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Model for microwave emission of a snow-covered ground with focus on L band

Model for microwave emission of a snow-covered ground with focus on L band
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  Model for microwave emission of a snow-covered ground with focus onL band Mike Schwank a,c, ⁎ , Kimmo Rautiainen b , Christian Mätzler c , Manfred Stähli a , Juha Lemmetyinen b , Jouni Pulliainen b , Juho Vehviläinen b , Anna Kontu b , Jaakko Ikonen b , Cécile B. Ménard b , Matthias Drusch d ,Andreas Wiesmann c , Urs Wegmüller c a Swiss Federal Institute WSL, Mountain Hydrology and Torrents, Zürcherstrasse 111, 8903 Birmensdorf, Switzerland b Finnish Meteorological Institute, Arctic Research, P.O. Box 503, FI-00101 Helsinki, Finland c Gamma Remote Sensing AG, Worbstrasse 225, 3073 Gümligen, Switzerland d European Space Agency, ESTEC, 2200 AG Noordwijk, The Netherlands a b s t r a c ta r t i c l e i n f o  Article history: Received 4 April 2014Received in revised form 6 June 2014Accepted 3 August 2014Available online xxxx Keywords: Microwave radiometryL-bandSMOSSMAPRadiative transferSnowSoil freeze/thaw Passive L-band (1 – 2 GHz)observablesare sensitive tosurface soil moistureandocean salinity,whichisthecoreof the  “ soil moisture and ocean salinity ”  (SMOS)mission of the European Space Agency (ESA). This work inves-tigatesmicrowaveemissionprocessesthatareimportanttolinkL-bandbrightnesstemperatureswithsoilfreeze/thawstatesandthepresenceandthestateofsnow.Tothisend,agroundsnowradiativetransfer(GSRT)modelhasbeendevelopedonthebasisofthe “ MicrowaveEmissionModelofLayeredSnowpacks ” (MEMLS).Ourmodelsensitivity study revealed that L-band emission of a freezing ground can be affected signi 󿬁 cantly by dry snow,whichhasbeenmostlydisregardedinpreviousstudies.Simulationssuggestthatevendrysnowwithmostlyneg-ligibleabsorptionattheL-bandcanimpactL-bandemissionofwinterlandscapessigni 󿬁 cantly.Wefoundthatim-pedancematchingandrefractioncausedbyadrysnowpackcanincreaseordecreaseL-bandemissiondependingon the polarization and the observation angle. Based on the performed sensitivity study, these RT processes canbecompensatoryatverticalpolarizationandtheobservationangleof50°.Thissuggeststheuseofverticalpolar-ized brightness temperatures measured at around 50° in order to achieve segregated information on soil-frost.Furthermore,our simulationsdemonstrate a signi 󿬁 cantsensitivity of L-bandemission athorizontalpolarizationwith respect to the density of the lowest snow layer as the result of refraction and impedance matching by thesnowpack.© 2014 Elsevier Inc. All rights reserved. 1. Introduction Radiation, heat and mass  󿬂 uxes through the terrestrial surface layerare major drivers of climate. For land areas, the quantities involved intheirassociatedexchangeratesarefundamentallylinkedtothepresenceandthestateofvegetationandsnowcover,theamountofwaterstoredinthe soil, as well as the freeze/thaw state of the ground. Soil moisture andocean salinity have been identi 󿬁 ed as essential climate variables by theGlobal Climate Observing System (Anonymous, 2006).The provision of these data products has been de 󿬁 ned as thefundamental objective (Anonymous, n d) of the second Earth ExplorerOpportunity mission  “ soil moisture and ocean salinity ”  (SMOS) (Fontet al., 2010; Kerr et al., 2010) launched on 2nd November 2009. Sincetheendof theSMOScommissioningphasein May2010, theL-bandra-diometer MIRAS (Martin-Neira & Goutoule, 1997) on board the SMOSsatellite has been in an excellent operational state and the full suite of multi-angular brightness temperature data  T  B  p at horizontal (  p  = H)and vertical (  p  = V) polarization has been delivered for more thanthreeyearsnow.Likewise,themainSMOSdataproductshavebeenpro-ducedfromtheSMOS T  B  p withglobalcoverageandarepeat-timeofap-proximately three days (Mecklenburg et al., 2012).However, being an explorer mission, a generally important aspectfor SMOS is the extended and advanced exploitation of   T  B  p in compli-ance with the challenges identi 󿬁 ed in ESA's scienti 󿬁 c strategy for theLiving Planet Programme,  “ The Changing Earth ”  (Anonymuos, 2006).Amongst other areas, this includes the characterization of snow coversand freeze/thaw cycles of grounds in high latitudes to understandtheir feedback regarding the water cycle and the climate system. Like-wise, monitoring soil freeze/thaw states has been considered to be akey component of the National Aeronautics and Space Administration(NASA) Soil Moisture Active and Passive (SMAP) mission (Entekhabiet al., 2010). Remote Sensing of Environment 154 (2014) 180 – 191 ⁎  Corresponding author at: Swiss Federal Institute WSL, Mountain Hydrology andTorrents, Zürcherstrasse 111, 8903 Birmensdorf, Switzerland. Tel.: +41 44 7392 559. E-mail address: (M. Schwank).© 2014 Elsevier Inc. All rights reserved. Contents lists available at ScienceDirect Remote Sensing of Environment  journal homepage:  Meanwhile, changes in the high-latitude annual cycles of soilfreezing and thawing and seasonal snow cover have an impact of ahighly complex nature on the global hydrological cycle, the release of trace gases, and high-latitude ecosystems. For example, increasingthickness of the seasonally thawed active layers (e.g. in Siberia) hasbeen found to cause large amounts of CH 4  releases from wetlands(Anisimov, 2007). Another example revealing the importance to moni- tor cycles of soil freeze/thaw at large spatial scales are boreal forests.Due to increased respiration, these large areas in the northern hemi-sphere can turn from carbon sinks into sources (Piao et al., 2008;Thum et al., 2009).Fromresearchalongtheselines,itiswellknownthatshiftsinlarge-scalefreeze/thawcyclesandseasonalsnowcovercauseclimaterelevantfeedbacks.Furthermore,thepotentialofSMOS(Druschetal.,2012)andSMAP (McDonald et al., 2010) to map freeze/thaw states is recognized,and their application to retrieve snow states could be a further optionas is outlined in this work. However, at present SMOS measurementsover Northern latitudes are often excluded from further analysis(Anonymous, 2010) due to thick snow covers present often duringmore than half of the year.Sensitivity of microwave brightness temperatures  T  B  p with respectto soil-frost results from the much smaller dielectric constant of ice incomparison to liquid water. Dielectric properties of soils in frozen andunfrozen conditions are reported in Hallikainen, Ulaby, Dobson andEl-Rayes (1984), (1985). According to these studies (summarized in Chapter 4.14.3 of (Mätzler, 2006)), real parts of dielectric constants ε  FG  of frozen soils are of the order of   ε  ′ FG ≈ 5 at L-band frequencies(1 – 2 GHz),and ε  ″ FG ≈ 0.5canbeassumedtobeanupperlimitofcorre-sponding imaginary parts for most soil types at temperatures between − 0.5 °C and − 10 °C. As a consequence, the power absorption coef  󿬁 -cient  γ   ¼ 4 π = λ  Im  ffiffiffiffiffiffiffi  ε  FG p   is  γ  ≈ 6.68 m − 1 , corresponding to L-bandemission depths  χ   = 1 /   γ  ≈ 15 cm in frozen soils. However, beyondthese theoretical considerations, the sensitivity of passive L-bandradiancewithrespecttosoilfreezinghasbeendemonstrated,forexam-ple, by means of a  󿬁 eld experiment performed on a grassland soil inSwitzerland (Schwank et al., 2004), and recently in a long-term  󿬁 eldcampaign performed in a forest opening in northern Finland(Rautiainen et al., 2012). The possibility of using L-band radiometry to derive temperature gradients and surfacetemperatures of active layersoccurring in permafrost regions was explored theoretically in Mironovet al. (2013). Field studies accompanied by radiative transfer modelinginvestigations were presented in Hao et al. (2011) and Zhao et al. (2012), to quantify sensitivities of passive microwave radiance atshorter wavelengths (C-, X-, K- and Ka-band) with respect to soil-froststates.Themajormotivationtodevelopthegroundsnowradiativetransfer(GS RT) model, in order to simulate the microwave emission fromwinter landscapes, was to foster new SMOS data products, such asmaps of frozen/thawed soil states or snow conditions. Our GS RTmodel makes use of a two-stream approach to compute RT across astackoflayersthatapproximate,forinstance,afreezinggroundcoveredwithalayeredsnowpack(SP).TheGSRTmodelherepresentedisdevel-oped on thebasis of the  “ Microwave EmissionModel of Layered Snow-packs ”  MEMLS (Mätzler, 1996, updated 2004; Mätzler & Wiesmann,1999,2012;Wiesmann&Mätzler,1999)complementedwithaddition-al modeling components to represent a multi-layered soil beneath thesnow. These additional modeling components allow for the consider-ation of roughness of layer interfaces, and the effective temperature of the soil beneath the SP. Accordingly, the parametric investigation of the GS RT model is a major aspect of this work, to explore the impactof soil freezing and of dry snow on L-band  T  B  p . Furthermore, the GS RTmodel is fed with realistic input data to compare simulated  T  B  p againstmeasurements performed with a tower-based L-band radiometer. Thecorresponding experimental data sets were taken from the precedingstudy (Rautiainen et al., 2012) carried out at the test-site of the Finnish Meteorological Institute (FMI) in Sodankylä (Finland). 2. Datasets used as inputs to the radiative transfer model This section describes the datasets used to drive the ground snowradiative transfer (GS RT) model for the comparison with measuredbrightness temperatures  T  B  p . To this aim, we use a number of in-situdata acquired in the preceding study (Rautiainen et al., 2012) from a two-month period during the early winter of 2010. This period wasselected to span the transition from wet fall to rather stable mid-winter states, including the occurrence of snow and soil freezing. Aspring period was not selected, since, for example, thawing and re-freezing cause highly transient snow and soil conditions. The subse-quent descriptions of the in-situ data used in the current study aredeliberately condensed. Detailed information on the correspondingexperimental issues, including the description of the FMI test-site inSodankylä (Finland) and its surrounding landscape, is provided inSection  “ III. Measurements and data ”  in Rautiainen et al. (2012).The in-situ data used include time-series of   󿬁 eld scale brightnesstemperatures  T  B  p observed with a tower-based L-band radiometer(Section 2.1), and a number of concurrent in-situ measured soil andsnow properties (Section 2.2), representative of the radiometer foot-print. Apart from the available time-series of ground (G) temperatures T  G 2cm  and ground liquid water-content  WC  G 2cm  measured 2 cmbelow the soil surface, corresponding information at greater ground-depth is required in Section 4.4. As such measurements were not avail-able, corresponding temperatures  T  COUP  and water-content  WC  COUP weresimulatedusingthenumericalsoil – vegetation – atmospheretrans-fer model COUP ( Jansson & Moon, 2001), as is outlined in Section 2.3.  2.1. Brightness temperature measurements As part of the calibration and validation activities of the SMOS mis-sion, ESA funded three ELBARA II L-Band radiometers (Schwank et al.,2010b) developed by GAMMA Remote Sensing AG (Switzerland). Thebrightness temperatures  T  B  p used in the present study were measuredwith the ELBARA II radiometer operated by the Finnish MeteorologicalInstitute Arctic Research Centre (FMI-ARC) in Sodankylä (Finland).FromSeptember2009toSeptember2012,ELBARAIIwasoperationalona5-mobservationtowerinstalledinaforestopeningwithintheinten-sive observation area (IOA, 67.368 N, 26.633 E) at the FMI-ARC station.Automated measurements of   T  B  p at horizontal (  p  = H) and vertical(  p  = V) polarization were performed every 3 or 4 hours at observationangles  θ 0  = 30° – 70° relative to nadir in steps of 5°. Fig. 3 and Table 1 inthe preceding study (Rautiainen et al., 2012) show the corresponding ELBARA II setup, and the main instrument characteristics, respectively.The observed footprints are considered to be representative for a borealforest on mineral soils composed of sand (70%), silt (29%) and clay (1%)with layers of humus, litter, vegetation and lichen on top.Diurnal averages of   T  BH and  T  BV  measured at  θ 0  = 50° for the two-monthperiodduringtheearlywinterof2010wereselectedtocompareagainst GS RT model results (Section 4.4). These measurements, takenbetween 8 October and 8 December 2010, include the formation of aSP (from 25 October, arrow 1) and the onset of soil-freezing (from 5November, arrow 2) as is depicted in Fig. 1.  2.2. In-situ measurements Manualobservationsofdepths d FG ofthefrozenground(FG)nexttotheELBARAIIfootprintswillbeusedinSection4.4asinputtotheGSRTmodel for the comparison with tower-based brightness temperatures T  B  p . The upper panel of  Fig. 1 shows corresponding frost-tube readings d FG  performed weekly during the comparative period of 8 October – 8December 2010. Furthermore, the GS RT model was driven by snowparameters derived from snow pit measurements performed approxi-mately every week next to the ELBARA II footprints. These snow pitobservations include the total snow depth  d SP  (upper panel of  Fig. 1),pro 󿬁 les at 10 cm depth-resolution of snow (S) temperatures  T  S , snow 181 M. Schwank et al. / Remote Sensing of Environment 154 (2014) 180 – 191  densities  ρ S , and liquid snow water-content  WC  S . However, the quanti-tativevaluesof  WC  S pro 󿬁 lesmeasuredwithsnowforks(Sihvola&Tiuri,1986;Toikka,2009)arequestionableforvaluessmallerthan1.0 – 1.5%of volumetric liquid water-content, as shown in Techel & Pielmeier(2011). Moreover, L-band emission computed with the GS RT model israther sensitive with respect to  WC  S . This is why snow liquid water-content is assumed to be  WC  S  = 0.0%, which is reasonable for snowtemperatures  T  S  b − 0.1 °C measured in-situ during the comparativeperiod.In addition to these manual snow observations, automated soilmeasurements, alsotaken from the datasets describedin the precedingstudy (Rautiainen et al., 2012), are used in the present study. Thesetime-series are liquid soil water-contents  WC  G 2cm  and temperatures T  G 2cm  measured 2 cm below the soil surface. Delta-T (ML2x) sensorswereusedtomeasurethesenear-surfacesoildatacontinuously,where-as three or four-hour averages were stored in accordance with theavailable ELBARA II measurements  T  B  p . Fig. 1 shows  WC  G 2cm  and T  G 2cm  (solid lines) spanning the two-month period selected.  2.3. COUP simulations Beyond the available near-surface soil temperatures  T  G 2cm  andliquid water-content  WC  G 2cm , further information on temperaturesand liquid water-content at greater soil depths was used as inputsto the GS RT model in Section 4.4. As these variables were not mea-sured, they were simulated with the state-of-the-art numericalsoil – vegetation – atmosphere transfer model COUP ( Jansson & Moon,2001). This model accounts for the temporal variation of snow- andsoil-frost (Stähli et al., 1999) and has been validated comprehensivelyat other high-latitude boreal forest sites (Mellander et al., 2006). TheCOUP model is driven by site-speci 󿬁 c meteorological hourly data andaccounts for local hydraulic and thermal soil properties. To achievethe most realistic estimates of deep-soil temperatures  T  COUP  and liquidwater-content  WC  COUP , the COUP simulations need to match closelywith measured soil-surface temperatures  T  G 2cm  (solid line in thebottom panel of  Fig. 1) and the snow depth  d SP  (top panel of  Fig. 1).The corresponding Pearson correlation coef  󿬁 cients  r  T  and  r  SP  achievedfor the selected two-month period are  r  T  = 0.919 and  r  SP  = 0.969.These high correlations con 󿬁 rm the good agreement betweenmeasured and modeled soil-surface temperatures and snow depths,respectively.Inaddition,theCOUPmodelcalculationsalsoconvincinglyreproducetheobserveddepthsofthesoil-frostboundariesforthethreewinter periods (2009 – 2012). This overall performance supports theassumption that the COUP model con 󿬁 guration also yields realisticsoil temperatures  T  COUP  and water-content  WC  COUP  for deeper soil-layers. For the soil-depth interval of 100 cm – 120 cm, the simulatedtime-series  T  COUP  and  WC  COUP  shown in Fig. 1 (dashed lines) are usedin Section 4.4 as inputs to the GS RT model. 3. Radiative transfer model This section explains the ground snow radiative transfer (GS RT)model developed to simulate brightness temperatures  T  B  p of a freezingsoil below a snowpack. Accordingly, the GS RT model must be able tocope with multiple layers. This is one of the reasons why we use thesame multi-layer two-stream RT approach as used in MEMLS(Mätzler, 1996, updated 2004; Mätzler & Wiesmann, 1999, 2012;Wiesmann & Mätzler, 1999). The zero-order  τ  – ω   RT model used inthe current version of the prototype level-2 processor L-MEB (L-bandMicrowave Emission of the Biosphere) (Wigneron et al., 2007) canhandle only a single layer (vegetation) on top of an in 󿬁 nite half space(soil). Hence, this approach is not appropriate to simulate the micro-wave radiance of snow above a ground with a frozen surface-layer.Section3.1 will provide anoverview of theGSRTmodel. Section 3.2explains the impedance matching approach implemented to considerthe effects of interface roughness, and Section 3.3 describes theapproach chosen to consider coherent effects as can be caused by athin frozen soil-layer on top of unfrozen soil.  3.1. Structure, inputs, and outputs of the GS RT model Fig. 2 shows an example of a ground snow (GS) system of   N   = N  G  +  N  S =5 layersto represent N  S  =2snow layersontopofagroundconsisting of   N  G  = 3 soil-layers. The mean height of the interface sepa-rating the ground from the snowpack (SP) is at  z   = 0, and  d SP  is thethickness of the SP (snow depth). The downwelling sky radiance is T  sky , and the  T  B  p are the modeled upwelling brightness temperaturesat polarization  p  = H and V for observations at the angle  θ 0  relative tonadir.Fig.3showsthe 󿬂 owchartoftheGSRTmodeltorepresentitsstruc-tureintermsofthemajormodelingblocks1) – 6).Theinputparametersrequired to represent the GS system are indicated on the left of thevertical dashed line in Fig. 2 and in block 1) of the  󿬂 ow chart. For eachsnow layer  j , the input parameters are: temperature  T  S  j [K], liquidwater-content  WC  S  j [m 3 m − 3 ], layer thickness  d S  j [cm], salinity S  S  j [ppt], mass density  ρ S  j [kg m − 3 ], correlation length  p S  j [mm], andthe  “ sub-resolution ”  roughness expressed by the standard deviation σ  S  j [mm] of the height of the interface above layer  j  at scales smallerthan the resolution limit. The input parameters used to de 󿬁 ne thestate of a ground-layer  i  are: temperature  T  G i [K], liquid water-content WC  G i [m 3 m − 3 ], layer thickness  d G i [cm], clay content  clay i [%], and thestandard deviation  σ  G i [mm] was used to characterize the  “ sub-resolu-tion ”  roughness of the interface  i  (Section 3.2). Other input parametersto the GS RT model include the temperature  T  G0 and the interfacere 󿬂 ectivities  s  p  0 below the lowest ground layer. However,  T  G0 and  s  p  0 Fig. 1.  Time-series used as input to the GSRT model (Section 4.4): measured snow depth d SP  and ground frost-depth  d FG  (top panel, same data as shown in Fig. 11). In-situ liquidwater-content  WC  G 2cm  and temperatures  T  G 2cm  (solid lines) measured 2 cm below thesoil surface. COUP simulations  WC  COUP  and  T  COUP  (dashed lines) for the soil-layer at adepth of 100 cm to 120 cm. The observed appearance of snow (arrow 1) and the onsetof soil freezing (arrow 2) are indicated (also indicated in Fig. 11).182  M. Schwank et al. / Remote Sensing of Environment 154 (2014) 180 – 191  donotaffectthemodelresultsifthetotaldepthofthesimulatedpro 󿬁 leis much larger than the emission depth  χ  .The parameterslistedtotherightof theverticaldashed linein Fig.2arethelayerre 󿬂 ectivities r   pk ,thelayertransmissivities t   pk ,andtheinter-face re 󿬂 ectivities  s  pk on top of each layer  k  at polarization  p  = H and V.As can be seen from the  󿬂 ow chart of the GS RT model (Fig. 3),  r   pk ,  t   pk ,and  s  pk are the primary layer RT properties fed into the two-stream RTmodel ((Wiesmann & Mätzler, 1999) and Section 3.1 in (Mätzler & Wiesmann, 2012)) used to derive  T  B  p (  p  = H, V). It is worth notingthat  r   pk and  t   pk are volume properties of the layers that include the ef-fects of absorption and volume scattering. Accordingly, layerre 󿬂 ectivities are  r   pk = 0.0 since volume scattering in soil- and snowlayers is neglected at L-band. In contrast, the interface re 󿬂 ectivities  s  pk are mainly given by the differences between effective permittivities ε  k  + 1 ,  ε  k of adjacent layers  k , the incidence angle  θ k  + 1 at the interface k , and interface  “ sub-resolution ”  roughness  σ  k , as explained in thenext section.Asillustratedinthe 󿬂 owchart(Fig.3),thesnow-layerRTproperties r   pk ,  t   pk , and  s *  pk are derived from corresponding snow-layer inputparameters  T  S  j ,  WC  S  j ,  S  S  j ,  ρ S  j , and  p S  j using the MEMLSmodels describedin Mätzler & Wiesmann (2012). As a consequence, the interfacere 󿬂 ectivities  s *  pk are understood to be specular re 󿬂 ectivities, meaningthat interface  “ sub-resolution ”  roughness  σ  S  j =  σ  k ( k  =  N  G  +  j ) is notconsidered at this stage. The modeling of the corresponding RTproperties  r   pk ,  t   pk , and  s *  pk of the ground layers requires the implemen-tation of a dielectric mixing model in the modeling chain (block 2b inthe  󿬂 ow chart). For this study, we use Mironov's dielectric mixingmodel (Mironov et al., 2004) that is used in the current SMOS level-2soil moisture processor. Inblock3, thin (coherent)layersand thick (in-coherent) layers are distinguished and treated following the strategyemployed in MEMLS. The treatment of a coherent soil-layer (block 3b)required some adaptations, as is outlined in Section 3.3. The specular in-terfacere 󿬂 ectivities s *  pk computedinblock4arereduceddueto “ sub-res-olution ”  roughness  σ  k . This instance is considered with impedancematching implemented in block 5 and outlined in Section 3.2.Finally,  r   pk ,  t   pk , and  s  pk are fed into the two-stream RT model((Wiesmann & Mätzler, 1999) and Section 3.1 in (Mätzler & Wiesmann, 2012)), which is identical to the one used in MEMLS.Additional input parameters to the block 6 are the layer temperatures T  k , the observation wavelength  λ , and the downwelling sky radiance T  sky . The two-stream RT model used in block 6 takes into account theeffects of multiple re 󿬂 ections at the layer interfaces, and generates the 󿬁 nal outputs of the GS RT model, which include the upwelling bright-ness temperatures  T  B  p and system emissivities  E   p at horizontal (  p  =H) and vertical (  p  = V) polarization.  3.2. Interface roughness Inblock5ofthe 󿬂 owchart(Fig.3),theinterfacere 󿬂 ectivities s  pk thattake  “ sub-resolution ”  interface roughness into account are computedbased on the specular interface re 󿬂 ectivities  s *  pk . To this end we haveadoptedtheconceptofimpedancematchingfromthe “ air-to-soil ” tran-sition model described in detail in Mätzler (2006). Nevertheless,it makes good sense to recall the concept of impedance matchingthat was used to correct the specular interface re 󿬂 ectivities  s *  pk for “ sub-resolution ”  interface roughness:First of all, the concept of impedance matching is only applicable toroughness at lateral scales smaller than the Bragg resolution limit Λ  ≈ λ k  + 1 / (2·sin θ k  + 1 ). With λ k  + 1 and θ k  + 1 beingthewavelengthand the propagation angle of the wave incident onto the interface  k (Fig. 4a), the Bragg limit is of the order of   Λ  ≈ 10 cm at the L-band.Hence,  “ sub-resolution ”  roughness is understood as the standard Fig. 2.  Pro 󿬁 le across the GS system. Inputs to the GS RT model (also shown in block 1 of Fig. 3) are indicated on the left side of the vertical dashed line; parameters fed into thetwo-stream RT approach (block 6 in Fig. 3) are indicated on the right. Fig.3. FlowchartoftheGSRTmodel.SolidblocksrepresentMEMLSroutines;dashedblocksare newly implemented to represent a layered ground beneath a SP, and  “ sub-resolution ” interface roughness.183 M. Schwank et al. / Remote Sensing of Environment 154 (2014) 180 – 191  deviation  σ  k of interface heights at interface areas ≈ Λ   ×  Λ   (Schwanket al., 2010a) smaller than the resolution cell. This is illustrated inFig. 4a, demonstrating how impedance matching can be used to simu-latethere 󿬂 ectivity s  pk ofaninterface k with “ sub-resolution ” roughness σ  k .In this so-called  “ quasi optical ”  regimeof electromagnetismtherealinterface topography at the scale ≈ Λ   ×  Λ   is replaced with a gradualtransition from zone  k  + 1 to zone  k , as is illustrated in panel b). Thissmooth transition zone is modeled with an empirical function  ν  k (  z  )describing the volumetric fraction of material from the region  k . At themean interface height of   z   = 0,  ν  k (  z  ) takes the value of   ν  k (  z   = 0) =0.5. Below  z   = − h k / 2, the volume fraction must approach  ν  k (  z  ) = 1to render the sole existence of material  k , and above  z   = + h k / 2, itmust reach  ν  k (  z  ) = 0 to render pure material from region  k  + 1. As isshown in Mätzler (2006) (equation (4.95))  h k = 4.0339 ·  σ  k relatesthe valley-to-peak distance  h k to the standard deviation  σ  k of interfaceheights within the  “ sub-resolution ”  area ≈ Λ   ×  Λ  , which we de 󿬁 ne asthe  “ sub-resolution ”  roughness of an interface  k .Theapparentdielectricpro 󿬁 le ε  eff  k (  z  )acrossaninterface k (Fig.4c)isrepresented by a refractive mixing model (Birchak et al., 1974; Sihvola,1999),weightingthepermittivities ε  k and ε  k  + 1 withthecorrespondentvolumetric fractions  ν  k (  z  ) and (1 − ν  k (  z  )), respectively. Finally,  ε  eff  k (  z  )is discretized (spatial sampling  ≪  wavelengths) and fed to a RTmodel (Bass et al., 1995), to compute re 󿬂 ectivities  s  pk (  p  = H, V) of the interface  k  with  “ sub-resolution ”  roughness  σ  k .Fig. 5 demonstrates the impact of interface  “ sub-resolution ”  rough-ness  σ  k on relative interface re 󿬂 ectivities 0 ≤ R  pk ≤ 1, de 󿬁 ned as: R  pk ≡  s  pk s   pk  ð 1 Þ where s *  pk arethespecularre 󿬂 ectivitiesoftheinterface k atpolarization  p  = H and V computedwiththeFresnelequationscalculated for  θ k  + 1 ,and thepermittivities  ε  in =  ε  k  + 1 and  ε  trans =  ε  k . The R  pk ( σ  k ) shown inFig.5aremodeledfortheincidenceangle θ k  + 1 =0°(hence R H k ( σ  k )= R V  k ( σ  k ))and0mm ≤ σ  k ≤ 40mmforfourtypicalinterfacesoccurringinaGSsystem.Thede 󿬁 nitionsofthesecharacteristicinterfacesseparatingregionswithpermittivities ε  in and ε  trans containingtheincidentandthetransmitted wave are provided in Table 1.The permittivity  ε  S  = 1.25 +  i  · 3.3 · 10 − 5 of the dry SP was esti-mated from MEMLS models (Section 3.7 in Mätzler & Wiesmann,2012) for  WC  S  = 0.0 m 3 m − 3 ,  T  S  = 268.15 K,  ρ S  = 150 kg m − 3 ,and  S  S  = 0.0 ppt. The permittivity  ε  UG  = 19.3 +  i  · 2.3 of the unfro-zen ground (UG) was computed for the water-content  WC  UG  =0.30 m 3 m − 3 , clay content  clay  = 1%, and the temperature  T  UG  =273.15 K using the Mironov dielectric mixing model (Mironovet al.,2004). Forthefrozenground(FG)at T  FG b  272.65K,weconsid-ered the permittivity  ε  FG  = 5.0 +  i  · 0.5. Independent of soil water-content and texture, this constant value is considered if soil tempera-turedropsbelow − 0.5 °C.Duetothetypicallylowclaycontentofarcticsoils, along with low speci 󿬁 c surface areas, the amount of bound soilwaterisexpectedtoberathersmall(Santamarinaetal.,2001).Accord-ingly,theabruptsoilfreezingat − 0.5 °Cseemsreasonabletorepresenta narrow temperature range around the freezing temperature of freewater, where liquid soil-water goes through a supercooled state (Or &Wraith, 1999; Santamarina et al., 2001). We are aware of this ratherpoor approach that may introduce noticeable quantitative errors onthe simulations, although at least the parametric model investigationsshown will not change qualitatively. Beyond that, a validated tempera-ture dependent dielectric mixingmodel for the respective effective soilpermittivity is still missing.As can be seen in Fig. 5, already small values of   “ sub-resolution ” roughness σ  k reducerelativeinterfacere 󿬂 ectivities R  pk ( σ  k )signi 󿬁 cantly.This results from the speci 󿬁 c meaning of   σ  k de 󿬁 ned as the standarddeviation of interface heights measured over areas smaller than aresolution cell  ≈  Λ   ×  Λ   which is of the order of 10 cm × 10 cm(Schwank et al., 2010a). Accordingly, values of   σ  k that reduce  R  pk Fig. 4.  Concept of the impedance matching approach applied in block 5 of the  󿬂 ow chart(Fig. 3) to consider  “ sub-resolution ”  roughness of interfaces. The symbols are explainedin the text. Fig. 5.  Relative interface re 󿬂 ectivities  R  pk ( σ  k ) for characteristic interfaces  k  within the GSsystem. Corresponding permittivities are given in Table 1.  σ  * FG → UG and  σ  * SP → FG arecharacteristic  “ sub-resolution ”  roughness for which  R  pk = 1 /   e ≈ 0.37 (Fig. 7).  Table 1 Interfacecharacteristicsconsideredinthesimulatedrelativeinterfacere 󿬂 ectivities R  pk ( σ  k )shown in Fig. 5.  ε  in and  ε  trans are the permittivities containing the incident and thetransmitted wave above and below the interface  k .interface  k  ε  in =  ε  k  + 1 ε  trans =  ε  k Air → UG 1 19.3 +  i  · 2.3Air → FG 1 5.0 +  i  · 0.5SP → FG 1.25 +  i  · 3.3 · 10 − 5 5.0 +  i  · 0.5FG → UG 5.0 +  i  · 0.5 19.3 +  i  · 2.3184  M. Schwank et al. / Remote Sensing of Environment 154 (2014) 180 – 191
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