Compensation segments Φ(,p)

4a6ff8d8-10d5-48e2-ad55-5820f11b2323 127 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

1500 NASA 84.6424

support x NASA x c * 2 1 sys

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 . . 20km]

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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 0.2394 p * - 60.62 + 5014 p 47.32 + 1506 p 25.56 - 186.1 p 26.36 - 93.75 p 15.79 - / + / + / + / - :

Unsourced Altimetric ... long segment
compensated [0..30km] < ± 0.05 m p Alt p ISO / log 2.28264 / 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 18497

Δm 26217 0.004 22781 0.149 - 19921 0.161 - 16453 0.109 - 13766 0.038 - 11115 0.038 8654 0.043 5563 0.028 2423 0.012 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 2.28264 / 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 101.400930905 22.707364717 2.522271418 0.027911516 41 mat

λ 1 el 101.400930905 : λ 2 el 22.707364717 : λ 3 el 2.522271418 : 13 mat

λ 1 el 101.4009 : λ 2 el 22.3135 : λ 3 el 2.52227 : 13 mat p NASA_alt 20001440714404100 p * 10129 / ln 5.256 / exp * - (* 649 / λ 2 el p ≤ (p λ 1 el ≤ (& 1000017310020 p * 453 / ln * - (* 157 / λ 3 el p ≤ (p λ 2 el ≤ (& 100014189 - 21660125 p * 311 / ln 11.388 / - exp * + (* 299 / cases : 21 sys

0.027911516

0.0279115 NASA_alt λ 3 el NASA_alt λ 2 el NASA_alt 101.325 NASA_alt 41 mat 600002500011111 6.3 41 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

Revisit the single segment

x c 12 x ≤ (x 101.401 ≤ (& cases : x Δm x NASA_alt x Alt - (x c * (: 21 sys

datum 0.027912 0 0.5344 0 1.0409 0 2.0539 0 3.5733 0 5.5993 0 9.6511 0 14.716 0 22.3133 0 32.443 0 50.17 0 75.4942 0 101.325 0132 mat :

Deviation for J(p) 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

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 :

dataum datum . 14 black plot :

Commercial Jet segment  ... [0..11115m] ± < 0.5 m p J 0.01838 p * 7.135 - 1329 p 8.984 + 218.2 p 17.96 - 54.05 p 19.51 - / + / + / + : ISO 101.325 : x log x log10 : 12 mat p Alt p ISO / log 2.28264 / exp 1 - 6.876 106 -^* - / 0.3048 * (1 p J * + : 31 sys

22.3133 Alt 11115

75.5 NASA_alt 75.5 Alt - 0.43

x c 1 22.3133 x ≤ (x 101.401 ≤ (& cases : x Δm x NASA_alt x Alt - (x c * (: 21 sys

22.3133 3 22.3133 3 - 2 2 mat x Δm 0.027912 0 . 14 black 0.5344 0 . 14 black 1.0409 0 . 14 black 2.0539 0 . 14 black 3.5733 0 . 14 black 5.5993 0 . 14 black 9.6511 0 . 14 black 14.716 0 . 14 black 22.3133 0 . 14 black 32.443 0 . 14 black 50.17 0 . 14 black 75.4942 0 . 14 black 101.325 0 . 14 black 13 5 mat 22.5 2 - Jet altitude 11000 m 12 blue 1 5 mat 4 1 sys

The glitch has no consequence on single Alt(x) segment, results from NASA_alt(x) discontinuities

In the air, as you fly along and encounter nonstandard temperatures, indicated altitude can differ from true altitude. ... When the temperature is colder than standard, you are at an altitude lower than your altimeter indicates. @ temperature warmer than standard, you are higher than your altimeter indicates. ...................... In such conditions, an approximate correction is 4 per cent height increase for every 10°C below standard temperature as measured at the altimeter setting source. This is safe for all altimeter setting source altitudes for temperatures above –15°C. 4.3.2 Tabulated corrections

http://code7700.com/altimeter_temperature_correction.htm#correction

Jet-Liner Segment . . .

corrected flight altitude

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t 30 - : (local temoerature °C ≡ L0 0.0065 : (standard ISA temperature laps rate °C/m ≡ H 0 : (sea level elevation [m] ≡ m 1 : (meter unit ≡ h NASA11(kPa)uncompensated wrt ISA ≡ 51 sys

Local °C flight altitude correction
add [m] to  altimeter reading [h] h C h 15 t L0 H * + (- 273 t L0 H * + (+ 0.5 L0 * h H + (* - / * :

sanity from  [h = m] uncompensated NASA11(kPa) h 60 :

60 C m 11.1

p[kPa] from altimeter

p NASA11 20001440714404100 p * 10129 / ln 5.256 / exp * - (* 649 / : p Flight p NASA11 h C + : 21 sys

raw altimeter reading

t < 15°C : flying too low !

... flight altitude [m]

101.325 NASA11 101.325 Flight 21 mat 6.3 17.4 21 mat

x c 1 22.3135 x ≤ (x 101.401 ≤ (& cases :

x correct 10 x ≤ (x 1500 ≤ (& cases :

x NASA11 x c * x C x correct * x C x correct * - 2 1 sys

t 15 °C * <

t 15 °C * >

h 609012015018021024027030045060090012001500114 mat transpose :

Local °C flight altitude correction
add [m] to  altimeter reading  NASA11(p) h C h 15 t L0 H * + (- 273 t L0 H * + (+ 0.5 L0 * h H + (* - / * :

h C vectorize transpose m 11172228333945505684112169226283114 mat

Conservative pilot chart . . .

OBSERVE: NASA11(kPa) is conservative by ~ 6 m

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..... -10,210 is in error .....

'h' is altimeter reading

t 30 - : (local temperature °C ≡ L0 0.0065 : (standard ISA temperature laps rate °C/m ≡ H 0 : (sea level elevation [m] ≡ m 1 : (meter unit ≡ h NASA11(kPa)altimeter reading ≡ 51 sys

otherwise formulated

'h' NASA(kPa) altitude

't' local °C from Pt sensor

Local °C flight altitude correction
add [m] to  altimeter reading [h] h t C h 15 t L0 H * + (- 273 t L0 H * + (+ 0.5 L0 * h H + (* - / * :

add to altimeter reading 30050 - C m 88

p[kPa] from altimeter

p NASA11 20001440714404100 p * 10129 / ln 5.256 / exp * - (* 649 / : p t Flight p NASA11 p NASA11 t C + : 31 sys

raw altimeter reading

t < 15°C flying too low !

... flight altitude [m]

't' [°C]from the airport "OR" as measured at aircraft approaching landing runway.

Altimeter reading

in short . . .

1. altimeter reads ISA flying altitude from NASA11(kPa)

2. Flight(kPa) adds/subtracts corrected altitude [m]

3. NASA11(kPa)=m approximates ISA as "kPa sensed".

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M x y Spline A M 112 M cols submatrix : B M 2 M rows 11 submatrix : C M 2 M rows 2 M cols submatrix : j 1 A length 2 range CC j el B C 1 C rows j j submatrix y linterp : for A transpose CC x linterp 51 line :

'x' direction

TableIII °C/Altimeter 60901201501802102402703004506009001200150005510101015151520253550708510 - 10101515202525303045609012015020 - 101520252530354045658513017021530 - 1520253035404555608511517023028540 - 15253040455060657511014522029036550 - 20304055565758090135180270360450715 mat :

'y'

TableIII 50035 - Spline TableIII 60030 - Spline TableIII 100015 - Spline 31 mat 10811512231 mat