Radar surveys of the cold-temperate transition in polythermal glaciers have been used to investigate the polythermal structure, as well as the stability of the temperature regime, in glaciers. The mapping is based on the detection of small water-filled pockets present in the temperate ice beneath the cold surface layer. I compare radar profiles recorded at different frequencies on a polythermal glacier to show that the apparent depth to the cold-temperate transition reflection at a center frequency of 155 MHz is significantly greater than that detected at frequencies above 345 MHz. This increase appears to be the result of a reduction of scattering efficiency from the uppermost part of the temperate ice, as the theory of electromagnetic scattering from small objects indicates a strong decrease in scattering efficiency from objects much smaller than the wavelength of the incident wave. Thus I interpret this apparent depth increase to indicate that the average radii of the water pockets of the cold-temperate transition is on a subdecimeter scale. Spatial variation in the difference between cold-temperate transition depths at 155 MHz and higher frequencies may arise from spatial variations in size or the number density of scatterers.
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 The cold-temperate transition surface (CTS) within polythermal glaciers can be detected using ground penetrating radar (GPR) because of the increased scatter from water inclusions present on the temperate side of the CTS [Dowdeswell et al., 1984; Bamber, 1988]. This scatter has been used to map the thermal structure in glaciers both for hydrological purposes [Björnsson et al., 1996; Moore et al., 1999] and for the detection of changes in the cold surface layer thickness, which influences the erosion potential and the ice flow properties of the glacier [Pettersson et al., 2003]. Radar studies on glaciers are often performed using frequencies from a few MHz up to 1–2 GHz. Ødegård et al.  compared profiles of the CTS at different GPR frequency bands between 30 and 1000 MHz and could not detect any change in CTS depth at frequencies above ∼300 MHz. However, at their lowest frequency band (30–80 MHz), the CTS reflection was lost or difficult to interpret. Similarly, Walford and Kennett  detected a strong decrease in scattering efficiency from the temperate ice just beneath the cold surface layer when the frequency was decreased from 65 MHz to 6 MHz. These results indicate that GPR detection of the CTS starts to change around a few hundred MHz. However, other studies of the frequency dependence of the CTS using GPR have not been able to limit the frequency range where the scattering efficiency decreases [e.g., Ødegård et al., 1997; Pettersson et al., 2003].
 Electromagnetic scattering theory [Stratton, 1941; van de Hulst, 1981; Bohren and Huffman, 1998] shows that the scattering efficiency from the CTS is a function of both frequency, and the size and shape of the objects causing the CTS reflection. The theory predicts a strong decrease in scattering efficiency when the wavelength becomes much larger than the scattering object, while the scattering efficiency is virtually invariable for objects comparable to or larger than the wavelength. This was first pointed out by Watts and England  and may explain the decreasing scattering efficiency at lower frequencies.
 This well known theory suggests that it should be possible to gain information about the scattering object sizes by inversion of the scattering theory at different frequencies. However, the inverse problem is difficult to solve because it requires that the amplitude and phase of the scattered vector field, as well as the internal field within the scattering object are completely known [Bohren and Huffman, 1998], which is impossible to obtain from surface based GPR. Nevertheless, one can estimate the order of magnitude of the scattering object sizes from the frequency at which the scattering efficiency starts to decrease in repeated GPR surveys of the CTS at different frequencies and by comparing these data to calculated scattering efficiencies at each frequency for objects of different sizes and with a prescribed shape.
 In this study I acquired five GPR profiles at different center frequencies between 130 and 1300 MHz along a transect across Storglaciären, a small polythermal valley glacier in northern Sweden (Figure 1a), to find the lower-frequency limit for accurate detection of the CTS. I obtain the order of magnitude for the size of the objects causing the CTS reflection by using Lorenz-Mie scattering theory, and, I compare the results with the data. In the calculations I assumed independent spherical scatterers and that the radius of a single object represents a size distribution, which is a common condition in most natural systems. For a wide size distribution, this assumption may not be correct [Mishchenko et al., 2002], but for my purpose the assumption provides a first approximation.
 The results show an increase in depth of the radar-inferred CTS at 155 MHz compared to the consistent radar-inferred CTS depths at the higher frequencies. This suggests a decrease in scattering efficiency and indicates that the scatterers are on the subdecimeter scale or much smaller than the corresponding wavelength at 155 MHz. The shift in depth of the CTS reflection may be explained by a combination of decrease in size and number of scatterers in the uppermost temperate ice compared with deeper in the ice.
2. Location and Fieldwork
 The radar profiles are located in the upper part of the ablation area, approximately 400 m downstream from the equilibrium line on Storglaciären (Figure 1b). The glacier surface is crevasse free and relatively flat because of a subglacial bedrock overdeepening in the upper ablation area [Herzfeld et al., 1993]. The surveys were made in late April 2001 under dry snow conditions.
 I used a continuous wave, stepped frequency ground-penetrating radar system [Hamran and Aarholt, 1993] using five different center frequencies; 155, 345, 600, 800, and 1150 MHz. The GPR system allows full control of the transmitted bandwidth and I use 50, 200 and 300 MHz (Table 1). In what follows I refer to the different center frequencies of the frequency bands instead of the frequency range. However, the frequency band must also be accounted for in any theoretical calculation.
Table 1. Parameters for the GPR Surveys at Different Center Frequencies
Center Frequency, MHz
Range resolution, m
Antenna separation, m
3 dB beam width, deg
 The GPR system was mounted on a sled and pulled by a snowmobile at an average travel speed of 5 km h−1. I collected 2 traces s−1 which corresponds to about 0.7 m of travel between each trace. The profiles were acquired in the upper part of the ablation area along the same path for all profiles. The instrument setup was not changed between the surveys except the frequency range. Every 20th GPR trace was positioned with differential GPS. The data were transformed to the time domain using an inverse Fourier transform [Hamran et al., 1995]. No gain function or filtering was applied to the transformed data. The interpreted CTS along each GPR profile was picked manually on the basis of a sharp increase in returned intensity. The accuracy in detection and digitization of the CTS reflection was ±0.7 m at center frequencies of 800 MHz and ±1.2 m at 345 MHz [Pettersson et al., 2003].
 The scattering efficiency of electromagnetic waves from a single, homogeneous spherical object embedded in a uniform dielectric media can be estimated from Lorenz-Mie scattering theory. The scattering efficiency Qsca, which is a projected area weighted radar cross section, is expressed as the infinite series
where x = 2πr/λ, r is the radius of the scatterer and λ is the wavelength in ice of the incident GPR wave. The coefficients an and bn are the Mie coefficients of nth order and for nonmagnetic media are given by
where jn are spherical Bessel functions of the first kind, hn(1) are Hankel functions (a linear combination of first and second kind of spherical Bessel functions) [Bowman, 1958], and the prime indicates their derivatives. The quantity m = ni/nm is the ratio between the refractive index of the inclusion and the medium. In ice and water, assuming a negligible conductivity, the refractive index can be approximated by n = , where ɛ is the complex permittivity of ice or water.
 The solution for Mie coefficients and the scattering efficiency requires higher orders of the spherical Bessel functions, which can be found using a recurrence relation in the form
where f(z) are Bessel functions of the argument z; in my case z = x or mx. In the calculations the infinite series of the Mie equations is truncated at nmax = x + 4x1/3 + 2 [Bohren and Huffman, 1998], resulting from a balance between roundoff errors in the recurrence equation and a reasonable convergence. I have used the code from Mätzler  to calculate the Mie coefficients and the scattering efficiency for radii from 0.01 m to 0.5 m at the five center frequencies used and their bandwidth limits. The refractive index ratio, m, is given by the complex values of the permittivity of ice tabulated by Warren  over a wide frequency range and at several temperatures, and the permittivity for water is based on analytical fits to the Debye theory [Ray, 1972].
 Lorenz-Mie scattering theory is simplified greatly by assuming a spherical shape of the scatterers and that they act independently of each other. In reality, all objects in temperate ice are probably not spherical in shape and the effects of arbitrary shapes upon the scattering efficiency can be profound. Nevertheless, Mie scattering serves as a first approximation [van de Hulst, 1981; Mishchenko et al., 2002]. The assumption of independent scattering requires that the distance between the scatterers must be at least three times the radii of the scatterers, which is not considered a restrictive criterion in glacier ice [Bamber, 1988].
 The objects causing the CTS reflection most likely consist of an ensemble of sizes. Therefore the scattering results from an unknown size distribution, which will influence the scattering characteristics. Adding a normalized size distribution to the calculations of the scattering efficiency is straightforward, but I have chosen to omit this calculation because the size distribution in natural ice is unknown. Any size distribution of low or moderate variance has a smoothing effect on the scattering efficiency over a range of frequencies. However, it retains the most important features such as the strong decrease in scattering efficiency for scatterer sizes much smaller than the wavelength [Mishchenko et al., 2002].
 The five GPR profiles and an interpretation of the depth of the CTS are shown in Figures 2a–2f. The interpretation for all but the lowest frequency is the same. At the lowest frequency, the interpreted depth is generally deeper and becoming increasingly deeper at both the northern and southern ends of the transect (Figure 2).
 The returned intensity averaged over 10 traces at one location indicated by a gray box in the background of the radargrams is shown to the right of each radargram (Figures 2a–2e). The sharp increase in intensity seen at approximately 34 m depth in the A-scope plots are present along the whole radargrams and is used to pick the CTS as shown in Figure 2f.
 Some scattering also occurs above the interpreted CTS reflection. Much of this scattering originates from larger, single-scattering objects that are probably rock fragments or water pockets that have not yet frozen as they are transported toward the surface. The “bumps” in the interpreted CTS picks at 345 MHz coincide with such larger single scattering events. These point scatterers coalesce with the CTS reflection at lower frequencies because of the decreasing range resolution. One of these single scatterers is also visible in the A-scope intensity plots in Figures 2a–2e.
 The scattering efficiencies for objects with a radius of 0.0–0.5 m were calculated with Lorenz-Mie theory at the five center frequencies and at their bandwidth limits used in the present study (Figure 3). The irregularities in the plots are caused by interference and resonance effects within the object and would be considerably reduced if any size distribution were introduced. For all the calculated frequencies in Figure 3 the scattering efficiency drops considerably for scattering radii less than about λ/10. Thus a radius of λ/10, seems to be a good approximation of an upper size limit for scatterers with reduced scattering efficiency at a given frequency.
 The location of the rightmost intensity plots in Figure 2 is where Pettersson et al.  compared temperature measurements with the interpreted CTS depth. They showed that the interpreted depth of the CTS reflection at center frequencies of 345 MHz and 800 MHz corresponds to within 1 m of the location of the measured pressure melting point of ice. The different CTS depth interpreted at 155 MHz in this study suggests that the reflection at 155 MHz deviates from the actual CTS by an average of 5 m.
 The difference in CTS depth at the lower frequency indicates that the degree of scattering is reduced, while the scattering efficiency remains high above 345 MHz. Theoretical studies suggest that this decrease in scattering efficiency is due to scatterer sizes much smaller than the wavelength at 155 MHz (Figure 3). This implies that the mean size of the scatterers that make up the CTS would be between 1/10th of the wavelengths at 155 and 345 MHz, or about 0.1 to 0.05 m in radius if the scattering objects were spherical in shape. The finite frequency bandwidth at each center frequency gives only small variations from these values (Figure 3). This size range compares with observations of ice cores [Nye and Frank, 1973; Raymond and Harrison, 1975; Jania et al., 1996], which indicate that scattering objects on the subdecimeter to meter scale exist in temperate ice and object sizes on the order of 0.1 m detected from the scatter characteristics of GPR waves [Watts and England, 1976; Jacobel and Anderson, 1987; Bamber, 1988].
 We might expect that the ice would become more transparent to the GPR wave and diminish the CTS reflection when the scattering efficiency is reduced rather than the CTS reflection becoming deeper as in Figure 2a. One possible explanation for why this is not the case is that the uppermost part of the temperate ice consists of smaller water pockets, while the pockets are larger farther down in the temperate ice. This would reduce the scattering efficiency and make the uppermost part of the temperate ice transparent to wavelengths much larger than the scattering objects. The fact that the disagreement seems to vary along the transect (Figure 2f) suggests that a spatial variation in the size of scatterers causing the CTS reflection in the radargrams.
 Another possible explanation for the decreasing scattering efficiency is a variation in number density of scatterers. An increase in the number density of scatterers amplifies the total intensity of scattering, suggesting that an increasing number of scatterers much smaller than the incident wavelength could become detectable even though the scattering efficiency is low. Hence increased water content instead of object size at some distance below the CTS could explain the depth difference in CTS interpretations in Figure 2f. Another possibility is that relatively lower signal-to-noise ratio at the lower frequency, causes the reflection to become more difficult to detect. However, the signal-to-noise ratio for continuous wave stepped-frequency waveforms is independent of the frequency and bandwidth and is only dependent on transmitted power and the integration time for the waveform [Eide, 2000], which is the same for all surveys in this study.
 At a center frequency of 155 MHz, the CTS reflection shows an increased depth along parts of the profile when compared with results at higher frequencies. This indicates frequency dependence in the radar-detected CTS on Storglaciären. The shift in interpreted CTS depth at the lowest frequency is explained either by a decrease in size of scatterers, a decrease in the number of scatterers (i.e., water content), or a combination of both in the uppermost part of the temperate ice. The difference in scattering efficiency between the lowest frequency and higher frequencies implies that the size of the scatterer is between 0.1 and 0.05 m if the scatterers are spherical.
 This fieldwork was generously supported by Göran Gustavssons Foundation. Peter Jansson, Cecilia Richardson-Näslund, and Robert W. Jacobel provided valuable comments on the manuscript. The reviews of Steven Arcone, associate editor Stephen Frasier, and one anonymous reviewer helped to improve the manuscript considerably.