Structural information in firn reflections

Here We examine whether the normal-incidence firn reflection response contains enough information to determine small changes in firn velocity structure, as parameterized by two parameters, dv0 and s (described below). We compute the reflection response, r(dv0,s,t), and its envelope, e(dv0,s,t), as follows:

• Calculating the reference reflection response of a vostok model, perturbed by 0.01*v(z)*dv0*exp(-z/s). Here dv0 is the amplitude of the perturbation (in percent) and s is a scale length over which perturbation occurs. It is largest at the top of the firn.
• Convolve it with the usual radar wavelet;
• Gating the interval -500 ns to 3596 ns, which contains the firn reflections;
• Tapering the end of the ends with a 1/8 cosine taper, to produce a syntehtic radar signal, r(dv0,s,t);
• Form the evvelope, e0(t)=|r(t)+iH[r(t)]|, where H is the Hilbert transpose. (The process of forming the envelope demodulates out the carrier).

  We then compute a reference signal r0(t)=r(dv0=1,s=30,t). How well can we estimate dv0 and s by comparing the reference envelope to a sequence of other envelopes calculated on a 2D grid of different dv0's and s's? An example of two envelopes are:

  

  Envelope functions for the reference signal (bottom) and one with slighty different v0 and s. (Postscript version).

  The rms error is used to quantify the difference between pairs of envelope functions. The resulting error surface is:

  

  Error surface between reference envelope (dv0=1,s=30) and envelopes with slightly different dv0's and s's. Note that a single minimum occurs at exactly the right value, suggesting that the firn reflection response contains usable information about firn structure. (Postscript version).

  

  Same error surface as above, but for larger variation of dv0 and s. (Postscript version).

  Finally, we test whether a fresh layer of snow can be detected. The reference model is vostok modified by adding a top h=0.5 m thick layer of material with velocity v0/n m/ns, where v0=300 and n=1.5. The error surface of the envelope for a suite of layers of varying n and h is:

  

  Note that there is a minimum at exactly the right place. However there are also other local minima along an arc, suggesting that h and n trade off to a considerable degree. (Postscript version).

  

  Fortunately, the error surface of the timeseries (as contrasted to its envelope) does not have this undesirable property. (Postscript version).