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One drawback or impediment to the storage of flight recorder data on the cloud is the amount of data that'd be transmitted and at cost.

How much data in GB or TB is stored in the flight data recorder per hour in a typical flight of an A380?

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I don't have the numbers for the A380 but have some for the Dreamliner. On a typical flight a Dreamliner will record about a half a terabyte of data:

This level of operational insight will involve generating large amounts of data from each 787 aircraft, he explained. “We can get upwards of half a terabyte of data from a single flight from all of the different devices which are internet connected," Bulman said.

source

Depending on the airline this data is stored/processed/compressed/uploaded to HQ.

The flight data recorder or "Black Box" stores far less parameters as only 88 are required by the FAA:

(1) Time; (2) Pressure altitude; (3) Indicated airspeed; (4) Heading--primary flight crew reference (if selectable, record discrete, true or magnetic); (5) Normal acceleration (Vertical); (6) Pitch attitude; (7) Roll attitude; (8) Manual radio transmitter keying, or CVR/DFDR synchronization reference; (9) Thrust/power of each engine--primary flight crew reference; (10) Autopilot engagement status; (11) Longitudinal acceleration; (12) Pitch control input; (13) Lateral control input; (14) Rudder pedal input; (15) Primary pitch control surface position; (16) Primary lateral control surface position; (17) Primary yaw control surface position; (18) Lateral acceleration; (19) Pitch trim surface position (20) Trailing edge flap or cockpit flap control selection (21) Leading edge flap or cockpit flap control selection (22) Each Thrust reverser position (or equivalent for propeller airplane); (23) Ground spoiler position or speed brake selection (24) Outside or total air temperature; (25) Automatic Flight Control System (AFCS) modes and engagement status, including autothrottle; (26) Radio altitude; (27) Localizer deviation, MLS Azimuth; (28) Glideslope deviation, MLS Elevation; (29) Marker beacon passage; (30) Master warning; (31) Air/ground sensor (primary airplane system reference nose or main gear); (32) Angle of attack ; (33) Hydraulic pressure low (each system); (34) Ground speed ; (35) Ground proximity warning system; (36) Landing gear position or landing gear cockpit control selection; (37) Drift angle (38) Wind speed and direction ; (39) Latitude and longitude (40) Stick shaker/pusher (41) Windshear
(42) Throttle/power lever position; (43) Additional engine parameters (44) Traffic alert and collision avoidance system; (45) DME 1 and 2 distances; (46) Nav 1 and 2 selected frequency; (47) Selected barometric setting
(48) Selected altitude ; (49) Selected speed (50) Selected mach (51) Selected vertical speed (52) Selected heading (53) Selected flight path (54) Selected decision height (55) EFIS display format; (56) Multi-function/engine/alerts display format; (57) Thrust command (58) Thrust target (59) Fuel quantity in CG trim tank (60) Primary Navigation System Reference; (61) Icing (62) Engine warning each engine vibration (63) Engine warning each engine over temp. (64) Engine warning each engine oil pressure low (65) Engine warning each engine over speed (66) Yaw trim surface position; (67) Roll trim surface position; (68) Brake pressure (selected system); (69) Brake pedal application (left and right); (70) Yaw of sideslip angle; (71) Engine bleed valve position
(72) De-icing or anti-icing system selection (73) Computed center of gravity
(74) AC electrical bus status; (75) DC electrical bus status; (76) APU bleed valve position (77) Hydraulic pressure (each system); (78) Loss of cabin pressure; (79) Computer failure; (80) Heads-up display (81) Para-visual display (82) Cockpit trim control input position-pitch; (83) Cockpit trim control input position--roll; (84) Cockpit trim control input position--yaw; (85) Trailing edge flap and cockpit flap control position; (86) Leading edge flap and cockpit flap control position; (87) Ground spoiler position and speed brake selection; and (88) All cockpit flight control input forces (control wheel, control column, rudder pedal)

Source

Some modern airliners such as the A330neo store up to 1200 parameters, making it easier to traceback what caused a crash.

Exact numbers are difficult to provide as the data from a black box is considered to be highly confidential.

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    $\begingroup$ Note the data size is not only a function of how many parameters but also how frequent they are recorded. Example: Pinpointing airspeed each second requires much more memory than recording it each ten seconds. $\endgroup$ – jean Apr 10 at 11:13

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