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This is a HW4 Tesla on FSD 14.3.2 trying to drive into a lake five days ago (a la The Office): https://www.reddit.com/r/TeslaFSD/comments/1t9rl2u/fsd_tried..., so I would not say Tesla has solved standing water yet.

That said, FSD seems quite capable of routing around standing water in many cases (e.g. https://xcancel.com/planoken/status/2030754820462633031, https://www.reddit.com/r/TeslaFSD/comments/1pw9f2m/fsd_navig..., https://xcancel.com/BLKMDL3/status/1991862465328779317, https://xcancel.com/JVTacoma/status/2046313902749921638), so handling the remaining cases seems more like a model intelligence / data issue rather than a sensor limitation. Lidar beams generally bounce off mirrorlike surfaces without returning to the sensor, so I think all lidar would tell you about standing water is "there's something shiny/reflective within this region of the image", which you already know from cameras+headlights.

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Waymo has LIDAR and cameras, so it is better equipped for every situation.
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Kind of unrelated. That issue was due to a misguided effort to be cautious by having vehicles requesting human-review when they didn't really need it. Waymo fixed the issue by allowing the vehicles to operate in their normal, independent, mode.
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part of the problem is that SFs traffic lights just turn off in a power outage, rather than flashing red battery power as I have seen in many other jurisdictions
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LIDAR isn't helpful for water. Standing water behaves like a mirror on LIDAR.
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This is one of the reasons why I'm suspicious of camera-only systems, here in Finland. Half the year there's a lot of snow and ice around. Which I imagine means most of the view is "white" and "shiny". Coupled with the dark winters it's gotta be a nightmare to deal with.
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Plenty of snow and ice in parts of the US. Hopefully you'll get a chance to try the self-driving for yourself.
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do humans drive in it?
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Not necessarily. Depending on angle and water depth, multi-return LIDAR can give you returns from both water surface and the road surface beneath, in the same way multi-return LIDAR can produce returns from vegetation and the ground beneath.
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Could you use a different spectrum of EM radiation to detect water? There are parts of the microwave band that attenuate the signal by absorption and I wonder if you could use that. The only clue a human driver has in that situation is in the visible spectrum. The lines of the road disappear from view, which can be challenging to see at night.
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If the LIDAR can sense the road close enough to the front of the car, then it could estimate how far underwater the car is.
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