This file is indexed.

/usr/lib/python3-escript/esys/downunder/magtel2d.py is in python3-escript 5.0-3.

This file is owned by root:root, with mode 0o644.

The actual contents of the file can be viewed below.

   1
   2
   3
   4
   5
   6
   7
   8
   9
  10
  11
  12
  13
  14
  15
  16
  17
  18
  19
  20
  21
  22
  23
  24
  25
  26
  27
  28
  29
  30
  31
  32
  33
  34
  35
  36
  37
  38
  39
  40
  41
  42
  43
  44
  45
  46
  47
  48
  49
  50
  51
  52
  53
  54
  55
  56
  57
  58
  59
  60
  61
  62
  63
  64
  65
  66
  67
  68
  69
  70
  71
  72
  73
  74
  75
  76
  77
  78
  79
  80
  81
  82
  83
  84
  85
  86
  87
  88
  89
  90
  91
  92
  93
  94
  95
  96
  97
  98
  99
 100
 101
 102
 103
 104
 105
 106
 107
 108
 109
 110
 111
 112
 113
 114
 115
 116
 117
 118
 119
 120
 121
 122
 123
 124
 125
 126
 127
 128
 129
 130
 131
 132
 133
 134
 135
 136
 137
 138
 139
 140
 141
 142
 143
 144
 145
 146
 147
 148
 149
 150
 151
 152
 153
 154
 155
 156
 157
 158
 159
 160
 161
 162
 163
 164
 165
 166
 167
 168
 169
 170
 171
 172
 173
 174
 175
 176
 177
 178
 179
 180
 181
 182
 183
 184
 185
 186
 187
 188
 189
 190
 191
 192
 193
 194
 195
 196
 197
 198
 199
 200
 201
 202
 203
 204
 205
 206
 207
 208
 209
 210
 211
 212
 213
 214
 215
 216
 217
 218
 219
 220
 221
 222
 223
 224
 225
 226
 227
 228
 229
 230
 231
 232
 233
 234
 235
 236
 237
 238
 239
 240
 241
 242
 243
 244
 245
 246
 247
 248
 249
 250
 251
 252
 253
 254
 255
 256
 257
 258
 259
 260
 261
 262
 263
 264
 265
 266
 267
 268
 269
 270
 271
 272
 273
 274
 275
 276
 277
 278
 279
 280
 281
 282
 283
 284
 285
 286
 287
 288
 289
 290
 291
 292
 293
 294
 295
 296
 297
 298
 299
 300
 301
 302
 303
 304
 305
 306
 307
 308
 309
 310
 311
 312
 313
 314
 315
 316
 317
 318
 319
 320
 321
 322
 323
 324
 325
 326
 327
 328
 329
 330
 331
 332
 333
 334
 335
 336
 337
 338
 339
 340
 341
 342
 343
 344
 345
 346
 347
 348
 349
 350
 351
 352
 353
 354
 355
 356
 357
 358
 359
 360
 361
 362
 363
 364
 365
 366
 367
 368
 369
 370
 371
 372
 373
 374
 375
 376
 377
 378
 379
 380
 381
 382
 383
 384
 385
 386
 387
 388
 389
 390
 391
 392
 393
 394
 395
 396
 397
 398
 399
 400
 401
 402
 403
 404
 405
 406
 407
 408
 409
 410
 411
 412
 413
 414
 415
 416
 417
 418
 419
 420
 421
 422
 423
 424
 425
 426
 427
 428
 429
 430
 431
 432
 433
 434
 435
 436
 437
 438
 439
 440
 441
 442
 443
 444
 445
 446
 447
 448
 449
 450
 451
 452
 453
 454
 455
 456
 457
 458
 459
 460
 461
 462
 463
 464
 465
 466
 467
 468
 469
 470
 471
 472
 473
 474
 475
 476
 477
 478
 479
 480
 481
 482
 483
 484
 485
 486
 487
 488
 489
 490
 491
 492
 493
 494
 495
 496
 497
 498
 499
 500
 501
 502
 503
 504
 505
 506
 507
 508
 509
 510
 511
 512
 513
 514
 515
 516
 517
 518
 519
 520
 521
 522
 523
 524
 525
 526
 527
 528
 529
 530
 531
 532
 533
 534
 535
 536
 537
 538
 539
 540
 541
 542
 543
 544
 545
 546
 547
 548
 549
 550
 551
 552
 553
 554
 555
 556
 557
 558
 559
 560
 561
 562
 563
 564
 565
 566
 567
 568
 569
 570
 571
 572
 573
 574
 575
 576
 577
 578
 579
 580
 581
 582
 583
 584
 585
 586
 587
 588
 589
 590
 591
 592
 593
 594
 595
 596
 597
 598
 599
 600
 601
 602
 603
 604
 605
 606
 607
 608
 609
 610
 611
 612
 613
 614
 615
 616
 617
 618
 619
 620
 621
 622
 623
 624
 625
 626
 627
 628
 629
 630
 631
 632
 633
 634
 635
 636
 637
 638
 639
 640
 641
 642
 643
 644
 645
 646
 647
 648
 649
 650
 651
 652
 653
 654
 655
 656
 657
 658
 659
 660
 661
 662
 663
 664
 665
 666
 667
 668
 669
 670
 671
 672
 673
 674
 675
 676
 677
 678
 679
 680
 681
 682
 683
 684
 685
 686
 687
 688
 689
 690
 691
 692
 693
 694
 695
 696
 697
 698
 699
 700
 701
 702
 703
 704
 705
 706
 707
 708
 709
 710
 711
 712
 713
 714
 715
 716
 717
 718
 719
 720
 721
 722
 723
 724
 725
 726
 727
 728
 729
 730
 731
 732
 733
 734
 735
 736
 737
 738
 739
 740
 741
 742
 743
 744
 745
 746
 747
 748
 749
 750
 751
 752
 753
 754
 755
 756
 757
 758
 759
 760
 761
 762
 763
 764
 765
 766
 767
 768
 769
 770
 771
 772
 773
 774
 775
 776
 777
 778
 779
 780
 781
 782
 783
 784
 785
 786
 787
 788
 789
 790
 791
 792
 793
 794
 795
 796
 797
 798
 799
 800
 801
 802
 803
 804
 805
 806
 807
 808
 809
 810
 811
 812
 813
 814
 815
 816
 817
 818
 819
 820
 821
 822
 823
 824
 825
 826
 827
 828
 829
 830
 831
 832
 833
 834
 835
 836
 837
 838
 839
 840
 841
 842
 843
 844
 845
 846
 847
 848
 849
 850
 851
 852
 853
 854
 855
 856
 857
 858
 859
 860
 861
 862
 863
 864
 865
 866
 867
 868
 869
 870
 871
 872
 873
 874
 875
 876
 877
 878
 879
 880
 881
 882
 883
 884
 885
 886
 887
 888
 889
 890
 891
 892
 893
 894
 895
 896
 897
 898
 899
 900
 901
 902
 903
 904
 905
 906
 907
 908
 909
 910
 911
 912
 913
 914
 915
 916
 917
 918
 919
 920
 921
 922
 923
 924
 925
 926
 927
 928
 929
 930
 931
 932
 933
 934
 935
 936
 937
 938
 939
 940
 941
 942
 943
 944
 945
 946
 947
 948
 949
 950
 951
 952
 953
 954
 955
 956
 957
 958
 959
 960
 961
 962
 963
 964
 965
 966
 967
 968
 969
 970
 971
 972
 973
 974
 975
 976
 977
 978
 979
 980
 981
 982
 983
 984
 985
 986
 987
 988
 989
 990
 991
 992
 993
 994
 995
 996
 997
 998
 999
1000
1001
1002
1003
1004
1005
# -*- coding: utf-8 -*-
##############################################################################
#
# Copyright (c) 2015 by University of Queensland
# http://www.uq.edu.au
#
# Primary Business: Queensland, Australia
# Licensed under the Apache License, version 2.0
# http://www.apache.org/licenses/LICENSE-2.0
#
# Development until 2012 by Earth Systems Science Computational Center (ESSCC)
# Development 2012-2013 by School of Earth Sciences
# Development from 2014 by Centre for Geoscience Computing (GeoComp)
#
##############################################################################

from __future__ import print_function, division

__copyright__="""Copyright (c) 2015 by University of Queensland
http://www.uq.edu.au
Primary Business: Queensland, Australia"""
__license__="""Licensed under the Apache License, version 2.0
http://www.apache.org/licenses/LICENSE-2.0"""
__url__="https://launchpad.net/escript-finley"

"""
2D Magnetotelluric modelling for TE and TM mode.

:var __author__: name of author
:var __copyright__: copyrights
:var __license__: licence agreement
:var __url__: url entry point on documentation
:var __version__: version
:var __date__: date of the version
"""

__author__="Ralf Schaa, r.schaa@uq.edu.au"

import os, sys
import numpy
import math
import cmath
import types
from . import magtel1d         as mt1d
import esys.weipa              as weipa
import esys.escript            as escript
try:
    import esys.finley         as finley
    HAVE_FINLEY = True
except ImportError:
    HAVE_FINLEY = False

import esys.escript.pdetools   as pdetools
import esys.escript.linearPDEs as pde

class MT_2D(object):

   # class options:
  _debug   = False    #
  _solver = "DEFAULT" #

   # 'private' field:
  __version = 0.1     #

  """
  DESCRIPTION:
  ------------
  solves the scalar 2-D electromagnetic diffusion equation,
  (where 'u' is the electric field E or magnetic field H).

  [1]  -div( k*grad(u) ) + q*u = 0  (+ Boundary Conditions)

  In 2D the equation is solved for the transverse electric
  field (TE mode) or transverse magnetic field (TM mode).
  These fields are parallel to the 2D strike direction.
  Based on the actual mode, the coefficients are given by:

  TE: k = 1/mu   , q = i*w*sigma
  TM: k = 1/sigma, q = i*w*mu

  'mu'    is the vacuum permeability,
  'i'     is the imaginary unit
  'w'     is the angular frequency
  'sigma' is the conductivity

  The EM diffusion equation is complex and is solved as
  a coupled PDE for the real and imaginary parts. The
  coupled PDE is given by the following equations, with
  Er, Ei and Hr, Hi are the real and imaginary components
  of the electric and magnetic field, respectively:

  TE:
  [2] div( grad(Er) ) + w*mu*sigma*Ei = 0
  [3] div( grad(Ei) ) - w*mu*sigma*Er = 0

     the complementary magnetic fields
     are calculated via Faraday's Law:

  [4] Hr =-d/dz(Ei) / (w*mu)
  [5] Hi = d/dz(Er) / (w*mu)


  TM:
  [6] div( rho*grad(Hr) ) + w*mu*Hi = 0
  [7] div( rho*grad(Hi) ) - w*mu*Hr = 0

     (resistivity 'rho' is 1/sigma)
     the complementary electric fields
     are calculated via Ampere's Law:

  [8] Er = d/dz(Hr) * rho
  [9] Ei = d/dz(Hi) * rho


  Based on the ratio of electric to magnetic field
  apparent resistivity and phase is calculated, viz:

  rho_a = (1/w*mu) * [ (Er)^2 + (Ei)^2 ] / [ (Hr)^2 + (Hi)^2 ]
  phase = arctan( [Ei*Hr - Er*Hi] / [Er*Hr + Ei*Hi] )


  Boundary conditions:
  --------------------
  the source term on the right-hand-side of equation [1] is zero,
  i.e. no artificial source is employed but instead the 'source'
  is provided via the boundary conditions of the PDE which are
  given as Dirichlet conditions at all boundaries. To calculate
  the Dirichlet values, a 1D response is calculated at the left
  and right boundary (based on the 1D recursion formula for MT).
  Interpolation from the left to the right sides then provides
  the values at the top and bottom boundary. See module 'mt1d'
  for details of the computation of the 1D response. Once the
  values on the boundaries have been calculated, the values
  inside the domain are solved in this class.
  """

  def __init__(self, domain, mode, freq_def, tags, rho, rho_1d, ifc_1d,
        xstep=100, zstep=100, maps=None, plot=False, limits=None):
    """
    DESCRIPTION:
    -----------
    Constructor which initialises the 2D magnetotelluric class:
    (*) check for argument type
    (*) check for valid argument values
    (*) initialises required values

    ARGUMENTS:
    ----------
    param  domain       :: the 2d mesh domain
    type   domain       :: ``escript data object``

    param  mode         :: TE or TM mode
    type   mode         :: ``string``

    param  freq_def     :: highest/lowest frequency & points per decade
    type   freq_def     :: ``dictionary``

    param  tags         :: the tag names of the regions defined in the mesh
    type   tags         :: ``list``

    param  rho          :: the resistivity values of the regions in the mesh
    type   rho          :: ``list``

    param  rho_1d       :: the resistivity values at the left & right boundary
    type   rho_1d       :: ``dictionary``

    param  ifc_1d       :: the layer interface depths of the left & right boundary
    type   ifc_1d       :: ``dictionary``

    param  xstep        :: user-defined step size for horizontal sample list
    type   xstep        :: ``number``  (optional)

    param  zstep        :: user-defined step size for vertical sample list
    type   zstep        :: ``number``  (optional)

    param  maps         :: list with user-defined  functions which map the resistivity for each region
    type   maps         :: ``list``    (optional)

    param  plot         :: user-defined flag to show a plot of apparent resistivity and phase at each frequency
    type   plot         :: ``boolean`` (optional)



    DATA ATTRIBUTES:
    ---------------
    self.domain         :: escript data object of mesh
    self.X              :: escript data object with all mesh coordinates
    self.mode           :: string with TE or TM mode
    self.xmin           :: float with x-coordinate minimum
    self.xmax           :: float with x-coordinate maximum
    self.zmin           :: float with z-coordinate minimum
    self.zmax           :: float with z-coordinate maximum
    self.zstep          :: number with sample step in vertical direction
    self.xstep          :: number with sample step in horizontal direction
    self.rho            :: list with resistivity values of all regions
    self.rho_1d         :: dictionary with resistivity values at boundaries left/right
    self.ifc_1d         :: dictionary with interface depths at boundaries left/right
    self.plot           :: boolean flag to show plots of apparent resistivity and phase
    self.sigma          :: escript data object with the conductivity model (based on 'rho' and 'maps')
    self.frequencies    :: list of sounding frequencies
    self.boundary_mask  :: Dirichlet mask at boundaries
    """

    if not HAVE_FINLEY:
        raise ImportError("Finley module not available")
    #make python3 compatible, since long disappeared in python 3
    if sys.version_info[0] == 3:
        long_type = int
    else:
        long_type = long
    # ---
    # Checks
    # ---

    # Types:
    if not isinstance(domain, escript.Domain):
      raise ValueError("Input parameter DOMAIN must be an Escript mesh")
    if not isinstance(mode, str):
      raise ValueError("Input parameter MODE must be a string")
    if not isinstance(freq_def, dict) or len(freq_def) != 3:
      raise ValueError("Input parameter FREQ_DEF must be a dictionary with length 3")
    if not isinstance(tags, list) or not all(isinstance(t,str) for t in tags):
      raise ValueError("Input parameter TAGS must be a list of strings")
    if not isinstance(rho, list) or not all(isinstance(d,(int,long_type,float)) for d in rho):
      raise ValueError("Input parameter RHO must be a list of numbers")
    if not isinstance(rho_1d, dict) or len(rho_1d) != 2:
      raise ValueError("Input parameter RHO_1D must be a dictionary with length 2")
    if not isinstance(ifc_1d, dict) or len(ifc_1d) != 2:
      raise ValueError("Input parameter IFC_1D must be a dictionary with length 2")
    if not isinstance(xstep, (int,long_type,float)):
        raise ValueError("Optional input parameter XSTEP must be a number")
    if not isinstance(zstep, (int,long_type,float)):
        raise ValueError("Optional input parameter ZSTEP must be a number")
    if maps is not None:
      if not isinstance(maps, list) or not all(isinstance(m,(types.FunctionType, types.NoneType)) for m in maps):
        raise ValueError("Optional input parameter MAPS must be a list of Functions or Nones")
    if plot is not None:
      if not isinstance(plot, bool):
        raise ValueError("Optional input parameter PLOT must be True or False")

    # Values:
    if mode.upper() != "TE" and mode.upper() != "TM": # Check mode:
      raise ValueError("Input parameter mode must be either 'TE' or 'TM'")
    if not 'high' in freq_def and not 'low' in freq_def and not 'step' in freq_def:
       raise ValueError("Input dictionary FREQ_DEF must have keys 'high', 'low' and 'step' defined" )
    if freq_def['high'] < freq_def['low']:
      raise ValueError("High frequency value is smaller than low frequency value in input parameter FREQ_DEF")
    if freq_def['step'] < 1:
      raise ValueError("Step frequency value is smaller than 1 in input parameter FREQ_DEF")
    if not all(r>0 for r in rho): # Check resistivity values:
      raise ValueError("Input parameter RHO must be all positive")
    if len(rho) != len(tags): # Check resistivity list-length:
      raise ValueError("Input parameter RHO must have the same length as input parameter TAGS")
    if not 'left' in rho_1d and not 'right' in rho_1d:
       raise ValueError("Input dictionary RHO_1D must have keys 'left' and 'right' defined" )
    if not 'left' in ifc_1d and not 'right' in ifc_1d:
      raise ValueError("Input dictionary IFC_1D must have keys 'left' and 'right' defined" )
    if len(ifc_1d['left'])-1 != len(rho_1d['left']) and len(ifc_1d['right'])-1 != len(rho_1d['right']):
      raise ValueError("Lists with values in input dictionary RHO_1D must have length equal to IFC_1D" )
    if xstep < 0.5: # Step size should be non-zero but should not be smaller than 0.5m:
      raise ValueError("Input parameter XSTEP must be at least 0.5" )
    if zstep < 0.5: # Step size should be non-zero but should not be smaller than 0.5m:
      raise ValueError("Input parameter ZSTEP must be at least 0.5" )



    # ---
    # Domain coordinates & tags:
    # ---

    # Extract the model coordinates..
    X = escript.Solution(domain).getX()

    # Get the Min/Max coordinates:
    xmin = escript.inf(X[0])
    xmax = escript.sup(X[0])
    zmin = escript.inf(X[1])
    zmax = escript.sup(X[1])

    # Get the tag names from the mesh file
    mesh_tags = escript.getTagNames(domain)

    if xmin >= xmax or zmin >= zmax: raise ValueError("The mesh limits are not valid (min >= max)" )
    if zmin >= 0                   : raise ValueError("The mesh must be defined with a negative vertical axis" )
    if not set(mesh_tags) == set(tags)       :
        print("user-tags:", tags)
        print("mesh-tags:", mesh_tags)
        raise ValueError("Input parameter TAGS does not match the tags defined in the mesh")



    # ---
    # Define the boundary mask:
    # ---

    boundary_mask = self.__setBoundaryMask(X)


    # ---
    # Calculate list of sounding frequencies:
    # ---

    frequencies = self.__getSoundingFrequencies(freq_def)



    # ---
    # Tag the domain with conductivity maps:
    # ---

    sigma_model = self.__tagDomain(domain, X, tags, rho, maps)

    # Check for valid values
    if  escript.inf(sigma_model) < 0 or escript.sup(sigma_model) < 0:
       raise ValueError("Negative conductivity encountered" )
    if cmath.isnan( escript.inf(sigma_model) ) or \
       cmath.isnan( escript.sup(sigma_model) ) or \
       cmath.isinf( escript.sup(sigma_model) ):
       raise ValueError("The conductivity model contains NaNs or INFs" )



    # ---
    # Projector and Locator objects.
    # ---

    print("Setting up Escript Locator and Projector objects...")

    # Setup a list with sample points along the vertical mesh extent, bottom to top:
    xsample = self.__getSamplePoints(escript.inf(X[0]),escript.sup(X[0]),xstep, constant=0.0)

    # Get the locations of mesh points at the surface via the Locator object
    # operating on the continuous function-space (i.e. nodes) of the domain.
    loc  = pdetools.Locator(escript.ContinuousFunction(domain),xsample )

    # Instantiate the Projector class with smoothing on (fast=False);
    # the Projector is used to calculate the gradient correctly.
    proj = pdetools.Projector(domain, reduce=False, fast=False)




    # ---
    # Print information:
    # ---

    print("")
    print("="*72)
    print("Escript MT2D, version", self.__version)
    print("="*72)
    print("Mesh XMin/XMax       : ", xmin, xmax)
    print("Mesh ZMin/ZMax       : ", zmin, zmax)
    print("Number of Tags       : ", len( tags ))
    print("Mapping              : ", {True: 'Yes', False: 'No'}[maps is not None])
    print("Conductivity Model   : ", sigma_model)
    print("Nr of Frequencies    : ", len( frequencies ))
    print("Start/End/Step (Hz)  : ", freq_def["high"], freq_def["low"], freq_def["step"])
    print("Mode                 : ", mode.upper())
    print("Solver               : ", MT_2D._solver)
    print("Show plots           : ", plot)
    print("="*72)
    print("")

    if self._debug:
      print("Mesh-Info     : ")
      print(domain.print_mesh_info(full=False))



    # ---
    # Set all required variables as data attributes
    # ---

    self.domain         = domain
    self.X              = X
    self.mode           = mode
    self.xmin           = xmin
    self.xmax           = xmax
    self.zmin           = zmin
    self.zmax           = zmax
    self.zstep          = zstep
    self.xstep          = xstep
    self.rho            = rho
    self.rho_1d         = rho_1d
    self.ifc_1d         = ifc_1d
    self.plot           = plot
    self.limits         = limits
    self.sigma          = sigma_model
    self.frequencies    = frequencies
    self.boundary_mask  = boundary_mask
    self.proj           = proj
    self.loc            = loc


#_______________________________________________________________________________


  def __interpolLinear(self,dx,x0,x1,y0,y1):
    """
    DESCRIPTION:
    -----------
    Function for simple 1D interpolation using the line-equation.

    ARGUMENTS:
    ----------
    dx :: interpolation step.
    x0 :: first coordinate point of known value y0.
    x1 :: last coordinate point of known value y1.
    y0 :: known value at first coordinate.
    y1 :: known value at last coordinate.

    RETURNS:
    --------
    y  :: list with interpolated values
    """
    # Initialise return lists.
    y = []

    # Test for long enough interval.
    if abs(x1-x0) <= dx: return y
    # Test for correct abscissae.
    if x0 >= x1: return y

    x = x0
    while x <= x1:
        y.append( y0 + (y1-y0)*(x-x0)/(x1-x0)  )
        x = x + dx

    return y

#_______________________________________________________________________________


  def __getSamplePoints(self, min,max,step,constant=None):
    """
    DESCRIPTION:
    -----------
    Function to setup a list with sample points. If a
    constant value was passed a 2D list is returned
    where the second column is set to the constant.

    ARGUMENTS:
    ----------
    min        :: minimum value.
    max        :: maximum value.
    step       :: step value.
    constant   :: optional constant value for 2nd column.

    RETURNS:
    --------
    sample     :: list with samples.
    """

    # Initialise return list.
    sample = []

    # Cycle with step-size and fill sample list.
    dp = min
    while dp <= max:
        if constant is not None:
            sample.append([dp,constant])
        else:
            sample.append(dp)
        # Increment the step.
        dp = dp + step

    # Return the list:
    return sample
    #___________________________________________________________________________


  def __getSoundingFrequencies(self, frequencies):
    """
    DESCRIPTION:
    -----------
    Defines the sounding frequencies in Hz.

    ARGUMENTS:
    ----------
    frequencies :: dictionary with frequency start/stop/step

    RETURNS:
    --------
    sounding_frequencies  :: list with frequency values
    """
    # Output list with frequencies in Hertz:
    sounding_frequencies = []

    # Period definition (from freq to time):
    tme_1 = 1.0/frequencies["high"]
    tme_n = 1.0/frequencies["low"]

    # Number of points per decade:
    tme_p = frequencies["step"]

    # Number of periods in range:
    nt = int(math.log10(tme_n/tme_1) * tme_p) + 1
    # Fill list with times:
    for n in range(nt):
      # Sounding period in seconds:
      period = tme_1*10**( (n)/float(tme_p))
      # And store as frequency in Hertz:
      sounding_frequencies.append( 1.0/period )

    return sounding_frequencies

#_______________________________________________________________________________


  def __getGradField(self, proj, mt2d_field, wm):
    """
    DESCRIPTION:
    -----------
    Calculates the complementary fields via Faraday's Law (TE-mode)
    or via Ampere's Law (TM-mode). Partial derivative w.r.t. the
    vertical coordinate are taken at the surface for which an Escript
    'Projector' object is used to calculate the gradient.

    ARGUMENTS:
    ----------
    proj       :: escript Projection object
    mt2d_field :: calculated magnetotelluric field
    wm         :: number with actual angular sounding frequency * mu

    RETURNS:
    --------
    mt2d_grad  :: dictionary with computed gradient fields
    """

    # Define the derived gradient fields:
    if self.mode.upper() == 'TE':
      # H = -(dE/dz) / iwm
      grad_real =-proj.getValue( escript.grad(mt2d_field["imag"])/wm )
      grad_imag = proj.getValue( escript.grad(mt2d_field["real"])/wm )
       #<Note the coupled dependency on real/imaginary part>:
    else:
      # E = (dH/dz) / sigma
      grad_real = proj.getValue( escript.grad(mt2d_field["real"])/self.sigma )
      grad_imag = proj.getValue( escript.grad(mt2d_field["imag"])/self.sigma )
      #<'sigma' is an Escript data-object and as such the division
      # will use the tagged sigma values of the associated domains>


    # And return as dictionary for real and imaginary parts:
    mt2d_grad = {"real": grad_real[1], "imag":grad_imag[1] }
    #<Note>: the derivative w.r.t. 'z' is used (i.e. '[1]').

    return mt2d_grad

#_______________________________________________________________________________


  def __tagDomain(self, domain, X, tags, rho, maps):
    """
    DESCRIPTION:
    -----------
    Defines the conductivity model. Conductivities of tagged regions can be mapped
    via user-defined functions passed in 'maps'. If no user-defined functions are
    passed, a constant value is applied as provided in list 'rho' for each region.
    User-defined functions have 3 arguments: x-coordinate, z-coordinate, resistivity.

    ARGUMENTS:
    ----------
    domain  :: escript object of mesh
    X       :: escript object with all coordinates
    tags    :: list with domain tags
    rho     :: list with domain resistivities
    maps    :: list with user-defined resistivity mappings

    RETURNS:
    --------
    sigma   :: escript object of conductivity model

    """
    # Setup the conductivity structure (acts on elements and can be discontinuous).
    sigma = escript.Scalar(0, escript.Function(domain))

    # Setup conductivity domains.
    for i in range( len(tags) ):

      # Default: assign conductivity which is the inverse of resistivity:
      m = 1.0/rho[i]

      # Map a user-defined conductivity distribution if given:
      if maps is not None:
            # Guard against undefined elements:
        if maps[i] is not None:
          # Map the conductivity according to the defined functions:
          m = maps[i]( X[0], X[1], rho[i] )

      # Tag the mesh with the conductivity distributions at each iteration:
      sigma += m * escript.insertTaggedValues(escript.Scalar(0,escript.Function(domain)),**{ tags[i] : 1})


    if self._debug == True:
      sigma.expand()
      mydir = os.getcwd()
      dbgfl = mydir + os.sep + "mt2d_sigma_dbg.silo"
      print("")
      print("DEBUG: writing SILO debug output of conductivity model:")
      print(dbgfl)
      print("")
      weipa.saveSilo(dbgfl, sigma = sigma)


    # All done:
    return sigma

#_______________________________________________________________________________


  def __setBoundaryMask(self, X):
    """
    DESCRIPTION:
    -----------
    Define Dirichlet model boundaries conditions.

    ARGUMENTS:
    ----------
    X :: escript object with all coordinates

    RETURNS:
    --------
    boundary_mask :: escript object with mask values at boundaries

    """
    # Boundaries are defined as masks (1 or 0) for all mesh coordinates;
    # values at the boundary are '1', whereas all other values are '0'.
    mask_l = escript.whereZero( X[0] - escript.inf(X[0]) )
    mask_r = escript.whereZero( X[0] - escript.sup(X[0]) )
    mask_t = escript.whereZero( X[1] - escript.inf(X[1]) )
    mask_b = escript.whereZero( X[1] - escript.sup(X[1]) )

    # Combine the mask for all boundaries:
    boundary_mask = mask_t + mask_b + mask_l + mask_r

    return boundary_mask
    #<Note>: this boundary mask is used later on as PDE coefficient 'q'.

#_______________________________________________________________________________


  def __getBoundaryValues(self, mode, X, rho_1d, ifc_1d, xstep, zstep, frequency):
    """
    DESCRIPTION:
    -----------
    Returns a list with boundary values along each Dirichlet boundary.
    Values at the left and right side of the domain are evaluated at
    sample points and interpolated across the domain. The subroutine
    expects that layers at the right- and left-hand-side are defined.

    ARGUMENTS:
    ----------
    mode      :: string with TE or TM mode
    X         :: escript object with all coordinates
    rho_1d    :: dictionary with resistivities at the left/right boundary
    ifc_1d    :: dictionary with layer interfaces at the left/right boundary
    xstep     :: number with step size for horizontal sample list
    zstep     :: number with step size for vertical sample list
    frequency :: number with actual sounding frequency

    RETURNS:
    --------
    bondary_value :: dictionary with lists of boundary values at sample points
    """

    # ---
    # Sample lists at vertical and horizontal boundaries.
    # ---

    # Horizontal extents:
    xmin = escript.inf(X[0])
    xmax = escript.sup(X[0])

    # Vertical extents:
    zmin = escript.inf(X[1])
    zmax = escript.sup(X[1])

    # Setup a list with sample points along the vertical mesh extent, bottom to top:
    zsamples = self.__getSamplePoints(-zmax,-zmin,zstep)


    # ---
    # Calculate the 1D response at the left- and right-hand-side boundaries
    # ---

    # Instantiate an 'mt1d' object for the left- and right-hand-sides:
    mt1d_left = mt1d.MT_1D( frequency, ifc_1d['left'] , rho_1d['left'] , zsamples )
    mt1d_rght = mt1d.MT_1D( frequency, ifc_1d['right'], rho_1d['right'], zsamples )

    # Compute the 1D field values at the sample nodes:
    te1d_left, tm1d_left  = mt1d_left.mt1d(  )
    te1d_rght, tm1d_rght  = mt1d_rght.mt1d(  )

    # Distinguish TE and TM mode and save 1D values in dictionary:
    if mode.upper() == "TE":
      mt_1d = {"left":te1d_left, "right":te1d_rght}
    else:
      mt_1d = {"left":tm1d_left, "right":tm1d_rght}


    # ---
    # Interpolation across mesh.
    # ---

    # Now setup a 2D-table from left to right at each sampled depth for mesh-interpolation.
    table2d_real = []
    table2d_imag = []

     # 1D-interpolation of values from left to right at different depths 'i':
    for i in range( len(zsamples)):
      table2d_real.append( self.__interpolLinear(xstep, xmin, xmax, mt_1d["left"].real[i], mt_1d["right"].real[i]) )
      table2d_imag.append( self.__interpolLinear(xstep, xmin, xmax, mt_1d["left"].imag[i], mt_1d["right"].imag[i]) )

    # 2D-interpolation to map the values on the mesh coordinates:
    bondary_value_real = escript.interpolateTable( table2d_real, X, (xmin,zmin), (xstep,zstep) )
    bondary_value_imag = escript.interpolateTable( table2d_imag, X, (xmin,zmin), (xstep,zstep) )

    # Return the real and imaginary values as a dictionary:
    boundary_value = {"real":bondary_value_real, "imag":bondary_value_imag}


    return boundary_value

#_______________________________________________________________________________


  def __getAppResPhase(self, mt2d_field, mt2d_grad, wm):
    """
    DESCRIPTION:
    -----------
    Calculates the apparent resistivity and phase.

    ARGUMENTS:
    ----------
    mt2d_field :: dictionary with real/imaginary field values
    mt2d_grad  :: dictionary with real/imaginary gradient values

    RETURNS:
    --------
    apparent resistivity and phase
    """

    # Define the associated modelled fields in readable variables:
    if self.mode.upper() == 'TE':
      # Transverse electric field:
      Er = mt2d_field["real"]
      Ei = mt2d_field["imag"]
      Hr = mt2d_grad["real"]
      Hi = mt2d_grad["imag"]
    else:
      # Transverse magnetic field :
      Hr = mt2d_field["real"]
      Hi = mt2d_field["imag"]
      Er = mt2d_grad["real"]
      Ei = mt2d_grad["imag"]


    # Return apparent Resistivity and Phase:
    arho_2d = ( (Er**2 + Ei**2)/(Hr**2 + Hi**2) ) / wm
    aphi_2d = escript.atan( (Ei*Hr - Er*Hi)/(Er*Hr + Ei*Hi) ) * 180.0/cmath.pi

    return arho_2d, aphi_2d
#_______________________________________________________________________________


  def __showPlot(self, loc, rho_2d, phi_2d, f, **kwargs):
    """
    DESCRIPTION:
    -----------
    Plot of apparent resistivity and phase. Requires matplotlib to be available.

    ARGUMENTS:
    ----------
    loc     :: escript Locator object
    rho_2d  :: list with computed apparent resistivities
    phi_2d  :: list with computed phase values
    f       :: sounding frequency

    RETURNS:
    --------
    Plot in window.

    """
    try:
        import matplotlib.pyplot as plt
    except ImportError:
        print("Warning: matplotlib not available, plot will not be shown")
        return

    # Abscissas/Ordinates:
    x  = numpy.array( loc.getX() )[:,0]
    y0 = numpy.array( loc.getValue(rho_2d) )
    y1 = numpy.array( loc.getValue(phi_2d) )

    # Plot labels:
    title = 'Escript MT-2D ' + '(' + self.mode.upper() + ')' + ' freq: ' + str(f) + ' Hz'
    ylbl0 = r'Apparent Resistivity $(\Omega\cdot\,m)$'
    ylbl1 = r'Phase $(^{\circ})$'
    xlbl1 = 'Easting (m)'


    # Setup the plot window with app. res. on top and phase on bottom:
    f, ax = plt.subplots(2, figsize=(8,8), sharex=True) # Mind shared axis
    f.subplots_adjust(top=0.9)  # Little extra space for 'suptitle'
    f.suptitle(title)           # This is actually the plot-title

    # Top: apparent resistivity on semi-log plot
    ax[0].plot(x,y0, color='red') # semilogy
    ax[0].grid(b=True, which='both', color='grey',linestyle=':')
    ax[0].set_title( ylbl0 )
    # Plot limits in **kwargs:
    if 'limits' in kwargs:
        ax[0].set_xlim(kwargs["limits"])

    # Bottom: phase on linear plot
    ax[1].plot(x,y1, color='blue')
    ax[1].grid(b=True, which='both', color='grey',linestyle=':')
    ax[1].set_xlabel( xlbl1 )
    ax[1].set_title( ylbl1 )
    # Plot limits in **kwargs:
    if 'limits' in kwargs:
        ax[1].set_xlim(kwargs["limits"])

    plt.show()


#_______________________________________________________________________________


  def __setSolver(self, mode, domain, sigma, boundary_mask, boundary_value, f):
    """
    DESCRIPTION:
    -----------
    Setups the coupled PDE for real and complex part.

    ARGUMENTS:
    ----------
    mode           :: string with TE or TM mode
    domain         :: escript object with mesh domain
    sigma          :: escript object with conductivity model
    boundary_mask  :: escript object with boundary mask
    boundary_value :: dictionary with real/imag boundary values
    f              :: sounding frequency

    RETURNS:
    --------
    mt2d_fields    :: dictionary with solved PDE, magnetotelluric fields real/imag
    """

    # Constants:
    pi  = cmath.pi     # Ratio circle circumference to diameter.
    mu0 = 4*pi*1e-7    # Free space permeability in V.s/(A.m).
    wm  = 2*pi*f*mu0   # Angular frequency times mu0.


    # ---
    # Setup the coupled PDE for real/imaginary parts:
    # ---

    # Initialise the PDE object for two coupled equations (real/imaginary).
    mtpde = pde.LinearPDE(domain, numEquations=2)

    # If set, solve the 2D case using the direct solver:
    if MT_2D._solver.upper() == "DIRECT":
       mtpde.getSolverOptions().setSolverMethod(pde.SolverOptions().DIRECT)
    else:
       mtpde.getSolverOptions().setSolverMethod(pde.SolverOptions().DEFAULT)

    # Now initialise the PDE coefficients 'A' and 'D',
    # as well as the Dirichlet variables 'q' and 'r':
    A = mtpde.createCoefficient("A")
    D = mtpde.createCoefficient("D")
    q = mtpde.createCoefficient("q")
    r = mtpde.createCoefficient("r")

    # Set the appropriate values for the coefficients depending on the mode:
    if mode.upper() == "TE":
        a_val = 1.0
        d_val = wm*sigma
    elif mode.upper() == "TM":
        a_val = 1.0/sigma
        d_val = wm


    # ---
    # Define the PDE parameters, mind boundary conditions.
    # ---


    # Now define the rank-4 coefficient A:
    for i in range(domain.getDim()):
        A[0,i,0,i] = a_val
        A[1,i,1,i] = a_val

    # And define the elements of 'D' which are decomposed into real/imaginary values:
    D[0,0] = 0     ; D[1,0] = d_val
    D[0,1] =-d_val ; D[1,1] = 0


    # Set Dirichlet boundaries and values:
    q[0] = boundary_mask ; r[0] = boundary_value['real']
    q[1] = boundary_mask ; r[1] = boundary_value['imag']

    # ---
    # Solve the PDE
    # ---

    mtpde.setValue(A=A, D=D, q=q, r=r  )
    pde_solution = mtpde.getSolution()

    # And return the real and imaginary parts individually:
    mt2d_fields = {"real":pde_solution[0], "imag":pde_solution[1] }
    #<Note>: the electric field is returned for TE-mode.
    #        the magnetic field is returned for TM-mode.

    return mt2d_fields

#_______________________________________________________________________________


  def pdeSolve(self):
    """
    DESCRIPTION:
    -----------
    Solves the PDE for either the TE or the TM mode.
    (TE mode is the transverse Electric field).
    (TM mode is the transverse Magnetic field).

    ARGUMENTS:
    ----------
    (uses `self`)

    RETURNS:
    --------
    mt2d  :: list with real/imag fields for each sounding frequency
    arho  :: list with apparent resistivities for each sounding frequency
    aphi  :: list with phase values for each sounding frequency

    """


    # ---
    # Constants.
    # ---

    # Pi & vacuum permeability:
    pi = cmath.pi
    mu = 4*pi*1e-7

    # Number of frequencies:
    nfreq = len(self.frequencies)


    # ---
    # Solve the PDE for all frequencies.
    # ---

    # Prepare lists to store the values at each frequency:
    arho = []
    aphi = []
    mt2d = []

    # Cycle over all frequencies:
    print("Solving for frequency: ...")
    for n in range( nfreq ):

      f = self.frequencies[n] # actual frequency (Hz)
      wm  = (2*pi*f)*mu       # angular frequency (rad/s)
      T = 1.0 / f             # sounding period (s)

      print(n+1,":", f, "(Hz)")

      # Calculate 1D Dirichlet boundary values:
      boundary_value = self.__getBoundaryValues(self.mode.upper(), self.X,
            self.rho_1d, self.ifc_1d, self.xstep, self.zstep, f)

      # Solve the 2D-MT PDE:
      fld_2d = self.__setSolver(self.mode.upper(),self.domain, self.sigma,
            self.boundary_mask, boundary_value, f)

      # Calculate the field gradients:
      grd_2d = self.__getGradField(self.proj, fld_2d, wm)

      # Calculate the apparent resistivity and phase:
      rho_2d, phi_2d = self.__getAppResPhase(fld_2d, grd_2d, wm)

      # Save in lists for each frequency:
      mt2d.append( fld_2d )
      arho.append( self.loc.getValue(rho_2d) )
      aphi.append( self.loc.getValue(phi_2d) )

      # Optionally plot the apparent resistivity and phase:
      if self.plot:
          self.__showPlot(self.loc, rho_2d, phi_2d, f, limits=self.limits)


    # ---
    # All done
    # ---

    print("field calculations finished.")
    return mt2d, arho, aphi

#_______________________________________________________________________________