How do researchers map root systems of plants?
You originally posted this question on the Garden and Landscaping Section - that's where I'm from, and though it was moved here instead, I'll answer it anyway:

That image you've shown isn't like a photograph, its an impression of what the roots are like. Usual method of working out root patterns on plants is to remove what's known as a monolith, or a block of soil where the plant is - the soil is then either carefully removed by hand, in the manner that archaeologists use, or air blasted, or soaked off in water to reveal as much of the roots as possible. For trees or plants with very large roots, newer methods include geophysical surveying by machine, or electric resistance surveying. Even this doesn't reveal everything - many roots may penetrate too far down, or run along cracks in rocks beneath the soil. The root length and distribution pattern vary according to local conditions, so its hard to be definite about where the roots of any particular plant are in a particular position in the landscape. 



Reference and further reading:

Plant Physiological Ecology (ISBN 978-94-009-2221-1; Google books link)

Especially, check the Introduction on page No.367