Simulation of seeing in large telescopes (D>>r0), tested against SNfactory ad-hoc PSF models.

GD71_07_253_090_B.txt

# lbda: 4327A
# radius[spx]   flux
0.716608613055 12.3552
0.745272338424 11.5852
0.745343175736 11.7769
0.834997936881 10.4475
1.55358944456 4.54829
1.57210666273 4.74318
1.61169010512 4.2645
1.68493686848 3.5319
1.74032685753 3.35422
1.75684694171 3.12297
1.79226673209 3.36554
1.85843829222 3.02449
2.22949482886 1.52675
2.22956586929 1.63293
2.26810158109 1.49356
2.3864965395 1.25747
2.50436608068 0.936273
2.53564800666 1.00553
2.57673311389 1.15053
2.64212091831 0.831548
2.84900419776 0.594853
2.8765223231 0.583048
2.91271272972 0.675153
2.95450069299 0.464054
2.97073230766 0.661472
2.98549666889 0.826283
3.03348947381 0.520949
3.15229100498 0.431523
3.15583097812 0.429738
3.18473553815 0.40955
3.22985083738 0.390899
3.34164711507 0.400866
3.48282339057 0.264776
3.51958836633 0.289079
3.56111809486 0.369776
3.62251206252 0.301754
3.71499467699 0.209787
3.71506573374 0.280393
3.80624506539 0.181903
3.81961937739 0.245014
3.85427253245 0.315292
3.90638349286 0.21185
3.93801478407 0.216523
3.97741102894 0.182418
4.00963045684 0.190295
4.02225203426 0.201376
4.04604214996 0.215724
4.10019441344 0.223117
4.19075895105 0.159204
4.23430276792 0.167069
4.28187742792 0.166068
4.38779345193 0.172258
4.41132978275 0.119673
4.43217301666 0.160119
4.55231486508 0.163044
4.55489274787 0.138129
4.61133216536 0.156659
4.61574725029 0.130903
4.63553478918 0.115133
4.67432513177 0.132926
4.75308449483 0.135578
4.77468775407 0.145954
4.86763479788 0.12417
4.94166816714 0.119244
5.11250019988 0.100622
5.14730787421 0.101647
5.18381181554 0.121259
5.20067742058 0.0851084
5.20074848183 0.110162
5.23572576045 0.120342
5.24276662586 0.113552
5.27072464047 0.0976773
5.3226859422 0.101061
5.37114048618 0.102004
5.37139631847 0.0972523
5.47300132818 0.102482
5.4895360163 0.0946428
5.494603756 0.0905185
5.54667280234 0.0987886
5.60412125453 0.104601
5.60565947748 0.0864304
5.63826802659 0.0823288
5.72187640056 0.087733
5.76272665222 0.0884338
5.87321571525 0.087444
5.88741514768 0.090112
5.89834272182 0.0884296
6.08820976309 0.0842796
6.10320793027 0.0701937
6.10975008792 0.0780519
6.2109695558 0.071304
6.25063231368 0.0781598
6.28707456055 0.0712531
6.29016204293 0.076421
6.32362546528 0.0949923
6.36647668914 0.0752858
6.37287235961 0.0686733
6.37408350084 0.0894448
6.40590213299 0.0689079
6.43020509584 0.0848073
6.47952226162 0.071904
6.54275077635 0.0773923
6.57796639746 0.0802376
6.59908825003 0.0705312
6.64504325816 0.0650343
6.67331319186 0.0732772
6.6844667654 0.0713967
6.68631708223 0.0718472
6.68649220901 0.069661
6.81477152573 0.0720635
6.8485932839 0.0717355
6.92172979656 0.07091
7.0012341749 0.0654785
7.01223861405 0.0660574
7.04105826117 0.0682567
7.04980125208 0.0667779
7.07574269414 0.0687417
7.27140842143 0.0642235
7.27310203124 0.0585062
7.37238482467 0.0664186
7.38586791292 0.0680497
7.39040165176 0.0707092
7.42796882965 0.0668018
7.48741291865 0.0565187
7.50171787322 0.0633702
7.53986712527 0.0627339
7.56589687247 0.0628192
7.59537730648 0.0637952
7.59811853646 0.0637862
7.65665216215 0.0593574
7.71392687641 0.0592076
7.75354246372 0.0620853
7.81951423141 0.0661176
7.83954189482 0.0555423
7.89228038364 0.0632143
7.93813168346 0.0649871
7.94932363649 0.0656511
8.05903117593 0.0696435
8.12372685071 0.0631257
8.17226365677 0.064036
8.19845933265 0.0591071
8.29727509571 0.0531699
8.36770178377 0.0604193
8.50830765398 0.0608632
8.52398615886 0.055592
8.53115203558 0.0583338
8.53765804024 0.0594891
8.56954033569 0.0563404
8.59252856405 0.0573073
8.59258097012 0.0594579
8.60953474605 0.0597631
8.63519245452 0.0559289
8.71203502457 0.0545608
8.72315814804 0.0583596
8.79695263499 0.0555135
8.80156929375 0.0556808
8.85036671934 0.0563155
8.88011142202 0.057412
8.92400234686 0.0583789
8.98609785093 0.0564457
9.04624865759 0.0603954
9.10006349 0.0585736
9.14090205762 0.0555651
9.26072865572 0.0593005
9.29909399773 0.0577292
9.33805436566 0.0581372
9.34914012608 0.0560803
9.49498230705 0.0562901
9.53591177379 0.0545264
9.58499289816 0.0566096
9.59027308371 0.0529429
9.60940640301 0.0549165
9.61814821684 0.0503754
9.65805003171 0.063911
9.67253636646 0.0561432
9.71155025843 0.0575919
9.73661702538 0.0582254
9.76118321868 0.0579715
9.77895436687 0.0560509
9.83355597488 0.0562717
9.85243406899 0.0592493
9.9017908702 0.0534869
9.97487957323 0.0558348
10.0555383063 0.0562221
10.0909219472 0.0512105
10.0944264535 0.0565847
10.163130117 0.0565575
10.1736796356 0.0571313
10.1852922705 0.0542905
10.3305975653 0.0535239
10.3641298066 0.0529833
10.4203145061 0.0548901
10.5134658945 0.0561336
10.5252947021 0.0495351
10.528141146 0.055008
10.534496756 0.054671
10.5884431583 0.0561611
10.6042916555 0.0506579
10.6265506055 0.0567964
10.764270591 0.0507111
10.7954280765 0.0549711
10.8514769538 0.0568192
10.9442549519 0.0561116
11.0528078648 0.0602084
11.0561657198 0.0491225
11.0580784016 0.0556615
11.1016403932 0.0562249
11.1438453632 0.0571678
11.1784741553 0.0519887
11.4464172469 0.0586112
11.4552593037 0.0509309
11.5137508099 0.0533362
11.6788378502 0.0546892
11.8122319215 0.051182
11.8155777402 0.0591309
11.9508165928 0.0568809
12.011004085 0.0511922
12.0208727733 0.052377
12.5248695186 0.055433
12.5313987639 0.0574275
12.6115499831 0.0611926
12.6296464902 0.0517462
13.2746333263 0.0621648
13.2948005553 0.0534115

hankel.py

#!/usr/bin/env python

"""
Based on Brian Borchers' MATLAB implementation
(http://infohost.nmt.edu/~borchers/hankel.html) of Walt Anderson's
Fortran code for numerical approximation of Hankel transforms of order
0 and 1 described in:

Anderson, W. L., 1979, Computer Program Numerical Integration of
Related Hankel Transforms of Orders 0 and 1 by Adaptive Digital
Filtering. Geophysics, 44(7):1287-1305.

Author: Yannick Copin <y.copin@ipnl.in2p3.fr>
"""

import numpy as N


def hankel_weights(order=0):
    """~30x faster than hankel_points_ascii"""

    if order == 0:
        return N.fromstring('\x8fC\xa7\xfbr\xaa\xfa9{\xd9\x8cj+y\x0c\xbd\xfb\xad\x9eC\xf8\xfb)=\xa3Ng\x98\xb6\x82\x1f\xbd\xc2\x9a\x84Y\xff\xc4.=\x92"\xfc\x1f\x8f\xde\x1d\xbd)p\xe2\x83i\xf2/=\xdbF\xa8W\xedI\x18\xbd+2\xef\xb8Ij0=\x18\xb0\x10x\xbdi\x11\xbd/\x8a\xee\x14\xbf\x0c1=\xed|\x19%`%\x03\xbd\x16\xdc\xd6W\r\xfb1=\x9a\xd4A\xe0\x1cd\xc4\xbc\x90\xc9#a\xddF3=\xe3\xabJg\xf0M\x03=\x1d\xc7u\x04&\x035=\xa5\x17\x1c\xc2o\xa0\x15=\x9d\x02\xe4\xfc\xb0F7=\x0c\x1c\x12\x94\x9e\xd8!=4\x14\xad\xf1[-:=\xe3\x93\xd5\xf4\x03,*=\xd9\xb2\x91\x1dJ\xd9==e\x0f_\xb7\xc4\x0b2=\x03/\xc6\xd8$:A=\x9c\xe2\x0eY\x96\xfc7=\x01\xef\xda\xbb\xc2\x18D=`~\xca\xf9\xc2!?=A\xc7\x8a\x01\xcb\xa7G=\r\xb9\xc3\xa7\x96\xe0C=\xa8I\xdd\x02\x84\rL=+\xfa\xe0\xc9\x0c\x18I=\xe9WEt\\\xbcP=\x15WzPKkO=]zq\xedM\x11T=\xc0\xc3\x12\x13\xea\x8cS=T\x10\x86~\x89(X=\xe6P0q\x978X=\xdd\x17\x10\xec\xc3,]=\x1a\x04\x8a\x0e\x02\xe8]=\x8e:\xae\x85\x15\xa9a=M\xdd\x104zjb=\xa0\x0f\x8e5?le=\x92\xc8\xc30\x00\xa3f=N\xd0\x8fh\xcf\x06j=\xaf\xde\xa1\xe6\xa9\xc8k=M\xbc!nW\xa8o=\x03)\xecV\x0b\x08q=\xf4]5;yEs=\x9cd\xf3\\\xc5\xdct=\xb1\x1a\xc4d\xcdzw=\t\xe7\x9c\x95\xc1\x89y=\xa7\xfe\xf7\x9c\x81\x9f|= \xe5\xb7!\xee>\x7f=UJ\'\x86Dt\x81=\xd4,\xcc\x85j\x1b\x83=\x9c{A[HK\x85=,\xc5\x96\xd0z\\\x87= \x7fd\x07L\xfc\x89=\xd3=\xcaxQ\x8e\x8c=_\xba.X\x8e\xb7\x8f=\x04#\x17:!s\x91=\n\x8b\xd0s\xfe[\x93=l\x1e\xdd\x01\xb8R\x95=\xb7\xf8\xdc\xca\xc8\xa2\x97=\\\xb1W\xee\xb3\r\x9a=\x8b\xc2\x0b\x08\x1b\xdc\x9c=I\xf6p<\xaa\xd4\x9f=WUn\xc5\xd2\x9e\xa1=\x8a\xd2\xd8\xd6xq\xa3=\x9cf\xcd\xed}\x84\xa5=\xddUz\x16\x82\xc0\xa7=\xaf\'\x9a6\x1bG\xaa=\'\xce\x85\xb6\xb3\x03\xad=\x0b\xe8\xc7b\xca\x0b\xb0=\xdb\xf7\xea\x98\x8e\xb8\xb1=\x11e\x8f\x9e\xd4\x98\xb3=\xb6\xc92\x08c\xa5\xb5=\x1ex\xe6\xc8(\xef\xb7=}\xe7\xd8%\xaap\xba=Hz\x94cW;\xbd=\x92\xa7n9\xd2%\xc0=\x12\xee\xd0j\xe7\xd9\xc1=\x9d[\xed\xfc<\xb9\xc3=@\xa6ET\x85\xcd\xc5=\x07\x95\x95.R\x17\xc8=\x89fRK\x1f\xa1\xca=\x1a\x00\xed\xdd\xf0l\xcd=\x9f,!5&C\xd0=\xc7\x81S\xaes\xf8\xd1=\xd8\xeaDD\xcf\xdc\xd3=\x13D1*\x13\xf3\xd5=^\x95iD\x88B\xd8=_9\xc6C8\xcf\xda=\x19/^7\x7f\xa1\xdd=\x9d-\xe0\xb5h_\xe0=Vmaoq\x18\xe2=\xc8\x0b\x9e\x91m\xff\xe3=\xc5\x88{\xd8\x0c\x1a\xe6=\x8ag\xa9H\xe9l\xe8=\xdfcQ*\xbb\xfe\xea=\x1a\x17riY\xd5\xed=\x1a\xc9b[a|\xf0=[N\xe3\xee&8\xf2=Io&\xab\xc5"\xf4=~2\xe8\xe2\xd1@\xf6=\xe1H=\xa6\n\x98\xf8=\x98cQ[\x1f.\xfb=t\x91~\xed\xfd\t\xfe=\xfbp\xfe\x15W\x99\x00>|\xea\x03\x9cHX\x02>\x15,\xb4#)F\x04>\xdcw\x8f\x8d\x0ch\x06>\xf6\xed\xea\xfcG\xc3\x08>\xd2\xf5\x11\x8a\x05^\x0b>\xe2 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    else:
        raise NotImplementedError("%d-order Hankel transform "
                                  "not implemented" % order)


def hankel_points():
    """~30x faster than hankel_points_ascii"""

    return N.fromstring('`\rC\x94r\x199=\xb5\x8dn\xc17\xbd;=1h34\x0f\xa8>=\x02\x02X7\xb9\xf0@=\x9cb\xeb\x18\xd2\xb8B=\x92-\xdb\xd3\xe2\xb0D=\xa1H\xcd\xe3\xf6\xddF=\xba\x02\xfa\x97\xa1EI=\xc9\\!\\\x0c\xeeK=\x04^\x1b\x82\x06\xdeN=\x82=;Z\x8b\x0eQ=\x8b\x11l\x1f\xc7\xd9R=\\\x8e\xc3.O\xd5T=[\xc2\\\xe57\x06W=u)\xffb\x1erY=dy9\xec6\x1f\\=\x89\x83\xbc\xcf\\\x14_=\x054\xd8\xfb\x91,a=\x96o\x03*\xf6\xfab=\x12\x9d\xc4\xa7\xfb\xf9d=O\x06N\xc3\xbf.g=\xd6M=~\xe9\x9ei=\xe5\x02\x08\t\xb8Pl=\xee\xfcQ\xc4\x12Ko=\xfa\xb5\x97x\xcdJq=qO\xd2\x9e_\x1cs=7\xd9\xbc\xaf\xe8\x1eu=dp^\xfa\x8eWw=\xb1A\x90s\x03\xccy=ow\xe8J\x90\x82|=\xcfP=\x08)\x82\x7f=\xfd?\x85->i\x81=\xc0K\xad\xe4\x03>\x83=\xa2rQ\xb8\x16D\x85=G\xf2&\x08\xa6\x80\x87=\xb3\x84\xc6\xcdl\xf9\x89=\xd0\x88BK\xc0\xb4\x8c=xn\x08E\xa0\xb9\x8f=<\x19Px\xe4\x87\x91=\x96\x03\x1ec\xe3_\x93=@\xa7\xef3\x86i\x95=\xf7\x95\x1dk\x05\xaa\x97=\x8c\xef\xa2\x18&\'\x9a=\xa1\xf4\x8b\xa4H\xe7\x9c=\xa4\xb7g%y\xf1\x9f=\xc4sL\xb7\xc0\xa6\xa1=\x1aYd\x82\xfe\x81\xa3=A#\xce\x957\x8f\xa5=\xfa\x02\x97\xa2\xad\xd3\xa7=\x17b\xde\xe0/U\xaa=\x9f_J\xf2)\x1a\xad=\xed\x06\x9e*\xda\x14\xb0=\xd1\x8etI\xd3\xc5\xb1=^\xb2w\xabU\xa4\xb3=\xadc\xeeQ+\xb5\xb5=A\x06\xc8.\x9f\xfd\xb7=yt)\xb4\x8a\x83\xba=b4\x15\xd1dM\xbd=\xbcq\xca@)1\xc0=&\xdbi\x8e\x1c\xe5\xc1=s<\x08H\xe9\xc6\xc3=n\x1b\x1e\xdda\xdb\xc5=\xb7\x1c\xc7\x90\xda\'\xc8=(+.!7\xb2\xca=9\x85\x97\xde\xf9\x80\xcd= 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def hankel_weights_ascii(order=0):

    if order == 0:
        return N.array([+0.21035620538389819885E-28,
                        -0.12644693616088940552E-13,
                        +0.46157312567885668321E-13,
                        -0.27987033742576678494E-13,
                        +0.54657649654108409156E-13,
                        -0.26529331099287291499E-13,
                        +0.56749134340673213135E-13,
                        -0.21572768289772080733E-13,
                        +0.58318460867739760925E-13,
                        -0.15465892848687829700E-13,
                        +0.60573024556529743179E-13,
                        -0.85025312590830646706E-14,
                        +0.63880180611476449908E-13,
                        -0.56596576350102877128E-15,
                        +0.68485006047914070374E-13,
                        +0.85728977321682762439E-14,
                        +0.74650681546818133979E-13,
                        +0.19208372932613381433E-13,
                        +0.82693454289757706437E-13,
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                        -0.70696084721646144402E-27,
                        +0.72149205056137611593E-27], dtype='d')
    else:
        raise NotImplementedError("%d-order Hankel transform "
                                  "not implemented" % order)


def hankel_points_ascii():

    return N.array([8.9170998013274418e-14,
                    9.8549193740052245e-14,
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                    1.0052615772688342e+18,
                    1.1109858602563712e+18,
                    1.2278292631485970e+18,
                    1.3569611939940810e+18,
                    1.4996740485594657e+18,
                    1.6573961450606881e+18,
                    1.8317060192517601e+18,
                    2.0243482229411579e+18,
                    2.2372507840526853e+18,
                    2.4725445029769687e+18,
                    2.7325842783379528e+18,
                    3.0199726756098365e+18,
                    3.3375859744670935e+18,
                    3.6886029555582029e+18,
                    4.0765367148108068e+18,
                    4.5052698236765440e+18,
                    4.9790931872111176e+18,
                    5.5027489888943135e+18,
                    6.0814781519961702e+18,
                    6.7210727924986010e+18,
                    7.4279341885389363e+18,
                    8.2091368465530675e+18,
                    9.0724993053136814e+18,
                    1.0026662386494198e+19,
                    1.1081175674916358e+19,
                    1.2246593094004847e+19,
                    1.3534578533000223e+19,
                    1.4958022583082809e+19,
                    1.6531171550741899e+19,
                    1.8269770039599454e+19,
                    2.0191218527695090e+19,
                    2.2314747517318812e+19,
                    2.4661610000341512e+19,
                    2.7255294165301002e+19,
                    3.0121758475087553e+19,
                    3.3289691467965432e+19,
                    3.6790798882106413e+19,
                    4.0660120977274061e+19,
                    4.4936383229520880e+19,
                    4.9662383908768727e+19,
                    5.4885422418279211e+19,
                    6.0657772682979369e+19,
                    6.7037206324472250e+19,
                    7.4087570858843619e+19,
                    8.1879428704062800e+19,
                    9.0490763392378634e+19,
                    1.0000776005572131e+20,
                    1.1052566799547061e+20,
                    1.2214975396947848e+20,
                    1.3499635573716302e+20,
                    1.4919404640690720e+20,
                    1.6488492123894242e+20,
                    1.8222601978247286e+20,
                    2.0139089758026668e+20,
                    2.2257136317086207e+20,
                    2.4597939777289005e+20,
                    2.7184927686435983e+20,
                    3.0043991489038549e+20,
                    3.3203745656597683e+20,
                    3.6695814070852361e+20,
                    4.0555146526217175e+20,
                    4.4820368519071852e+20,
                    4.9534167824711496e+20,
                    5.4743721730949612e+20,
                    6.0501169204271369e+20,
                    6.6864132714134700e+20,
                    7.3896294938012195e+20,
                    8.1668036119031775e+20,
                    9.0257138455105503e+20,
                    9.9749564569309793e+20,
                    1.1024031785271021e+21,
                    1.2183439329023096e+21,
                    1.3464782828575407e+21,
                    1.4880886400345899e+21,
                    1.6445922884849697e+21,
                    1.8175555693250643e+21,
                    2.0087095572044880e+21,
                    2.2199673854830119e+21,
                    2.4534433935122555e+21,
                    2.7114742876545719e+21,
                    2.9966425278257163e+21,
                    3.3118021736216768e+21,
                    3.6601074487063940e+21,
                    4.0450443093423623e+21,
                    4.4704653330125727e+21,
                    4.9406282763108614e+21], dtype='d')


def hankel(f, k, order=0):
    """0th/1st-order Hankel-transform of f:

    F_n(k) = int_0^oo r f(r) J_n(kr) dr,     n=0,1

    Note that Anderson's implementation includes the r factor in input
    function, but this is *not* the case for this procedure.
    """

    w = hankel_weights(order=order)
    p = hankel_points()

    # Anderson's implementation requires function g to include the r factor
    def g(r): return r*f(r)

    # N.dot(w, g(p/k[:,N.newaxis]).T) / k is only valid for 1D arrays
    return N.dot(w, g(p/k[..., N.newaxis]).swapaxes(-1, -2)) / k


if __name__ == '__main__':

    import matplotlib.pyplot as P

    # Same test functions as in Borchers' MATLAB implementation, but does not
    # include the extra r factor.

    # Test functions (see http://en.wikipedia.org/wiki/Hankel_transform)
    def f1(r): return N.exp(-r**2)

    def f2(r): return r*N.exp(-r**2)

    def f3(r): return N.exp(-r)/r

    def f4(r): return N.exp(-2*r)

    def f5(r): return N.exp(-2*r)/r

    # Analytic transforms
    def F1(k): return N.exp(-k**2/4)/2     # 0th-order

    def F2(k): return k*N.exp(-k**2/4)/4   # 1st-order

    def F3(k): return (N.sqrt(1+k**2)-1)/(k*N.sqrt(1+k**2))  # 1st-order

    def F4(k): return k/N.sqrt((4+k**2)**3)  # 1st-order

    def F5(k): return 1/N.sqrt(4+k**2)       # 0th-order

    fs = [f1, f2, f3, f4, f5]               # Test functions
    Fs = [F1, F2, F3, F4, F5]               # Analytic transforms
    orders = [0, 1, 1, 1, 0]                # Orders

    k = N.concatenate(([0.0001, 0.001, 0.01, 0.02, 0.03, 0.04],
                       N.linspace(0.05, 5, 100))).astype('d')

    fig = P.figure()
    ax1 = fig.add_subplot(2, 1, 1,
                          title="0th/1st-order Hankel tranforms",
                          ylabel='Hankel transform')
    ax2 = fig.add_subplot(2, 1, 2,
                          xlabel='k',
                          ylabel='Relative error', yscale='log')

    for i, (f, F, order) in enumerate(zip(fs, Fs, orders)):
        # if order!=0: continue
        h = hankel(f, k, order=order)   # Numeric
        t = F(k)                        # Analytic
        l, = ax1.plot(k, h, label='H%d(f%d)' % (order, i+1))
        col = l.get_color()
        ax1.plot(k, t, c=col, ls='--')

        ax2.plot(k, N.abs(h-t)/N.abs(t), c=col, ls='-', label='_')

    ax1.legend(loc='best', prop=dict(size='small'))

    fig.tight_layout()

    P.show()

HR7950_07_253_041_R.txt

# lbda: 7603A
# radius[spx]   flux
0.418593273815 18.1837
0.5349266477 16.3112
0.919587991512 9.29802
0.971844008792 9.15756
1.22670232071 5.66755
1.27585746046 5.48179
1.28633172146 7.10789
1.40976163455 5.59343
1.52877680815 3.99235
1.62286001831 3.41529
1.72398957832 2.70682
1.82788979249 2.72601
1.87511701485 2.20257
1.89810827837 2.46238
2.17357618332 1.56991
2.19872163731 1.40708
2.22923260348 1.38511
2.24046150617 1.12481
2.29762145977 0.996929
2.29780795798 1.19263
2.3278810413 1.03583
2.43171208388 0.821884
2.4555725823 0.93069
2.50791364179 0.71838
2.57777134088 0.674734
2.58745711476 0.741269
2.84844288343 0.460966
2.86120351681 0.623048
2.8742215112 0.704532
2.9235907623 0.46231
3.05536249719 0.34627
3.06342499664 0.3748
3.11483177589 0.404891
3.15058219771 0.453676
3.16397626403 0.330685
3.17667387593 0.318302
3.1876411399 0.272309
3.18809584127 0.37018
3.21109696855 0.404542
3.26770385838 0.2804
3.28376732383 0.237104
3.39690804064 0.197484
3.40692260011 0.254243
3.47112194369 0.296218
3.57742101492 0.184147
3.57885785942 0.225812
3.63063936958 0.214682
3.661311191 0.168465
3.73780741908 0.169168
3.82630935387 0.162878
3.85446589165 0.252423
3.86224189251 0.136527
3.86257473124 0.272684
3.92129908485 0.171978
3.95412027557 0.128086
4.04398798761 0.129368
4.04780017091 0.185285
4.07092310874 0.185173
4.07840690215 0.112517
4.11294243377 0.111917
4.1525157644 0.101422
4.18252714849 0.162912
4.18385292786 0.104878
4.20152466164 0.178031
4.24539969208 0.0883317
4.24585344043 0.106553
4.35024279457 0.111205
4.36722000066 0.0891081
4.38913660053 0.092006
4.40482017513 0.131164
4.41619975722 0.109993
4.45148306955 0.115698
4.45571223124 0.0770115
4.4941655258 0.0776948
4.5049795378 0.0874144
4.62746520586 0.0598339
4.68466077254 0.0774905
4.76890209863 0.0870922
4.84519522369 0.0633349
4.85049781519 0.124598
4.85569951845 0.113245
4.8776410512 0.0595956
4.92003820434 0.0642767
4.92049891027 0.0606985
4.94604446013 0.0569694
4.96481049366 0.0716581
4.96951756618 0.0603814
4.97444042875 0.0567405
5.00671503634 0.095196
5.02181856802 0.0877579
5.0293142344 0.0584706
5.04764817346 0.0530278
5.10864052491 0.0463022
5.15342952743 0.0523819
5.17720483634 0.0427152
5.17910335895 0.0883288
5.19014788889 0.0477646
5.19562077363 0.0913007
5.26049131125 0.0565143
5.28127811199 0.0398706
5.29284407504 0.0404179
5.31012401111 0.0636266
5.31432998036 0.0629292
5.33384109296 0.0632562
5.36248020998 0.0714763
5.43141889242 0.034259
5.52359972433 0.0344777
5.57434660886 0.0406432
5.59029961952 0.0419796
5.62747672452 0.0338062
5.66641579305 0.0514407
5.69998204593 0.0343423
5.72363349568 0.0342854
5.73647291599 0.0393544
5.73742047701 0.0396858
5.76813791775 0.0476922
5.76917145059 0.0306585
5.8008678815 0.0340075
5.84501281891 0.0340159
5.84751790108 0.0321971
5.84788430121 0.0645667
5.85116672745 0.0579969
5.8608158118 0.0367026
5.90782653411 0.0246767
5.91058130148 0.0369088
5.97910567615 0.0552103
5.98024687102 0.0267481
5.98758312595 0.0314273
5.9887317942 0.0479029
6.03266233444 0.0287829
6.08692837443 0.0308964
6.11728928452 0.0245978
6.12542085599 0.0268558
6.17678636974 0.0550516
6.19161860618 0.0504805
6.19672118636 0.0247216
6.23634618436 0.0381883
6.25172137027 0.0363088
6.2520150473 0.0280512
6.2591966974 0.0238032
6.2632998974 0.023682
6.28964179494 0.0360326
6.29947017055 0.0297034
6.33324144729 0.0429346
6.36808291794 0.0200404
6.41373226008 0.0201326
6.48391923936 0.0206196
6.48895908482 0.0215115
6.50419230415 0.0190411
6.52356138304 0.0229855
6.53286166452 0.0199665
6.59348986905 0.0288672
6.60177864589 0.0196494
6.60489802601 0.0279993
6.62521598429 0.0294355
6.6278715614 0.0281915
6.67770424287 0.0174782
6.68458791426 0.0239929
6.7211343749 0.0259512
6.86514476453 0.015365
6.86618289053 0.0177741
6.8875793812 0.0197852
6.90547528991 0.0195681
6.91945866139 0.017058
6.95834844892 0.0172877
6.9590678424 0.0180712
6.97799393422 0.0210625
7.05844323299 0.0148512
7.06649381696 0.0149227
7.06736848445 0.0188029
7.09177754472 0.022784
7.17511450167 0.0321384
7.1887277529 0.031752
7.1907315832 0.0145086
7.21151333727 0.0120302
7.22452655067 0.01683
7.27166313007 0.0226572
7.3022213752 0.0140301
7.31188761244 0.0245256
7.32554291419 0.0143192
7.41441143823 0.0121539
7.47410572581 0.0132615
7.52474948616 0.0119247
7.539227372 0.0166344
7.56823682819 0.0176403
7.60664655512 0.0114279
7.63436217423 0.0187077
7.67781336256 0.0128527
7.69365896062 0.0111624
7.77417412219 0.0105209
7.80439266302 0.013102
7.81605793843 0.0109135
7.83709865142 0.0102302
7.86171440812 0.0100137
7.94409994108 0.00996457
8.16110686869 0.00901483
8.17385137906 0.021416
8.18654215444 0.0209207
8.20760631927 0.00945191
8.22823505858 0.0101361
8.25800069301 0.0142736
8.26822306553 0.00766784
8.29562858885 0.0145939
8.37576925958 0.00988022
8.3934165594 0.00899557
8.43609267865 0.00797652
8.49739901854 0.0104597
8.53430172484 0.00977896
8.62321880277 0.00827871
8.70236169544 0.00597591
8.74704676139 0.00694602
8.78546991541 0.00552142
8.85770209073 0.00976665
9.0221104261 0.0074013
9.0490272202 0.0057893
9.14912536131 0.0073833
9.15169538104 0.00507639
9.28700686923 0.00642731
9.46726166027 0.00552872
9.58712314609 0.00491504
9.72848613052 0.00694392
9.94804243126 0.00547246
10.0827916396 0.00495822
10.4827672035 0.00603948

seeing.py

#/usr/bin/env python
# -*- coding: utf-8 -*-

"""
Simulation of seeing in large telescopes (D>>r0), tested against SNfactory
ad-hoc PSF models.

References:
* Tokovinin 2002PASP..114.1156T
* Fried 1966JOSA...56.1372F
* Racine 1996PASP..108..699R

Author: Yannick Copin <y.copin@ipnl.in2p3.fr>
"""

from __future__ import division

import numpy as N

from scipy import special as S
from scipy import fftpack as F
from scipy import interpolate as I
from scipy.optimize import fsolve
from scipy.integrate import quad

import matplotlib.pyplot as P

import hankel as H
import fractions

RAD2ARCS = 206264.80624709636           # From radians to arcsecs
# Color names
BLUE = 'C0'
GREEN = 'C2'
RED = 'C3'

def otf_Kolmogorov(r, r0=0.15, expo=5/3):
    """
    Kolmogorov optical transfer function. r [m/rad], r0 is the Fried
    parameter [m].

    Note: lower exponent (e.g. 3/2=1.5 instead of 5/3=1.67) means more wings.
    1.55 corresponds roughly to long exposure ad-hoc SNfactory PSFs,
    short SNfactory PSFs are closer to 1.5.
    """

    # (Half) Phase structure function
    hdphi = (r/r0)**expo
    if expo == (5/3):
        # 2*( 24/5*gamma(6/5) )**(5/6) = 6.8838771822938112...
        # Including the 0.5 of otf: 3.4419385911469056...
        hdphi *= 3.4419385911469056
    else:
        # Possible generalisation, including the 0.5 of otf
        hdphi *= (8/expo*S.gamma(2/expo))**(expo/2)

    # otf = exp(-0.5*phase), the 0.5 factor has already been included
    return N.exp(-hdphi)


def otf_VonKarman(r, r0=0.15, L0=20., desaturate=True):
    """
    Von Karman optical transfer function.

    Note that Tokovinin's formula has a typo -- 2**(11/6) should be 2**(5/6)
    --, see Conan (2000).
    """

    #raise NotImplementedError("Von Karman phase does not work properly...")
    print "WARNING: Von Karman phase does not work properly..."

    # (Half) Phase structure function
    def hphase(r):
        x = 2*N.pi*r/L0
        # gamma(11/6)/(2**(5/6)*pi**(8/3)) * ( 24/5*gamma(6/5) )**(5/6) =
        # 0.08583068106228546...
        # gamma(5/6)/2**(1/6) = 1.0056349179985902...
        return 0.08583068106228546 * (r0/L0)**(-5/3) * \
            ( 1.0056349179985902 - x**(5/6) * S.kv(5/6, x) )

    hdphi = hphase(r)
    hdphi[r == 0] = 0.                    # Extension to 0 (NaN otherwise)
    otf = N.exp(-hdphi)                 # Optical transfert function

    if desaturate:                      # See Tokovinin (2002)
        otf -= N.exp(-hphase(N.max(r)*2))  # Remove constant level
        otf /= otf.max()                # Renormalization

    return otf


def psf2D_FFT(otfarr, normed=True):
    """
    FFT-computed 2D-PSF from input OTF 2D-array (e.g. from otf_Kolmogorov).
    """

    assert otfarr.ndim == 2, "Input OTF array is not 2D"

    psf = F.fftshift(N.absolute(F.fft2(otfarr)))  # PSF=FT(OTF), otfarr.shape
    if normed:
        psf /= psf.max()

    return psf


def friedParamater(lbda, r0ref=0.10, lref=5e-7, expo=5/3):
    """Implement r0 chromaticity."""

    return r0ref*(lbda/lref)**(2/expo)


def seeing_fwhm(lbda, r0ref=0.10, lref=5e-7, expo=5/3, verbose=False):
    """
    Seeing FWHM [arcsec] at wavelength lbda [m] for Fried parameter r0ref
    [m] at reference wavelength lref [m].

    Note: the 'traditional' seeing is dependant on exponent 5/3 used in
    otf_Kolmogorov through constant 0.976 (actually 0.97586...).
    """

    r0 = friedParamater(lbda, r0ref, lref)
    s = 0.975863432579*lbda/r0 * RAD2ARCS        # [arcsec] Valid for expo=5/3
    if expo != (5/3):
        s = 2*fsolve(lambda r:
                     psf_Kolmogorov_Hankel(r, lbda, r0=r0, expo=expo) - 0.5,
                     s/2)

    if verbose:
        print "r0 = %.0f cm @%.0fÅ = %.0f cm @%.0fÅ: " \
              "seeing=%.2f\" (FWHM) (expo=%.2f)" % \
              (r0ref*100, lref*1e10, r0*100, lbda*1e10, s, expo)

    return s


def psf_ES(r, alpha, coeffs, decompose=False):
    """
    Compute (max-normalized) extract_star PSF given free parameter
    alpha and set of correlation coefficients coeffs for radius array r
    [arcsec]. If decompose=True, return individual components of PSF
    (gaussian + moffat).
    """

    spx = 0.43                          # Spaxel size [arcsec]

    name, s1, s0, b1, b0, e1, e0 = coeffs
    sigma = s0 + s1*alpha
    beta = b0 + b1*alpha
    eta = e0 + e1*alpha
    print "PSF %s: sigma,beta,eta =" % name, sigma, beta, eta
    r2 = (r/spx)**2                     # (Radius [spx])**2
    gaussian = N.exp(-r2/2/sigma**2)
    moffat = (1 + r2/alpha**2)**(-beta)

    s = eta*gaussian + moffat
    if decompose:
        return eta*gaussian/s.max(), moffat/s.max()
    else:
        return s/s.max()


def psf_ES_long(r, alpha, decompose=False):

    return psf_ES(r, alpha,
                  ['long', 0.215, 0.545, 0.345, 1.685, 0.0, 1.04],
                  decompose=decompose)


def psf_ES_short(r, alpha, decompose=False):

    return psf_ES(r, alpha,
                  ['short', 0.2, 0.56, 0.415, 1.395, 0.16, 0.6],
                  decompose=decompose)


def psf_Kolmogorov_FFT(r, lbda, scale=None, r0=0.15, expo=5/3):
    """
    Compute (max-normalized) Kolmogorov (FFT-based) PSF given free
    parameters r0 and expo for radius array r [arcsec] at
    wavelength lbda [m].
    """

    assert scale is not None, "Pixel scale [arcsec/px] should be set"

    amax = N.max(r)                     # Image half width [arcsec]
    if amax < 20:
        # print "WARNING: extending PSF modeling from %.2f'' to 20''" % amax
        amax = 20

    # 2D-array construction
    xmax = int(amax / scale)            # Image half width [px]
    amax = xmax * scale                 # Image half width [arcsec]
    # print "Image half width: %d px = %.2f\"" % (xmax,amax)

    x = N.arange(-xmax, xmax+0.1, dtype='d')  # (n,) [px]
    rpx = N.hypot(*N.meshgrid(x, x))     # Radius 2D-array (n,n) [px]
    px2f = RAD2ARCS/scale/len(x)        # [px/rad]
    f = rpx * px2f                      # Spatial freq 2D-array (n,n) [1/rad]

    # FFT-based PSF computation
    psf = psf2D_FFT(otf_Kolmogorov(lbda*f, r0=r0, expo=expo))  # (n,n)

    a = rpx * scale                     # Radius (n,n) [arcsec]
    j = N.argsort(a.ravel())            # Such that a.ravel()[j] is sorted
    i = j[:N.searchsorted(a.ravel()[j], amax)]

    x = a.ravel()[i]
    y = N.log(psf.ravel()[i])

    # Spline interpolation
    n, = N.nonzero(N.diff(x))  # Discard duplicated radii (and last point)
    spl = I.UnivariateSpline(x[n], y[n], k=3, s=0)

    # Estimate PSF at given radii
    return N.exp(spl(r))


def psf_Kolmogorov_Hankel(r, lbda, r0=0.15, expo=5/3, normed=True):
    """
    Compute (max-normalized) Kolmogorov (Hankel-based) PSF given free
    parameters r0 and expo for radius array r [arcsec] at
    wavelength lbda [m].
    """

    # One should compute psf(r) = H.hankel(lambda
    # r:otf_Kolmogorov(lbda*r/(2*N.pi)), r/RAD2ARCS), but for some numerical
    # reasons, one computes lbda**2/(2*pi) * psf(r) = H.hankel(lambda
    # r:otf_Kolmogorov(r, r0=r0, expo=expo), 2*pi/lambda*r/RAD2ARCS)
    tmp = (6.2831853071795862/RAD2ARCS/lbda)*N.atleast_1d(r)
    psf = H.hankel(lambda r: otf_Kolmogorov(r, r0=r0, expo=expo), tmp)

    # psf(0) = 2*pi/expo*fO**2*Gamma(2/expo)
    # where f0 = r0/lbda * beta**(-1/expo)
    # and beta = (8/expo*Gamma(2/expo))**(expo/2) = 3.44... for expo=5/3
    # Therefore f0**2 = (r0/lbda)**2 / ( 8/expo*Gamma(2/expo) )
    # and psf(0) = pi/4 * (r0/lbda)**2. Given we actually compute
    # lbda**2/(2*pi)*psf(r), this gives psf(0) = r0**2/8
    pmax = r0**2/8
    psf[tmp == 0] = pmax                # Remove NaNs at r=0
    if normed:
        psf /= pmax                     # Max normalisation

    return psf.reshape(N.shape(r))      # PSF should have the same shape as r


def integ_radial_laguerre(f, accuracy=1e-3, imin=1, imax=40):
    """
    Compute s = 2*pi int_0^infinity g(r) dr with g(r) = r*f(r), using
    Gauss-Laguerre quadrature: int_0^infinity exp(-r)*exp(r)*g(r) dr = sum_i
    w_i exp(r_i) g(r_i) where r_i and w_i are the roots and weights of
    ith-order Laguerre polynomial.
    """

    def g(r): return r*f(r)

    sprev = 0
    errprev = N.inf
    for i in xrange(imin, imax):
        w = S.laguerre(i+1).weights     # Laguerre polynomial roots & weights
        s = N.dot(w[:, 2], g(w[:, 0]))  # sum_i w_i exp(x_i) g(x_i)

        err = abs(1-sprev/s)
        print "Order %d (%f-%f): s=%f err=%f" % \
              (i+1, min(w[:, 0]), max(w[:, 0]), 2*N.pi*s, err)
        if err < accuracy:
            print "Requested accuracy is reached"
            break
        elif err > errprev:
            print "Relative error is increasing"
            # break

        sprev = s
        errprev = err

    s *= 6.2831853071795862              # 2*pi

    return s, s*err                      # Integral and error estimate


def integ_radial_quad(f, epsabs=1e-6, epsrel=1e-6):
    """Compute s = 2*pi int_0^infinity g(r) dr with g(r) = r*f(r)."""

    s, ds = quad(lambda r: r*f(r), 0, N.inf, epsabs=epsabs, epsrel=epsrel)

    s *= 6.2831853071795862
    ds *= 6.2831853071795862

    return s, ds


if __name__ == '__main__':

    ftel = 22.5                         # Telescope focal length [m]
    pixsize = 15e-6                     # Pixel size [m]
    px2arcs = pixsize/ftel*RAD2ARCS     # Pixel scale [arcsec/px]
    print "Focal length=%.2f m, pixel size=%.0f microns, scale=%.3f\"/px" % \
          (ftel, pixsize*1e6, px2arcs)

    amax = 20.                          # Image half width [arcsec]
    xmax = int(amax / px2arcs)          # Image half width [px]
    amax = xmax * px2arcs               # Image half width [arcsec]
    print "Image half width: %d px = %.2f\"" % (xmax, amax)

    x = N.arange(-xmax, xmax+0.1, dtype='d')  # (n,) [px]
    r = N.hypot(*N.meshgrid(x, x))      # Radius 2D-array (n,n) [px]
    a = r * px2arcs                     # Radius (n,n) [arcsec]
    px2f = ftel/pixsize/len(x)          # [px/rad]
    f = r * px2f                        # Spatial freq 2D-array (n,n) [1/rad]

    # Reference medium seeing
    lref = 5e-7                         # Reference wavelength [m]
    r0ref = 0.10                        # Fried parameter at ref. lbda [m]
    expo = 5/3
    # expo = 1.5                        # Kolmogorov requires 5/3

    # Medium seeing at 5000A
    lbda = 5e-7                         # Current wavelength [m]
    r0m = friedParamater(lbda, r0ref, lref, expo)
    s = seeing_fwhm(lbda, r0ref, lref, expo,
                    verbose=True)       # Seeing FWHM [arcsec]

    psf = psf2D_FFT(otf_Kolmogorov(lbda*f, r0=r0m, expo=expo))

    ar = a.ravel()                      # (n**2,)
    j = N.argsort(ar)
    i = j[:N.searchsorted(ar[j], 5)]    # ar[i]: r up to 5 arcsec

    r1 = N.logspace(-1, N.log10(5/px2arcs), 100)  # Radius (n,) [px]
    a1 = r1 * px2arcs                   # Radius (n,) [arcsec]
    psf1 = psf_Kolmogorov_Hankel(a1, lbda, r0=r0m, expo=expo)

    psf2 = psf_Kolmogorov_FFT(a1, lbda, scale=px2arcs, r0=r0m, expo=expo)

    print "Integral(Laguerre)", \
          integ_radial_laguerre(
              lambda r:
              psf_Kolmogorov_Hankel(r, lbda, r0=r0m, expo=expo))

    print "Integral(quad)", \
          integ_radial_quad(
              lambda r:
              psf_Kolmogorov_Hankel(r, lbda, r0=r0m, expo=expo))

    if False:
        fig = P.figure()
        ax = fig.add_subplot(1, 1, 1,
                             title="Kolmogorov PSF (expo=%.2f)" % expo,
                             xlabel="r [arcsec]", xscale='log',
                             ylabel="Normalized flux", yscale='log',
                             )
        ax.plot(ar[i], psf.ravel()[i],
                color=RED, ls='None', marker='.', ms=5,
                label='2D-FFT')
        ax.plot(a1, psf2,
                color=GREEN, ls='None', marker='.', ms=5,
                label='Interpolated 2D-FFT')
        ax.plot(a1, psf1, label='Hankel')
        #ax.plot(a1,a1*psf1*N.exp(a1), label='Hankel')
        ax.axhline(0.5, c='0.8', label='_')
        ax.axvline(s/2, c='0.8', label='_')
        ax.set_ylim(psf1.min()/1.1, psf1.max()*1.1)
        ax.legend(loc='best', fontsize='small')

    # Plots ==============================

    if False:
        fig = P.figure()
        axI = fig.add_subplot(1, 1, 1,
                              title=u"PSF (r₀=%.0f cm@%.0f Å)" %
                              (r0m*100, lbda*1e10),
                              xlabel='x [arcsec]',
                              ylabel='y [arcsec]',
                              aspect='equal')
        psf = psf_Kolmogorov_Hankel(a, lbda, r0=r0m)
        print psf.sum()/len(a)**2
        print psf.max(), lbda, S.gamma(6/5)
        axI.imshow(psf,
                   norm=P.matplotlib.colors.LogNorm(vmin=(psf[psf > 0]).min()),
                   interpolation='nearest',
                   extent=[-amax, amax, -amax, amax])

    # Radial profile ------------------------------

    if True:
        fig = P.figure()
        axR = fig.add_subplot(1, 1, 1,
                              title=u"Kolmogorov seeing radial profile "
                              u"(r₀=%.0f cm@%.0f Å)" %
                              (r0ref*1e2, lref*1e10),
                              xlabel='r [arcsec]',  # xscale='log',
                              ylabel='Normalized flux', yscale='log')

        axR.plot(ar[i], psf.ravel()[i],
                 color=GREEN, ls='-', marker='None',
                 label=u'%.0f Å: seeing=%.2f"' % (lbda*1e10, s))

        lbda = 3e-7                         # Medium seeing @3000A
        r0 = friedParamater(lbda, r0ref, lref)
        s = seeing_fwhm(lbda, r0ref, lref)
        axR.plot(ar[i], psf2D_FFT(otf_Kolmogorov(lbda*f, r0)).ravel()[i],
                 color=BLUE, ls='-', marker='None',
                 label=u'%.0f Å: seeing=%.2f"' % (lbda*1e10, s))

        lbda = 10e-7                        # Medium seeing @10000A
        r0 = friedParamater(lbda, r0ref, lref)
        s = seeing_fwhm(lbda, r0ref, lref)
        axR.plot(ar[i], psf2D_FFT(otf_Kolmogorov(lbda*f, r0)).ravel()[i],
                 color=RED, ls='-', marker='None',
                 label=u'%.0f Å: seeing=%.2f"' % (lbda*1e10, s))

        lbda = 5e-7
        r0ref = 0.15                        # Good seeing @5000A
        r0 = friedParamater(lbda, r0ref, lref)
        s = seeing_fwhm(lbda, r0ref, lref)
        axR.plot(ar[i], psf2D_FFT(otf_Kolmogorov(lbda*f, r0)).ravel()[i],
                 color=GREEN, ls=':', marker='None',
                 label=u'r₀=%.0f cm: seeing=%.2f"' % (r0ref*100, s))

        r0ref = 0.07                        # Bad seeing @5000A
        r0 = friedParamater(lbda, r0ref, lref)
        s = seeing_fwhm(lbda, r0ref, lref)
        axR.plot(ar[i], psf2D_FFT(otf_Kolmogorov(lbda*f, r0)).ravel()[i],
                 color=GREEN, ls='--', marker='None',
                 label=u'r₀=%.0f cm: seeing=%.2f"' % (r0ref*100, s))

        axR.set_ylim(1e-4, 1)
        axR.legend(loc='best', fontsize='small')

    # Comparison w/ extract_star PSF ------------------------------

    if True:
        lbda = 5e-7
        r0ref = 0.10
        s = seeing_fwhm(lbda, r0ref, lref)

        fig = P.figure()
        ax = fig.add_subplot(1, 1, 1,
                             title=u"Kolmogorov (r₀=%.0f cm@%.0f Å) "
                             "vs. ad-hoc seeing PSF" % (r0ref*1e2, lref*1e10),
                             xlabel='r [arcsec]',  # xscale='log',
                             ylabel='Normalized flux', yscale='log')

        # Long exposures
        gl, ml = psf_ES_long(ar[i], 2.03, decompose=True)
        ax.plot(ar[i], gl+ml,
                color=BLUE, ls='-', marker='None',
                label="Ad-hoc PSF (long)")
        ax.set_autoscale_on(False)
        ax.plot(ar[i], gl,
                color=BLUE, ls='--', marker='None',
                label="_")
        ax.plot(ar[i], ml,
                color=BLUE, ls='--', marker='None',
                label="_")

        # Short exposures
        gs, ms = psf_ES_short(ar[i], 2, decompose=True)
        ax.plot(ar[i], gs+ms,
                color=RED, ls='-', marker='None',
                label="Ad-hoc PSF (short)")
        ax.plot(ar[i], gs,
                color=RED, ls='--', marker='None',
                label="_")              # [Individual components]
        ax.plot(ar[i], ms,
                color=RED, ls='--', marker='None',
                label="_")

        # Kolmogorov
        ax.plot(ar[i], psf.ravel()[i],
                color=GREEN, ls='-', lw=2, marker='None',
                label=u'%.0f Å: seeing=%.2f"' % (lbda*1e10, s))

        for n in range(8, 12):
            y = psf_Kolmogorov_Hankel(a1, lbda, r0=r0m, expo=n/6)
            if n != 10:
                ax.plot(a1[72:], y[72:],
                        color=GREEN, ls='-', marker='None',
                        label='_')
            ax.annotate('n=%s' % (fractions.Fraction(n, 6)), (a1[90], y[90]),
                        fontsize='small', rotation=-25)

        ax.set_ylim(1e-4, 1)
        ax.legend(loc='best', fontsize='small')

    # Ad-hoc comparison w/ observations ------------------------------

    if True:
        # Long exposure @4330A: GD71_07_253_090_B.txt
        Rl, Fl = N.loadtxt("GD71_07_253_090_B.txt", unpack=True)
        Fl /= Fl.max()/0.8                  # /0.85
        ll = 4.33e-7
        rl = 0.038
        psfl = psf2D_FFT(otf_Kolmogorov(ll*f, rl)) + 0.0035
        psfl2 = psf2D_FFT(otf_Kolmogorov(ll*f, rl, expo=1.52)) + 0.0035

        # Short exposure @7600: HR7950_07_253_041_R.txt
        Rs, Fs = N.loadtxt("HR7950_07_253_041_R.txt", unpack=True)
        Fs /= Fs.max()/0.9
        ls = 7.6e-7
        rs = 0.075
        psfs = psf2D_FFT(otf_Kolmogorov(ls*f, rs))
        psfs2 = psf2D_FFT(otf_Kolmogorov(ls*f, rs, expo=1.5))

        fig = P.figure(figsize=(10, 5))
        axl = fig.add_subplot(1, 2, 1,
                              title=u"Long exposure @%.0f Å" % (ll*1e10),
                              xlabel="r [arcsec]",
                              ylabel="Normalized flux", yscale='log')
        axs = fig.add_subplot(1, 2, 2,
                              title=u"Short exposure @%.0f Å" % (ls*1e10),
                              xlabel="r [arcsec]",
                              yscale='log')

        i = j[:N.searchsorted(ar[j], 10)]     # Keep r up to 10 arcsec

        axl.plot(Rl, Fl,
                 color=BLUE, ls='None', marker='.', ms=5,
                 label='Observed')
        axl.plot(ar[i], psfl.ravel()[i],
                 color=GREEN, ls='-', lw=2, marker='None',
                 label='Kolmogorov (n=5/3)', zorder=0)
        axl.plot(ar[i], psfl2.ravel()[i],
                 color=GREEN, ls='-', marker='None',
                 label='Kolmogorov (n=3/2)', zorder=0)
        axs.plot(Rs, Fs,
                 color=RED, ls='None', marker='.', ms=5,
                 label='Observed')
        axs.plot(ar[i], psfs.ravel()[i],
                 color=GREEN, ls='-', lw=2, marker='None',
                 label='Kolmogorov (n=5/3)', zorder=0)
        axs.plot(ar[i], psfs2.ravel()[i],
                 color=GREEN, ls='-', marker='None',
                 label='Kolmogorov (n=3/2)', zorder=0)

        axl.set_ylim(1e-4, 1.1)
        axs.set_ylim(1e-4, 1.1)
        axl.legend(loc='best', fontsize='small')
        axs.legend(loc='best', fontsize='small')

    if True:
        for i in P.get_fignums():
            fig = P.figure(i)
            fig.savefig("seeing_%d.png" % i)

    P.show()