dep h. An example is gi en in Figu e 101. The p o ile shows ou clea echoes loca ed a ∼5,90,130, and 240 me e dep h. The igu e also displays he in eg a ed spec a ob ained wi h he MRSR model and he gi en densi y p o ile. Some– bu no all–o he e lec ing laye s ag ee wi h he da a. The disc epancies could be due o inaccu acies in he densi y p o ile assumed by he model, o by locally il ed in e aces (no conside ed in his ini ial model). Some snow laye s appea consis en ly in many o he lag-holog ams as e lec ing el- emen s. Those a e iden i ied in Table 21. Howe e , in spi e o he high empo al e- pea abili y ound in each GPS sa elli e da a, he laye s iden i ied by di e en sa elli es a e no always coinciden , ha is, he esul s p esen high empo al epea abili y, bu poo geog aphic consis ency. This could be due o a di e si y o causes, among hem possible inhomogenei ies in he snow ac oss he scanned a ea, ∼500 me e long; and he limi ed capabili y o he simple MRSR model o explain all he ea u es in he da a. To accoun o il ed laye s in he sub-su ace s uc u e could be a possible way o imp o e he o wa d model. Dep h [m] 10 70 130 240 Table 21.: Lis o snow sub-s uc u al laye s ha e lec signal owa ds he ecei e p oducing in e e ence pa e ns, as consis en ly appea in mos o he lag-holog ams a ∼45◦ele- a ion angle o obse a ion. 173 emo e sensing o d y snow 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Accumula ed spec um (a.u.) 0 50 100 150 200 250 300 Snow dep h (me e s) Da a−128 Model−128 Model−256 Figu e 101.: (Ci cles:) The equencies in he lag-holog am shown a he igh panel in Figu e 84 ha e been con e ed in o snow dep h using Equa ions (65) and (101). This igu e shows he sum o all i s lag (in eg a ion o he lag-holog am along he lag-axis). Ob- se a ion co esponding o PRN13,16 Decembe 2009, a 45◦ele a ion. (T iangles:) In eg a ed spec al powe ob ained wi h he MRSR model un in 128-s eps, he ge- ome y o which is iden ical o he geome y in he 128 samples used o gene a e he da a in ci cles. (In e ed iangles:) Same as he iangles, bu using a se ies o 256 syn he ic obse a ions o imp o e he equency esolu ion. Figu e om Ca dellach e al. (2012). 174 5.2 expe imen al esul s 5.2.4TOTAL INVERSION The analysis p esen ed in Sec ion 5.1 elies on he knowledge o he p o ile o pe mi i i y laye s. Tha is, he ela ionship o link equency s ipes wi h dep h o he laye s is based on a gi en p o ile o snow densi ies. The possibili y o pe o m di ec in e sion o he lag-holog ams o e ie e he snow densi y p o iles is he e in es iga ed. 5.2.4.1Model sensi i i y The i s s ep o asses he easibili y o o al in e sion consis in analyzing he sensi i i y o he model. Mos o he in e sion app oaches ely on a cos unc ion o be minimized. This cos unc ion usually akes he squa ed di e ences be ween he da a and he model, e alua ed a di e en unknown-pa ame e s. To gauge he cos unc ion app op ia ely, he sum o he squa ed di e ences a e weigh ed wi h he in e se o he da a noise. In his i s simple sensi i i y exe cise we only ake in o accoun he model, ha is, he squa ed di e ences scan he model space o compa e wi h a syn he ic u h model (pa icula case o he model). Each lag-holog am Wwi hin a s udy window o K×Jcomponen s (τw om 5 o 60 lags and I om -20 o 20 cycles/deg-ele a ion) is a anged as a 1-D a ay o N=K×Jelemen s: YW=                W(τw1, I1) W(τw1, I2) . . . W(τw1, IJ) W(τw2, I1) W(τw2, I2) . . . W(τwK, IJ)                (107) Since we a e in e es ed in e alua ing how sensi i e he model is a ound a pa icula pe u ba ion case Yp e W, compa ed o a se o o he model pe u ba ions Yp W, he cos unc ion o e alua e becomes: SC(p;p e ) = N ∑ i (Yp W[i]−Yp e W[i])2(108) As explained be o e, he lag-by-lag no maliza ion pe mi s he weak signals a he end o he ailing edge o eme ge, bu i masks he seconda y signals in he cen al lags. A no maliza ion using he o al lag-holog am powe wi hin he s udy window, masks he weak signals in he ailing edge bu allows o eme ge he seconda y con ibu ions in he cen al lags. The sensi i i y o bo h no maliza ion app oaches has been es ed. The densi y p o ile conside ed o his sensi i i y p o ile is he smoo h analy ical ex- p ession om Equa ion (105), sampled a 1me e laye s. The densi y pe u ba ions conside ed a e ∆ρs=0.05 g /cm3added a one pa icula laye (dep h), as illus a ed in Figu e 102. The e o e, he cos unc ion alue SC(12;20)e alua es he o e all di e - ence be ween he lag-holog am esul ing om a p o ile wi h a 0.05 g /cm3pe u ba ion 175 emo e sensing o d y snow 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Snow densi y [g /cm3] 0 50 100 150 200 Snow dep h [me e ] Figu e 102.: Examples o densi y p o iles used o he sensi i i y analysis: (black) smoo h analy - ical exp ession; (g een o b own) 10 m o 150 m deep 0.05 g /cm3pe u ba ions. loca ed a 12 m dep h, compa ed o a e e ence (o syn he ic u h) densi y p o ile whe e he pe u bed laye is 20 m deep. Some examples o he lag-holog ams esul ing om hese pe u bed p o iles (wi h lag- o-lag and o al powe no maliza ion) a e displayed in Figu e 103. Figu e 104 compiles he cos unc ion SCe alua ed in s eps o 1me e deep pe u - ba ions, o e e ence p o iles wi h pe u ba ions a 10 o 90 me e s dep h (in s eps o 10 me e ), using he lag-by-lag no maliza ion. Figu e 105 epea s he same exe cise o lag-holog ams no malized by he o al powe . I he model we e a linea combina ion o i s pa ame e s, o any hing close o linea , he SC unc ions would ollow a pa abolic well wi h i s minimum a he syn he ic u h. The loca ion o he ough is he solu ion, whe eas i s cu a u e ela es o he unce ain y. Non-linea i y in oduce a a ie y o e ec s, such as mul iple oughs (po en ial mul i- ple solu ions–degene a ed solu ion), o wide la oughs (la ge unce ain y a ound he solu ion). The ac ha his exe cise deals only wi h he model, wi h no assumed mis- modelling e o s, nei he noise, b ings he minimum cos unc ion alue o ze o. When e alua ing SCwi h eal da a ( ha is, when Yp e Wis eplaced by da a-obse ables), he noise will mask he lowe le els o ou syn he ic unc ional cos , and any sys ema ic mis- modelling migh comple ely change he shape o hem. This exe cise mus hus be seen as he sensi i i y analysis o he model i sel . As shown in Figu es 104 and 105, he unc ional cos s migh p esen some deg ee o deg ada ion, wi h some mul iple oughs appea ing along he unc ion. Howe e , a minimum a ound he e e ence (syn he ic ue) is always p esen , wi h a ypical wid h o ≤10 me e . The lag-by-lag no maliza ion seems o be e disce n be ween ea u es coming om deep laye s, while he o al powe no maliza ion pe o ms be e nea he su ace le els. In spi e o he non-linea aspec o he cos unc ionals e alua ed in his sec ion, and p o ided ha he in e sion app oach scans only a small po ion o he unc ional cos a ound he solu ion (a-p io i close o he solu ion), we p oceed wi h a linea ized in e - sion o he laye s in he nex sec ion. 176 5.2 expe imen al esul s −20 −15 −10 −5 0 5 10 15 20 F equency (cycle/deg−ele .) 10 20 30 40 50 60 Delay−lag (15 m.−lag) −20 −15 −10 −5 0 5 10 15 20 F equency (cycle/deg−ele .) 10 20 30 40 50 60 Delay−lag (15 m.−lag) −20 −15 −10 −5 0 5 10 15 20 F equency (cycle/deg−ele .) 10 20 30 40 50 60 Delay−lag (15 m.−lag) −20 −15 −10 −5 0 5 10 15 20 F equency (cycle/deg−ele .) 10 20 30 40 50 60 Delay−lag (15 m.−lag) −20 −15 −10 −5 0 5 10 15 20 F equency (cycle/deg−ele .) 10 20 30 40 50 60 Delay−lag (15 m.−lag) −20 −15 −10 −5 0 5 10 15 20 F equency (cycle/deg−ele .) 10 20 30 40 50 60 Delay−lag (15 m.−lag) −20 −15 −10 −5 0 5 10 15 20 F equency (cycle/deg−ele .) 10 20 30 40 50 60 Delay−lag (15 m.−lag) −20 −15 −10 −5 0 5 10 15 20 F equency (cycle/deg−ele .) 10 20 30 40 50 60 Delay−lag (15 m.−lag) Figu e 103.: Examples o lag-holog ams p oduced wi h a smoo h analy ical p o ile, wi h a single sha p pe u bed laye , as hose shown in Figu e 102. On he le , lag-by-lag no mal- iza ion, on he igh no maliza ion by o al powe . Top o bo om, pe u bed laye a 20,40,60, and 90 m espec i ely. The sa u a ion o he colo scale has been lowe ed o highligh he seconda y ea u es. Some o he equency bands a e lag 45 appea - ing on he lag-by-lag no maliza ion a e a i ac s o he laye disc e iza ion (1me e esolu ion). They a e much weake in he o al powe no maliza ion app oach. 177 emo e sensing o d y snow 186 188 190 192 194 B igh ness Tempe a u e [K] 26 28 30 32 34 Ele a ion [deg] 0.00 0.01 0.02 0.03 0.04 0.05 No malized Powe [a.u.] 206 208 210 212 214 216 B igh ness Tempe a u e [K] 26 28 30 32 34 Ele a ion [deg] 0.00 0.01 0.02 0.03 0.04 0.05 No malized Powe [a.u.] Figu e 107.: [Le ] Ho izon al componen o b igh ness empe a u e (Thin ed) measu ed wi h RaDomeX du ing 15/01/2010 compa ed wi h simula ed Sun’s e lec ed powe using adap ed MRSR and ho izon al pola iza ion (g een) and applying he adiome e ’s an enna gain-pa e n. [Righ ] The same compa ison using Ve ical componen o b igh ness empe a u e (T in blue) and e ical pola iza ion in he adap ed MRSR model (o ange). No e ha he empe a u e’s le els di e while keeping he same esolu ion. The ele a ion ange chosen co esponds o he ime momen s when he Sun’s Azimu h lies in he in e al ±35◦wi h espec o he an enna’s line-o -sigh o ien a ion (Azimu h=315◦). No inges we e ound ou side his in e al. 188 190 192 194 196 B igh ness Tempe a u e [K] 16 18 20 22 24 26 Ele a ion [deg] 0.00 0.01 0.02 0.03 0.04 0.05 No malized Powe [a.u.] 208 210 212 214 216 B igh ness Tempe a u e [K] 16 18 20 22 24 26 Ele a ion [deg] 0.00 0.01 0.02 0.03 0.04 0.05 No malized Powe [a.u.] Figu e 108.: The same ype o ep esen a ion as in Figu e 107 bu o 16/02/2010. pe a u e measu ed, he lowe he impac p oduced by Sun e lec ions om he d y snow laye s. No ice ha his s a emen p o ides an addi ional a gumen o explaining he be e obus ness shown by he e ical componen o b igh ness empe a u e du ing he Sun inges e en s. In spi e o he p omising p elimina y esul s ob ained, he e a e s ill some inconsis en- cies ha should be analyzed and u he esea ch is equi ed, which is ou o he scope o his wo k. Howe e , o ind e idences o sub-su ace in e e ome ic beha io om he measu emen s made by a di e en a chi ec u e/sys em (a L-band, bu no ela ed o GPS) a he same expe imen al si e, ep esen s a supplemen a y jus i ica ion o he in es iga ion desc ibed along his Chap e owa ds emo e sensing o d y snow. 184 5.2 expe imen al esul s 186 188 190 192 194 B igh ness Tempe a u e [K] 6 8 10 12 14 16 Ele a ion [deg] 0.00 0.01 0.02 0.03 0.04 0.05 No malized Powe [a.u.] 206 208 210 212 214 216 B igh ness Tempe a u e [K] 6 8 10 12 14 16 Ele a ion [deg] 0.00 0.01 0.02 0.03 0.04 0.05 No malized Powe [a.u.] Figu e 109.: The same ype o ep esen a ion as in Figu e 107 bu o 16/03/2010. 184 186 188 190 192 B igh ness Tempe a u e [K] 2 4 6 8 10 12 Ele a ion [deg] 0.00 0.01 0.02 0.03 0.04 0.05 No malized Powe [a.u.] 206 208 210 212 214 B igh ness Tempe a u e [K] 2 4 6 8 10 12 Ele a ion [deg] 0.00 0.01 0.02 0.03 0.04 0.05 No malized Powe [a.u.] Figu e 110.: The same ype o ep esen a ion as in Figu e 107 bu o 16/09/2010. 182 184 186 188 190 192 B igh ness Tempe a u e [K] 12 14 16 18 20 22 Ele a ion [deg] 0.00 0.01 0.02 0.03 0.04 0.05 No malized Powe [a.u.] 206 208 210 212 214 B igh ness Tempe a u e [K] 12 14 16 18 20 22 Ele a ion [deg] 0.00 0.01 0.02 0.03 0.04 0.05 No malized Powe [a.u.] Figu e 111.: The same ype o ep esen a ion as in Figu e 107 bu o 16/10/2010. 185 emo e sensing o d y snow 184 186 188 190 192 194 B igh ness Tempe a u e [K] 18 20 22 24 26 28 Ele a ion [deg] 0.00 0.01 0.02 0.03 0.04 0.05 No malized Powe [a.u.] 206 208 210 212 214 216 B igh ness Tempe a u e [K] 18 20 22 24 26 28 Ele a ion [deg] 0.00 0.01 0.02 0.03 0.04 0.05 No malized Powe [a.u.] Figu e 112.: The same ype o ep esen a ion as in Figu e 107 bu o 01/11/2010. 186 5.2 expe imen al esul s 5.2.6EXTRAPOLATION TO A SPACEBORNE SCENARIO E en hough he esul s o his expe imen show he po en ial sensing o d y snow sub- su ace signa u es wi h e lec ed GPS signals om a ixed pla o m, he inal applica ion o his echnique should be pe o med om sa elli e ecei e s in o de o achie e wide co e age. The ex apola ion o his esul s o a spacebo ne scena io is he e o e needed. A i s conside a ion is ha he peak powe o he wa e o ms will be subs an ially lowe , due o highe p opaga ion losses and he signal’s sp eading o e ange delay and equency shi s (inc ease o he glis ening a ea). In Wiehl e al. (2003), a dec ease o ∼20 dB is ob ained when going om an ai c a scena io (4km heigh and 200 m/s speed) o a LEO sa elli e case (400 km heigh and 7.6km/s speed) in simula ions o e An a c ica. Despi e ha his wo k is done using P-code wa e o ms and conside ing he subsu ace con ibu ions as olume sca e ing, simila esul s can be expec ed om ou case ( he in luence o he di ec signal impai s us o pe o m he same analysis o e he da ase ob ained). The immedia e conclusion ha we can ge is ha a high gain an enna is equi ed, which means a high di ec i i y. Taking in o accoun ha a he same ime, di e en ele a ion angles o obse a ion a e desi ed o achie ing spa ial co e age (in gene al) o o gene a ing he lag-holog ams (in ou pa icula case), a beam o ming s a egy like in Ma ín-Nei a e al. (2011) seems o be he bes op ion. Wi h hund eds o kilome e s o dis ance om he su ace le el, he di ec and e lec ed signals do no o e lap, and will also ha e la gely di e en Dopple alues. This impedes he use o he di ec signal o s op he e lec ed one. In absence o a be e e e ence o s op he e lec ed signals, du ing a i s i e a ion, a coun e - o a o phaso should be gene a ed using he su ace- e lec ed pa h delay compu ed om he posi ions o e- cei e & ansmi e plus a model o he Ea h’s su ace (e.g. geoids o su ace ele a ion models). No e ha his is he app oach aken in he Sun inges s udy. Ano he aspec o be conside ed in dynamic scena ios–wi h espec o ixed-pla o ms– is ime- a ying opog aphy. Simula ions done in Wiehl e al. (2003) show how he wa e- o m’s shape is pe u bed by he e ec o a opog aphic slope, whe e he e is a shi owa ds a equency sense depending on he di ec ions o he slope and he ajec o y o he sa elli e, which can be de ec ed by means o delay-Dopple maps. Rega ding Dopple e ec s, he eloci y o he ecei e in a LEO sa elli e (∼7.5km/s) migh lead o di e en Dopple - equency con ibu ions o e he e lec ing g ound e- gion (non-specula e lec ions). Howe e , p e ious expe imen s wi h eal GPS e lec- ions (Lowe e al., 2002a; Gleason, 2010) show how hese equencies can be p ope ly de e mined om space. Ano he conce n is he spa ial esolu ion o he p esen echnique in a sa elli e pla o m. Basically, he analysis shown along Sec ion 5.1is based on compu ing Fou ie ans o ms o wa e o m se ies, long enough o include a ia ion in he ele a ion angle, in o de o sepa a e he con ibu ions o he e lec ed signal coming below he su ace le el. The dep h esolu ion depends on he a e o ele a ion’s a ia ion and on he leng h o he da a se ies (which de e mines he esolu ion in he equency domain o a gi en sampling a e). Wi h he ecei e inside a LEO sa elli e, we can assume ha he ele a ion angle and ele a ion- a e is domina ed by he GPS ansmi e . Howe e , he speed o he specula oo p in de e mines he minimum g ound ack (spa ial ange) o ob ain he numbe o da a samples o he FFT algo i hm, and i depends on he ecei e , wi h a ypical alue o 187 emo e sensing o d y snow 7.5km/s. No ice ha in ou analysis, we assume ha he in e nal laye ing is cons an o he whole da a se ies, which is a alid s a emen wi h local measu emen s om a ixed pla o m, bu i seems un ealis ic om space due o he la ge spa ial anges equi ed. This e ec would make he e ie al much mo e challenging, o di e en app oaches should be in es iga ed. Finally, aking in o accoun ha ou me hodology equi es phase de e mina ion, he cohe ence o he signal om GPS e lec ions o e d y snow masses collec ed om space- bo ne ecei e s should be p ope ly s udied. Tha includes also he e ec o speckle noise and he impac o oughness. To asses his p oblem is no s aigh o wa d and i emains as an open ques ion ha will equi e a deepe analysis. 188