Documentation and digital files in support of "Classification of aircraft types using seismic data in Alaska'"

dopplerD_partial_derivatives.nb Partial derivatives of Doppler equation Evaluate dopplerC_estimate.nb first. Carl Tape and Bella Seppi October 20, 2025 In[257]:= ClearParms Partial derivatives In[258]:= Clear[partialv, partialfs, partialt0, partiald0, partialc] Clear[partialvX, partialfsX, partialt0X, partiald0X, partialcX] In[260]:= Clear[vv, t0s, d0s, fss, cc] partialv[v_, t0_, d0_, fs_, c_, tp_] :=D[Ft[vv, t0, d0, fs, c, tp], vv] /. vv →v partialfs[v_, t0_, d0_, fs_, c_, tp_] :=D[Ft[v, t0, d0, fss, c, tp], fss] /. fss →fs partialt0[v_, t0_, d0_, fs_, c_, tp_] :=D[Ft[v, t0s, d0, fs, c, tp], t0s] /. t0s →t0 partiald0[v_, t0_, d0_, fs_, c_, tp_] :=D[Ft[v, t0, d0s, fs, c, tp], d0s] /. d0s →d0 partialc[v_, t0_, d0_, fs_, c_, tp_] :=D[Ft[v, t0, d0, fs, cc, tp], cc] /. cc →c This factor will help shorten the explicit expressions In[266]:= G[v_, t0_, d0_, c_, tp_] :=1-v2 c2+(t0 -tp)2v2 d02 Partial derivative expressions, fully simplified. They can be commented out, since we define explicit expressions. Speed of source, v In[267]:= (*FullSimplify[D[Ft[v,t0,d0,fs,c,tp],v]]*) In[268]:= partialvX[v_, t0_, d0_, fs_, c_, tp_] := fs v 2 d04v4-d02(t0 -tp)2v6+3 c d03(-t0 +tp)v4G[v, t0, d0, c, tp]+ c5d0 (t0 -tp)2 d02+(t0 -tp)2v2G[v, t0, d0, c, tp]+ c3d0 (t0 -tp)v2d02+3(t0 -tp)2v2G[v, t0, d0, c, tp]+ c2v2-4 d04-5 d02(t0 -tp)2v2+(t0 -tp)4v4+c42 d04+6 d02(t0 -tp)2v2+3(t0 -tp)4v4 d0 (c-v)(c+v)G[v, t0, d0, c, tp]d02+c2v2(c t0 -c tp +d0 G[v, t0, d0, c, tp])2 c2-v22 c(t0 -tp)v2+d0 v2G[v, t0, d0, c, tp]-c2d02+c2v2(c t0 -c tp +d0 G[v, t0, d0, c, tp])2 c2-v22+ v2d02+c2v2(c t0 -c tp +d0 G[v, t0, d0, c, tp])2 c2-v22 2 Unshifted frequency, fs (note: it does not depend on fs) In[269]:= (*FullSimplify[D[Ft[v,t0,d0,fs,c,tp],fs]]*) In[270]:= partialfsX[v_, t0_, d0_, fs_, c_, tp_] :=1-v2(c t0 -c tp +d0 G[v, t0, d0, c, tp]) c2-v2d02+c2v2(c t0-c tp+d0 G[v,t0,d0,c,tp])2 c2-v22 -1 Arrival time, t0 In[271]:= (*FullSimplify[D[Ft[v,t0,d0,fs,c,tp],t0]]*) In[272]:= partialt0X[v_, t0_, d0_, fs_, c_, tp_] := d0 fs (c-v)v2(c+v)(t0 -tp)v2+c d0 G[v, t0, d0, c, tp] G[v, t0, d0, c, tp]d02+c2v2(c t0 -c tp +d0 G[v, t0, d0, c, tp])2 c2-v22 c(t0 -tp)v2+d0 v2G[v, t0, d0, c, tp]-c2d02+c2v2(c t0 -c tp +d0 G[v, t0, d0, c, tp])2 c2-v22+ v2d02+c2v2(c t0 -c tp +d0 G[v, t0, d0, c, tp])2 c2-v22 2 Distance to object, d0 In[273]:= (*FullSimplify[D[Ft[v,t0,d0,fs,c,tp],d0]]*) 2 dopplerD_partial_derivatives.nb In[274]:= partiald0X[v_, t0_, d0_, fs_, c_, tp_] := fs (t0 -tp)v2-c2+v2(t0 -tp)v2+c d0 G[v, t0, d0, c, tp] G[v, t0, d0, c, tp]d02+c2v2(c t0 -c tp +d0 G[v, t0, d0, c, tp])2 c2-v22 c(t0 -tp)v2+d0 v21-v2 c2+(t0 -tp)2v2 d02-c2d02+c2v2(c t0 -c tp +d0 G[v, t0, d0, c, tp])2 c2-v22+ v2d02+c2v2(c t0 -c tp +d0 G[v, t0, d0, c, tp])2 c2-v22 2 Acoustic wavespeed, c In[275]:= (*FullSimplify[D[Ft[v,t0,d0,fs,c,tp],c]]*) In[276]:= partialcX[v_, t0_, d0_, fs_, c_, tp_] := - fs v22 d04v4-2 d02(t0 -tp)2v6+4 c d03(-t0 +tp)v4G[v, t0, d0, c, tp]+ c5d0 (t0 -tp)d02+(t0 -tp)2v2G[v, t0, d0, c, tp]+ 3 c3d0 (t0 -tp)v2d02+(t0 -tp)2v2G[v, t0, d0, c, tp]+ c2v2-4 d04-3 d02(t0 -tp)2v2+(t0 -tp)4v4+c42 d04+5 d02(t0 -tp)2v2+3(t0 -tp)4v4 c d0 (c-v)(c+v)G[v, t0, d0, c, tp]d02+c2v2(c t0 -c tp +d0 G[v, t0, d0, c, tp])2 c2-v22 c(t0 -tp)v2+d0 v2G[v, t0, d0, c, tp]-c2d02+c2v2(c t0 -c tp +d0 G[v, t0, d0, c, tp])2 c2-v22+ v2d02+c2v2(c t0 -c tp +d0 G[v, t0, d0, c, tp])2 c2-v22 2 Unused Find the maximal slope of the Doppler curve. That means finding the root for the second derivative. In[277]:= (* Solve[D[Ft[v,t0,d0,fs,c,tp],{tp,2}]⩵0,tp]*) (* not sure if this will run *) Find the inflection points by setting the third derivative to zero and solving. In[278]:= (* Solve[D[Ft[v,t0,d0,fs,c,tp],{tp,3}]⩵0,tp]*) (* this probably does not finish *) Example values for a Doppler curve dopplerD_partial_derivatives.nb 3 Example values for a Doppler curve In[279]:= SetParms In[280]:= thwid =100; In[281]:= Plot[Ft[v, t0, d0, fs, c, tp],{tp, t0 -thwid, t0 +thwid}] Out[281]= 150 200 250 300 95 100 105 110 115 120 125 130 Partial derivatives for v, t0, d0, fs, c, each plotted as a function of tp In[282]:= Plot[partialv[v, t0, d0, fs, c, tp],{tp, t0 -thwid, t0 +thwid}] Out[282]= 150 200 250 300 -0.2 0.2 0.4 4 dopplerD_partial_derivatives.nb In[283]:= Plot[partialt0[v, t0, d0, fs, c, tp],{tp, t0 -thwid, t0 +thwid}] Out[283]= 150 200 250 300 0.2 0.4 0.6 0.8 In[284]:= Plot[partiald0[v, t0, d0, fs, c, tp],{tp, t0 -thwid, t0 +thwid}] Out[284]= 150 200 250 300 -0.006 -0.004 -0.002 0.002 0.004 0.006 In[285]:= Plot[partialfs[v, t0, d0, fs, c, tp],{tp, t0 -thwid, t0 +thwid}] Out[285]= 150 200 250 300 0.90 0.95 1.00 1.05 1.10 1.15 dopplerD_partial_derivatives.nb 5 In[286]:= Plot[partialc[v, t0, d0, fs, c, tp],{tp, t0 -thwid, t0 +thwid}] Out[286]= 150 200 250 300 -0.06 -0.04 -0.02 0.02 0.04 Test values (for checking Python) first check In[287]:= Ft[v, t0, d0, fs, c, t0] Ft[v, t0, d0, fs, c, tp0] fs Out[287]= 112.388 Out[288]= 110. Out[289]= 110. choose a (sensor) time point: In[290]:= tp =200.; t[v, t0, d0, c, tp] Ft[v, t0, d0, fs, c, tp] Out[291]= -19.0237 Out[292]= 122.286 6 dopplerD_partial_derivatives.nb compare explicit expressions with expressions based on D[ ]: In[293]:= partialv[v, t0, d0, fs, c, tp] partialvX[v, t0, d0, fs, c, tp] partialt0[v, t0, d0, fs, c, tp] partialt0X[v, t0, d0, fs, c, tp] partiald0[v, t0, d0, fs, c, tp] partiald0X[v, t0, d0, fs, c, tp] partialfs[v, t0, d0, fs, c, tp] partialfsX[v, t0, d0, fs, c, tp] partialc[v, t0, d0, fs, c, tp] partialcX[v, t0, d0, fs, c, tp] Out[293]= 0.432579 Out[294]= 0.432579 Out[295]= 0.419014 Out[296]= 0.419014 Out[297]= -0.00628521 Out[298]= -0.00628521 Out[299]= 1.11169 Out[300]= 1.11169 Out[301]= -0.044734 Out[302]= -0.044734 dopplerD_partial_derivatives.nb 7