Berechnung Schichtdicken Refraktionsseismik

by Andreas Messer <andreas.messer@stud-mail.uni-wuerzburg.de>

Messwerte

Geschwindigkeiten und Knickpunkte Schuss

u = {641, 1150, 2770};

x = {0, 22.6, 33.5};

Geschwindigkeiten und Knickpunkte Gegenschuss

v = {653, 1017, 2220};

y = {0, 60 - 45.1, 60 - 28.7};

Berechnung

Allgemeines

Geschwindigkeit in der obersten Schicht ausgleichen

w[1] = ((u[[1]] +v[[1]] )/2)

Zeit direkte Welle

(Subscript[t, direkt])[z_] =  z/w[1];

1.Schicht

Laufzeit Mintropwelle 1-te Schicht

r1 = 
        t -> (1/w)[1] (2 H1 Cos[g1] Cos[th1] + 
                z Cos[th1] Sin[g1] + z Cos[g1] Sin[th1]);

Schuss

T1s[z_]  =  t /. r1;

Gegenschuss

T1g[z_]  =  T1s[z] /. g1 -> -g1;

Winkel 1-te Schichtgrenze

Ableitungen bilden

f1 = D[T1s[z], z];

f2 = D[T1g[z], z];

Winkel berechnen

r2 = 
      FindRoot[{f1 ==  1/ u[[2]], 
          f2  ==  1/v[[2]]}, {g1, 0}, {th1, 
          ArcSin[((2  w[1])/(u[[2]] + v[[2]]))]}]

γ1 = g1 /. r2; θ1 = th1 /. r2;

Geschwindigkeit 1-te Schicht

w[2] = N[w[1]/Sin[θ1]]

Hoehen 1-te Schicht

hs[1] = 
      H1 /. Solve[
            T1s[x[[2]]] == 
                (Subscript[t, direkt])[x[[2]]] /. {g1 -> γ1, 
                th1 -> θ1}, H1][[1]]

hg[1] = 
      H1 /. Solve[
            T1g[y[[2]]] == 
                (Subscript[t, direkt])[y[[2]]] /. {g1 -> γ1, 
                th1 -> θ1}, H1][[1]]

2.Schicht

Laufzeit 2-te Schichtgrenze

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r2 = 
        t -> (v1 Cos[g2] Sec[
                      th2] (2 (v1^2) Sin[g1 - g2 + th2]^2 (2 H2 - 
                              z Tan[g1]) - 
                        (v2^2) (4 H2 + (-2 z + 
                                    H1 (Sqrt[4 - ((2 (v1^2))/(v2^2)) + ((2 (v1^2) Cos[2 (g1 - g2 + th2)])/(v2^2))]) Cos[
                                        g1 - 
                                        ArcSin[((v1 Sin[g1 - g2 + th2])/v2)]] Sin[g1] + 
                                    H1 (Sqrt[4 - ((2 (v1^2))/(v2^2)) + ((2 (v1^2) Cos[2 (g1 - g2 + th2)])/(v2^2))]) Cos[
                                        g1 + 
                                        ArcSin[((v1 Sin[g1 - g2 + th2])/v2)]] Sin[g1] + 
                                    z (Sqrt[4 - ((2 (v1^2))/(v2^2)) + ((2 (v1^2) Cos[2 (g1 - g2 + th2)])/(v2^2))]) Sin[g1] Sin[
                                        g1 + 
                                        ArcSin[((v1 Sin[g1 - g2 + th2])/v2)]]) Tan[g1])) + 
                  2 (-2 H1 (v2^3) Cos[
                            g1] (Sqrt[1 - (((v1^2) Sin[g1 - g2 + th2]^2)/(v2^2))]) - (v2^3) z Sin[
                            g1] (Sqrt[1 - (((v1^2) Sin[g1 - g2 + th2]^2)/(v2^2))]) + v1 Sec[g2] Sin[th2] ((v1^2) z Sin[g1 - g2 + th2]^2 + 
                              2 H1 v1 v2 Cos[g1]^2 Sin[
                                  g1 - g2 + 
                                    th2] (Sqrt[1 - (((v1^2) Sin[g1 - g2 + th2]^2)/(v2^2))]) + (v2^2) z (-1 + 
                                    Sin[g1] (Sqrt[1 - (((v1^2) Sin[g1 - g2 + th2]^2)/(v2^2))]) Sin[g1 + ArcSin[((v1 Sin[g1 - g2 + th2])/v2)]])) + 
                        v1 Sec[th2] Sin[
                            g2] (H1 Sin[
                                  2 g1] ((v2^2) - 
                                    (v1^2) Sin[g1 - g2 + th2]^2) + 
                              z ((v1^2) Sin[g1 - g2 + th2]^2 + 
                                    (v2^2) (-1 + 
                                        Sin[g1] (Sqrt[1 - (((v1^2) Sin[g1 - g2 + th2]^2)/(v2^2))]) Sin[g1 + 
                                        ArcSin[((v1 Sin[g1 - g2 + th2])/v2)]]))) - H2 v1 (v2^2) Sin[g2 - th2] Tan[th2] + 
                        H2 (v1^3) Sin[g2 - th2] Sin[g1 - g2 + th2]^2 Tan[
                            th2] + H2 v1 (v2^2) Sin[g2 + th2] Tan[th2] - 
                        H2 (v1^3) Sin[g1 - g2 + th2]^2 Sin[g2 + th2] Tan[
                            th2] - 
                        H1 v1 (v2^2) Sin[g1] Sin[
                            g2 - th2] (Sqrt[1 - (((v1^2) Sin[g1 - g2 + th2]^2)/(v2^2))]) Sin[g1 - ArcSin[((v1 Sin[g1 - g2 + th2])/v2)]] Tan[th2] + 
                        H1 v1 (v2^2) Sin[
                            g1] (Sqrt[1 - (((v1^2) Sin[g1 - g2 + th2]^2)/(v2^2))]) Sin[g2 + th2] Sin[g1 + ArcSin[((v1 Sin[g1 - g2 + th2])/v2)]] Tan[
                            th2] - 
                        v1 (v2^2) z Sin[g2 + th2] Tan[g1] Tan[th2] + 
                        (v1^3) z Sin[g1 - g2 + th2]^2 Sin[g2 + th2] Tan[
                            g1] Tan[th2] + 
                        v1 (v2^2) z Sin[
                            g1] (Sqrt[1 - (((v1^2) Sin[g1 - g2 + th2]^2)/(v2^2))]) Sin[g2 + th2] Sin[g1 + ArcSin[((v1 Sin[g1 - g2 + th2])/v2)]] Tan[
                            g1] Tan[th2] + 
                        v1 (v2^2) z Sin[g2 + th2] Tan[g2] Tan[th2] - 
                        (v1^3) z Sin[g1 - g2 + th2]^2 Sin[g2 + th2] Tan[
                            g2] Tan[th2] + 
                        H1 v1 (v2^2) Cos[g1] Sin[
                            g2 - th2] (Sqrt[1 - (((v1^2) Sin[g1 - g2 + th2]^2)/(v2^2))]) Sin[g1 - ArcSin[((v1 Sin[g1 - g2 + th2])/v2)]] Tan[g2] Tan[
                            th2] - 
                        H1 v1 (v2^2) Cos[
                            g1] (Sqrt[1 - (((v1^2) Sin[g1 - g2 + th2]^2)/(v2^2))]) Sin[g2 + th2] Sin[g1 + ArcSin[((v1 Sin[g1 - g2 + th2])/v2)]] Tan[
                            g2] Tan[th2] - 
                        v1 (v2^2) z Sin[
                            g1] (Sqrt[1 - (((v1^2) Sin[g1 - g2 + th2]^2)/(v2^2))]) Sin[g2 + th2] Sin[g1 + ArcSin[((v1 Sin[g1 - g2 + th2])/v2)]] Tan[
                            g2] Tan[th2]))/(-2 v1 (v2^3) + 
                  2 (v1^3) v2 Sin[g1 - g2 + th2]^2) /. {v1 -> w[1], 
            v2 -> w[2]};

Laufzeit Schuss

T2s[
          z_] = t /. r2 /. {g1 -> γ1, th1 -> θ1, 
            H1 -> hs[1]};

Laufzeit Gegenschuss

T2g[
          z_] = t /. r2 /. {g1 -> -g1, 
              g2 -> -g2} /. {g1 -> γ1, 
            th1 -> θ1, H1 -> hg[1]};

Winkel 2-te Schichtgrenze

Ableitungen bilden

f3 = D[T2s[z], z];

f4 = D[T2g[z], z];

Winkel berechnen

r3 = 
      FindRoot[{f3 ==  1/ u[[3]], 
          f4  ==  1/v[[3]]}, {g2, γ1}, {th2, 
          ArcSin[((2 w[2])/(v[[3]] + u[[3]]))]}]

γ2 = g2 /. r3 ;θ2 = 
        th2 /. r3 ;

Geschwindigkeit 2-te Schicht

w[3] = N[w[2]/Sin[θ2]]

Hoehen 2-te Schicht

hs[2] = 
      N[H2 /. Solve[
              T2s[x[[3]]] == 
                  (Subscript[t, direkt])[x[[3]]] /. {g2 -> γ2, 
                  th2 -> θ2}, H2][[1]]]

hg[2] = 
      N[H2 /. Solve[
              T2g[y[[3]]] == 
                  (Subscript[t, direkt])[y[[3]]] /. {g2 -> γ2, 
                  th2 -> θ2}, H2][[1]]]

Ergebnisse

1.Schicht

Print["Dicke: Schusspunkt: ", hs[1], "m Gegenschusspunkt: ", 
        hg[1],  "m; Geschwindigkeit: ", w[1],  "m/s"];

2.Schicht

Print["Dicke: Schusspunkt: ", hs[2], "m Gegenschusspunkt: ", 
        hg[2],  "m; Geschwindigkeit: ", w[2],  "m/s"];

3.Schicht

Print["Geschwindigkeit: ", w[3],  "m/s"];

Plot

H1[z_] = -(hs[1] + z Tan[γ1]);
H2[z_] = -(hs[1] + hs[2] + z Tan[γ2]); 
    zmax = z /. Solve[H1[z] == -hg[1], z][[1]]; 
    Plot[{H1[x], H2[x]}, {x, 0, zmax}, 
      PlotRange -> {0, -Max[hs[1] + hs[2], 
              hg[1] + hg[2]]}];