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We examine the use of ground penetrating radar (GPR) to simultaneously estimate snow and ice thickness. Since velocity is essential for depth inversion, we developed an automatic common mid-point (CMP) measurement system, which can work in an ultra-wide band. Envelope velocity spectrum, robust in reflection identification, is used to estimate the dielectric permittivity and layer thickness from CMP datasets. We tested our system on two 50-m-long test lines on a brackish lagoon, i.e., Lake Saroma, in Hokkaido, Japan, in February 2012. The first test line failed due to the existence of a radar-absorbing layer of a mixture of snow and seawater between the snow cover and sea ice. The estimated dielectric permittivity of both snow and ice on the second test line seems to be highly correlated with their surface temperature. Compared with the ground truth, the snow and ice thickness estimations possess a good accuracy, with a respective mean absolute error of about 2 cm (12%) and less than 2 cm (4%), which verifies the accuracy of their dielectric permittivity estimation. Then the estimated two-layer velocity model obtained by interpolation was used to continuously estimate the snow and ice thickness from the common-offset (CO) GPR profile acquired by a commercial GPR system. Our system and method appear capable of accurately measuring the dielectric permittivity and thickness of other layered media.