2. Equipment

The wide-angle reflection measurements were made with a system consisting of the 60 MHz ice radar RLS-60-98, a digital processing interface (DPI) and a satellite navigation system (Reference MasolovMasolov and others, 1999; Reference Lukin, Masolov, Mironov, Popkov, Popov and Sheremet’yevLukin and others, 2000; Reference Popov, Mironov and Sheremet’yevPopov and others, 2000, Reference Popov, Mironov, Sheremet’yev and Luchininov2001). The DPI is based on the industrial computer “Favorite-IPC” with a SBC-8259 processor (AxiomTechnology Co). The analog–digital transformation used a 12-bit analog–digital converter AD9042AST (Analog Devices Inc.). The digitizing of the radar output data and its stacking were carried out in real time. The ice radar system specifications are given below:

The very widely spaced receiver and transmitter required a special system to trigger the receiver. We used a sound system whereby the leading edge of a sound pulse provided the trigger.

3. RES Technique

The wide-angle reflection soundings were made along two lines (northern with an azimuth 335° and western with an azimuth 245°), each of them divided into 18 segments of 200 m (Fig. 2). The lines were laid out with a theodolite, with a maximal deviation of ±0.05° from a straight line. Marking was done with a 100 m metal tape with a margin of error of ±0.1 m. The transmitting antenna was fixed near a snowbound metal beam (point 0, Fig. 2). The receiving antenna was fixed on the roof of a mobile beam and, during the moving, was positioned strictly above marks at 3 m height from the ice surface. The velocity was calculated with a one-layer model which assumed a horizontal subsurface interface. The correctness of the applied model was confirmed by the natural measurements. It was provided with a RES survey around Vostok station and mapping of the ice base of the area (Reference Popov, Mironov, Sheremet’yev and LuchininovPopov and others, 2001). One of the RES records is shown in Figure 3.

Fig. 2 Wide-angle reflection technique location chart.

Fig. 3. RES record (western leg).

It is necessary to note that it would have been more correct to use the common depth point (CDP) technique, i.e. both antennas move equally from a center point (Bogorodsky and others, 1983). Unfortunately, this was technically impossible. The CDP technique is necessary when dealing with an inclined ice base, because then the point of reflection is fixed. In our case we dealt with horizontal layers, for which the points of reflections are fixed because of the extremely simple geometry of radio-wave propagation.

4. Processing of the Data and Results

The average velocity of radio-wave propagation in ice was calculated following Reference Bogorodsky, Bentley and GudmandsenBogorodsky and others (1985). The mathematics of the geometry of radio-wave propagation are given as b ² + (2T )² = L ², where b is the distance between the antennas, T is the ice thickness and L is the radio ray distance in ice, for horizontal layers. L = vτ, where v is the average velocity of the radio-wave propagation in ice and τ is the delay of the reflected signal. By simple substitution

(1)

and we fit a straight line by the least-squares regression method (LSM). The regression coefficients are given as:

(2)

where a ₁ and a ₀ are the coefficients at the first and free members accordingly. Equation (1) and the coefficients (2) were calculated for each line (Fig. 4).

Fig. 4. Plots of the squares of the reflected signal delays vs squares of the distance between antennas, trend lines and best-fit equations. 1.The northern leg; 2. the western leg.

The average velocity of the radio-wave propagation in ice is found to be 168.36 m μs⁻¹ for the northern line and 168.43 m μs⁻¹ for the western line. Therefore, the average englacial velocity in the lake Vostok area is 168.4 ± 0.5 m μs⁻¹. An account for the air layer only changes the second decimal and is not important.

5. Hyperbolic Diffraction Processing

It is also possible to determine the average velocity of electromagnetic wave propagation using the hyperbolic diffractions (Reference MacheretMacheret, 2000; Reference VasilenkoVasilenko and others, 2001). Radio-wave rays from a point diffractor are shown in Figure 5. If x ₀ is the value of the abscissa at the apex of the hyperbola, ξ_i is the distance between x ₀ and x_i , T is the ice thickness, d is the lateral distance between the reflector and the RES route, and τ ₀ and τi are delays of the reflected signal (they correspond to distances L ₀ and L_i ), then, for the ray at x ₀

and at x_i we have

where

(3)

Fig. 5. Ray geometry of electromagnetic waves propagating from a point reflector.

We can calculate a linear fit using regression coefficients calculated by LSM (Equation (3)). The velocity v can be found through Equation (2), as described in Reference MacheretMacheret (2000) and Reference VasilenkoVasilenko and others (2001).

We estimate our error, δv, is

(4)

Z-record modeling (Fig. 6a) gives the relative height value δv. The errors δτ ₀ and δτ_i can be estimated as approximately 0.25 μs (τ ₀ ≈ 45 μs). We estimate the error δξ_i (ξ_i ≈ 2000 m) as approximately 30 m. Then, according to Equation (4), δv ≈ 2.8 m μs⁻¹, at v = 168 m μs⁻¹, but the level of accuracy is unacceptable.

Fig. 6. Hyperbolic diffractions and their modeling; (a) Z record; (b) RES record formed using the described method; (c) modeling.

We can reduce the error (Reference PopovPopov, 2002) by considering the RES record amplitude A = A(τ). The digitizing was done on the leading edges (LES), i.e. on the maximum of the first derivative τ ^′ = dA/dτ. With the understanding that the LES of the reflections must be positive, we redefine τ ^′ as such that

(5)

We then created a synthetic binary Z record that consists of two different values: black points (corresponding to peaks ) and white points (Fig. 6b). For obvious reasons, there will be a significant number of such peaks. We establish a limit ς, which allows us to plot only peaks with . Therefore, we can reduce the number of the peaks by varying ς, which allows us to process the data more precisely (Fig. 6c).

We now estimate δv again (Equation (4)). The δξ_i can be estimated as 1 point (≈1.5 m). Practically the same value is obtained for the carrier position determination. Hence, it is possible to accept δξ_i ≈ 3 m. The error in determining the position of the hyperbola apex can be estimated as 1 point (≈0.05 μs). According to Figure 6c, the δτ_i could be estimated as 0.1 μs. Finally, δv could be as little as 0:75 m μs⁻¹ (≤0.5%).

There were three hyperbolic reflections in the RES data. The velocities were 168.46, 168.52 and 168.46 m μs⁻¹. The average value is 168.5 m μs⁻¹. This is close to the value defined by the wide-angle reflection technique, so velocity determination by hyperbolic diffractions could be used in future RES investigations.

6. Comparison of Results With Other Geophysical Measurements

In addition to our work, vertical seismic profiling (VSP), thermometry, ice density determination and other geophysical investigations were carried out in borehole 5G-1 and its vicinity. We estimate ice thickness from the convergence of all available data on the ice thickness. Based on RES data, the ice thickness in the vicinity of 5G-1 borehole is 3775 ± 15 m. Based on VSP data, the ice thickness is 3760 ± 30 m (Reference Popkov, Verkulich, Masolov and LukinPopkov and others, 1999). The thermometry data give a value of 3776 ± 3 m (Reference Salamatin, Vostretsov, Petit, Lipenkov and BarkovSalamatin and others, 1998). The divergence between all the data is <1%, which is quite good for so many methods.

7. Discussion: Firn Correction

The precision of the velocity of electromagnetic wave propagation in ice by the wide-angle reflection method was discussed by Reference Babenko and MacheretBabenko and Macheret (1997). They used Rasmussen’s approach by elliptic functions (Reference RasmussenRasmussen, 1986) for the firn layer. Ice density, ρ, down the ice sheet was converted to refractive index n with the following dependence: n = 1 + Kρ, where K = (8.51 ± 0.1) × 10⁻⁴ m³ kg⁻¹ (Reference Babenko and MacheretBabenko and Macheret, 1997). We believe that the approach of Reference Rees and DonovanRees and Donovan (1992) was more correct because they used K = (8.4 ± 0.1) × 10⁻⁴ m³ kg⁻¹ (Reference Rees and DonovanRees and Donovan, 1992), but for our estimation this difference is not important.

We estimated the firn correction for the vicinity of Vostok station following Reference Babenko and MacheretBabenko and Macheret (1997). For an ice thickness of T ≈ 4000 m, firn thickness of T _firn = 105 m (Reference Salamatin, Lipenkov, Smirnov and ZhilovaSalamatin and others, 1985) and surface snow density of ρ ₀ = 320 kg m⁻³ (Reference Ekaykin, Lipenkov and BarkovEkaykin and others,1998), the error of the velocity definition is v ≈ 0.04 m μs⁻¹ (∼0.02%).Therefore, accounting for the firn layer impact on the velocity of electromagnetic wave propagation in ice is not important for RES investigations.