grmdoc.ps

su 的源代码库

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1 0 bop -94 340 a Fq(GRM)54 b(-)g(Generalized)g(Recipro)t(cal)h(Metho)t
(d)-94 506 y(with)f(automated)e(optim)l(um)g(parameters)h(and)g(depth)
-94 672 y(estimation)-94 1005 y Fp(Stev)m(en)38 b(D.)h(Shea\013er)-94
1104 y Fo(Center)25 b(for)h(Wave)g(Phenomena)-94 1187
y(Color)l(ado)j(Scho)l(ol)e(of)f(Mines)952 1488 y Fn(ABSTRA)n(CT)952
1712 y Fm(Key)32 b(w)m(ords:)-94 2110 y(In)m(tro)s(duction)-94
2260 y Fl(Refraction)37 b(metho)r(ds)e(for)i(determination)f(of)g(the)g
(depth)f(to)-94 2359 y(shallo)n(w)j(in)n(terfaces)f(are)f(utilized)g
(in)g(b)r(oth)f(near-surface)i(in-)-94 2459 y(v)n(estigations)g(and)d
(deep)h(re\015ection)g(surv)n(eys.)g(F)-6 b(or)35 b(re\015ection)-94
2558 y(data)27 b(pro)r(cessing,)h(the)f(depth)e(to)i(the)f(base)h(of)g
(the)g(lo)n(w)g(v)n(elo)r(c-)-94 2658 y(it)n(y)20 b(surface)h(la)n(y)n
(er)f(\(the)f("w)n(eathering")j(la)n(y)n(er\))e(is)g(useful)h(for)g
(the)-94 2758 y(determination)e(of)i(static)g(corrections.)g(One)e
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h(\014rst)f(arriv)l(al)h(times)e(of)i(refracted)g("head")-94
2957 y(w)n(a)n(v)n(es)f(generated)g(b)n(y)f(an)g(increase)i(in)e(v)n
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3156 y(F)-6 b(or)42 b(the)g(simple)f(case)i(of)g(a)g(plane)f(dipping)g
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3355 y(surface)31 b(top)r(ograph)n(y)-6 b(,)29 b(the)g(w)n(ell)h(kno)n
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b(1959\))h(will)g(yield)e(the)g(depth)f(normal)h(to)h(the)-94
3555 y(in)n(terface)h(at)f(all)g(surface)h(p)r(oin)n(ts.)f(The)g(metho)
r(d)e(requires)i(a)-94 3654 y(forw)n(ard)31 b(and)d(rev)n(erse)i(shot,)
f(pro)r(ducing)g(a)h(tra)n(v)n(el)f(time)f(from)-94 3754
y(eac)n(h)33 b(at)f(ev)n(ery)f(geophone)i(lo)r(cation)g(b)r(et)n(w)n
(een)f(them.)g(Figure)-94 3854 y(1.)26 b(sho)n(ws)f(the)f(ra)n(ypath)g
(for)i(\014xed)e(shots)h(A)f(and)g(B)h(and)g(a)g(par-)-94
3953 y(ticular)i(midp)r(oin)n(t)d Fk(i)p Fl(.)i(If)g(the)f(function)445
4103 y Fk(T)501 4071 y Fj(\000)553 4103 y Fl(\()p Fk(i)p
Fl(\))c(=)g Fk(T)786 4111 y Fi(ai)863 4103 y Fh(\000)c
Fk(T)985 4112 y Fi(bi)1055 4103 y Fl(+)g Fk(T)1177 4112
y Fi(ab)-94 4252 y Fl(is)32 b(ev)l(aluated)g(at)f(eac)n(h)h(midp)r(oin)
n(t)e Fk(i)i Fl(for)g(\014xed)e(A)h(and)g(B,)i(the)-94
4352 y(result)25 b(will)h(b)r(e)f(a)g(line)g(with)g(slop)r(e)h(equal)f
(to)g(the)f(recipro)r(cal)i(of)-94 4451 y(the)f(apparen)n(t)h
(refractor)h(v)n(elo)r(cit)n(y)e(1)p Fk(=v)1074 4420
y Fj(0)1071 4465 y Fi(r)1107 4451 y Fl(.)h(A)f(second)h(function,)446
4601 y Fk(T)502 4569 y Fg(+)552 4601 y Fl(\()p Fk(i)p
Fl(\))c(=)f Fk(T)786 4609 y Fi(ai)862 4601 y Fl(+)c Fk(T)984
4610 y Fi(bi)1055 4601 y Fh(\000)g Fk(T)1177 4610 y Fi(ab)-94
4750 y Fl(can)25 b(b)r(e)f(seen)g(to)g(b)r(e)g(equiv)l(alen)n(t)g(to)g
(a)h(solution)g(o)n(v)n(er)f(the)f(inner)-94 4850 y(triangle,)k(whic)n
(h)f(is)g(ev)l(aluated)g(to)g(b)r(e,)175 5023 y Fk(T)231
4991 y Fg(+)281 5023 y Fl(\()p Fk(i)p Fl(\))21 b(=)g
Fk(T)514 5035 y Fi(a)548 5021 y Ff(0)570 5035 y Fi(i)614
5023 y Fl(+)16 b Fk(T)735 5035 y Fi(b)762 5021 y Ff(0)785
5035 y Fi(i)828 5023 y Fh(\000)h Fk(T)950 5035 y Fi(a)984
5021 y Ff(0)1006 5035 y Fi(b)1033 5021 y Ff(0)1081 5023
y Fl(=)1198 4975 y(2)p Fk(d)c Fl(cos)g Fk(\022)1440 4983
y Fi(cr)p 1198 5006 306 4 v 1314 5074 a Fk(v)1351 5082
y Fg(0)-94 5192 y Fl(where)31 b Fk(v)170 5200 y Fg(0)236
5192 y Fl(is)g(the)f(kno)n(wn)g(near)h(surface)h(v)n(elo)r(cit)n(y)-6
b(,)30 b(and)h Fk(\022)1637 5200 y Fi(cr)1730 5192 y
Fl(is)-94 5292 y(the)g(critical)h(angle,)g(whic)n(h)f(can)g(b)r(e)g
(estimated)f(from)h Fk(v)1592 5300 y Fg(0)1658 5292 y
Fl(and)1931 2110 y Fk(v)1971 2078 y Fj(0)1968 2123 y
Fi(r)2027 2110 y Fl(\(since)25 b(the)e(dip)h(is)h(not)f(kno)n(wn)f(but)
h(is)g(assumed)g(small,)h(the)1931 2210 y(apparen)n(t)k(v)n(elo)r(cit)n
(y)g(is)g(tak)n(en)f(to)h(b)r(e)g(the)g(true)f(v)n(elo)r(cit)n(y)-6
b(.\))29 b(The)1931 2309 y(normal)35 b(depth)f Fe(d)43
b Fl(is)36 b(found)e(b)n(y)g(m)n(ultiplying)g Fk(T)3412
2278 y Fg(+)3463 2309 y Fl(\()p Fk(i)p Fl(\))h(with)g(a)1931
2409 y(v)n(elo)r(cit)n(y)26 b(function)g(that)f(accoutns)h(for)h(the)e
(cosine,)2516 2552 y Fk(d)p Fl(\()p Fk(i)p Fl(\))c(=)2755
2521 y Fg(1)p 2755 2535 31 4 v 2755 2577 a(2)2795 2552
y Fk(T)2851 2520 y Fg(+)2901 2552 y Fl(\()p Fk(i)p Fl(\))3068
2504 y Fk(v)3105 2512 y Fg(0)p 2997 2535 215 4 v 2997
2603 a Fl(cos)14 b Fk(\022)3149 2611 y Fi(cr)1931 2714
y Fl(In)19 b(realit)n(y)-6 b(,)20 b(ho)n(w)n(ev)n(er,)g(w)n(e)g(do)g
(not)f(w)n(an)n(t)h(to)g(b)r(e)f(limited)g(to)h(plane)1931
2814 y(refractors,)27 b(but)d(are)h(primarily)f(in)n(terested)h(in)g
(imaging)g(irreg-)1931 2914 y(ular)34 b(refractor)h(surfaces.)g(It)f
(is)g(easlily)g(seen)g(that)g(the)f(plus-)1931 3013 y(min)n(us)23
b(metho)r(d)g(will)j(break)d(do)n(wn)i(for)f(a)h(non-planar)f(surface.)
1931 3113 y(Since)30 b(the)g(refractor)i(surface)f(at)g
Fk(a)2995 3081 y Fj(0)3047 3113 y Fl(and)f Fk(b)3234
3081 y Fj(0)3287 3113 y Fl(will)h(most)f(lik)n(ely)1931
3212 y(ha)n(v)n(e)24 b(di\013eren)n(t)f(dips,)h(triangle)h(geometry)e
(will)j(not)d(form,)i(and)1931 3312 y(the)e(metho)r(d)f(will)j(pro)r
(duce)e(erroneous)h(results.)g(The)f(problem)1931 3412
y(of)28 b(dip)f(con\015ict)h(can)f(b)r(e)g(o)n(v)n(ercome)g(b)n(y)f
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b(sho)n(wn)g(in)g(Figure)h(2.,)h(where)f(b)r(oth)f(ra)n(ys)g(lea)n(v)n
(e)h(the)1931 3611 y(irregular)e(surface)f(at)g(the)f(same)g(p)r(oin)n
(t,)g(and)g(are)h(sub)t(ject)g(to)1931 3711 y(the)25
b(same)g(dip.)g(The)g(geometry)g(of)g(the)g(inner)g(triangle)h(is)g(no)
n(w)1931 3810 y(oblique,)31 b(but)f(still)i(suggests)h(solution.)f(It)e
(is)i(this)e("optimal")1931 3910 y(geometry)23 b(that)h(is)g(the)f
(basis)i(of)f(the)f(Generalized)i(Recipro)r(cal)1931
4009 y(Metho)r(d)h(\(GRM\))g(dev)n(elop)r(ed)f(b)n(y)g(P)n(almer)g
(\(1982\).)2081 4109 y(T)-6 b(o)20 b(use)h(the)e(GRM)i(geometry)-6
b(,)19 b(some)h(mo)r(di\014cation)h(of)g(the)1931 4209
y(previous)28 b(theory)g(is)h(required,)e(not)h(only)g(to)h(accoun)n(t)
f(for)h(the)1931 4308 y(new)21 b(geometry)e(but)g(also)j(to)e(address)g
(the)g(fact)h(that)f(w)n(e)g(do)g(not)1931 4408 y(kno)n(w)k(what)f
(separation)i(XY)e(on)g(the)g(surface)i(corresp)r(onds)f(to)1931
4508 y(the)33 b(ra)n(ys)h(lea)n(ving)g(the)f(refractor)j(at)d(a)h
(common)e(p)r(oin)n(t)i(\(the)1931 4607 y(optimium)g(XY\).)h(The)h
(geophone)g(lo)r(cation)h Fe(i)42 b Fl(is)36 b(de\014ned)f(as)1931
4707 y(halfw)n(a)n(y)30 b(b)r(et)n(w)n(een)f(X)g(and)f(Y,)h(as)h
(measured)e(on)h(the)g(surface.)1931 4806 y(The)34 b(V)-6
b(elo)r(cit)n(y)33 b(Analysis)h(F)-6 b(unction)32 b(is)i(analogous)h
(to)f(a)f(one-)1931 4906 y(w)n(a)n(y)26 b(v)n(ersion)g(of)g
Fk(T)2491 4874 y Fj(\000)2543 4906 y Fl(,)2416 5049 y
Fk(T)2461 5057 y Fi(v)2498 5049 y Fl(\()p Fk(i)p Fl(\))c(=)2697
5018 y Fg(1)p 2697 5032 31 4 v 2697 5074 a(2)2737 5049
y Fl(\()p Fk(T)2812 5057 y Fi(ay)2900 5049 y Fh(\000)17
b Fk(T)3022 5058 y Fi(bx)3105 5049 y Fl(+)f Fk(T)3226
5058 y Fi(ab)3291 5049 y Fl(\))1931 5192 y(The)j(recipro)r(cal)i(of)f
(its)f(slop)r(e)h(will)g(giv)n(e)f Fk(V)3148 5160 y Fj(0)3131
5205 y Fi(r)3170 5192 y Fl(,)h(as)f(b)r(efore.)h(Observ-)1931
5292 y(ing)j(Figure)g(2.,)g(it)g(should)f(b)r(e)h(clear)g(that)f(if)h
(the)f(ra)n(ys)h(lea)n(v)n(e)g(the)p eop
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2 1 bop -94 -83 a Fd(2)146 b Fc(She)-5 b(a\013er)-94
116 y Fl(refractor)31 b(at)e(di\013eren)n(t)g(p)r(oin)n(ts,)h
Fk(T)929 124 y Fi(v)966 116 y Fl(\()p Fk(i)p Fl(\))f(will)i(b)r(e)e(p)r
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b(this)f(additional)g(path)g(will)g(v)l(ary)f(in)h(length)f(along)-94
415 y(the)37 b(refractor,)j(causing)e(the)f(function)h
Fk(T)1185 423 y Fi(v)1222 415 y Fl(\()p Fk(i)p Fl(\))g(to)f(b)r(e)h
(rough.)-94 515 y(Therefore,)25 b(if)f(it)f(is)g(ev)l(aluated)g(for)g
(v)l(arious)h(v)l(alues)f(of)g(XY,)g(the)-94 614 y(smo)r(othest)29
b(will)h(giv)n(e)f(the)g(optim)n(um)d(XY.)i(This)i(is)f(the)f(v)l(alue)
-94 714 y(of)f(the)f(o\013set)h(that,)f(on)g(a)n(v)n(erage,)h(will)h
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(common)e(tak)n(e)i(o\013)g(p)r(oin)n(t)f(for)i(eac)n(h)e(midp)r(oin)n
(t)g Fe(i)p Fl(.)56 913 y(Once)c(XY)f(is)i(determined,)e(the)h
(Time-Depth)f(function)i(is)-94 1013 y(a)k(mo)r(di\014ed)f
Fk(T)340 981 y Fg(+)390 1013 y Fl(:)222 1164 y Fk(T)267
1172 y Fi(g)303 1164 y Fl(\()p Fk(i)p Fl(\))d(=)502 1134
y Fg(1)p 502 1148 31 4 v 502 1190 a(2)542 1164 y Fl(\()p
Fk(T)617 1172 y Fi(ay)705 1164 y Fl(+)17 b Fk(T)827 1173
y Fi(bx)910 1164 y Fh(\000)g Fk(T)1032 1173 y Fi(ab)1114
1164 y Fh(\000)f Fk(X)6 b(Y)j(=V)1413 1132 y Fj(0)1396
1177 y Fi(r)1435 1164 y Fl(\))-94 1315 y(F)-6 b(or)25
b(zero)g(dip,)f(this)h(is)g(analogous)h(to)f(a)f(1-w)n(a)n(y)h(v)n
(ersion)f(of)i Fk(T)1710 1284 y Fg(+)1760 1315 y Fl(,)-94
1415 y(but)i(in)h(the)g(presence)g(of)g(dip,)g(its)g(geometry)g(is)g
(oblique,)h(and)-94 1515 y(do)r(es)24 b(not)e(giv)n(e)h(the)g(exact)g
(normal)f(depth.)g(Again,)i(GRM)e(con-)-94 1614 y(v)n(erts)31
b Fk(T)145 1622 y Fi(g)181 1614 y Fl(\()p Fk(i)p Fl(\))g(in)n(to)h(its)
f(estimate)h(of)g(depth)e(b)n(y)g(m)n(ultiplication)-94
1714 y(with)h(the)f(cosine)h(scaled)g(v)n(elo)r(cit)n(y)g(function)f
(as)h(in)f(the)g(Plus-)-94 1813 y(min)n(us)25 b(calculation,)i(so)g
(that)469 1965 y Fk(d)509 1973 y Fi(g)r(r)r(m)630 1965
y Fl(\()p Fk(i)p Fl(\))21 b(=)g Fk(T)863 1973 y Fi(g)899
1965 y Fl(\()p Fk(i)p Fl(\))1066 1916 y Fk(v)1103 1924
y Fg(0)p 995 1948 215 4 v 995 2015 a Fl(cos)14 b Fk(\022)1147
2023 y Fi(cr)-94 2135 y Fl(T)-6 b(o)36 b(understand)f(the)g
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2135 y Fl(and)g(the)-94 2235 y(actual)28 b(depth)e(normal)h(to)g(the)g
(refractor,)h Fk(d)p Fl(,)f(a)h(reference)g(line)-94
2334 y Fk(d)-54 2303 y Fj(0)-8 2334 y Fl(can)c(b)r(e)g(dra)n(wn)g(on)g
(Figure)h(2.)f(connecting)g(the)g(midp)r(oin)n(t)f Fk(i)p
Fl(,)-94 2434 y(halfw)n(a)n(y)28 b(b)r(et)n(w)n(een)f(X)f(and)g(Y)g
(and)h(the)f(tak)n(e-o\013)g(p)r(oin)n(t)h(on)g(the)-94
2534 y(refractor.)33 b(Calculation)g(of)e Fk(d)794 2542
y Fi(g)r(r)r(m)946 2534 y Fl(o)n(v)n(er)g(the)f(oblique)h(triangle)-94
2633 y(giv)n(es)26 b(the)g(result)g(that)265 2826 y Fk(d)305
2834 y Fi(g)r(r)r(m)448 2826 y Fl(=)21 b Fk(d)569 2794
y Fj(0)604 2711 y Fb(\024)657 2778 y Fl(cos)760 2746
y Fg(2)808 2778 y Fk(\013)c Fh(\000)g Fl(sin)1045 2746
y Fg(2)1092 2778 y Fk(\022)1128 2786 y Fi(cr)1204 2778
y Fl(cos)c Fk(\013)p 657 2809 712 4 v 746 2881 a Fl(cos)849
2859 y Fg(2)896 2881 y Fk(\013)18 b Fh(\000)f Fl(sin)1134
2851 y Fg(2)1181 2881 y Fk(\022)1217 2889 y Fi(cr)1378
2711 y Fb(\025)-94 3019 y Fl(where)26 b Fk(\013)g Fl(is)g(the)g(lo)r
(cal)h(dip.)f(The)g(p)r(ole)g(o)r(ccurs)g(at)638 3170
y Fk(\022)674 3178 y Fi(cr)754 3170 y Fl(+)17 b Fk(\013)k
Fl(=)992 3122 y Fk(\031)p 992 3153 47 4 v 996 3221 a
Fl(2)-94 3332 y(whic)n(h)34 b(is)g(the)g(con\014guration)g(where)g(the)
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3531 y(itely)26 b(long.)56 3631 y(It)f(is)g(clear)i(from)e(this)g(that)
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(small)g(dips,)g(since:)681 3882 y Fk(d)721 3890 y Fi(g)r(r)r(m)863
3882 y Fh(\031)21 b Fk(d)984 3850 y Fj(0)-94 4033 y Fl(and,)742
4184 y Fk(d)782 4153 y Fj(0)825 4184 y Fh(\031)g Fk(d)-94
4335 y Fl(only)26 b(when)f Fk(\013)h Fl(is)g(small.)-94
4644 y Fm(SU)33 b(Implemen)m(tation)28 b(of)k(GRM)-94
4793 y Fl(The)27 b(curren)n(t)f(v)n(ersion)h(of)g(GRM)g(will)h
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g(deter-)-94 4993 y(mined)e(\014rst)h(arriv)l(al)h(times.)f(The)g
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b(refractors,)j(giv)n(en)e(a)g(set)h(of)f(\014rst)g(arriv)l(als)h(for)g
(eac)n(h,)-94 5192 y(where)19 b(eac)n(h)g(in)n(terface)h(is)f(mo)r
(deled)g(as)g(a)g(single)h(in)n(terface)g(o)n(v)n(er-)-94
5292 y(lain)27 b(b)n(y)f(a)g(la)n(y)n(er)h(of)g(constan)n(t)f(v)n(elo)r