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EFFECT OF A THIN DRY SNOW LAYER ON THE LAKE ICE THICKNESS MEASUREMENT USING WIDEBAND AUTOCORRELATION RADIOMETRY

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Wideband autocorrelation radiometry (WiBAR) is a new method to remotely sense the microwave propagation time τ delay of multi-path microwave emission of low loss layered surfaces such as dry snowpack and freshwater lake icepack. The microwave
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  EFFECT OF A THIN DRY SNOW LAYER ON THE LAKE ICE THICKNESS MEASUREMENTUSING WIDEBAND AUTOCORRELATION RADIOMETRY Seyedmohammad Mousavi 1  , Roger De Roo 2  , Kamal Sarabandi 1  , and Anthony W. England  31 Electrical Engineering and Computer Science Department, University of Michigan, Ann Arbor, MI 2 Climate and Space Sciences and Engineering Department, University of Michigan, Ann Arbor, MI 3 College of Engineering and Computer Science, University of Michigan, Dearborn, MI ABSTRACT Wideband autocorrelation radiometry (WiBAR) is a newmethod to remotely sense the microwave propagation time τ  delay  of multi-path microwave emission of low loss layeredsurfaces such as dry snowpack and freshwater lake icepack.The microwave propagation time  τ  delay  through the pack yields a measure of its vertical extent; thus, this techniqueis adirect measurement of depth. However, the presence of a dif-ferentlowloss layeronthelakeicepacksuchas drysnowpack introduces another multi-path interference, which can effectthe lake icepack thickness measurement. We present a sim-ple geophysical forward model for the multipath interferencephenomenon and derive the WiBAR system requirementsneeded to correctly measure the icepack thickness. An X-band instrument fabricated from commercial-off-the-shelf (COTS) components are used to measure the thickness freshwater lake ice at the University of Michigan Biological Sta-tion. Ice thickness retrieval is demonstrated from nadir to 73 . 9 ◦ .  Index Terms —  Snowpack, icepack, remote sensing, mi-crowave radiometry, autocorrelation 1. INTRODUCTION Wideband autocorrelation radiometric sensing of microwavetravel time in single layer pack has been previously discussedand investigated in [1–4]. This method directly measures themicrowave propagation time  τ  delay  through the low loss ter-rain covers and other layered surfaces [4]. This  τ  delay  relatesto the vertical extent of the pack,  d , as given by τ  delay  = 2 c    d 0   n  p ( z ) 2 − sin 2 θ dz  (1)where n  p  is the refractiveindex of the pack,  θ  is the incidenceangle,  d  is the pack thickness in the  z  direction, and  c  is thespeed of light in free space. Fig. 1 . Remote sensing of microwave travel time through theicepack with thickness  d ice  in the presence of a snowpack with thickness  d snow . 2. WIDEBAND AUTOCORRELATIONRADIOMETRY FOR MULTILAYERED MEDIA2.1. Physics of Operation The presenceof a snowpackon the icepackadds anothermul-tipath. The configuration of this layered media is shown inFig. 1. The first delayed ray of icepack and snowpack arrivesat the radiometer with the time delays  τ  ice  and  τ  snow , respec-tively, relative to the direct ray. These time delays are similarto (1). The freshwater lake icepack has an almost constantrefractive index,  n ice  = √  3 . 15 , over microwave frequencies,while the refractive index of the dry snowpack ( n snow ) maydepend on the vertical extent of the pack [4].The transfer matrix method can be used to calculate theemissivity of this medium [5], as given e  = (1 −| R 01 | 2 )(1 −| R 12 | 2 )(1 −| R 23 | 2 ) C  0  +  C  d  +  C  d −  +  C  d + (2) 7109978-1-5386-7150-4/18/$31.00 ©2018 IEEE IGARSS 2018  012345678910 Time (ns) -15-10-50    A  u   t  o  c  o  r  r  e   l  a   t   i  o  n   (   d   B   ) Rectangular Window (4 GHz Bandwidth)Rectangular Window (8 GHz Bandwidth)Hamming Window (4 GHz Bandwidth)Hamming Window (8 GHz Bandwidth) τ  ice  = 4.2 ns τ  ice  + τ  snow  = 4.5ns Fig. 2 . Simulated autocorrelation response of a 35.5 cmicepack with a 4 cm snowpack on top using rectangular andHamming window functions ( θ  = 0 ◦ ,  τ  ice  = 4 . 2  ns, and τ  snow  = 0 . 3  ns).where C  0  =1 + | R 01 | 2 | R 12 | 2 + | R 01 | 2 | R 23 | 2 + | R 12 | 2 | R 23 | 2 C  d  =2 R 12 R 23 (1 + | R 01 | 2 )cos(2 k 2 z d ice )+2 R 01 R 12 (1 + | R 23 | 2 )cos(2 k 1 z d snow ) C  d −  =2 R 01 R 23 | R 12 | 2 cos  2  k 1 z d snow − k 2 z d ice  C  d +  =2 R 01 R 23  cos  2  k 1 z d snow − k 2 z d ice   (3)where  R 01  is the Fresnel reflection coefficient of air to snow, R 12  is the Fresnel reflection coefficient of snow to ice,  R 12 is the Fresnel reflection coefficient of snow to ice,  R 23  isthe Fresnel reflection coefficient of ice to water,  k 1 z  is thevertical component of the wavenumber for snow, and  k 2 z  isthe vertical component of the wavenumber for ice. It shouldbe noted that this model is for horizontal polarization as ex-plained in [4]. 2.2. Minimum and Maximum Detectable Time Delays The lower limit of the sensed time delay depends on theWiBAR ability to resolve autocorrelation delay peaks in timedomain. In order to resolve two equal amplitude autocorrela-tion delay peaks, the requirement | ∆ τ  |  =  | τ  2 − τ  1 |  > τ  6 . 0 dB should be satisfied, where  τ  6 . 0 dB  =  ζ  6 . 0 dB F  s , where  ζ  6 . 0 dB  isa factor depending on the Fourier window  w ( f  ) , and  1 F  s isthe IDFT time bin. Detecting minimum snowpack thick-nesses approaching 4 cm and density of   ρ s  = 0 . 21  g/cm 3 atincidence angle of   θ  = 0 ◦ requires resolving  τ  snow  at au-tocorrelation lag times of around 0.3 ns. Thus, the system’sbandwidth should be greater than 3 GHz for the rectangularand 6 GHz for the Hamming window function to resolve themultiple interfaces. This effect is shown in Fig. 2.On the other hand, in the case of resolving two unequalamplitude delay peaks, if the weak amplitude delay peak re-sides after the first side lobe level of the strong amplitudedelay peak, in order to resolve the two peaks, their ampli-tude difference  | ∆ A |  (dB) should satisfy inequality (4), orequivalently the time difference between the two delay peaks | ∆ τ  | (s) should satisfy inequality (5). | ∆ A | < | SLL | +  SLF × log 2   ∆ τ τ  SLL   (4) | ∆ τ  | > τ  SLL × 2( | ∆ A |−| SLL | SLF  ) (5)WhereSLF (dB/Octave)is the sidelobefall offof thewindowfunction, SLL (dB) is the first side lobe level of the windowfunction, and  τ  SLL  =  12  ( τ  main lobe  +  τ  side lobe )  is the locationof the SLL peak, where  τ  main lobe  =  2 ζ  main lobe F  s is the main lobenull-to-null width, and  ζ  main lobe  is a factor depending on thewindow function, and  τ  side lobe  is the null-to-null width of thefirst side lobe. For instance, to detect a delay peak at 1 nsawayfromthezerodelaypeakwith  F  s  = 3 GHz, | ∆ A | <  3 . 5 dB and | ∆ A | <  21 . 5  dB is required using the rectangular andthe Hamming window functions, respectively. This effect isshown in Fig. 3. 00.511.522.53 Frequency (GHz) -30-25-20-15-10-50    A  u   t  o  c  o  r  r  e   l  a   t   i  o  n   R  e  s  p  o  n  s  e   (   d   B   ) Rectangular WindowHamming Window τ  delay  = 1 ns Fig. 3 . Simulated autocorrelation response with a delay peak at 1 ns using the rectangular window (black solid line) andthe Hamming window (dashed red line). The bandwidth is F  s  = 3  GHz.The minimum resolution of the IDFT is  12 τ  main lobe , so theclosesttheweakamplitudedelaypeak,withamplitudegreaterthan  | SLL | , can reside in time is between the  12 τ  main lobe  and τ  SLL  of the strong amplitude delay peak. In this case, the twopeaks can be resolved if   | ∆ A |  <  | SLL | . As an example, todetect a delay peak at 0.4 ns using the rectangular window, | ∆ A | should be less than 7 dB, while to detect a delay peak at0.8 ns using the Hamming window, | ∆ A | should be less than21 dB. This effect is shown in Fig. 4.The theoretical upper limit to the maximum detectabletime delay is similar to [4] and determined by the resolutionbandwidth (RBW) of the spectrum analyzer and the numberof frequency bins, as given by ( τ  ice  +  τ  snow )  <  12 min  N  f  F  s ,  1 RBW   (6) 7110  00.511.522.53 Frequency (GHz) -30-25-20-15-10-50    A  u   t  o  c  o  r  r  e   l  a   t   i  o  n   R  e  s  p  o  n  s  e   (   d   B   ) Rectangular WindowHamming Window τ  delay = 0.4 ns τ  delay = 0.8 ns Fig. 4 . Simulated autocorrelation response with delay peaksat 0.4 ns for the rectangular window (black solid line) andat 0.8 ns for the Hamming window (red dashed line). Thebandwidth is  F  s  = 3  GHz.where  N  f   is the number of frequency bins. 3. FIELD MEASUREMENTS AND RESULTS Field measurementsforlake icepack were conductedonDou-glas Lake at the University of Michigan Biological Station(UMBS) in March 2016. The measurements were in H-poland conducted at incidence angles from 0.8 ◦ to 73.9 ◦ . Themeasurement setup is shown in Fig. 5(a). Using the groundtruth measurements, the icepack and snowpack thicknesseswerefoundto be d ice  = 35.5cmand d snow  = 3.1cmas shownin Fig. 5(b).The power spectra of the sky, absorber, and lake ice foroneoftheobservationsareshowninFig. 6. Usingthecalibra-tion procedure mentioned in [4], the emissivity of this obser-vation is shown in Fig. 7. To find the time delay, the standardIDFT method can be used as explained in [4]. The autocorre-lation response of the measurement and the expected value of the autocorrelationresponse for a  35 . 5  cm icepack with about 3 . 1  cm top dry snowpack are shown in Fig. 8. The Hammingwindow was used. The first delay peak after the zero delaypeak is at 4.23 ns. The incidence angle is 0.8 ◦ . There is about5 dB difference in the first delay peak after zero between themodel and measurement autocorrelation response. Gain vari-ations due to temperature drift of electronics, as well as im-perfections in the ice such as surface and volume scattering,can cause a decrease in the delayed autocorrelation peak.On the other hand, high resolution methods such as Gen-eralized Pencil-of-Function (GPOF) method [6] can be usedto find the time delays. This method is able to resolve delaypeaks seperated in time by less than resolution limit of thestandard IDFT method,  1 F  s . The calibrated emissivity, canbe represented by a sum of exponentials in the frequency do- (a) (b) (c) Fig. 5 . (a) Measurement setup of the lake icepack measure-ment using a wideband autocorrelation radiometer (WiBAR)on a tripod (a motorcyclebattery was used as a power source)and groundtruth measurementof the (b) icepack, d ice  = 35 . 5 cm, and (c) snowpack,  d snow  = 3 . 1  cm. 77.588.599.510 Frequency (GHz) -49-48-47-46-45-44-43-42-41-40    P  o  w  e  r   (   d   B  m   ) Lake ice ObservationAbs ObservationSky Observation Fig. 6 . Sky, load, and lake icepack observation on DouglasLake on March 02, 2016. The incidence angle is 0.8 ◦ .main, as given by ˜ e ( f  ) = M   m =0 A m e j 2 πfτ  m +  n ( f  )  (7)where  A m  is the estimated amplitude of the delay peaks,  τ  m is the estimated time delays,  n ( f  )  is the noise in the system,and  M   is the number of exponentials. The  A m ’s and  τ  m ’s arecalculated using GPOF method.Finally, the retrievedmicrowavepropagationtimes for theicepack with respect to incidence angles from 0.8 ◦ to 73.9 ◦ isplotted in Fig. 9. It has been shown that the presence of a thindry snow layer on the lake icepack can cause uncertaintiesin the measured thickness of the icepack by WiBAR. Thisuncertaintycan be reducedby either using a window functionwith lower side lobe levels in standard IFFT method or usinga high resolution method such as GPOF. For this ice pack, theaccuracy is within 2 cm of the pack thickness. 7111  77.588.599.510 Frequency (GHz) 0.60.650.70.750.80.850.9    E  m   i  s  s   i  v   i   t  y Fig. 7 . The emissivity of the lake icepack measured on Dou-glas Lake on March 02, 2016. The incidence angle is 0.8 ◦ . 051015 Time Delay (ns) -50-45-40-35-30-25-20-15-10-50    A  u   t  o  c  o  r  r  e   l  a   t   i  o  n   R  e  s  p  o  n  s  e   (   d   B   ) MeasurementExpected Value of the Autocorrelation Response Fig. 8 . The autocorrelation function of the lake icepack mea-sured on Douglas Lake on March 02, 2016 (blue solid line)with the expected value of the autocorrelation response of the lake icepack model. The Hamming window was used( θ  = 0 . 8 ◦ ,  d ice  = 35 . 5  cm,  d snow  = 3 . 1  cm). 0102030405060708090 Incidence Angle (deg) 33.23.43.63.844.24.44.64.85    T   i  m  e   D  e   l  a  y   (  n  s   ) Expected Delay (d ice =35.56 cm)Expected Delay (d ice =36.83 cm)Expected Delay (d ice =38.10 cm) Measured delay using WiBAR (Hamming) Measured delay using WiBAR (Kaiser, α k =3.02) Measured delay using WiBAR (GPOF) Fig. 9 . Measured and expected microwave propagation timethrough the icepack with respect to the incidence angle. 4. CONCLUSIONS Wideband autocorrelation radiometry is a passive remotesensing methodof measuringthe roundtrip propagationtime, τ  delay , of microwaves through a low-loss dielectric slab, suchas a freshwater lake icepack or terrestrial snowpack. Thispaper describes the effect of a dry snowpack on lake icepack thickness measurement using WiBAR. In fact, it is shownthat the presence of a thin snowpack ( τ  snow  <  ζ F  s ) not onlycan widen the autocorrelation delay peak at  τ  ice  but also canshift its location. This uncertainty can be reduced either byusing a window function with lower side lobe levels such asKaiser-Bessel ( α k  = 3 . 02 ) window, or by using a high res-olution method such as GPOF. Experiments were conductedin H-Pol at X-band for the lake icepack. The measurementswere performed for incidence angles from 0.8 ◦ to 73.9 ◦ . 5. ACKNOWLEDGMENT The authors hereby express their gratitude to NASA for itssupport via contracts NNX15AB36G and NNX17AD66G.We also thank Bob Vande Kopple and the University of Michigan Biological Station for their assistance in the fieldcampaign. 6. REFERENCES [1] S. Mousavi, R. De Roo, K. Sarabandi, A. England, andH. Nejati, “Remote sensing using coherent multipath in-terference of wideband Planck radiation,” in  2016 IEEE  International Symposium on Antennas and Propagation(APSURSI) , June 2016, pp. 2051–2052.[2] S. Mousavi, R. De Roo, K. Sarabandi, A. England, andH. Nejati, “Dry snowpack and freshwater icepack remotesensing using wideband autocorrelation radiometry,” in 2016IEEEInternationalGeoscienceandRemoteSensingSymposium (IGARSS) , July 2016, pp. 5288–5291.[3] S. Mousavi, R. De Roo, K. Sarabandi, and A. W. Eng-land, “Sampling requirements for wideband autocorre-lation radiometric (wibar) remote sensing of dry snow-pack and lake icepack,” in  2017 IEEE InternationalGeo-science and Remote Sensing Symposium (IGARSS) , July2017, pp. 1004–1007.[4] S. Mousavi, R. D. De Roo, K. Sarabandi, A. W. Eng-land, S. Y. E. Wong, and H. Nejati, “Lake icepack anddry snowpack thickness measurement using widebandautocorrelation radiometry,”  IEEE Transactions on Geo-science and Remote Sensing , vol. 56, no. 3, pp. 1637–1651, March 2018.[5] L. Tsang, J. A. Kong, and R. T. Shin,  Theory of Mi-crowave Remote Sensing , John Wiley and Sons, NewYork, 1985.[6] B. Yektakhah and M. Dehmollaian, “A method for can-cellation of clutter due to an object in transceiver sideof a wall for through-wall sensing applications,”  IEEE Geoscience and Remote Sensing Letters , vol. 9, no. 4, pp.559–563, July 2012. 7112
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