[5675] | 1 | # |
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| 2 | # slide_tsunami function |
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| 3 | # |
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| 4 | |
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| 5 | """This function returns a callable object representing an initial water |
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| 6 | displacement generated by a submarine sediment slide. |
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| 7 | |
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| 8 | Using input parameters: |
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| 9 | |
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| 10 | Required |
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| 11 | length downslope slide length |
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| 12 | depth water depth to slide centre of mass |
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| 13 | slope bathymetric slope |
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| 14 | |
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| 15 | Optional |
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| 16 | x0 x origin (0) |
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| 17 | y0 y origin (0) |
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| 18 | alpha angular orientation of slide in xy plane (0) |
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| 19 | w slide width (0.25*length) |
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| 20 | T slide thickness (0.01*length) |
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| 21 | g acceleration due to gravity (9.8) |
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| 22 | gamma specific density of sediments (1.85) |
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| 23 | Cm added mass coefficient (1) |
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| 24 | Cd drag coefficient (1) |
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| 25 | Cn friction coefficient (0) |
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| 26 | psi (0) |
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| 27 | dx offset of second Gaussian (0.2*width of first Gaussian) |
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| 28 | kappa multiplier for sech^2 function (3.0) |
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| 29 | kappad multiplier for second Gaussian function (0.8) |
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| 30 | zsmall an amount near to zero (0.01) |
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| 31 | |
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| 32 | The following parameters are calculated: |
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| 33 | |
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| 34 | a0 initial acceleration |
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| 35 | ut theoretical terminal velocity |
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| 36 | s0 charactistic distance of motion |
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| 37 | t0 characteristic time of motion |
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| 38 | w initial wavelength of tsunami |
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| 39 | a2D 2D initial amplitude of tsunami |
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| 40 | a3D 3D initial amplitude of tsunami |
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| 41 | |
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| 42 | The returned object is a callable double Gaussian function that represents |
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| 43 | the initial water displacement generated by a submarine sediment slide. |
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| 44 | |
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| 45 | Adrian Hitchman |
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| 46 | Geoscience Australia, June 2005 |
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| 47 | """ |
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[6157] | 48 | |
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[6304] | 49 | import numpy as num |
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[6157] | 50 | |
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| 51 | |
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[5675] | 52 | def find_min(x0, wa, kappad, dx): |
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| 53 | """Determine eta_min to scale eta(x,y) |
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| 54 | |
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| 55 | eta(x,y) = n03d/nmin*[-f(y)]*g(x) |
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| 56 | |
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| 57 | nmin = min (-f(y)*g(x) ) |
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| 58 | = -f(ystar)*g(xstar) |
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| 59 | |
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| 60 | ystar = min (-f(y) ), i.e. diff(-f(y))=0 |
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| 61 | xstar = min ( g(x) ), i.e. diff(g(x))=0 |
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| 62 | |
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| 63 | ystar = y0 and -f(ystar)=1.0 |
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| 64 | """ |
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| 65 | from math import exp, cosh |
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| 66 | |
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| 67 | step = 0.05 |
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| 68 | x = x0+50. |
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| 69 | deriv = 10.0 |
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| 70 | count_max = 1000000 |
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| 71 | c = 0 |
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| 72 | deriv = 10. |
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| 73 | f_ystar = 1. |
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| 74 | |
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| 75 | while c < count_max and deriv > 0: |
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| 76 | deriv = (x-x0)*exp(-((x-x0)/wa)**2.0) - \ |
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| 77 | kappad*(x-dx-x0)*exp(-((x-dx-x0)/wa)**2.0) |
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| 78 | |
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| 79 | if deriv <= 0: xstar = x |
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| 80 | x -= step |
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| 81 | c += 1 |
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| 82 | |
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| 83 | g_xstar = exp(-((xstar-x0)/wa)**2)-kappad*exp(-((xstar-dx-x0)/wa)**2) |
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| 84 | |
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| 85 | etastar = g_xstar*f_ystar |
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| 86 | |
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| 87 | return etastar |
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| 88 | |
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| 89 | def slide_tsunami(length, depth, slope, width=None, thickness=None, \ |
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| 90 | x0=0.0, y0=0.0, alpha=0.0, \ |
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| 91 | gravity=9.8, gamma=1.85, \ |
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| 92 | massco=1, dragco=1, frictionco=0, psi=0, \ |
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| 93 | dx=None, kappa=3.0, kappad=0.8, zsmall=0.01, \ |
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| 94 | scale=None, |
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| 95 | domain=None, verbose=False): |
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| 96 | |
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| 97 | from math import sin, tan, radians, pi, sqrt, exp |
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| 98 | |
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| 99 | if domain is not None: |
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| 100 | xllcorner = domain.geo_reference.get_xllcorner() |
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| 101 | yllcorner = domain.geo_reference.get_yllcorner() |
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| 102 | x0 = x0 - xllcorner # slump origin (relative) |
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| 103 | y0 = y0 - yllcorner |
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| 104 | |
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| 105 | #if width not provided, set to typical value |
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| 106 | if width is None: |
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| 107 | width = 0.25 * length |
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| 108 | |
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| 109 | #if thickness not provided, set to typical value |
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| 110 | if thickness is None: |
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| 111 | thickness = 0.01 * length |
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| 112 | |
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| 113 | #calculate some parameters of the slide |
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| 114 | |
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| 115 | sint = sin(radians(slope)) |
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| 116 | tant = tan(radians(slope)) |
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| 117 | tanp = tan(radians(psi)) |
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| 118 | |
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| 119 | a0 = gravity * sint * ((gamma-1)/(gamma+massco)) * (1-(tanp/tant)) |
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| 120 | ut = sqrt((gravity*depth) * (length*sint/depth) \ |
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| 121 | * (pi*(gamma-1)/(2*dragco)) * (1-(tanp/tant))) |
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| 122 | s0 = ut**2 / a0 |
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| 123 | t0 = ut / a0 |
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| 124 | |
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| 125 | #calculate some parameters of the water displacement produced by the slide |
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| 126 | |
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| 127 | w = t0 * sqrt(gravity*depth) |
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| 128 | a2D = s0 * (0.0574 - (0.0431*sint)) \ |
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| 129 | * (thickness/length) \ |
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| 130 | * ((length*sint/depth)**1.25) \ |
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| 131 | * (1 - exp(-2.2*(gamma-1))) |
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| 132 | a3D = a2D / (1 + (15.5*sqrt(depth/(length*sint)))) |
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| 133 | |
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| 134 | from math import sqrt, log, e |
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| 135 | dx = 2.0 * (w * sqrt(-log((zsmall/a3D),e))) / 5.0 |
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| 136 | |
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| 137 | # determine nmin for scaling of eta(x,y) |
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| 138 | nmin = find_min(x0,w,kappad,dx) |
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| 139 | |
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| 140 | if scale is None: |
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| 141 | scale = a3D/nmin |
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| 142 | |
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| 143 | #a few temporary print statements |
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| 144 | if verbose is True: |
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| 145 | print '\nThe slide ...' |
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| 146 | print '\tLength: ', length |
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| 147 | print '\tDepth: ', depth |
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| 148 | print '\tSlope: ', slope |
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| 149 | print '\tWidth: ', width |
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| 150 | print '\tThickness: ', thickness |
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| 151 | print '\tx0: ', x0 |
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| 152 | print '\ty0: ', y0 |
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| 153 | print '\tAlpha: ', alpha |
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| 154 | print '\tAcceleration: ', a0 |
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| 155 | print '\tTerminal velocity: ', ut |
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| 156 | print '\tChar time: ', t0 |
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| 157 | print '\tChar distance: ', s0 |
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| 158 | print '\nThe tsunami ...' |
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| 159 | print '\tWavelength: ', w |
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| 160 | print '\t2D amplitude: ', a2D |
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| 161 | print '\t3D amplitude: ', a3D |
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| 162 | print '\tscale for eta(x,y):', scale |
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| 163 | |
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| 164 | #keep an eye on some of the assumptions built into the maths |
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| 165 | |
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| 166 | if ((slope < 5) or (slope > 30)): |
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| 167 | if verbose is True: |
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| 168 | print 'WARNING: slope out of range (5 - 30 degrees) ', slope |
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| 169 | if ((depth/length < 0.06) or (depth/length > 1.5)): |
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| 170 | if verbose is True: |
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| 171 | print 'WARNING: d/b out of range (0.06 - 1.5) ', depth/length |
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| 172 | if ((thickness/length < 0.008) or (thickness/length > 0.2)): |
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| 173 | if verbose is True: |
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| 174 | print 'WARNING: T/b out of range (0.008 - 0.2) ', thickness/length |
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| 175 | if ((gamma < 1.46) or (gamma > 2.93)): |
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| 176 | if verbose is True: |
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| 177 | print 'WARNING: gamma out of range (1.46 - 2.93) ', gamma |
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| 178 | |
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| 179 | return Double_gaussian(a3D, w, width, x0, y0, alpha, kappa, kappad, zsmall, dx, scale) |
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| 180 | |
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| 181 | # |
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| 182 | # slump_tsunami function |
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| 183 | # |
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| 184 | |
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| 185 | """This function returns a callable object representing an initial water |
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| 186 | displacement generated by a submarine sediment slump. |
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| 187 | |
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| 188 | Using input parameters: |
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| 189 | |
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| 190 | Required |
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| 191 | length downslope slump length |
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| 192 | depth water depth to slump centre of mass |
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| 193 | slope bathymetric slope |
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| 194 | |
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| 195 | Optional |
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| 196 | x0 x origin (0) |
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| 197 | y0 y origin (0) |
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| 198 | alpha angular orientation of slide in xy plane (0) |
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| 199 | w slump width (1.0*length) |
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| 200 | T slump thickness (0.1*length) |
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| 201 | R slump radius of curvature (b^2/(8*T)) |
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| 202 | del_phi slump angular displacement (0.48) |
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| 203 | g acceleration due to gravity (9.8) |
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| 204 | gamma specific density of sediments (1.85) |
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| 205 | Cm added mass coefficient (1) |
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| 206 | Cd drag coefficient (1) |
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| 207 | Cn friction coefficient (0) |
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| 208 | dx offset of second Gaussian (0.2*width of first Gaussian) |
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| 209 | kappa multiplier for sech^2 function (3.0) |
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| 210 | kappad multiplier for second Gaussian function (0.8) |
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| 211 | zsmall an amount near to zero (0.01) |
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| 212 | |
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| 213 | The following parameters are calculated: |
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| 214 | |
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| 215 | a0 initial acceleration |
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| 216 | um maximum velocity |
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| 217 | s0 charactistic distance of motion |
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| 218 | t0 characteristic time of motion |
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| 219 | w initial wavelength of tsunami |
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| 220 | a2D 2D initial amplitude of tsunami |
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| 221 | a3D 3D initial amplitude of tsunami |
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| 222 | |
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| 223 | The returned object is a callable double Gaussian function that represents |
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| 224 | the initial water displacement generated by a submarine sediment slump. |
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| 225 | |
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| 226 | Adrian Hitchman |
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| 227 | Geoscience Australia, June 2005 |
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| 228 | """ |
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| 229 | |
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| 230 | def slump_tsunami(length, depth, slope, width=None, thickness=None, \ |
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| 231 | radius=None, dphi=0.48, x0=0.0, y0=0.0, alpha=0.0, \ |
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| 232 | gravity=9.8, gamma=1.85, \ |
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| 233 | massco=1, dragco=1, frictionco=0, \ |
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| 234 | dx=None, kappa=3.0, kappad=1.0, zsmall=0.01, scale=None, \ |
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| 235 | domain=None, |
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| 236 | verbose=False): |
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| 237 | |
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| 238 | from math import sin, radians, sqrt |
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| 239 | |
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| 240 | if domain is not None: |
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| 241 | xllcorner = domain.geo_reference.get_xllcorner() |
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| 242 | yllcorner = domain.geo_reference.get_yllcorner() |
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| 243 | x0 = x0 - xllcorner # slump origin (relative) |
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| 244 | y0 = y0 - yllcorner |
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| 245 | |
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| 246 | #if width not provided, set to typical value |
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| 247 | if width is None: |
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| 248 | width = length |
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| 249 | |
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| 250 | #if thickness not provided, set to typical value |
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| 251 | if thickness is None: |
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| 252 | thickness = 0.1 * length |
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| 253 | |
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| 254 | #if radius not provided, set to typical value |
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| 255 | if radius is None: |
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| 256 | radius = length**2 / (8.0 * thickness) |
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| 257 | |
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| 258 | #calculate some parameters of the slump |
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| 259 | |
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| 260 | sint = sin(radians(slope)) |
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| 261 | |
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| 262 | s0 = radius * dphi / 2 |
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| 263 | t0 = sqrt((radius*(gamma+massco)) / (gravity*(gamma-1))) |
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| 264 | a0 = s0 / t0**2 |
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| 265 | um = s0 / t0 |
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| 266 | |
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| 267 | #calculate some parameters of the water displacement produced by the slump |
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| 268 | |
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| 269 | w = t0 * sqrt(gravity*depth) |
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| 270 | a2D = s0 * (0.131/sint) \ |
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| 271 | * (thickness/length) \ |
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| 272 | * (length*sint/depth)**1.25 \ |
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| 273 | * (length/radius)**0.63 * dphi**0.39 \ |
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| 274 | * (1.47 - (0.35*(gamma-1))) * (gamma-1) |
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| 275 | a3D = a2D / (1 + (2.06*sqrt(depth/length))) |
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| 276 | |
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| 277 | from math import sqrt, log, e |
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| 278 | dx = 2.0 * (w * sqrt(-log((zsmall/a3D),e))) / 5.0 |
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| 279 | |
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| 280 | # determine nmin for scaling of eta(x,y) |
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| 281 | nmin = find_min(x0,w,kappad,dx) |
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| 282 | |
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| 283 | if scale is None: |
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| 284 | scale = a3D/nmin |
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| 285 | |
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| 286 | #a few temporary print statements |
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| 287 | if verbose is True: |
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| 288 | print '\nThe slump ...' |
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| 289 | print '\tLength: ', length |
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| 290 | print '\tDepth: ', depth |
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| 291 | print '\tSlope: ', slope |
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| 292 | print '\tWidth: ', width |
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| 293 | print '\tThickness: ', thickness |
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| 294 | print '\tRadius: ', radius |
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| 295 | print '\tDphi: ', dphi |
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| 296 | print '\tx0: ', x0 |
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| 297 | print '\ty0: ', y0 |
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| 298 | print '\tAlpha: ', alpha |
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| 299 | print '\tAcceleration: ', a0 |
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| 300 | print '\tMaximum velocity: ', um |
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| 301 | print '\tChar time: ', t0 |
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| 302 | print '\tChar distance: ', s0 |
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| 303 | print '\nThe tsunami ...' |
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| 304 | print '\tWavelength: ', w |
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| 305 | print '\t2D amplitude: ', a2D |
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| 306 | print '\t3D amplitude: ', a3D |
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| 307 | print '\tDelta x ', dx |
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| 308 | print '\tsmall ', zsmall |
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| 309 | print '\tKappa d ', kappad |
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| 310 | print '\tscale for eta(x,y):', scale |
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| 311 | |
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| 312 | #keep an eye on some of the assumptions built into the maths |
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| 313 | |
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| 314 | if ((slope < 10) or (slope > 30)): |
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| 315 | if verbose is True: |
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| 316 | print 'WARNING: slope out of range (10 - 30 degrees) ', slope |
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| 317 | if ((depth/length < 0.34) or (depth/length > 0.5)): |
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| 318 | if verbose is True: |
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| 319 | print 'WARNING: d/b out of range (0.34 - 0.5) ', depth/length |
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| 320 | if ((thickness/length < 0.10) or (thickness/length > 0.15)): |
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| 321 | if verbose is True: |
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| 322 | print 'WARNING: T/b out of range (0.10 - 0.15) ', thickness/length |
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| 323 | if ((radius/length < 1.0) or (radius/length > 2.0)): |
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| 324 | if verbose is True: |
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| 325 | print 'WARNING: R/b out of range (1 - 2) ', radius/length |
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| 326 | if ((dphi < 0.10) or (dphi > 0.52)): |
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| 327 | if verbose is True: |
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| 328 | print 'WARNING: del_phi out of range (0.10 - 0.52) ', dphi |
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| 329 | if ((gamma < 1.46) or (gamma > 2.93)): |
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| 330 | if verbose is True: |
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| 331 | print 'WARNING: gamma out of range (1.46 - 2.93) ', gamma |
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| 332 | |
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| 333 | return Double_gaussian(a3D, w, width, x0, y0, alpha, kappa, kappad, zsmall, dx, scale) |
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| 334 | |
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| 335 | # |
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| 336 | # Double_gaussian class |
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| 337 | # |
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| 338 | |
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| 339 | """This is a callable class representing the initial water displacment |
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| 340 | generated by a sediment slide or slump. |
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| 341 | |
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| 342 | Using input parameters: |
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| 343 | |
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| 344 | Required |
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| 345 | w initial wavelength of tsunami |
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| 346 | a3D 3D initial amplitude of tsunami |
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| 347 | width width of smf |
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| 348 | |
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| 349 | Optional |
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| 350 | x0 x origin of smf |
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| 351 | y0 y origin of smf |
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| 352 | alpha angular orientation of smf in xy plane (0) |
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| 353 | dx offset of second Gaussian (0.2*width of first Gaussian) |
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| 354 | kappa multiplier for sech^2 function (3.0) |
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| 355 | kappad multiplier for second Gaussian function (0.8) |
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| 356 | zsmall an amount near to zero (0.01) |
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| 357 | |
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| 358 | Adrian Hitchman |
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| 359 | Geoscience Australia, June 2005 |
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| 360 | """ |
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| 361 | |
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| 362 | class Double_gaussian: |
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| 363 | |
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| 364 | def __init__(self, a3D, wavelength, width, x0, y0, alpha, \ |
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| 365 | kappa, kappad, zsmall, dx, scale): |
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| 366 | self.a3D = a3D |
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| 367 | self.wavelength = wavelength |
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| 368 | self.width = width |
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| 369 | self.x0 = x0 |
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| 370 | self.y0 = y0 |
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| 371 | self.alpha = alpha |
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| 372 | self.kappa = kappa |
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| 373 | self.kappad = kappad |
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| 374 | self.scale = scale |
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| 375 | |
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| 376 | if dx is None: |
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| 377 | from math import sqrt, log, e |
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| 378 | dx = 2.0 * (self.wavelength * sqrt(-log((zsmall/self.a3D),e))) / 5.0 |
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| 379 | self.dx = dx |
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| 380 | |
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| 381 | def __call__(self, x, y): |
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| 382 | """Make Double_gaussian a callable object. |
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| 383 | |
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| 384 | If called as a function, this object returns z values representing |
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| 385 | the initial 3D distribution of water heights at the points (x,y) |
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| 386 | produced by a submarine mass failure. |
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| 387 | """ |
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| 388 | |
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| 389 | from math import sin, cos, radians, exp, cosh |
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| 390 | |
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| 391 | #ensure vectors x and y have the same length |
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| 392 | N = len(x) |
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| 393 | assert N == len(y) |
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| 394 | |
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| 395 | am = self.a3D |
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| 396 | am2 = 1.0 |
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| 397 | wa = self.wavelength |
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| 398 | wi = self.width |
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| 399 | x0 = self.x0 |
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| 400 | y0 = self.y0 |
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| 401 | alpha = self.alpha |
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| 402 | dx = self.dx |
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| 403 | kappa = self.kappa |
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| 404 | kappad = self.kappad |
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| 405 | scale = self.scale |
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| 406 | |
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| 407 | #double Gaussian calculation assumes water displacement is oriented |
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| 408 | #E-W, so, for displacement at some angle alpha clockwise from the E-W |
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| 409 | #direction, rotate (x,y) coordinates anti-clockwise by alpha |
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| 410 | |
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| 411 | cosa = cos(radians(alpha)) |
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| 412 | sina = sin(radians(alpha)) |
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| 413 | |
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| 414 | xr = ((x-x0) * cosa - (y-y0) * sina) + x0 |
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| 415 | yr = ((x-x0) * sina + (y-y0) * cosa) + y0 |
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| 416 | |
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[6304] | 417 | z = num.zeros(N, num.float) |
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[5675] | 418 | maxz = 0.0 |
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| 419 | minz = 0.0 |
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| 420 | for i in range(N): |
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| 421 | try: |
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| 422 | z[i] = -scale / ((cosh(kappa*(yr[i]-y0)/(wi+wa)))**2) \ |
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| 423 | * (exp(-((xr[i]-x0)/wa)**2) - \ |
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| 424 | kappad*exp(-((xr[i]-dx-x0)/wa)**2)) |
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| 425 | if z[i] > maxz: maxz = z[i] |
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| 426 | if z[i] < minz: minz = z[i] |
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| 427 | |
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| 428 | except OverflowError: |
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| 429 | pass |
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| 430 | |
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| 431 | return z |
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| 432 | |
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| 433 | def determineDX(self, zsmall): |
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| 434 | """Determine a suitable offset for the second Gaussian function. |
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| 435 | |
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| 436 | A suitable offset for the second Gaussian function is taken to |
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| 437 | be some fraction of the 'width' of the first Gaussian function. |
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| 438 | |
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| 439 | The 'width' of the first Gaussian is obtained from the range of |
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| 440 | the x coordinates over which the function takes values from |
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| 441 | 'near zero', through 1, and back to 'near zero'. |
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| 442 | |
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| 443 | The parameter zsmall passed to this function specifies how much |
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| 444 | 'near zero' is. |
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| 445 | """ |
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| 446 | |
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| 447 | from math import sqrt, log, e |
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| 448 | |
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| 449 | a = self.a3D |
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| 450 | c = self.wavelength |
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| 451 | |
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| 452 | self.dx = 2.0 * (c * sqrt(-log((zsmall/a),e))) / 5.0 |
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| 453 | |
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| 454 | return self.dx |
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| 455 | |
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