def transformDerivU(self, u, m):
self.setModel(m)
alpha = self.alpha
I = self.I
n = self.n
Ks = self.Ks
m = 1.0 - 1.0 / n
g = I * alpha * n * np.exp(Ks) * abs(alpha * u)**(n - 1.0) * np.sign(
alpha * u) * (1.0 / n - 1.0) * (
(abs(alpha * u)**n + 1)**(1.0 / n - 1))**(I - 1) * (
(1 - 1.0 / ((abs(alpha * u)**n + 1)**(1.0 / n - 1))**
(1.0 / (1.0 / n - 1)))**(1 - 1.0 / n) -
1)**2 * (abs(alpha * u)**n + 1)**(1.0 / n - 2) - (
2 * alpha * n * np.exp(Ks) * abs(alpha * u)**
(n - 1) * np.sign(alpha * u) * (1.0 / n - 1) *
((abs(alpha * u)**n + 1)**(1.0 / n - 1))**I *
((1 - 1.0 / ((abs(alpha * u)**n + 1)**(1.0 / n - 1))**
(1.0 / (1.0 / n - 1)))**(1 - 1.0 / n) - 1) *
(abs(alpha * u)**n + 1)**(1.0 / n - 2)) / (
((abs(alpha * u)**n + 1)**
(1.0 / n - 1))**(1.0 / (1.0 / n - 1) + 1) *
(1 - 1.0 /
((abs(alpha * u)**n + 1)**(1.0 / n - 1))**
(1.0 / (1.0 / n - 1)))**(1.0 / n))
g[u >= 0] = 0
g = Utils.sdiag(g)
return g
def transformDerivU(self, u, m):
self.setModel(m)
alpha = self.alpha
I = self.I
n = self.n
Ks = self.Ks
m = 1.0 - 1.0 / n
g = I * alpha * n * np.exp(Ks) * abs(alpha * u) ** (n - 1.0) * np.sign(alpha * u) * (1.0 / n - 1.0) * (
(abs(alpha * u) ** n + 1) ** (1.0 / n - 1)
) ** (I - 1) * (
(1 - 1.0 / ((abs(alpha * u) ** n + 1) ** (1.0 / n - 1)) ** (1.0 / (1.0 / n - 1))) ** (1 - 1.0 / n) - 1
) ** 2 * (
abs(alpha * u) ** n + 1
) ** (
1.0 / n - 2
) - (
2
* alpha
* n
* np.exp(Ks)
* abs(alpha * u) ** (n - 1)
* np.sign(alpha * u)
* (1.0 / n - 1)
* ((abs(alpha * u) ** n + 1) ** (1.0 / n - 1)) ** I
* ((1 - 1.0 / ((abs(alpha * u) ** n + 1) ** (1.0 / n - 1)) ** (1.0 / (1.0 / n - 1))) ** (1 - 1.0 / n) - 1)
* (abs(alpha * u) ** n + 1) ** (1.0 / n - 2)
) / (
((abs(alpha * u) ** n + 1) ** (1.0 / n - 1)) ** (1.0 / (1.0 / n - 1) + 1)
* (1 - 1.0 / ((abs(alpha * u) ** n + 1) ** (1.0 / n - 1)) ** (1.0 / (1.0 / n - 1))) ** (1.0 / n)
)
g[u >= 0] = 0
g = Utils.sdiag(g)
return g
def transform(self, u, m):
self.setModel(m)
f = np.exp(self.Ks)*self.A/(self.A+abs(u)**self.gamma)
if Utils.isScalar(self.Ks):
f[u >= 0] = np.exp(self.Ks)
else:
f[u >= 0] = np.exp(self.Ks[u >= 0])
return f
def transform(self, u, m):
self.setModel(m)
f = np.exp(self.Ks)*self.A/(self.A+abs(u)**self.gamma)
if Utils.isScalar(self.Ks):
f[u >= 0] = np.exp(self.Ks)
else:
f[u >= 0] = np.exp(self.Ks[u >= 0])
return f
def transform(self, u, m):
self.setModel(m)
alpha = self.alpha
I = self.I
n = self.n
Ks = self.Ks
m = 1.0 - 1.0/n
theta_e = 1.0/((1.0+abs(alpha*u)**n)**m)
f = np.exp(Ks)*theta_e**I* ( ( 1.0 - ( 1.0 - theta_e**(1.0/m) )**m )**2 )
if Utils.isScalar(self.Ks):
f[u >= 0] = np.exp(self.Ks)
else:
f[u >= 0] = np.exp(self.Ks[u >= 0])
return f
def transform(self, u, m):
self.setModel(m)
alpha = self.alpha
I = self.I
n = self.n
Ks = self.Ks
m = 1.0 - 1.0/n
theta_e = 1.0/((1.0+abs(alpha*u)**n)**m)
f = np.exp(Ks)*theta_e**I* ( ( 1.0 - ( 1.0 - theta_e**(1.0/m) )**m )**2 )
if Utils.isScalar(self.Ks):
f[u >= 0] = np.exp(self.Ks)
else:
f[u >= 0] = np.exp(self.Ks[u >= 0])
return f
def transformDerivU(self, u, m):
self.setModel(m)
g = -(np.exp(self.Ks) * self.A * self.gamma * abs(u) ** (self.gamma - 1) * np.sign(u)) / (
(self.A + abs(u) ** self.gamma) ** 2
)
g[u >= 0] = 0
g = Utils.sdiag(g)
return g
def transformDerivU(self, u, m):
self.setModel(m)
g = -(np.exp(self.Ks) * self.A * self.gamma * abs(u)**
(self.gamma - 1) * np.sign(u)) / (
(self.A + abs(u)**self.gamma)**2)
g[u >= 0] = 0
g = Utils.sdiag(g)
return g
def _getCasingHertzMagDipole(srcloc,obsloc,freq,sigma,a,b,mu=mu_0*np.ones(3),eps=epsilon_0,moment=1.):
Kc1 = getKc(freq,sigma[1],a,b,mu[1],eps)
nobs = obsloc.shape[0]
dxyz = obsloc - np.c_[np.ones(nobs)]*np.r_[srcloc]
r2 = _r2(dxyz[:,:2])
sqrtr2z2 = np.sqrt(r2 + dxyz[:,2]**2)
k2 = k(freq,sigma[2],mu[2],eps)
return Kc1 * moment / (4.*np.pi) *np.exp(-1j*k2*sqrtr2z2) / sqrtr2z2
def setFrequency(self, time=np.logspace(-7, -1, 256)):
self.Nch = self.time.size
wt = np.array([7.214369775966785e-20, 5.997984537445829e-20, 1.383536819510307e-20, 6.127201193993877e-20, 2.735622069700930e-20, 6.567948836420383e-20, 4.144963335850363e-20, 7.316414067200350e-20, 5.682375914662966e-20, 8.391977074915078e-20, 7.418756524583309e-20, 9.829637687190485e-20, 9.430643800653847e-20, 1.168146262188112e-19, 1.180370735968097e-19, 1.401723019040171e-19, 1.463726071463266e-19, 1.692722072070252e-19, 1.804796158499069e-19, 2.052560499147526e-19, 2.217507732438609e-19, 2.495469564846162e-19, 2.718603842873614e-19, 3.039069705922034e-19, 3.328334008394297e-19, 3.705052796297763e-19, 4.071277819975917e-19, 4.520053409594589e-19, 4.977334107366132e-19, 5.516707191291291e-19, 6.082931168675559e-19, 6.734956703766505e-19, 7.432489554623685e-19, 8.223651399147256e-19, 9.080210233648037e-19, 1.004250388267800e-18, 1.109225156214032e-18, 1.226448534750949e-18, 1.354938655056596e-18, 1.497875155579711e-18, 1.655024636692164e-18, 1.829422009902478e-18, 2.021527957180686e-18, 2.234394042862191e-18, 2.469158736824458e-18, 2.729043278909879e-18, 3.015882778812807e-18, 3.333221019045560e-18, 3.683642665131121e-18, 4.071174485366807e-18, 4.499238428427072e-18, 4.972519918024098e-18, 5.495403162992602e-18, 6.073431145514256e-18, 6.712116746365455e-18, 7.418091347704607e-18, 8.198210388921290e-18, 9.060466264497684e-18, 1.001332641867938e-17, 1.106647001686341e-17, 1.223031194783507e-17, 1.351661046246575e-17, 1.493814249254853e-17, 1.650922025025269e-17, 1.824549287949245e-17, 2.016440324953847e-17, 2.228509875325462e-17, 2.462885473506622e-17, 2.721908372832262e-17, 3.008174877960754e-17, 3.324546598231868e-17, 3.674192913569353e-17, 4.060610542324258e-17, 4.487669220181069e-17, 4.959641037849226e-17, 5.481251456381401e-17, 6.057719336989671e-17, 6.694815564512041e-17, 7.398915178848498e-17, 8.177066132132114e-17, 9.037055462918574e-17, 9.987491078055815e-17, 1.103788451159722e-16, 1.219874911140742e-16, 1.348170262066998e-16, 1.489958578076007e-16, 1.646658879212839e-16, 1.819839514458913e-16, 2.011233698894207e-16, 2.222757000537238e-16, 2.456526388749016e-16, 2.714881529754608e-16, 3.000408107960083e-16, 3.315963787425073e-16, 3.664706739627943e-16, 4.050127315080793e-16, 4.476082920363670e-16, 4.946836672898304e-16, 5.467100025245505e-16, 6.042079955957903e-16, 6.677531050397348e-16, 7.379813122861424e-16, 8.155954842977402e-16, 9.013724102689123e-16, 9.961705740887021e-16, 1.100938748010566e-15, 1.216725486808607e-15, 1.344689623369201e-15, 1.486111865526057e-15, 1.642407614840039e-15, 1.815141131499014e-15, 2.006041190779248e-15, 2.217018384471440e-15, 2.450184243392977e-15, 2.707872369692257e-15, 2.992661792874233e-15, 3.307402781094011e-15, 3.655245368051253e-15, 4.039670879180488e-15, 4.464526774284602e-15, 4.934065153895433e-15, 5.452985315986473e-15, 6.026480787914038e-15, 6.660291305149181e-15, 7.360760256360466e-15, 8.134898170257041e-15, 8.990452879276204e-15, 9.935987062502841e-15, 1.098096394385775e-14, 1.213584200318437e-14, 1.341217964828528e-14, 1.482275089528562e-14, 1.638167321535499e-14, 1.810454882702344e-14, 2.000862084851265e-14, 2.211294587257239e-14, 2.443858469135401e-14, 2.700881307980678e-14, 2.984935474755050e-14, 3.298863879030854e-14, 3.645808421795958e-14, 4.029241440643229e-14, 4.453000462105175e-14, 4.921326608894885e-14, 5.438907046503769e-14, 6.010921893911273e-14, 6.643096067976429e-14, 7.341756580308676e-14, 8.113895860149252e-14, 8.967241736929777e-14, 9.910334783010448e-14, 1.095261379057530e-13, 1.210451023825933e-13, 1.337755269287210e-13, 1.478448219118764e-13, 1.633937975650728e-13, 1.805780732628623e-13, 1.995696350122467e-13, 2.205585567465074e-13, 2.437549026489779e-13, 2.693908295460095e-13, 2.977229104105259e-13, 3.290347022305518e-13, 3.636395839428896e-13, 4.018838928348062e-13, 4.441503908040617e-13, 4.908620951685787e-13, 5.424865123659980e-13, 5.995403169151822e-13, 6.625945224685207e-13, 7.322801967084261e-13, 8.092947772848716e-13, 8.944090520057436e-13, 9.884748731403624e-13, 1.092433683043238e-12, 1.207325936425662e-12, 1.334301513576084e-12, 1.474631228748613e-12, 1.629719548899119e-12, 1.801118650062676e-12, 1.990543952052933e-12, 2.199891286960273e-12, 2.431255873276498e-12, 2.686953285545802e-12, 2.969542629413028e-12, 3.281852154013172e-12, 3.627007558039277e-12, 4.008463272785582e-12, 4.430037035256956e-12, 4.895948097364050e-12, 5.410859453614547e-12, 5.979924509929487e-12, 6.608838660661838e-12, 7.303896290017477e-12, 8.072053768367932e-12, 8.920999073943177e-12, 9.859228736701785e-12, 1.089613287445852e-11, 1.204208917233957e-11, 1.330856674614333e-11, 1.470824092910627e-11, 1.625512013089818e-11, 1.796468603849469e-11, 1.985404856210394e-11, 2.194211707689892e-11, 2.424978967439970e-11, 2.680016231759770e-11, 2.961875999311579e-11, 3.273379217385409e-11, 3.617643514887572e-11, 3.998114404618718e-11, 4.418599767123930e-11, 4.883307961241208e-11, 5.396889942771051e-11, 5.964485812805529e-11, 6.591776261587440e-11, 7.285039422767879e-11, 8.051213707077629e-11, 8.897967244274265e-11, 9.833774628361575e-11, 1.086800173417544e-10, 1.201099945420632e-10, 1.327420729381141e-10, 1.467026786162787e-10, 1.621315340105112e-10, 1.791830562914075e-10, 1.980279028251780e-10, 2.188546791698937e-10, 2.418718267033471e-10, 2.673097087743666e-10, 2.954229162567076e-10, 3.264928155800021e-10, 3.608303647396648e-10, 3.987792254688925e-10, 4.407192027209688e-10, 4.870700458846789e-10, 5.382956497775456e-10, 5.949086974607432e-10, 6.574757913439202e-10, 7.266231239320192e-10, 8.030427449710128e-10, 8.874994877135167e-10, 9.808386236281220e-10, 1.083994322159010e-09, 1.197999000209434e-09, 1.323993654914953e-09, 1.463239283128961e-09, 1.617129501899646e-09, 1.787204496262075e-09, 1.975166433922344e-09, 2.182896501130837e-09, 2.412473730218034e-09, 2.666195807259519e-09, 2.946602068077095e-09, 3.256498912782063e-09, 3.598987893149563e-09, 3.977496754017933e-09, 4.395813739277522e-09, 4.858125505931142e-09, 5.369059025511281e-09, 5.933727892433384e-09, 6.557783502483194e-09, 7.247471613991360e-09, 8.009694857348590e-09, 8.852081819018630e-09, 9.783063390784292e-09, 1.081195714921208e-08, 1.194906060875559e-08, 1.320575428316232e-08, 1.459461558495058e-08, 1.612954470504804e-08, 1.782590372973567e-08, 1.970067039062624e-08, 2.177260798218037e-08, 2.406245315273551e-08, 2.659312344174916e-08, 2.938994664888302e-08, 3.248091431980495e-08, 3.589696189917651e-08, 3.967227833770833e-08, 4.384464827330457e-08, 4.845583018407081e-08, 5.355197433170284e-08, 5.918408463559961e-08, 6.540852915386353e-08, 7.228760421284378e-08, 7.989015791604288e-08, 8.829227916594097e-08, 9.757805922900159e-08, 1.078404332968648e-07, 1.191821106789995e-07, 1.317166026689236e-07, 1.455693587079098e-07, 1.608790217936311e-07, 1.777988162313823e-07, 1.964980809461758e-07, 2.171639645456637e-07, 2.400032980365736e-07, 2.652446652738443e-07, 2.931406901825997e-07, 3.239705657602287e-07, 3.580428475071237e-07, 3.956985425939703e-07, 4.373145214673157e-07, 4.833072913425415e-07, 5.341371626757850e-07, 5.903128587132423e-07, 6.523966036832935e-07, 7.210097538541495e-07, 7.968390110811429e-07, 8.806433020866372e-07, 9.732613658282036e-07, 1.075620158230134e-06, 1.188744116446123e-06, 1.313765428158270e-06, 1.451935342270991e-06, 1.604636717777632e-06, 1.773397831228256e-06, 1.959907713317686e-06, 2.166033001576880e-06, 2.393836687356070e-06, 2.645598681084377e-06, 2.923838733370935e-06, 3.231341523918154e-06, 3.571184694601016e-06, 3.946769446344899e-06, 4.361854837678969e-06, 4.820595081762782e-06, 5.327581531949061e-06, 5.887888119174313e-06, 6.507122780830562e-06, 7.191482772393097e-06, 7.947817716468041e-06, 8.783696866498923e-06, 9.707486485040472e-06, 1.072843153422521e-05, 1.185675077161778e-05, 1.310373578995573e-05, 1.448186809502301e-05, 1.600493890578862e-05, 1.768819362222417e-05, 1.954847630132444e-05, 2.160440843572022e-05, 2.387656249074371e-05, 2.638768394666778e-05, 2.916289862392297e-05, 3.222998971512441e-05, 3.561964367629314e-05, 3.936579782365431e-05, 4.350592904974602e-05, 4.808149299156779e-05, 5.313825827671661e-05, 5.872686606041739e-05, 6.490320915255368e-05, 7.172915206849267e-05, 7.927294798468421e-05, 8.761017620761336e-05, 9.682417843295337e-05, 1.070072955978771e-04, 1.182612851235724e-04, 1.306989769939818e-04, 1.444446003274482e-04, 1.596360362963627e-04, 1.764249271609239e-04, 1.949797924244976e-04, 2.154857030671910e-04, 2.381486646105023e-04, 2.631944925246626e-04, 2.908750792099106e-04, 3.214658697246949e-04, 3.552749625435381e-04, 3.926382043270680e-04, 4.339325952975191e-04, 4.795674127479124e-04, 5.300042093562213e-04, 5.857414026355948e-04, 6.473444397414629e-04, 7.154197401707392e-04, 7.906606243262904e-04, 8.738040302727717e-04, 9.657009935888906e-04, 1.067245638145834e-03, 1.179484028621435e-03, 1.303498707764836e-03, 1.440577691237741e-03, 1.592027938865682e-03, 1.759438818176274e-03, 1.944382214020240e-03, 2.148824632015574e-03, 2.374646777242952e-03, 2.624289840901410e-03, 2.899987938462482e-03, 3.204783728012370e-03, 3.541304571287609e-03, 3.913361077715114e-03, 4.323998734848948e-03, 4.778017035442578e-03, 5.278871213895021e-03, 5.832645828904957e-03, 6.443132211847618e-03, 7.118100704687155e-03, 7.861484687059508e-03, 8.683286454219962e-03, 9.587172959576953e-03, 1.058612645311708e-02, 1.168276512339872e-02, 1.289407692301174e-02, 1.422020567085629e-02, 1.568354709989395e-02, 1.727924763496293e-02, 1.903701004445868e-02, 2.094259894090355e-02, 2.303555498203885e-02, 2.528473397535577e-02, 2.774280095909549e-02, 3.034889679856765e-02, 3.317292189089636e-02, 3.610269051747732e-02, 3.923023471609136e-02, 4.235591398256915e-02, 4.559945470018810e-02, 4.861418172220856e-02, 5.155399423688033e-02, 5.382905665985834e-02, 5.563737547309198e-02, 5.599656739496778e-02, 5.517328802198061e-02, 5.157565446188783e-02, 4.561585237274122e-02, 3.481744626013846e-02, 1.997678484763328e-02, -2.511444299727086e-03, -3.078890380569448e-02, -6.952663437748715e-02, -1.140926319655417e-01, -1.692861783153246e-01, -2.240265004914591e-01, -2.809223452446239e-01, -3.165386782849084e-01, -3.295050746499982e-01, -2.805919713655642e-01, -1.744060875765448e-01, 2.722628846693606e-02, 2.668949880744598e-01, 5.262102231394616e-01, 6.256684356927903e-01, 4.995016301447683e-01, -1.002368152582941e-02, -6.114010724740713e-01, -9.727382503860407e-01, -3.838420705230950e-01, 7.198704705669955e-01, 1.262041888009595e+00, -2.998397076312483e-01, -1.479978761932394e+00, 1.886890549669046e-01, 1.961538671802124e+00, -2.104506074490929e+00, 7.701373097387101e-01, 4.062497351127477e-01, -8.229740504000808e-01, 7.307456920106093e-01, -4.903037312539515e-01, 2.839808721720737e-01, -1.517915989046718e-01, 7.860615976683388e-02, -4.139435902417716e-02, 2.340179865400356e-02, -1.488928090494461e-02, 1.080204283974104e-02, -8.695630540330540e-03, 7.448774255862835e-03, -6.571253694245813e-03, 5.859704720178251e-03, -5.235154219023063e-03, 4.669537109654244e-03, -4.153880559277143e-03, 3.685278478886407e-03, -3.262012231674279e-03, 2.882025619739767e-03, -2.542670610556139e-03, 2.240859550470028e-03, -1.973292341858488e-03, 1.736649256291777e-03, -1.527725614465373e-03, 1.343513590939351e-03, -1.181244115916277e-03, 1.038401885876272e-03, -9.127236961818876e-04, 8.021869803583510e-04, -7.049929363136232e-04, 6.195471678105551e-04, -5.444398377266471e-04, 4.784265058211163e-04, -4.204101656165671e-04, 3.694246665626042e-04, -3.246196272200836e-04, 2.852468930079681e-04, -2.506484828993674e-04, 2.202458813636377e-04, -1.935305291014704e-04, 1.700554065180346e-04, -1.494276181460851e-04, 1.313018693894386e-04, -1.153747197310416e-04, 1.013795159657149e-04, -8.908193308740761e-05, 7.827605834070905e-05, -6.878095175364698e-05, 6.043762035968366e-05, -5.310635544925448e-05, 4.666439257514449e-05, -4.100385733848758e-05, 3.602996086454923e-05, -3.165941281357187e-05, 2.781902585783293e-05, -2.444448983707277e-05, 2.147929539928508e-05, -1.887378820764930e-05, 1.658433732127529e-05, -1.457260421932255e-05, 1.280490076321861e-05, -1.125162529600942e-05, 9.886767060922143e-06, -8.687470528401389e-06, 7.633652544743499e-06, -6.707666049011719e-06, 5.894004649462370e-06, -5.179042976527735e-06, 4.550808463155193e-06, -3.998780821517730e-06, 3.513715894699684e-06, -3.087490887193285e-06, 2.712968339794580e-06, -2.383876585436306e-06, 2.094704718364733e-06, -1.840610321658703e-06, 1.617338386451554e-06, -1.421150054108675e-06, 1.248759998169063e-06, -1.097281408784448e-06, 9.641776560691427e-07, -8.472198144303782e-07, 7.444493348026655e-07, -6.541452446960252e-07, 5.747953299561648e-07, -5.050708135463763e-07, 4.438041043830051e-07, -3.899692438669907e-07, 3.426647247474678e-07, -3.010983952488172e-07, 2.645741945310525e-07, -2.324804964806138e-07, 2.042798670572760e-07, -1.795000644284691e-07, 1.577261311964345e-07, -1.385934457978807e-07, 1.217816165104314e-07, -1.070091160694850e-07, 9.402856728542481e-08, -8.262260063861433e-08, 7.260021429510923e-08, -6.379357556568284e-08, 5.605521036390507e-08, -4.925553366022761e-08, 4.328067952283362e-08, -3.803059434473390e-08, 3.341736133872807e-08, -2.936372828943425e-08, 2.580181391877248e-08, -2.267197117462836e-08, 1.992178838990867e-08, -1.750521159700888e-08, 1.538177331647477e-08, -1.351591490408599e-08, 1.187639109711096e-08, -1.043574678473461e-08, 9.169857246988264e-09, -8.057524168015392e-09, 7.080120656351000e-09, -6.221279323637567e-09, 5.466618198282681e-09, -4.803499887007487e-09, 4.220819952418786e-09, -3.708820961440479e-09, 3.258929089378474e-09, -2.863610543832675e-09, 2.516245405206862e-09, -2.211016771314404e-09, 1.942813349072686e-09, -1.707143861761559e-09, 1.500061838825614e-09, -1.318099529115311e-09, 1.158209830829835e-09, -1.017715265474667e-09, 8.942631413031228e-10, -7.857861555682931e-10, 6.904677759378704e-10, -6.067118212948776e-10, 5.331157324405220e-10, -4.684470851019305e-10, 4.116229519995125e-10, -3.616917683963526e-10, 3.178173974200357e-10, -2.792651282909316e-10, 2.453893729983475e-10, -2.156228554187546e-10, 1.894671118403349e-10, -1.664841438026135e-10, 1.462890834648734e-10, -1.285437486867098e-10, 1.129509799028343e-10, -9.924966395649082e-11, 8.721036155589632e-11, -7.663146513121262e-11, 6.733582275589511e-11, -5.916777159927192e-11, 5.199053123157253e-11, -4.568391312831626e-11, 4.014230801823632e-11, -3.527291737272422e-11, 3.099419942203976e-11, -2.723450367478780e-11, 2.393087107406673e-11, -2.102797969829416e-11, 1.847721835214320e-11, -1.623587253411553e-11, 1.426640914879302e-11, -1.253584798559500e-11, 1.101520943914518e-11, -9.679029223044069e-12, 8.504931950483330e-12, -7.473256440847272e-12, 6.566726477759826e-12, -5.770161505244737e-12, 5.070222417415932e-12, -4.455188184705872e-12, 3.914759576021876e-12, -3.439886690015256e-12, 3.022617407370815e-12, -2.655964226336168e-12, 2.333787251527801e-12, -2.050691376558676e-12, 1.801935938738539e-12, -1.583355332954674e-12, 1.391289255348560e-12, -1.222521408656179e-12, 1.074225642782795e-12, -9.439186286983481e-13, 8.294182731437524e-13, -7.288071777679118e-13, 6.404005307872462e-13, -5.627178934884672e-13, 4.944584091193514e-13, -4.344790190215893e-13, 3.817753212167385e-13, -3.354647509064286e-13, 2.947718012314345e-13, -2.590150368003291e-13, 2.275956825191015e-13, -1.999875966322778e-13, 1.757284600660376e-13, -1.544120345321836e-13, 1.356813597490631e-13, -1.192227758615040e-13, 1.047606709603187e-13, -9.205286574443716e-14, 8.088655803832021e-14, -7.107475925247343e-14, 6.245316311274798e-14, -5.487739422278246e-14, 4.822059038461202e-14, -4.237127819154857e-14, 3.723150631847816e-14, -3.271520525003134e-14, 2.874674597896990e-14, -2.525967353907224e-14, 2.219559416454687e-14, -1.950319744058413e-14, 1.713739707017873e-14, -1.505857586868545e-14, 1.323192234295437e-14, -1.162684774554722e-14, 1.021647384214807e-14, -8.977182814427699e-15, 7.888221761131355e-15, -6.931355174452619e-15, 6.090559572138626e-15, -5.351755171700232e-15, 4.702570113399845e-15, -4.132133283746073e-15, 3.630892270162917e-15, -3.190453398341683e-15, 2.803441173574897e-15, -2.463374772306652e-15, 2.164559515653694e-15, -1.901991507536473e-15, 1.671273840510759e-15, -1.468542966100438e-15, 1.290403996644843e-15, -1.133873855239388e-15, 9.963313217707399e-16, -8.754731385280971e-16, 7.692754403444974e-16, -6.759598633855865e-16, 5.939637650509565e-16, -5.219140562969427e-16, 4.586042081825544e-16, -4.029740475950767e-16, 3.540920038189227e-16, -3.111395086526072e-16, 2.733972888415509e-16, -2.402333212827215e-16, 2.110922493015534e-16, -1.854860827684001e-16, 1.629860263206755e-16, -1.432152988478505e-16, 1.258428239959391e-16, -1.105776860340182e-16, 9.716425824191095e-17, -8.537792224005708e-17, 7.502130657839168e-17, -6.592098159645411e-17, 5.792455520756561e-17, -5.089812097369260e-17, 4.472401573699795e-17, -3.929884925786503e-17, 3.453177286415005e-17, -3.034295811884372e-17, 2.666226003023219e-17, -2.342804241895063e-17, 2.058614577177349e-17, -1.808898029804633e-17, 1.589472900128195e-17, -1.396664742072978e-17, 1.227244831653922e-17, -1.078376099458355e-17, 9.475656216910048e-18, -8.326228742065685e-18, 7.316230504610306e-18, -6.428748291129759e-18, 5.648920515191333e-18, -4.963688348418389e-18, 4.361577040171507e-18, -3.832503763852548e-18, 3.367608772061774e-18, -2.959107033168789e-18, 2.600157864838119e-18, -2.284750381424515e-18, 2.007602836968745e-18, -1.764074178215185e-18, 1.550086326535024e-18, -1.362055887301009e-18, 1.196834143131819e-18, -1.051654326148005e-18, 9.240852862763527e-19, -8.119907797435101e-19, 7.134936960083838e-19, -6.269446240781204e-19, 5.508942318228495e-19, -4.840689957627215e-19, 4.253498749090647e-19, -3.737535715383304e-19, 3.284160650943604e-19, -2.885781434802982e-19, 2.535726894517719e-19, -2.228135092144265e-19, 1.957855161528666e-19, -1.720361053077579e-19, 1.511675741544441e-19, -1.328304627571508e-19, 1.167177017717951e-19, -1.025594703000911e-19, 9.011867747602604e-20, -7.918699208456320e-20, 6.958135363559505e-20, -6.114090626414241e-20, 5.372430364847189e-20, -4.720733874362162e-20, 4.148085614846149e-20, -3.644890635898519e-20, 3.202709755606534e-20, -2.814108611035396e-20, 2.472510802483146e-20, -2.172035832750181e-20, 1.907280017594962e-20, -7.276969157651721e-21])
ab = 0.7866057737580476e0
#------- Compute Frequency components reqired for transform -------#
# This is for Digital filtering and here we evalute frequency domain responses
# ritght at this bases.
# a. Generate time base
n = np.ceil(-10*np.log(time.min()/time.max()))
tbase = time.max()*np.exp(-0.1*np.arange(0, n+1))
self.wt = wt
self.ab = ab
self.n = n
self.tbase = tbase
# b. Determine required frequencies
omega_int = (ab/tbase[0])*np.exp(0.1*(np.r_[1:786+tbase.size:(786+tbase.size)*1j]-425))
# Case1: Compute frequency domain reponses right at filter coefficient values
if self.switchInterp == False:
self.frequency = omega_int/(2*np.pi)
self.Nfreq = self.frequency.size
# Case2: Compute frequency domain reponses in logarithmic then intepolate
elif self.switchInterp == True:
# This is tested decision: works well 1e-4-1e0 S/m
self.frequency = np.logspace(-3, 8, 81)
self.omega_int = omega_int
self.Nfreq = self.frequency.size
else:
raise Exception('Not implemented!!')
if self.offset is not None and np.isscalar(self.offset):
self.offset = self.offset*np.ones(self.Nfreq)
elif self.offset is not None and not np.isscalar(self.offset):
self.offset = self.offset[0]*np.ones(self.Nfreq)
def _getCasingHertzMagDipole(srcloc,
obsloc,
freq,
sigma,
a,
b,
mu=mu_0 * np.ones(3),
eps=epsilon_0,
moment=1.):
Kc1 = getKc(freq, sigma[1], a, b, mu[1], eps)
nobs = obsloc.shape[0]
dxyz = obsloc - np.c_[np.ones(nobs)] * np.r_[srcloc]
r2 = _r2(dxyz[:, :2])
sqrtr2z2 = np.sqrt(r2 + dxyz[:, 2]**2)
k2 = k(freq, sigma[2], mu[2], eps)
return Kc1 * moment / (4. * np.pi) * np.exp(-1j * k2 * sqrtr2z2) / sqrtr2z2
def run(plotIt=True):
"""
1D FDEM and TDEM inversions
===========================
This example is used in the paper Heagy et al 2016 (in prep)
"""
# Set up cylindrically symmeric mesh
cs, ncx, ncz, npad = 10., 15, 25, 13 # padded cyl mesh
hx = [(cs, ncx), (cs, npad, 1.3)]
hz = [(cs, npad, -1.3), (cs, ncz), (cs, npad, 1.3)]
mesh = Mesh.CylMesh([hx, 1, hz], '00C')
# Conductivity model
layerz = np.r_[-200., -100.]
layer = (mesh.vectorCCz >= layerz[0]) & (mesh.vectorCCz <= layerz[1])
active = mesh.vectorCCz < 0.
sig_half = 1e-2 # Half-space conductivity
sig_air = 1e-8 # Air conductivity
sig_layer = 5e-2 # Layer conductivity
sigma = np.ones(mesh.nCz) * sig_air
sigma[active] = sig_half
sigma[layer] = sig_layer
# Mapping
actMap = Maps.InjectActiveCells(mesh, active, np.log(1e-8), nC=mesh.nCz)
mapping = Maps.ExpMap(mesh) * Maps.SurjectVertical1D(mesh) * actMap
mtrue = np.log(sigma[active])
# ----- FDEM problem & survey -----
rxlocs = Utils.ndgrid([np.r_[50.], np.r_[0], np.r_[0.]])
bzi = FDEM.Rx.Point_bSecondary(rxlocs, 'z', 'real')
bzr = FDEM.Rx.Point_bSecondary(rxlocs, 'z', 'imag')
freqs = np.logspace(2, 3, 5)
srcLoc = np.array([0., 0., 0.])
print('min skin depth = ', 500. / np.sqrt(freqs.max() * sig_half),
'max skin depth = ', 500. / np.sqrt(freqs.min() * sig_half))
print('max x ', mesh.vectorCCx.max(), 'min z ', mesh.vectorCCz.min(),
'max z ', mesh.vectorCCz.max())
srcList = [
FDEM.Src.MagDipole([bzr, bzi], freq, srcLoc, orientation='Z')
for freq in freqs
]
surveyFD = FDEM.Survey(srcList)
prbFD = FDEM.Problem3D_b(mesh, sigmaMap=mapping, Solver=Solver)
prbFD.pair(surveyFD)
std = 0.03
surveyFD.makeSyntheticData(mtrue, std)
surveyFD.eps = np.linalg.norm(surveyFD.dtrue) * 1e-5
# FDEM inversion
np.random.seed(1)
dmisfit = DataMisfit.l2_DataMisfit(surveyFD)
regMesh = Mesh.TensorMesh([mesh.hz[mapping.maps[-1].indActive]])
reg = Regularization.Simple(regMesh)
opt = Optimization.InexactGaussNewton(maxIterCG=10)
invProb = InvProblem.BaseInvProblem(dmisfit, reg, opt)
# Inversion Directives
beta = Directives.BetaSchedule(coolingFactor=4, coolingRate=3)
betaest = Directives.BetaEstimate_ByEig(beta0_ratio=2.)
target = Directives.TargetMisfit()
directiveList = [beta, betaest, target]
inv = Inversion.BaseInversion(invProb, directiveList=directiveList)
m0 = np.log(np.ones(mtrue.size) * sig_half)
reg.alpha_s = 5e-1
reg.alpha_x = 1.
prbFD.counter = opt.counter = Utils.Counter()
opt.remember('xc')
moptFD = inv.run(m0)
# TDEM problem
times = np.logspace(-4, np.log10(2e-3), 10)
print('min diffusion distance ',
1.28 * np.sqrt(times.min() / (sig_half * mu_0)),
'max diffusion distance ',
1.28 * np.sqrt(times.max() / (sig_half * mu_0)))
rx = TDEM.Rx.Point_b(rxlocs, times, 'z')
src = TDEM.Src.MagDipole(
[rx],
waveform=TDEM.Src.StepOffWaveform(),
loc=srcLoc # same src location as FDEM problem
)
surveyTD = TDEM.Survey([src])
prbTD = TDEM.Problem3D_b(mesh, sigmaMap=mapping, Solver=Solver)
prbTD.timeSteps = [(5e-5, 10), (1e-4, 10), (5e-4, 10)]
prbTD.pair(surveyTD)
std = 0.03
surveyTD.makeSyntheticData(mtrue, std)
surveyTD.std = std
surveyTD.eps = np.linalg.norm(surveyTD.dtrue) * 1e-5
# TDEM inversion
dmisfit = DataMisfit.l2_DataMisfit(surveyTD)
regMesh = Mesh.TensorMesh([mesh.hz[mapping.maps[-1].indActive]])
reg = Regularization.Simple(regMesh)
opt = Optimization.InexactGaussNewton(maxIterCG=10)
invProb = InvProblem.BaseInvProblem(dmisfit, reg, opt)
inv = Inversion.BaseInversion(invProb, directiveList=directiveList)
m0 = np.log(np.ones(mtrue.size) * sig_half)
reg.alpha_s = 5e-1
reg.alpha_x = 1.
prbTD.counter = opt.counter = Utils.Counter()
opt.remember('xc')
moptTD = inv.run(m0)
if plotIt:
plt.figure(figsize=(10, 8))
ax0 = plt.subplot2grid((2, 2), (0, 0), rowspan=2)
ax1 = plt.subplot2grid((2, 2), (0, 1))
ax2 = plt.subplot2grid((2, 2), (1, 1))
fs = 13 # fontsize
matplotlib.rcParams['font.size'] = fs
# Plot the model
ax0.semilogx(sigma[active], mesh.vectorCCz[active], 'k-', lw=2)
ax0.semilogx(np.exp(moptFD), mesh.vectorCCz[active], 'bo', ms=6)
ax0.semilogx(np.exp(moptTD), mesh.vectorCCz[active], 'r*', ms=10)
ax0.set_ylim(-700, 0)
ax0.set_xlim(5e-3, 1e-1)
ax0.set_xlabel('Conductivity (S/m)', fontsize=fs)
ax0.set_ylabel('Depth (m)', fontsize=fs)
ax0.grid(which='both',
color='k',
alpha=0.5,
linestyle='-',
linewidth=0.2)
ax0.legend(['True', 'FDEM', 'TDEM'], fontsize=fs, loc=4)
# plot the data misfits - negative b/c we choose positive to be in the
# direction of primary
ax1.plot(freqs, -surveyFD.dobs[::2], 'k-', lw=2)
ax1.plot(freqs, -surveyFD.dobs[1::2], 'k--', lw=2)
dpredFD = surveyFD.dpred(moptTD)
ax1.loglog(freqs, -dpredFD[::2], 'bo', ms=6)
ax1.loglog(freqs, -dpredFD[1::2], 'b+', markeredgewidth=2., ms=10)
ax2.loglog(times, surveyTD.dobs, 'k-', lw=2)
ax2.loglog(times, surveyTD.dpred(moptTD), 'r*', ms=10)
ax2.set_xlim(times.min(), times.max())
# Labels, gridlines, etc
ax2.grid(which='both', alpha=0.5, linestyle='-', linewidth=0.2)
ax1.grid(which='both', alpha=0.5, linestyle='-', linewidth=0.2)
ax1.set_xlabel('Frequency (Hz)', fontsize=fs)
ax1.set_ylabel('Vertical magnetic field (-T)', fontsize=fs)
ax2.set_xlabel('Time (s)', fontsize=fs)
ax2.set_ylabel('Vertical magnetic field (-T)', fontsize=fs)
ax2.legend(("Obs", "Pred"), fontsize=fs)
ax1.legend(("Obs (real)", "Obs (imag)", "Pred (real)", "Pred (imag)"),
fontsize=fs)
ax1.set_xlim(freqs.max(), freqs.min())
ax0.set_title("(a) Recovered Models", fontsize=fs)
ax1.set_title("(b) FDEM observed vs. predicted", fontsize=fs)
ax2.set_title("(c) TDEM observed vs. predicted", fontsize=fs)
plt.tight_layout(pad=1.5)
def getKc(freq, sigma, a, b, mu=mu_0, eps=epsilon_0):
a = float(a)
b = float(b)
# return 1./(2*np.pi) * np.sqrt(b / a) * np.exp(-1j*k(freq,sigma,mu,eps)*(b-a))
return np.sqrt(b / a) * np.exp(-1j * k(freq, sigma, mu, eps) * (b - a))
def deriv(self, m):
return Utils.sdiag(np.exp(m))
def _transform(self, m):
"""
"""
return np.exp(m)
def run(plotIt=True):
"""
1D FDEM and TDEM inversions
===========================
This example is used in the paper Heagy et al 2016 (in prep)
"""
# Set up cylindrically symmeric mesh
cs, ncx, ncz, npad = 10., 15, 25, 13 # padded cyl mesh
hx = [(cs, ncx), (cs, npad, 1.3)]
hz = [(cs, npad, -1.3), (cs, ncz), (cs, npad, 1.3)]
mesh = Mesh.CylMesh([hx, 1, hz], '00C')
# Conductivity model
layerz = np.r_[-200., -100.]
layer = (mesh.vectorCCz >= layerz[0]) & (mesh.vectorCCz <= layerz[1])
active = mesh.vectorCCz < 0.
sig_half = 1e-2 # Half-space conductivity
sig_air = 1e-8 # Air conductivity
sig_layer = 5e-2 # Layer conductivity
sigma = np.ones(mesh.nCz)*sig_air
sigma[active] = sig_half
sigma[layer] = sig_layer
# Mapping
actMap = Maps.InjectActiveCells(mesh, active, np.log(1e-8), nC=mesh.nCz)
mapping = Maps.ExpMap(mesh) * Maps.SurjectVertical1D(mesh) * actMap
mtrue = np.log(sigma[active])
# ----- FDEM problem & survey -----
rxlocs = Utils.ndgrid([np.r_[50.], np.r_[0], np.r_[0.]])
bzi = FDEM.Rx.Point_bSecondary(rxlocs, 'z', 'real')
bzr = FDEM.Rx.Point_bSecondary(rxlocs, 'z', 'imag')
freqs = np.logspace(2, 3, 5)
srcLoc = np.array([0., 0., 0.])
print('min skin depth = ', 500./np.sqrt(freqs.max() * sig_half),
'max skin depth = ', 500./np.sqrt(freqs.min() * sig_half))
print('max x ', mesh.vectorCCx.max(), 'min z ', mesh.vectorCCz.min(),
'max z ', mesh.vectorCCz.max())
srcList = []
[srcList.append(FDEM.Src.MagDipole([bzr, bzi], freq, srcLoc,
orientation='Z')) for freq in freqs]
surveyFD = FDEM.Survey(srcList)
prbFD = FDEM.Problem3D_b(mesh, mapping=mapping)
prbFD.pair(surveyFD)
std = 0.03
surveyFD.makeSyntheticData(mtrue, std)
surveyFD.eps = np.linalg.norm(surveyFD.dtrue)*1e-5
# FDEM inversion
np.random.seed(1)
dmisfit = DataMisfit.l2_DataMisfit(surveyFD)
regMesh = Mesh.TensorMesh([mesh.hz[mapping.maps[-1].indActive]])
reg = Regularization.Simple(regMesh)
opt = Optimization.InexactGaussNewton(maxIterCG=10)
invProb = InvProblem.BaseInvProblem(dmisfit, reg, opt)
# Inversion Directives
beta = Directives.BetaSchedule(coolingFactor=4, coolingRate=3)
betaest = Directives.BetaEstimate_ByEig(beta0_ratio=2.)
target = Directives.TargetMisfit()
inv = Inversion.BaseInversion(invProb, directiveList=[beta, betaest, target])
m0 = np.log(np.ones(mtrue.size)*sig_half)
reg.alpha_s = 5e-1
reg.alpha_x = 1.
prbFD.counter = opt.counter = Utils.Counter()
opt.remember('xc')
moptFD = inv.run(m0)
# TDEM problem
times = np.logspace(-4, np.log10(2e-3), 10)
print('min diffusion distance ', 1.28*np.sqrt(times.min()/(sig_half*mu_0)),
'max diffusion distance ', 1.28*np.sqrt(times.max()/(sig_half*mu_0)))
rx = TDEM.Rx(rxlocs, times, 'bz')
src = TDEM.Src.MagDipole([rx], waveform=TDEM.Src.StepOffWaveform(),
loc=srcLoc) # same src location as FDEM problem
surveyTD = TDEM.Survey([src])
prbTD = TDEM.Problem3D_b(mesh, mapping=mapping)
prbTD.timeSteps = [(5e-5, 10), (1e-4, 10), (5e-4, 10)]
prbTD.pair(surveyTD)
prbTD.Solver = SolverLU
std = 0.03
surveyTD.makeSyntheticData(mtrue, std)
surveyTD.std = std
surveyTD.eps = np.linalg.norm(surveyTD.dtrue)*1e-5
# TDEM inversion
dmisfit = DataMisfit.l2_DataMisfit(surveyTD)
regMesh = Mesh.TensorMesh([mesh.hz[mapping.maps[-1].indActive]])
reg = Regularization.Simple(regMesh)
opt = Optimization.InexactGaussNewton(maxIterCG=10)
invProb = InvProblem.BaseInvProblem(dmisfit, reg, opt)
# Inversion Directives
beta = Directives.BetaSchedule(coolingFactor=4, coolingRate=3)
betaest = Directives.BetaEstimate_ByEig(beta0_ratio=2.)
target = Directives.TargetMisfit()
inv = Inversion.BaseInversion(invProb, directiveList=[beta, betaest, target])
m0 = np.log(np.ones(mtrue.size)*sig_half)
reg.alpha_s = 5e-1
reg.alpha_x = 1.
prbTD.counter = opt.counter = Utils.Counter()
opt.remember('xc')
moptTD = inv.run(m0)
if plotIt:
import matplotlib
fig = plt.figure(figsize = (10, 8))
ax0 = plt.subplot2grid((2, 2), (0, 0), rowspan=2)
ax1 = plt.subplot2grid((2, 2), (0, 1))
ax2 = plt.subplot2grid((2, 2), (1, 1))
fs = 13 # fontsize
matplotlib.rcParams['font.size'] = fs
# Plot the model
ax0.semilogx(sigma[active], mesh.vectorCCz[active], 'k-', lw=2)
ax0.semilogx(np.exp(moptFD), mesh.vectorCCz[active], 'bo', ms=6)
ax0.semilogx(np.exp(moptTD), mesh.vectorCCz[active], 'r*', ms=10)
ax0.set_ylim(-700, 0)
ax0.set_xlim(5e-3, 1e-1)
ax0.set_xlabel('Conductivity (S/m)', fontsize=fs)
ax0.set_ylabel('Depth (m)', fontsize=fs)
ax0.grid(which='both', color='k', alpha=0.5, linestyle='-',
linewidth=0.2)
ax0.legend(['True', 'FDEM', 'TDEM'], fontsize=fs, loc=4)
# plot the data misfits - negative b/c we choose positive to be in the
# direction of primary
ax1.plot(freqs, -surveyFD.dobs[::2], 'k-', lw=2)
ax1.plot(freqs, -surveyFD.dobs[1::2], 'k--', lw=2)
dpredFD = surveyFD.dpred(moptTD)
ax1.loglog(freqs, -dpredFD[::2], 'bo', ms=6)
ax1.loglog(freqs, -dpredFD[1::2], 'b+', markeredgewidth=2., ms=10)
ax2.loglog(times, surveyTD.dobs, 'k-', lw=2)
ax2.loglog(times, surveyTD.dpred(moptTD), 'r*', ms=10)
ax2.set_xlim(times.min(), times.max())
# Labels, gridlines, etc
ax2.grid(which='both', alpha=0.5, linestyle='-', linewidth=0.2)
ax1.grid(which='both', alpha=0.5, linestyle='-', linewidth=0.2)
ax1.set_xlabel('Frequency (Hz)', fontsize=fs)
ax1.set_ylabel('Vertical magnetic field (-T)', fontsize=fs)
ax2.set_xlabel('Time (s)', fontsize=fs)
ax2.set_ylabel('Vertical magnetic field (-T)', fontsize=fs)
ax2.legend(("Obs", "Pred"), fontsize=fs)
ax1.legend(("Obs (real)", "Obs (imag)", "Pred (real)", "Pred (imag)"),
fontsize=fs)
ax1.set_xlim(freqs.max(), freqs.min())
ax0.set_title("(a) Recovered Models", fontsize=fs)
ax1.set_title("(b) FDEM observed vs. predicted", fontsize=fs)
ax2.set_title("(c) TDEM observed vs. predicted", fontsize=fs)
plt.tight_layout(pad=1.5)
plt.show()
def getKc(freq,sigma,a,b,mu=mu_0,eps=epsilon_0):
a = float(a)
b = float(b)
# return 1./(2*np.pi) * np.sqrt(b / a) * np.exp(-1j*k(freq,sigma,mu,eps)*(b-a))
return np.sqrt(b / a) * np.exp(-1j*k(freq,sigma,mu,eps)*(b-a))