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The phrase "at a random depth" is correct and usable in written English.
It can be used when referring to a specific location or measurement that is chosen without a particular pattern or order, often in contexts like diving, drilling, or data analysis.
Example: "The researchers decided to collect samples at a random depth to ensure a diverse range of data."
Alternatives: "at an arbitrary depth" or "at a varied depth".
Exact(2)
An optical model was brought forward to describe the solar radiance distribution at a random depth and θ.
Each subreflector is placed at a random depth from the baseline smooth wall.
Similar(58)
Create a new canvas at a random position.
Look at a random article.
Keep the probe at a consistent depth.
Plant at a shallow depth.
To obtain a better depth sequence, a Random Sequence Diffuser (RSD) was designed.
In the Gulf of Lions (northwest Mediterranean), similar patterns were observed in the vast majority of 76 images taken at random at depths between 35 and 88 m in four different areas.
Cortical neurons were usually searched for by monitoring resistance increase during current pulse application and at random cortical depths (Bruno and Sakmann 2006; de Kock et al. 2007).
at an average depth of 1017×.
at an average depth of 218x.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com