Compensation segments Φ(,p)

4a6ff8d8-10d5-48e2-ad55-5820f11b2323 63 9 12 decimal

&[DATE] &[TIME] - &[FILENAME]

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

Explain this document . . . [m] altitude vs measured [kPa]

1. Barometric NASA(h) . . . kPa vs altitude

2. Solve recursive NASA(h) . . . select/collect support for Altimetric m_kPa

support points are needed for compensation of the Alt(p)

3. Install/compensate Altimetric [m vs kPa]

6. The final m(kPa) is the altitude [m] vs measured kPa.

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U 0.027912 60000 0.5344 35462 1.0409 30747 2.0539 26217 3.5733 22781 5.5993 19921 9.6511 16453 14.716 13766 22.3133 11115 32.443 8654 50.17 5563 75.4942 2423 101.325 6.5 132 mat :

1. NASA Barometric Formula . . .

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https://www.grc.nasa.gov/WWW/K-12/rocket/atmosmet.html

Atmospheric kPa <= m ...dressed as given ... h NASA 101.29 288.14 0.00649 h * - 288.08 / (5.256^* 1000 - h ≤ (h 11000 ≤ (& 22.65 1.73 0.000157 h * - exp * 11000 h ≤ (h 25000 ≤ (& 2.488 141.89 0.00299 h * + 216.6 / (11.388 -^* cases :

point solve sol 18498 NASA p - 0 ≡ p 105 - 105 solve : 7

S U 2 col U 1 col augment : support S o 5 black plot : x c 10 x ≤ (x 60000 ≤ (& cases : 31 sys

0 NASA 101.4009

101.29 288.14 0.00649 h * - 288.08 / (5.256^* 1000 - h ≤ (h 11000 ≤ (& 22.65 1.73 0.000157 h * - exp * 11000 h ≤ (h 25000 ≤ (& 2.488 141.89 0.00299 h * + 216.6 / (11.388 -^* cases

2. Altimetric Formula [0 . . 30km]

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data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

ISO 101.325 : x log x log10 : 12 mat

p J 0.2394141 p * - 60.62223 + 5013.924 p 47.31633 + 1505.831 p 25.55854 - 186.1156 p 26.36295 - 93.75324 p 15.79446 - / + / + / + / - :

Unsourced Altimetric ... long segment
compensated [0..30km] < ± 0.05 m p Alt p ISO / log 2.2826518 / exp 1 - 6.876 106 -^* - / 0.3048 * (1 p J * + :

NASA 0.027912 60000 0.5344 35462 1.0409 30747 2.0539 26217 3.5733 22781 5.5993 19921 9.6511 16453 14.716 13766 22.3133 11115 32.443 8654 50.17 5563 75.4942 2423 101.325 6.5 132 mat :

x c 10 x ≤ (x 101.325 ≤ (& cases : data NASA o 5 black plot : NASA NASA 413 range 12 range el : X NASA 1 col : Y NASA 2 col : 12 mat Δm Y X Alt vectorize Y - 3 round vectorize augment : 51 sys

m 1 :

7 Alt m 18498

Δm 26217 0.015 22781 0.006 19921 0.002 164530137660111150865405563024230 6.5 0102 mat

Compensated Altimetric Formula [0...30 km]

data x Alt x c * 2.0533 26217 + 50 blue 1 5 mat 3 1 sys

p ISO / log 2.2826518 / exp 1 - 6.876 106 -^* - / 0.3048 * (p J +

x c 1 1.5 x ≤ (x 101.325 ≤ (& cases :

Δev 2.0539 3001.65699 3.5733 1911.40366 5.5993 1144.19907 9.6511 464.80215 14.716 146.23102 22.3133 26.06087 32.443 18.48313 50.17 14.05603 75.4942 9.32128 101.325 6.5 102 mat :

Deviation from no compensation ... fitted by J(p)

Δev o 5 black plot x J x c * 2 1 sys

3. Altimetric Companion . . .

[0 . . . 60 km] ± 0.5m

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This Altimetric Companion recovers NASA(h) In the respective ranges Δm ± 0.5 m OBSERVE: NASA(h)is freak 6.5 m @ 101.325

Compensation segments Φ(,p)

β 2778.222709329 - 1107.877636153 141.185317434 - 7.909338858 0.200159963 - 0.001820543 1.127892206 0.13175127 - 0.004546095 0.000043771 - 101 mat :

λ 49.317588800077 8.451235910877 - 0.429359839792 0.009024915165 - 0.000083784758 0.000000285412 - 0.174226750638 - 0.014115687391 0.000245307741 - 0.000001210142 101 mat :

α 3481.01545628946 - 26918.3222500304 58216.9240964423 - 47781.17783335 12213.9372498084 - 719.270672385553 - 23.072671042616 49.777292499046 - 17.345050954428 8.857047486732 101 mat :

λ p Φ11 λ 1 el λ 2 el p * + λ 3 el p 2^* + λ 4 el p 3^* + λ 5 el p 4^* + λ 6 el p 5^* + 1 λ 7 el p * + λ 8 el p 2^* + λ 9 el p 3^* + λ 10 el p 4^* + / :

β p Φ25 β 1 el β 2 el p * + β 3 el p 2^* + β 4 el p 3^* + β 5 el p 4^* + β 6 el p 5^* + 1 β 7 el p * + β 8 el p 2^* + β 9 el p 3^* + β 10 el p 4^* + / :

α p Φ60 α 1 el α 2 el p * + α 3 el p 2^* + α 4 el p 3^* + α 5 el p 4^* + α 6 el p 5^* + 1 α 7 el p * + α 8 el p 2^* + α 9 el p 3^* + α 10 el p 4^* + / :

x log x log10 : ISO 101.325 : 12 mat p Alt 22.798 p ≤ (p 101.4009 ≤ (& p ISO / log 2.24646 / exp 1 - 6.95972 106 -^* - / 0.3048 * (λ p Φ11 - 2.533 p ≤ (p 22.798 ≤ (& p ISO / log 5.79476 / exp 1 - 2.97734 106 -^* - / 0.3048 * (β p Φ25 - 0.027912 p ≤ (p 2.432 ≤ (& p ISO / log 16.84916 - / exp 1 - 1.23656 106 -^* / 0.3048 * (α p Φ60 - N/A if if if : 21 sys

2251000 range 11 mat

Duplet coefficients are optimized Mathcad Minerr LM wrt their [...] respective ranges

25225 range 11 mat

Vice-versa sanity check 59717 NASA 25131 NASA 10975 NASA 0 NASA 41 mat 0.029 2.432 22.797 101.401 41 mat

124 range 11 mat

x c 1 0.027912 x ≤ (x ISO ≤ (& cases : support U o 5 black plot : 21 sys

2.533 24973 + 50 blue 1 5 mat x Alt x c * 22.798 10975 + 50 blue 1 5 mat 0.03 60000 o 5 black 0.53 35462 o 5 black 1.04 30747 o 5 black 2.05 26217 o 5 black 3.57 22781 o 5 black 5.6 19921 o 5 black 9.65 16453 o 5 black 14.72 13766 o 5 black 22.31 11115 o 5 black 32.44 8654 o 5 black 50.17 5563 o 5 black 75.49 2423 o 5 black 101.325 0 o 5 black 13 5 mat 4 1 sys

0.027912 Alt 0.101 Alt 2.432 Alt 22.798 Alt 101.4009 Alt 51 mat 59717485222513110975051 mat

101.325 Alt 6.5

Solving NASA(h)

altitude[m] from measured kPa

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Smath 'solve' meets Mathcad hyper-robust/fast seeded root scanner TOL e-6. The 3 MCD ranges are passed to MCD Minerr LM for optimizing the duplets. The deviation are then passed to MCD GenfitMatrix 5/4 rational compensat'n ± 0.5 m RANGES => [1..24], [25..225], [225..1000]

1000 points solutions ... Disabled

h NASA 101.29 288.14 0.00649 h * - 288.08 / (5.256^* 1000 - h ≤ (h 11000 ≤ (& 22.65 1.73 0.000157 h * - exp * 11000 h ≤ (h 25000 ≤ (& 2.488 141.89 0.00299 h * + 216.6 / (11.388 -^* cases :

L H N xd U 0 : dx H L - N 1 - / : 12 mat i 1 N range U i el L dx - (dx i * + eval : for U 31 line :

60000 NASA 0.027911516

L 60000 NASA : H 101.325 : N 1001 : 13 mat

u L H N xd :

sol i 1 u rows range m_altitude i el h NASA u i el - 0 ≡ h 0100000 solve : for u 4 round vectorize m_altitude 0 round vectorize augment 21 line :

t0 1 time :

selected support 0.027912 60000 0.5344 35462 1.0409 30747 2.0539 26217 3.5733 22781 5.5993 19921 9.6511 16453 14.716 13766 22.3133 11115 32.443 8654 50.17 5563 75.4942 2423 101.325 6.5 132 mat

NASA(h) solved [m_altitude] from measured kPa

sol 101.325 10000 101.325 10000 - 2 2 mat 2 1 sys

1 time t0 - min 4

4. NASA Altimetric Formula . . .

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Atmospheric kPa <= m ...dressed as given ... h NASA 101.29 288.14 0.00649 h * - 288.08 / (5.256^* 1000 - h ≤ (h 11000 ≤ (& 22.65 1.73 0.000157 h * - exp * 11000 h ≤ (h 25000 ≤ (& 2.488 141.89 0.00299 h * + 216.6 / (11.388 -^* cases :

maple InverseFunction

y 101.29 288.14 0.00649 h * - 288.08 / (n^* ≡ h solve maple 20001440714404100 y * 10129 / ln n / exp * - (* 649 /

y 22.65 1.73 0.000157 h * - exp * ≡ h solve maple 1000017310020 y * 453 / ln * - (* 157 /

y 2.488 141.89 0.00299 h * + 216.6 / (m -^* ≡ h solve maple 401347252 - 354725 y * + (* 299 y * / -

0 NASA 11000 NASA 25000 NASA 60000 NASA 41 mat 101.400931 22.707365 2.522271 0.027912 41 mat

p NASA_alt 20001440714404100 p * 10129 / ln 5.256 / exp * - (* 649 / 22.707365 p ≤ (p 101.400931 ≤ (& 1000017310020 p * 453 / ln * - (* 157 / 2.522271 p ≤ (p 22.707365 ≤ (& 100014189 - 21660125 p * 311 / ln 11.388 / - exp * + (* 299 / cases :

0.027912 NASA_alt 2.522271418 NASA_alt 22.707365 NASA_alt 101.325 NASA_alt 41 mat 600002500011000641 mat

x c 1 0.027912 x ≤ (x 101.400931 ≤ (& cases :

x NASA_alt x c * 101.401 5000 101.401 5000 - 2 2 mat 2 1 sys

x c 10 x ≤ (x 60000 ≤ (& cases :

Reflexive plot: native over inverse

60000 16 60000 16 - 2 2 mat x NASA x c * 2 1 sys