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Layer (i) Vi (m/s) Hi (m) Dt0i T0i True RMS Velocity (m/s) Zi (m)1 2000 600 0.6000 0.6000 2000 600

2 4000 400 0.2000 0.8000 2646 1000

3 3000 1000 0.6667 1.4667 2812 2000

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Layer (i) 1 2 3

Vi 2000 4000 3000

Hi 600 400 1000

Dt0i 0.6000 0.2000 0.6667T

0i0.6000 0.8000 1.4667

LAYER 1 LAYER 2 LAYER 3

X Te X Te X Te

0 0.600 0 0.800 0 1.467

21 0.600 49 0.800 101 1.467

42 0.600 98 0.801 203 1.468

63 0.601 147 0.802 305 1.471

84 0.601 197 0.803 407 1.474

105 0.602 247 0.805 510 1.478

126 0.603 297 0.808 615 1.483

147 0.605 348 0.811 720 1.489169 0.606 400 0.814 827 1.496

190 0.607 454 0.818 936 1.504

212 0.609 508 0.823 1047 1.513

233 0.611 564 0.828 1161 1.523

255 0.613 621 0.833 1277 1.535

277 0.616 680 0.840 1397 1.548

299 0.618 741 0.847 1520 1.562

322 0.621 806 0.855 1648 1.578

344 0.624 873 0.864 1781 1.596

367 0.627 944 0.874 1920 1.616390 0.631 1019 0.885 2065 1.638

413 0.635 1100 0.898 2219 1.662

437 0.639 1187 0.913 2382 1.689

461 0.643 1283 0.929 2558 1.720

485 0.647 1390 0.949 2748 1.755

509 0.652 1511 0.972 2958 1.795

534 0.657 1653 1.001 3193 1.842

560 0.662 1825 1.036 3464 1.898

585 0.668 2044 1.083 3789 1.968

611 0.673 2345 1.151 4205 2.061638 0.680 2821 1.261 4805 2.200

665 0.686 3836 1.504 5955 2.475

693 0.693 4019 1.548 6152 2.523

721 0.700 4235 1.600 6382 2.579

750 0.708 4493 1.664 6655 2.645

779 0.715 4813 1.742 6989 2.727

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 2000

T

(s)

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809 0.724 5222 1.843 7413 2.832

840 0.732 5774 1.979 7980 2.971

872 0.742 6580 2.178 8801 3.174

904 0.751 7926 2.512 10162 3.512

938 0.761 10953 3.266

972 0.772

1007 0.783

1043 0.795

1080 0.807

1119 0.820

1159 0.834

1200 0.849

1243 0.864

1287 0.880

1333 0.897

1380 0.915

1430 0.933

1482 0.953

1536 0.975

1592 0.997

1652 1.021

1714 1.046

1779 1.073

1848 1.102

1920 1.132

1997 1.165

2078 1.200

2165 1.238

2257 1.278

2355 1.322

2460 1.369

2573 1.420

2695 1.475

2827 1.536

2970 1.602

3126 1.674

3297 1.754

3485 1.843

3693 1.942

3925 2.052

4185 2.177

4478 2.318

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4813 2.480

5198 2.667

5646 2.886

6173 3.145

6806 3.455

7577 3.835

8538 4.311

9773 4.923

11417 5.740

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4000 6000 8000 10000

X (m)

LAYER 1

LAYER 2

LAYER 3

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X /Z T3(X) X T32

0 0.0 1.467 0 2.1511111111

101 0.1 1.467 10254 2.1524075470

203 0.1 1.468 41097 2.1563064405

305 0.2 1.471 92771 2.1628367606407 0.2 1.474 165689 2.1720475054

510 0.3 1.478 260441 2.1840088120

615 0.3 1.483 377810 2.1988135865

720 0.4 1.489 518789 2.2165797367

827 0.4 1.496 684614 2.2374531269

936 0.5 1.504 876788 2.2616114225

1047 0.5 1.513 1097133 2.2892690570

1161 0.6 1.523 1347844 2.3206836469

1277 0.6 1.535 1631562 2.3561643107

1397 0.7 1.548 1951476 2.39608254471520 0.8 1.562 2311445 2.4408865991

1648 0.8 1.578 2716177 2.4911207521

1781 0.9 1.596 3171468 2.5474515941

1920 1.0 1.616 3684528 2.6107046076

2065 1.0 1.638 4264469 2.6819163024

2219 1.1 1.662 4922995 2.7624106349

2382 1.2 1.689 5675482 2.8539147846

2558 1.3 1.720 6542671 2.9587415904

2748 1.4 1.755 7553487 3.0800909256

2958 1.5 1.795 8750015 3.2225770933

3193 1.6 1.842 10196916 3.3932204650

3464 1.7 1.898 12001050 3.6034917273

3789 1.9 1.968 14357718 3.8740761487

4205 2.1 2.061 17680864 4.2480822716

4805 2.4 2.200 23086862 4.8390744448

5955 3.0 2.475 35463157 6.1248639182

6152 3.1 2.523 37851321 6.3642023068

6382 3.2 2.579 40730115 6.6495690602

6655 3.3 2.645 44292183 6.99830069096989 3.5 2.727 48851418 7.4382928540

7413 3.7 2.832 54958993 8.0178251997

7980 4.0 2.971 63687175 8.8292366830

8801 4.4 3.174 77459752 10.0769717688

10162 5.1 3.512 103275478 12.3357843635

y = 2.4894071810E-48x6 - 7.0064843333E-40x5 +

25x3 - 4.6490373668E-16x2 + 1.2646238

R = 9.999999979

0

2

4

6

8

10

12

14

0.0E+00 2.0E+07 4.0E+07 6.

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.1839640395E-32x4 + 2.3518865606E-

19E-07x + 2.1511111111E+00

7E-01

.0E+07 8.0E+07 1.0E+08 1.2E+08

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# Terms Ploynomial Degree (N) True RMS Velocity (m/s) C1

2 2 2812 1.0462E-07

3 4 2812 1.1887E-07

4 6 2812 1.2539E-07

5 8 2812 1.2685E-07

6 10 2812 1.2676E-077 12 2812 1.2646E-07

Notes:

We find C1 by fitting a polynomial of degree N and copying the coefficient of.

(2)Calculated RMS velocity=1/(C1)^(1/2.)(3)E=|True_RMS - Calculated_RMS|/True_RMS * 100.

Note the decrease in E as N increases. This indicates that as we use more terms

RMS velocity from C1 becomes better.Here we are using al l offsets. Therefore, you see that we get relatively high err

offset assumption. For example, compare the 2-term error with that in short of.

0%

2%

4%

6%

8%

10%12%

0 2 4 6 8 10 12 14

Error

Polynomial Degree (N)

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Calculated RMS Velocity (m/s) E

3092 9.94%

2900 3.13%

2824 0.42%

2808 0.16%

2809 0.13%2812 0.01%

(1)X and pasting it into the C1 column

(4)of the expansion, the estimation of the

(5)ors because we are violating the short-

fsets

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X X/Z3 T3(X) X T32

0 0.0 1.467 0 2.1511111111

101 0.1 1.467 10254 2.1524075470

203 0.1 1.468 41097 2.1563064405

305 0.2 1.471 92771 2.1628367606407 0.2 1.474 165689 2.1720475054

510 0.3 1.478 260441 2.1840088120

615 0.3 1.483 377810 2.1988135865

720 0.4 1.489 518789 2.2165797367

827 0.4 1.496 684614 2.2374531269

936 0.5 1.504 876788 2.2616114225

1047 0.5 1.513 1097133 2.2892690570

1161 0.6 1.523 1347844 2.3206836469

1277 0.6 1.535 1631562 2.3561643107

1397 0.7 1.548 1951476 2.39608254471520 0.8 1.562 2311445 2.4408865991

1648 0.8 1.578 2716177 2.4911207521

1781 0.9 1.596 3171468 2.5474515941

1920 1.0 1.616 3684528 2.6107046076

2065 1.0 1.638 4264469 2.6819163024

y = 4.3544682657E-47x6 + 8.5970221306E-40x5 -

24x3 - 4.6687049018E-16x2 + 1.26436781

R = 1.000000000

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0.0E+00 1.0E+06 2.0E+06

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.7325425102E-32x4 + 1.5569100820E-

82E-07x + 2.1511111111E+00

0E+00

3.0E+06 4.0E+06 5.0E+06

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# Terms Ploynomial Degree (N) True RMS Velocity (m/s) C1

2 2 2812 1.2497E-07

3 4 2812 1.2643E-07

4 6 2812 1.2644E-07

5 8 2812 1.2644E-07

6 10 2812 1.2644E-077 12 2812 1.2644E-07

Notes:

We find C1 by fitting a ploynomial of degree N and copying the coefficient of.

(2)Calculated RMS velocity=1/(C1)^(1/2.)(3)E=|True_RMS - Calculated_RMS|/True_RMS * 100.

Note the decrease in E as N increases. This indicates that as we use more terms

RMS velocity from C1 becomes better.Here we are using shortoffsets only (maxX/Z=

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Calculated RMS Velocity (m/s) E

2829 0.58%

2812 0.00%

2812 0.00%

2812 0.00%

2812 0.00%2812 0.00%

(1)X and pasting it into the C1 column

(4)of the expansion, the estimation of the

(5)e get relatively low errors because we

r with that in all offsets

onclusione can achieve the same accuracy in

imating the true T-X curve through a

ies expansion by limiting the maximum

ed offset to less than the reflector depth

ter than including more terms in the

ries