Influence of saturated water content on estimating soil hydraulic properties from cumulative disc infiltrometer measurements

In luence o sa u a ed wa e con en on es ima ing soil hyd aulic p ope ies om cumula i e disc in il ome e measu emen s D. Mo e -Fe n´ andez a,* , F. Le a b , D. Yilmaz c , L. Lassaba e e d , J.J. Jim´ enez e , B. La o e a a Depa amen o de Suelo y Agua, Es aci´ on Expe imen al de Aula Dei, Consejo Supe io de In es igaciones Cien í icas (CSIC), PO Box 13034, 50080 Za agoza, Spain b Ins i u o de In es igaci´ on en Ingenie ía de A ag´ on, Uni e si y o Za agoza, 50018 Za agoza, Spain c Uni . G enoble Alpes, CNRS, IRD, G enoble INP, IGE, 38000 G enoble, F ance d Uni e si ´ e Claude Be na d Lyon 1, LEHNA UMR 5023, CNRS, ENTPE, F-69518, Vaulx-en-Velin, F ance e Ins i u o Pi enaico de Ecología, Consejo Supe io de In es igaciones Cien í icas (IPE-CSIC), A da. N a. S a. De la Vic o ia, 16, Jaca 22700, Spain ARTICLE INFO Keywo ds: So p i i y Hyd aulic conduc i i y In il a ion cu e In e se analysis Sa u a ed wa e con en ABSTRACT The soil so p i i y, S, and sa u a ed hyd aulic conduc i i y, K s , a e undamen al soil hyd aulic p ope ies ha can be es ima ed om he cumula i e in il a ion cu e measu ed wi h a disc in il ome e . The Ha e kamp in il- a ion model is widely used o es ima e Sand K s . This model includes as inpu s he cons an s βand γand he di e ence be ween he ini ial, θ i , and inal, θ s , olume ic wa e con en s, Δθ. Since Δθ would be exp essi e o he possible measu emen e o s, and assuming β,γ, and θ i as known alues, he i s objec i e o his wo k is o analyze he in luence o θ s on he op imiza ion o K s and S. To his end, a sensi i i y analysis, which consis s o es ima ing K s and S o a ange o θ s was applied on syn he ic in il a ion cu es simula ed o homogeneous columns o sand and loam soil. Then, and wo king on eal soils unde di e en illage managemen , we e alua ed di e en p ocedu es o measu e θ s and analyzed i s impac on K s and Ses ima ion. Fou di e en echniques we e compa ed: he g a ime ic-co e me hod and wo TDR in asi e (3 and 5 cm) and a non-in asi e (NiP) p obes. All TDR p obes we e connec ed o a low-cos NanoVNA. The sensi i i y analysis showed ha θ s ,K s and Scan be op imized simul aneously om he in e se analysis o an in il a ion cu e when βand γa e known alues and he in il a ion cu e is nea he s eady-s a e zone. Howe e , due o he in insic complexi ies o eal soils and he ac ha βand γa e unknown a iables, we ecommended o op imize K s and Susing measu ed θ s . The NiP senso connec ed o a NanoVNA p o ided a as , inexpensi e, clean, accu a e and obus al e na i e o measu e θ s a he end o he in il a ion expe imen s. 1. In oduc ion The cha ac e iza ion o he hyd aulic p ope ies o he soil su ace (so p i i y, S, and hyd aulic conduc i i y, K) is o pa amoun impo - ance o sol e many hyd ological and en i onmen al issues linked o soil wa e s o age and anspo in he adose zone. These soil p ope ies can be es ima ed om he in e se analysis o he ansien cumula i e in il a ion cu es measu ed wi h a ension disc in il ome e (Angulo- Ja amillo e al., 2000, 2016), which can wo ks om unsa u a ed o sa u a ed condi ions. The ension disc in il ome e (Pe oux and Whi e, 1988) consis s o a base disc a ached o a g adua ed wa e supply ese oi and a bubble owe ha can impose a nega i e p essu e head a he base disc. The diame e o he disc base can ange om he 25 cm p oposed by Pe oux and Whi e (1988) o he 3.2 cm used by Madsen and Chandle (2007). The co ec use o he ension disc in il ome e equi es he memb ane o he disc base o be comple ely in con ac wi h he soil su ace. To achie e his con ac , a hin laye o sand is commonly placed be ween he soil su ace and he disc base. The cumula i e in il a ion cu e is de e mined om he dec ease o he wa e le el inside he ese oi . The 1D analy ical solu ion, QEI, o Ha e kamp e al. (1994) com- bined wi h he e m o Sme em e al. (1994) o disk in il ome e measu emen s is one o he mos widely used models o es ima ing hyd aulic p ope ies. (e.g. Lassaba e e e al., 2009; La o e e al., 2015; Fe nandez-Gal ez e al., 2019). This model in ol es he ollowing pa- ame e s: K s ,S, he adius o he disc, d , he βand γcons an s, and he soil wa e con en inc ease Δθ, he la e de ined as he di e ence be- ween he ini ial, θ i , and inal, θ s , soil olume ic wa e con en s. While β * Co esponding au ho . E-mail add ess: [email p o ec ed] (D. Mo e -Fe n´ andez). Con en s lis s a ailable a ScienceDi ec Geode ma jou nal homepage: www.else ie .com/loca e/geode ma h ps://doi.o g/10.1016/j.geode ma.2024.117089 Recei ed 11 Janua y 2024; Recei ed in e ised o m 15 Oc obe 2024; Accep ed 28 Oc obe 2024 Geode ma 452 (2024) 117089 A ailable online 16 No embe 2024 0016-7061/© 2024 The Au ho s. Published by Else ie B.V. This is an open access a icle unde he CC BY license ( h p://c ea i ecommons.o g/licenses/by/4.0/ ). is ela ed o he soil di usi i y, D(θ), and he soil hyd aulic conduc i i y unc ions, γis a unc ion o he app oxima e es ima ion o so p i i y (Fuen es e al., 1992; Ha e kamp e al., 1994; Sme em e al., 1994). Gi en ha di ec o mula ions a e mo e con enien han complex implici equa ions, Ha e kamp e al. (1994) p oposed using he simpli ied wo-Te ms (2T) expansion. Howe e , since his app oxima- ion emains alid only o sho o in e media e in il a ion imes, Mo e -Fe n´ andez e al. (2020) sugges ed es ima ing K s and Susing he mo e accu a e h ee-Te m (3T) and ou -Te m (4T) expansions o he QEI model, alid o e longe ime in e als (Yilmaz e al., 2022). Al hough he QEI and 4T models ha e ou deg ees o eedom, and heo e ically ha e he po en ial o es ima e S,K s ,βand γ,La o e e al. (2018) and Mo e -Fe n´ andez e al. (2020) demons a ed ha , assuming Δθas a known alue, es ima ion o K s and S om ansien in il a ion cu es equi es using ixed and known βand γ alues. This is explained by he ac ha K s ,βand γa e closely linked h ough he second e m o he 4T expansion, which makes ha , a sho -medium in il a ion e m, hese h ee pa ame e s canno be simul aneously op imized. On he o he hand, Lassaba e e e al. (2009) and Yilmaz e al. (2023) showed ha he βand γdepend on he ype o soil and he ini ial wa e condi ion, and sugges ed ha hese pa ame e s o ela i ely d y soils can be app oxima ed om soil ex u al cha ac e is ics. In summa y, acco ding o p e ious s udies, K s and Sa e op imized a iables, βand γa e app oxima ed om soil ex u al p ope ies, d is a measu able inpu , and Δθ is a pa ame e o be measu ed. Howe e , despi e he p og ess in e alua ing K s ,S,βand γ, he in luence o Δθ on he es ima ion o K s and S emains unin es iga ed. The θ s and θ i alues used in he Ha e kamp e al. (1994) -Sme em e al. (1994) model a e commonly ob ained by g a ime ic measu e- men s, using he co e me hod (G ossman and Reinsch, 2002). The olume ic wa e con en , θ,is calcula ed as he p oduc be ween he g a ime ic wa e con en , W, and he soil bulk densi y, ρ b . While Wis calcula ed as he di e ence be ween he we and d y weigh o a soil sample, ρ b is he a io be ween he d y weigh and he co esponding olume o he sampled co e. Al hough he co e p ocedu e is aken as he e e ence me hod, he easibili y o his echnique o measu ing θ s a he end o a disc in il a ion expe imen could be ques ionable by di e en easons. Fo example, he inse ion o a co e in sa u a ed soil may a o soil compac ion, and hus al e he es ima e o θ s . In con as , an un- de es ima ion o θ s is ob ained i , o ins ance, he dep h o he we ing on ad ance is less han he o al heigh o he employed soil co e. Al hough his limi a ion could be sol ed by using hinne co es, he ep esen a i eness o ρ b dec eases wi h e y na ow co es. In addi ion o he ac ha soil managemen in sa u a ed soil condi ions is cumbe - some, he co e me hod equi es a u he labo a o y p ocessing wi h he use o o ens and balances. Las ly, some au ho s sugges equa ing he sa u a ed wa e con en o he soil po osi y, easily accessible om he d y bulk densi y; howe e , his calcula ion equi es an adequa e es i- ma ion o he speci ic densi y o soil pa icles, ha ing also he p oblem ha i may include apped ai in sa u a ed soils (Faye and Hillel, 1986). As al e na i e, he soil olume ic wa e con en can be measu ed by indi ec dielec ic me hods, such as he Time Domain Re lec ome y, TDR, echnique, a non-des uc i e me hod ha also allows eal- ime measu emen s o wa e con en . De e mina ion o θis based on he ime equi ed by an elec ical signal o a el and e lec back along he p obe’s ods (Topp e al., 1980). Ano he ad an age o he TDR ech- nique is he simplici y o he p obes, which allows o design and manu ac u e own p obes. This g ea e sa ili y allows, o example, o make discon inuous p obes o soil wa e p o iles (Topp e al., 1982), soil senso s o measu e he ma ix wa e po en ial (W ai h and O , 1999) o he elec ical conduc i i y o he soil wa e solu ion (Mo e -Fe n´ an- dez e al., 2012) o non-in asi e p obes (Selke e al., 1993; Pe sson and Be nd sson, 1998). The non-in asi e senso s a e especially sui able o soil su ace measu emen s, since hey allow apid measu emen s wi hou dis u bing he soil su ace (Pe sson and Be nd sson, 1998). This would be he case, o example, o he measu emen o wa e con en in he soil su ace c us . Selke e al. (1993) p esen ed a i s design o nonin asi e TDR, in which he wo p obe ods we e pa ially embedded in a se pen ine pa e n wi hin an ac ylic pad. This design was u he imp o ed by Pe sson and Be nd sson (1998), who longi udinally inse - ed a h ee- od TDR p obe in o a poly inyl chlo ide (PVC) block, so ha he su ace o he od su ounded hal o he measu emen olume. Nissen e al. (2003) s udied he spa ial sensi i i y o wo- and h ee- od p obes placed ho izon ally h ough he walls o an expe imen al box, and ound ha wo- od ins ead o h ee- od p obes should be used i sha p changes in ε a a e expec ed in he di ec ion ans e se o he plane con aining he p obe ods, owing o sepa a ion o he a eling elec- omagne ic wa es in he h ee- od case. These same au ho s epo ed ha ho izon al p obe o ien a ion is mo e app op ia e o moni o ing ac oss sha p e ical bounda ies, such as we ing on s. Since i s i s applica ion o θmeasu emen s in he 1980s, he TDR echnology has e ol ed owa ds mo e po able and accu a e in- s umen s. This is he case, o example, o he e olu ion om hea y and bulky 1502C Me allic Cable Tes e (Tek onix o Bea e on, O egon) o he small, ugged and po able TDR100 (Campbell Scien i ic). Howe e , al hough hese new de elopmen s make he TDR echnique mo e po able, he high cos o he TDR ins umen s (≅4000 € ) may limi hei use in some scena ios. To o e come his limi a ion, Qiwei e al. (2019) p oposed o measu e θusing a small size and low-cos mini ec o ne wo k analyze (miniVNA). This echnique was la e imp o ed by Mo e -Fe n´ andez e al. (2022) who, using a low-cos FDR-TDR de- ices NanoVNA (≅60 € ), om 50 kHz o 1.5 GHz, allowed accu a e measu emen s o θ. Since Δθ is a c i ical inpu o op imizing K s and S, he i s objec i e o his pape is o conduc a sensi i i y analysis o de e mine he in lu- ence o Δθ on K s and Ses ima ions. In he ollowing and wo king on soils unde di e en illage sys ems, we e alua ed di e en p ocedu es o measu e θ s and analyzed i s impac on K s and Ses ima ion. Fou di e en echniques we e compa ed: he g a ime ic-co e me hod and wo TDR in asi e (3 and 5 cm) and a non-in asi e (NiP) p obes. 2. Theo y 2.1. Cumula i e in il a ion cu e The quasi-analy ical 3D cumula i e in il a ion cu e, I3D, QEI, o disc in il ome e measu emen s and inal sa u a ion soil condi ions can be desc ibed as (Ha e kamp e al., 1994; Sme em e al. 1994): 2(Ks−Ki)2 S2 =2 1−β (Ks−Ki)(I3D−Ki −γS2 d(θs−θi) ) S2 −1 1−βln⎡ ⎢ ⎢ ⎢ ⎣ 1 βexp⎛ ⎜ ⎜ ⎜ ⎝ 2β(Ks−Ki)(I3D−Ki −γS2 d(θs−θi) ) S2⎞ ⎟ ⎟ ⎟ ⎠ +β−1 β⎤ ⎥ ⎥ ⎥ ⎦(1) whe e is he ime [T], Sis he so p i i y [L T -0.5 ], K i and K s [L T -1 ] a e he hyd aulic conduc i i y alues co esponding o ini ial, θ i , and sa u- a ion, θ s , olume ic wa e con en s [L 3 L -3 ], βis he in eg al shape pa ame e [-], γ[-] is a p opo ionali y cons an ha accoun s o he co ec ion o he we ing on shape (Sme em e al., 1994) and d is he adius o he disc [L]. Yilmaz e al. (2023) p oposed a simple ela ionship o es ima e βand γpa ame e s based on soil ex u al classes. The implici Eq. (1) o negligible ini ial hyd aulic conduc i i y can D. Mo e -Fe n´ andez e al. Geode ma 452 (2024) 117089 2 be simpli ied o a 4-Te m expansion acco ding o (Mo e -Fe n´ andez e al., 2020): I3D( ) = S 1 2+(2−β 3K+γS2 d(θs−θi)) +K2 9S(β2−β+1) 3 2+2(β−2)(β +1)(1−2β) 135 K3 S2 2 (2) This exp ession is alid o long in il a ion imes (be ween 2000 and > 50000 s o coa se and ine soil ex u es, espec i ely). Fo sho - in e media e in il a ion da a, Eq. (1) educes o a 2-Te m expansion (Ha e kamp e al., 1994): I3D( ) = S 1 2+(2−β 3K+γS2 d(θs−θi)) (3) Fo long- e m beha io Eq. (1) gi es he ollowing esul : I3D( ) = (Ks+γS2 d(θs−θi)) +S2 2(Ks−Ki)(1−β)ln(1 β)(4) 2.2. Time Domain Re lec ome y In Time Domain Re lec ome y, a as - ise s ep ol age elec omag- ne ic pulse is p opaga ed in he medium along a ansmission line. The cable es e eco ds a TDR signal exp essed by he e lec ion coe icien as a unc ion o ime. The ansi ime, L [T], o he TDR pulse p opa- ga ing one e u n ip in a ansmission line (e.g., TDR p obe) o leng h L [L] can be app oached (Topp e al., 1980) as: L=2L ε a √ c(5) whe e ε a is he appa en dielec ic cons an and c(3 x 10 8 m/s) is he speed o ligh cons an . The L alue is calcula ed as he di e ence be- ween he ime a which he signal en e s he TDR p obe’s ods ( i s peak) and he ime when he ace a i es a he end o he TDR p obe (second e lec ion poin ). This las poin is calcula ed wi h he widely accep ed angen me hod (Heimo aa a, 1993). Once L alue is ob- ained, ε a is calcula ed om Eq. (5). Fo he case o a non-in asi e TDR p obe, Maheshwa la e al. (1995) showed ha he measu ed e ec i e appa en dielec ic cons an , ε e , o a TDR p obe placed be ween wo semi-in ini e media wi h di e en ε a alues ( ε a1 and ε a2 ) can be exp essed as: ε e = ε a1+ ε a2 2(6) In Eq. (6) he subsc ip e e s o he ma e ials, i.e., ei he 1 o 2. Fo he pa icula case o a TDR p obe, in which he longi udinal hal o he ods a e co e ed by a block o known dielec ic cons an , ε b , he ε a alue when he block is placed on a soil su ace is calcula ed as (Pe sson and Be nd sson, 1998): ε a=2 ε e − ε b(7) whe e ε e is calcula ed om Eq. (5). Once ε a is ob ained, θcan be calcula ed by using he Topp e al. (1980) unc ion: θ= − 5.3⋅10−2+2.92⋅10−2⋅ ε a−5.5⋅10−4⋅ ε a2+4.3⋅10−6⋅ ε a3(8) 3. Ma e ial and me hods 3.1. In luence o Δθ on es ima ion o K s and S: sensi i i y analysis This nume ical expe imen aims o e alua e whe he Δθ can be es ima ed om he in e se analysis o an in il a ion cu e simula ed in a homogeneous soil column. To his end, nume ically gene a ed da a o elimina e unce ain ies associa ed wi h eal-wo ld expe imen al e o s was used. Assuming βand γas known alues, he sensi i i y analysis consis ed on op imizing K s and S o a ange o θ s alues and in il a ion cu es o di e en leng hs. Taking a ela i ely homogeneous ini ial wa e con en (θ i ) h oughou he opsoil laye , and he ease o measu ing θ i wi h g a ime ic o dielec ic me hods, θ i was se as a known alue and he analysis ocused only on a ia ions o Δθ, de i ed in changes in inal wa e con en (θ s ). The sensi i i y analysis was conduc ed on syn he ic in il a ion cu es gene a ed wi h he HYDRUS-3D model (ˇ Simunek e al., 1999). Wa e e en ion cu es we e cha ac e ized acco ding o he an Gen- uch en (1980) model wi h Mualem condi ion. The cu es we e gene - a ed on homogeneous sand and loam soil columns (Table 1) (Ca sel and Pa ish, 1988). The soil olume was disc e ized as a cylinde ( adius o 25 cm and dep h o 25 cm), co e ing he axisymme ic plane wi h a 2-D ec angula mesh o 100 x 900 cells. The base o he disc in il ome e o 10 cm adius was ep esen ed as a cons an p essu e head bounda y on he co esponding cells, whe eas he es o he soil su ace was ea ed as a mosphe ic bounda y wi h no lux. The ini ial soil wa e con en was e y close o he esidual wa e con en . Mo e de ails abou he in il- a ion cu e gene a ed wi h HYDRUS-3D can be ound in La o e e al. (2015). The sensi i i y analysis consis ed o calcula ing he di e ences be- ween he HYDRUS-3D syn he ic in il a ion cu e and he in il a ion cu es simula ed wi h he 4-Te ms expansion, Eq. (2), once he Sand K s we e op imized o di e en alues o θ s . In all cases, β,γ(Table 1), θ i and d (10 cm), Eq. (2), we e assumed as known alues. The in e se analysis was pe o med using he Le enbe g-Ma qua d (Mo e, 1978) op imi- za ion algo i hm. Once he Sand K s we e op imized o a gi en alue o θ s , he e o o objec i e unc ion, Q, ha ep esen s he di e ence be- ween he ac ual, I i , and he simula ed in il a ion cu es, I(S,Ks,θs), was calcula ed as Q= ∑ N i=1[(Ii−I(S,Ks,θs))Δ ]2 √ N−1(9) whe e N is he numbe o measu ed (I, ) alues. The sensi i i y analysis was applied o h ee in il a ion cu es o inc easing imes: 500, 3000, and 6000 s o he loam column and 75, 150, and 250 s o sand. Δθ alues explo ed a ange o ±10% a ound he ac ual θ s , wi h 1% in- c emen s. Finally, o each combina ion o θ s and in il a ion ime, he analysis iden i ied he minimum Q alue (indica ing he bes i ) and he co esponding op imal K s and S. Table 1 Theo e ical alues o ini ial (θ i ), sa u a ed (θ s ) and esidual (θ ) wa e con en , α and n pa ame e s o he an Genuch en (1980) wa e e en ion cu e, sa u a ed hyd aulic conduc i i y (Ks), so p i i y, S and γand βpa ame e s o he syn he ic soils. θ i θ θ s α n K s S a β b γ b cm 3 cm −3 cm -1 cm s −1 cm s -0.5 - - Sand 0.045 0.045 0.43 0.145 2.68 8.25 10 -3 0.1521 0.53 1.03 Loam 0.078 0.078 0.43 0.036 1.56 2.88 10 -4 0.0367 1.25 0.76 a Mo e -Fe n´ andez e al. (2017). b Lassaba e e e al. (2009). D. Mo e -Fe n´ andez e al. Geode ma 452 (2024) 117089 3 3.2. Field expe imen s In his sec ion, we i s desc ibe he TDR p obes ha we e used o measu e θ s a he end o an in il a ion expe imen , which p o ocol is desc ibed in ollowing sec ions. 3.2.1. TDR p obes Th ee TDR p obes o di e en geome ies we e compa ed: wo h ee- od TDR p obe o (i) 3 cm, P3, and (ii) 5cm, P5, leng hs, espec i ely and (iii) a wo- od non-in asi e TDR p obe, NiP. The geome ies o he di e en p obes a e summa ized in Table 2. A di e en manu ac u ing p ocess was employed in P3 and P5. While he ex e nal and cen al ods o P3 we e di ec ly welded o he inne e minal and connec o housing o a emale-BNC connec o , he co e- sponding ods in P5 we e di ec ly welded o he cen al cable and he ex e nal mesh o a coaxial cable, espec i ely. Fo he case o P5, he head o he TDR p obe was comple ely coa ed wi h epoxy and he end o he coaxial cable was welded o a male-BNC connec o . Measu emen s o soil wa e con en wi h P3 and P5 we e pe o med by e ically inse ing he p obes in o he soil. The NiP p obe consis ed o wo ods inse ed longi udinally in o a ci cula Te lon block o 10 cm adius and 1.9 cm hickness, so ha hal o he od su ace was con ained in he Te lon block. Bo h ods we e placed in he cen al pa o he Te lon block, such ha 1 cm o Te lon p o- uding om each end o he p obe. One end o each o he ods was welded o he cen al cable and he ex e nal mesh o a coaxial cable, espec i ely, which end was welded o a male-BNC connec o . Howe e , while he dep h explo ed by he 3- od TDR p obes is clea ly de ined by he leng h o he TDR wi es e ically inse ed in o he soil, he measu ed dep h allowed by NiP is mo e con using. To cla i y his unce ain y, he heo e ical olume explo ed by NiP was calcula ed nume ically ollowing he p ocedu e ou lined in Nissen e al. (2003). The c oss-sec ion p obe geome y elec os a ic p oblem was sol ed by ini e elemen s analysis wi h a comme cial so wa e (COMSOL Mul i- physics) in a 1m x 1 m domain. The sensi i i y unc ion was hen compu ed and in eg a ed o es ima e he p obe olume sensi i i y. 3.2.2. FDR-TDR ins umen The olume ic wa e con en was measu ed by connec ing he di e en TDR p obes o a low-cos Vec o Ne wo k Analyze s (VNA) comme cially a ailable (NanoVNA), wi h 1.5 GHz maximum ope a ing equency (Owo ech, 2019;Mo e -Fe n´ andez e al., 2022). The Nano- VNA can be used o measu emen s o F equency Domain Re lec ome y (FDR) o , a e sui able pos p ocessing, o TDR measu es. Al hough TDR and FDR a e dual p ocedu es, TDR p ocedu e was selec ed because i is easie o in e p e o soil expe imen s. Acco ding o Mo e - Fe n´ andez e al. (2022), he ime esolu ion o his de ice is app oxi- ma ely 0.333 ns. Al hough he ela i ely low equency supplied by his de ice limi s he numbe o poin s pe TDR wa e o m, p e ious expe - imen s demons a ed ha his ins umen can be sa is ac o ily employed as a TDR cable es e o measu ing he olume ic wa e con en in soils (Mo e -Fe n´ andez e al., 2022). The NanoVNA was connec ed o a sma mobile phone, ha allowed downloading he FDR signal using he ee a ailable NanoVNA WebApp (h ps://play.google.com/s o e/apps/de- ails?id=ne .low eal.nano nawebapp&hl=es_PA&pli=1). The FDR signal was hen ans o med o he ime domain using Fas Fou ie T ans o m (FFT) and In e se Fas Fou ie T ans o m (IFFT) e icien algo i hms (Mo e -Fe n´ andez e al., 2022). Gi en ha he eco ded TDR signal p esen s a limi ed densi y o poin s (100 poin s pe signal), he wa e o ms we e in e pola ed up o 2400 poin s using a cubic in e po- la ion me hod. Once he TDR signal was ob ained, θwas calcula ed acco ding o p ocedu e desc ibed in sec ion 2.2. The use o he Topp e al. (1980), Eq. (8), was accompanied by a p e ious calib a ion expe imen . The NanoVNA de ice was calib a ed ollowing he Open-Sho -Load s anda d calib a ion p ocedu e (Sayed and Ma ens, 2013) wi h he assis ance o he NanoVNA i mwa e. Finally, he e ec i e leng h o he di e en TDR p obes was de e mined by imme sing he co esponding TDR p obes in dis illed wa e , and compa ing he wa e dielec ic con- s an measu ed a known empe a u e wi h he co esponding Table 2 Rod diame e , ϕ , sepa a ion o ex e nal ods, S , and e ec i e leng h, L, o he h ee di e en TDR p obes used in he in il a ion expe imen o measu e he sa u a ed wa e con en . ϕ S L mm  P3 1 10 30 P5 2 20 50 NiP 5 20 78.5 Fig. 1. A schema ic o he p ocedu e used o measu e he olume ic wa e con en a he sa u a ed soil su ace using he h ee- od TDR p obes o 5 cm, P5, and 3 cm, P3, long, he no-in asi e p obe wi hou con ac sand laye , NiP, and he co e me hod, W. D. Mo e -Fe n´ andez e al. Geode ma 452 (2024) 117089 4 heo e ical alue (Jones e al., 2002). This calib a ion expe imen was also used o check he shape o he TDR signals when he P3, P5 and NiP p obes we e imme sed in wa e . 3.2.3. In il a ion expe imen s and θ s measu emen s The di e en TDR p obes designed o wa e con en measu es a he end o an in il a ion we e es ed in nine in il a ion expe imen s. To his end, he θ s measu ed wi h he di e en TDR p obes inse ed in he sa u a ed soil a he end o he in il a ion we e compa ed wi h he co esponding wa e con en measu ed g a ime ically using he co e me hod, θ s_w . A schema ic o he p ocedu e is summa ized in Figu e 1 The in il a ion expe imen s we e pe o med on a d yland esea ch a m o he Es aci´ on Expe imen al de Aula Dei (CSIC) in he p o ince o Za agoza (la i ude 418440N; longi ude 08460W; al i ude 270 m). Soil a he esea ch si e is a loam ( ine-loamy, mixed he mic Xe ollic Calcio - hid) acco ding o he USDA soil classi ica ion (Soil Su ey S a , 1975). Selec ed physical and chemical p ope ies o he soil o his laye we e gi en in L´ opez e al. (1996). The measu emen s we e conduc ed in Ap il 2023, in h ee adjacen and nea ly le el plo s (slope 0–2%). The plo s we e in he allow pe iod o a win e ba ley (Ho deum ulga e L.) - allow o a ion. Th ee di e en soil illage managemen s we e conside ed: con en ional illage (CT), educed illage (RT) and no- illage (NT). While CT consis ed o moldboa d ploughing o allow plo s, RT used chisel ploughing as p ima y illage. NT used exclusi ely he bicides (glyphosa e) o weed con ol h oughou he allow season. The ini ial wa e con en o he soil, θ i , was in all cases lowe han 0.1 cm 3 cm -3 . Wi hin each plo , h ee in il a ion measu emen s we e pe - o med. A compac design o ension disc in il ome e (10 cm diame e and heigh ) was used, whe e he wa e le el d op was moni o ed by a sma phone came a (La o e e al., 2021). A hin con ac sand laye , wi h a hickness be ween 1 o 2 mm, was placed be ween he disc base and he soil su ace (Pe oux and Whi e, 1988). The o se calcula ed o he con ac sand laye (Reynolds, 2006) was less han 2 mm. The sand was le elled using a le ele o me hac yla e ube closed a he bo om, wi h he same diame e and heigh as he in il ome e . Excess sand ou side he le ele base was emo ed. The ension a he disc base was ixed o 0 cm. Assuming ha wa e con en o he d y opsoil laye was homogeneously dis ibu ed, θ i was measu ed wi h TDR by e ically inse ing he P5 p obe igh nex o he in il a ion poin . A e illing he in il ome e wi h wa e , he sma phone was posi- ioned in on o he le ele o cap u e he en i e ese oi on i s sc een. Came a dis ance and angle we e adjus ed o ensu e a clea ocus. Video eco ding was hen ini ia ed. The le ele was swi ly eplaced wi h he wa e - illed disc in il ome e , ma king he s a o in il a ion ime. Fig. 2. Rela ionship be ween he sa u a ed olume ic wa e con en θ s and (a) he objec i e unc ion, Q, Eq. (9), (b) he so p i i y, S, and (c) he sa u a ed hyd aulic conduc i i y, K s , es ima ed om he in e se analysis o a syn he ic in il a ion cu e simula ed in a loam soil column, a in il a ion imes o 500, 3000 and 7000 s o , espec i ely. The dashed g ey lines indica e he heo e ical alues o θ s ,K s and S, and e ical blue lines a e he op imized θ s . The ed dashed line in Fig. 2a.2, 2b.2 and 2c.2 is he eg ession line be ween θ s and S. D. Mo e -Fe n´ andez e al. Geode ma 452 (2024) 117089 5 In il a ion p oceeded un il app oxima ely 30 mm o wa e had in il- a ed. A his poin , he ideo eco ding was s opped. Following he p ocedu e ou lined by La o e e al. (2021), each ideo was used o gene a e an expe imen al in il a ion cu e. A he end o he in il a ion expe imen , he in il ome e was li ed o he soil su ace and he P5 p obe, which was p e iously connec ed o he NanoVNA plus sma - phone, was inse ed h ee imes in o he soil, inside he sand laye ci cum e ence, bu making su e ha he p obe head wen comple ely h ough he sand laye and ha he ods we e ully inse ed in o he soil ma ix. Du ing each inse ion, a FDR signal was eco ded. This p ocess ook less han 10 s. In o de o eplenish he d ained wa e du ing TDR measu emen s, a he end o he TDR measu emen s he in il ome e was placed again on he sand laye o 2 minu es. A e his ime, he p ocedu e applied o P5 was epea ed o he P3 p obe. A e P3 measu emen s, he in il ome e was again li ed o he soil, and he sand laye was emo ed wi h a mason y owel. Immedia ely a e , he NiP p obe was placed on he ba e and sa u a ed soil, and he FDR signal was eco ded. This p ocess ook less han 5 s. A e ha , he in il ome e was placed again on he ba e soil o 2 minu es. Finally, he in il ome e was again li ed o and a s ain s eel cylinde (2.5 cm heigh and 5 cm diame e ) was inse ed on he cen e o he sa u a ed ci cum e ence. The co e was nex ex ac ed, he excess o soil emo ed and he sample pou ed in o plas ic lask wi h ai igh closu e. Once in he labo a o y, he TDR wa e o ms we e download and analyzed. The collec ed co es we e weigh ed, d ied a 105 ◦C and eweigh ed o ob ain he g a ime ic sa u a ed wa e con en o he soil, W s . The olume ic sa u a ed wa e con en ob ained g a ime ically, θ s_w , was calcula ed as he p oduc o W s and he soil bulk densi y, ρ b , he Fig. 3. Rela ionship be ween he sa u a ed olume ic wa e con en θ s and he objec i e unc ion, Q, Eq. (9), es ima ed om he in e se analysis o a syn he ic in il a ion cu e simula ed in a sand column, a in il a ion imes o 75, 150 and 250 s o , espec i ely. The dashed g ey lines indica e he heo e ical alues o θ s , and e ical blue lines a e he op imized θ s . Fig. 4. TDR wa e o ms measu ed wi h he 5 cm, P5, and 3 cm, P3, long h ee- od TDR p obes and he no-in asi e p obe, NiP, when imme sed in dis illed wa e and inse ed in he sa u a ed soil sampled in eplica ion one o he educed illage ea men . D. Mo e -Fe n´ andez e al. Geode ma 452 (2024) 117089 6 la e calcula ed as he a io o W s o he cylinde olume. A e a i s isual inspec ion o he da a and he absence o ex eme alues, o simplici y, he a i hme ic mean and s anda d de ia ion o he eplica es we e used o ob ain a ep esen a i e alue o each ea men as well as he in insic a iabili y o he measu emen s and analysis. To e alua e he accu acy o he TDR p obes, he ela i e e o (RE) was calcula ed as he quo ien be ween he θ s alues measu ed by he TDR p obes (θ s ) and he g a ime ic e e ence alues (θ s_w ). In addi ion, an analysis o a i- ance (ANOVA) was pe o med be ween he θes ima ed by he di e en p ocedu es. Fig. 5. Sensi i i y dis ibu ion o NiP wo- od p obe (colo ed) and sample a eas con aining 50%, 75% and 50% o he o al sensi i i y. Fig. 6. Rela ionship be ween he sa u a ed olume ic wa e con en measu ed wi h he co e me hod, θ s_w , and co esponding alues, θ s , measu ed wi h he 5 cm, P5, and 3 cm, P3, long h ee- od TDR p obes and he no-in asi e p obe, NiP, a he di e en sampling poin s. Table 3 A e age, s anda d de ia ion and ela i e e o , RE, o he olume ic wa e con en measu ed on he in il a ion expe imen s using he co e me hod, W, me hod, he h ee- od P3 and P5 TDR p obe and he non-in asi e, NiP, TDR senso . Rows wi h he di e en le e indica e signi ican di e ences (p <0.05) be ween ea men s. T ea men A e age S anda d de ia ion RE cm 3 cm -3 % W 0.345 a0.021 - P3 0.362 a0.025 4.9 P5 0.232 b0.058 -33.0 NiP 0.356 a0.015 3.2 Fig. 7. In il a ion cu es measu ed in he con en ional illage, CT, educed illage, RT, and, no- illage, NT, ea men s. D. Mo e -Fe n´ andez e al. Geode ma 452 (2024) 117089 7 Using he SIA p ocedu e (Mo e -Fe n´ andez e al., 2021), he S,K s and he op imal in il a ion ime, o , we e es ima ed om he in e se analysis o he measu ed in il a ion cu e. The op imiza ion p ocess in ol ed wo consecu i e sequen ial analyses. In a i s s ep, he e ec o he con ac sand laye on he in e se analysis was emo ed using La o e e al. (2015) p ocedu e. The sand e ec was conside ed as a gap, in ime and olume, be o e wa e in il a es in o he soil, and he con ac sand laye in luence was emo ed by inding he sand in il a ion ime, sand , (and i s co esponding wa e olume) and shi ing he expe imen al da a o he o igin. In a nex s ep, he op imiza ion o Sand K s as unc ion o θ s was pe o med using he Sequen ial In il a ion Analysis (SIA) p ocedu e (Mo e -Fe n´ andez e al., 2021). The SIA me hod es ima es S and K s o he uppe soil laye by i ing he 4-Te m expansion, Eq. (2), o inc easing in il a ion ime se ies, and compu es he objec i e unc ion Q, Eq. (9), as a unc ion o he numbe o da a poin s conside ed. A o al o 20 inc easing imes we e conside ed. The o is de ined as he ime wi h a minimum Q alue, and he ac ual Sand K s a e he co esponding alues calcula ed o an in il a ion cu e o ime o . The Le enbe g- Ma qua d (Mo e, 1978) op imiza ion algo i hm was used in he in- e se analysis. The γand βpa ame e s we e ixed o 0.75 and 1.25, espec i ely, which co espond o he heo e ical alues o a loam soil (Lassaba e e e al., 2009). Fig. 8. (a) Expe imen al, I exp , and op imized, I op , in il a ion cu es measu ed in he hi d eplica ion o he no- illage ea men , and ime e olu ion o he sa u a ed hyd aulic conduc i i y, K s , so p i i y, S, sa u a ed olume ic wa e con en , θ s , and he objec i e unc ion o he in il a ion cu e, Q, op imized wi h SIA me hod using (b) he measu ed olume ic wa e con en , θ s , and (c) he op imized wa e con en , θ s *. Ho izon al dashed lines a e he espec i e heo e ical alues K s ,S and θ s . D. Mo e -Fe n´ andez e al. Geode ma 452 (2024) 117089 8 To e alua e he in luence o θ s on he op imiza ion p ocess, he SIA p ocedu e was i s applied using he measu ed θ s , and hen applied on a simul aneous op imiza ion o S,K s and θ s ( aking θ s as unknown alue), whose op imized θ s was de ined as θ s *. The Sand K s es ima ed wi h bo h p ocedu es we e compa ed wi h each o he , and he op imized θ s *was hen compa ed wi h he measu ed θ s . Since his pape is no ocused on o compa e illage sys ems, no s a is ical analysis was pe o med among illage ea men s. 4. Resul s and discussion 4.1. In luence o Δθ on es ima ion o K s and S: sensi i i y analysis The sensi i i y analysis showed ha in il a ion ime has an e ec on he es ima ion o θ s ,K s and S(Fig. 2). O e all, he la shape obse ed in he θ s s Q ela ionship wi hin he in e al [θ s ±10%] and in il a ion ime o 500 s (Fig. 2a.1) indica es ha simila in il a ion cu es can be ob ained o di e en alues o θ s . On he o he hand, he op imized θ s (blue e ical line in Fig. 2a.1) was a om i s heo e ical alue. The limi ed sensi i i y obse ed in sho in il a ion cu es can be explained wi h he 2-Te m expansion, Eq. (3). This model, while alid o sho o in e media e in il a ion imes, migh no ully accoun o all he ele- an p ocesses go e ning in il a ion, pa icula ly o longe du a ions. The in e dependence be ween he h ee op imized pa ame e s exis ing in he second e m o Eq. (3) implies ha simila in il a ion cu es can be ob ained o di e en combina ions o S,K s and θ s . Simila ly, he in e dependence be ween γ,βand K s wi hin he 2-T expansion explains why op imiza ion o K s a medium in il a ion imes also equi es aking γand βas ixed alues. O e all, an inc easing ela ionship was obse ed be ween θ s and he op imized Sand K s . Howe e , while a linea ela- ionship, wi h ela i ely small slope, was obse ed be ween θ s and S (Fig. 2b.1), a mo e ab up beha io was ound o K s (Fig. 2c.1). The analysis e ealed ha a ia ions in K s we e la ge o θ s below he heo e ical alue compa ed o scena ios whe e θ s exceeded he heo- e ical alue. These indings sugges ha when op imizing Sand K s using sho - o-in e media e in il a ion cu es, i is c ucial o ea θ s as a known alue. This is pa icula ly impo an o expe imen al measu e- men s, as soil he e ogenei y o en makes i di icul o ob ain pe ec ly homogeneous in il a ion cu es wi h su icien ly long du a ions. The beha io obse ed o sho -in e media e in il a ion cu es changed signi ican ly wi h inc easing in il a ion imes, whe e he θ s s. Q ela ionship o longe in il a ions (e.g. 3000 and 600 s) exhibi ed a clea e and unique minimum (Fig. 2a.2 and 2a.3). In his case, longe in il a ion imes enabled he op imized θ s o con e ge close o i s heo e ical alue. These esul s could be explained by he addi ional coe icien s o he 4-Te m expansion, Eq. (2), which compensa e o he pa ame e in e ac ion desc ibed in he 2-Te m model, Eq. (3), and al- lows op imizing an addi ional a iable. The esul s sugges ha he close is he in il a ion ime o he s eady-s a e, Eq. (4), simul aneous and mo e accu a e es ima es o θ s ,K s and Scan be achie ed (Fig. 2). Fo example, his hypo hesis is suppo ed by he sensi i i y analysis ob- ained in he sand column (Fig. 3), whe e, compa ed o loam soil, signi ican ly sho e in il a ion imes (e.g. 250 s) allowed accu a e op imiza ion o θ s . Thus, hese esul s indica e ha op imiza ion o θ s depends on how close he in il a ion ime is o he s eady-s a e egion. Howe e , due o he in e dependence be ween Δθ,γand βwi hin he s eady-s a e zone, Eq. (4), hese h ee pa ame e s canno be simul a- neously es ima ed wi h Eq. (4), since di e en combina ion o hem esul s in simila s eady-s a e cu es. Thus, gi en ha βand γa e app oxima e alues ob ained om he soil ex u al cha ac e is ics, o educe he unce ain y in he op imiza ion o K s and S, as a as possible, i is p e e able o es ima e K s and S om measu ed alues o Δθ. As obse ed in sho -in e media e in il a ion cu es, an inc easing ela- ionship was obse ed be ween θ s and he op imized Sand K s (Fig. 2b and c). Howe e , he dispe sion o K s wi hin he [θ s ±10%] in e al dec eased as in il a ion imes app oached o he s eady-s a e zone. In conclusion, hese esul s indica e ha op imiza ion o Sand K s is no e y sensible o θ s when he o al in il a ion ime is close o i s s eady- s a e. 4.2. Field expe imen s 4.2.1. TDR p obes O e all, he empo al esolu ion o he NanoVNA connec ed o P3, P5 and NiP was enough o cha ac e ize he dielec ic cons an o wa e (Fig. 4), whe e he well-de ined shape o he TDR wa e o m allows de ec ing he i s peak and he second e lec ion poin . Fo example, o he mos es ic i e case o P3, se en poin s we e de ined be ween he i s peak and he second e lec ion poin . Al hough de ec ion o i s peak and second e lec ion poin is main ained when he p obes a e inse ed in sa u a ed soil, he low numbe o poin s ob ained in P3 (be ween 3 and 4) would p obably be a he limi o accu a e es ima ion o wa e con en (Fig. 4). Howe e , his is no he case o NiP, whe e he numbe o poin s wi hin hese ime in e als doubles wi h espec o P3. The low numbe o poin s ound in P5 inse ed in sa u a ed soil (Fig. 4) is due o he ac (as will be desc ibed in sec ion 4.2.1) ha he soil a he ime o he measu emen was no sa u a ed. The simula ed soil olume explo ed wi h NiP showed ha he 90% cu e o o al sensi i i y eaches 2 cm dep h in o he soil. Howe e , al hough he 90% cu e has a leakage o 3 mm ou side he Te lon sup- po in he ai egion, and he e o e, he ε a alue ob ained wi h Eq. (6) is unde es ima ed, he de ia ion o he expec ed ε a in sa u a ed soils is less han 0.5%, and can be neglec ed. Thus, he esul s showed ha he olume explo ed wi h NiP ex ends 2.5 cm o each side o he p obe cen e and eaches 2 cm below he soil su ace along he p obe leng h (Fig. 5). In conclusion, al hough he esul s sugges ha P3 connec ed o a NanoVNA de ice is no a p io i a consis en design o measu ing he Table 4 A e age soil so p i i y, S, sa u a ed hyd aulic conduc i i y, K s and objec i e unc ion, Q, Eq. (7), calcula ed wi hin he op imum in il a ion ime, o , using he olume ic wa e con en , θ s , measu ed wi h non-in asi e TDR p obe, and he co esponding alues ob ained using he op imized olume ic wa e con en , θ s *. Da a wi hin he pa en heses deno es he s anda d de ia ion. CT, RT and NT indica e con en ional illage, educes illage and no illage, espec i ely. Measu ed θ s Op imized θ s * θ s S K s Q o θ s * S K s Q o cm 3 cm -3 mm s -0.5 mm s -1 mm s cm 3 cm -3 mm s -0.5 mm s -1 mm s CT1 0.35 0.60 (0.015) 0.044 (0.006) 0.19 (0.014) 277 0.38 (0.04) 0.61 (0.042) 0.042 (0.015) 0.19 (0.014) 277 CT2 0.38 0.80 (0.004) 0.048 (0.002) 0.22 (0.016) 378 0.35 (0.03) 0.79 (0.010) 0.045 (0.004) 0.22 (0.016) 378 CT3 0.38 0.67 (0.013) 0.053 (0.002) 0.16 (0.002) 301 0.37 (0.05) 0.66 (0.081) 0.052 (0.004) 0.16 (0.002) 301 RT1 0.36 0.69 (0.000) 0.141 (0.000) 0.19 (0.000) 50 0.31 (0.00) 0.68 (0.000) 0.140 (0.000) 0.19 (0.000) 50 RT2 0.35 0.92 (0.030) 0.047 (0.009) 0.09 (0.005) 230 0.31 (0.03) 0.89 (0.032) 0.045 (0.011) 0.08 (0.055) 230 RT3 0.37 1.00 (0.004) 0.037 (0.003) 0.09 (0.005) 157 0.32 (0.00) 1.00 (0.003) 0.019 (0.003) 0.09 (0.005) 157 NT1 0.33 0.60 (0.003) 0.001 (0.001) 0.16 (0.008) 720 0.37 (0.01) 0.63 (0.003) 3.8 10 -4 (2 10 -4 ) 0.14 (0.006) 720 NT2 0.35 0.35 (0.004) 0.028 (0.001) 0.17 (0.006) 240 0.34 (0.05) 0.38 (0.001) 0.027 (0.002) 0.17 (0.006) 240 NT3 0.35 0.50 (0.000) 0.022 (0.002) 0.09 (0.008) 509 0.34 (0.04) 0.50 (0.006) 0.021 (0.004) 0.09 (0.008) 509 D. Mo e -Fe n´ andez e al. Geode ma 452 (2024) 117089 9