Could you take core soil samples at a series of defined radii from the tree trunk and check for roots within the sample at different depths? It would be laborious, but fairly straightforward.

The literature suggests ground penetrating radar is effective though high clay and water contents may interfere. You do not need complete architecture; identifying major roots will allow you to infer root ball dimensions and geometry and approximate the torque resistance. Routine foliar pruning may be more cost-effective? Google "Imaging tree root systems in situ" for more.

As Stanton suggested, please read this article "Using GPR (Ground-Penetrating Radar) to Estimate Tree Root Biomass" from the USDA Forest Service Southern Research Station website: http://www.srs.fs.usda.gov/news/85

http://www.treeradar.com/Root-Mapping-Procedure.pdf this might help.

Thank a lot for all your suggestions.

"GPR works very accurately on well-drained soils and may become a standard tool in forest research". But in the above-mentioned case, it might be not an appropriate tool because materials in the underground are not only soil and sand but also stones, debris, bitumen, concrete,...

How about the other tools  or methods? I am still looking for

I guess if minirhizotron could be of some use for your studies. It could be utilized in both shallow as well as deep rooted plants. Root turn over could also be very nicely calculated.    

An inexpensive and potentially sensitive method of detecting root presence at any point in the soil is to place Strontium (chloride or nitrate) at a location in the soil (or a number of points, of equal depth and distance from the tree), and see if elevated levels of Strontium turn up in the leaves. Strontium is taken up like Calcium, so accumulates over time in mature leaves. Like Ca, it doesn't move in the soil much either. To get a good idea of the profile of root activity, you'd need a plantation of many similar trees to test different distances and depths. Leaf sampling needs to be carefully standardised to establish background levels. Expressing Sr as a ratio to Ca can help with the variability. See http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=1842044&fileId=S0014479708006169

There are tables for failure curves of trees under wind stress. Root profile is just one parameter and in a street is just a cylinder volume under the crown, but much more important is the root elastic modulus, trunk removal force, TRF and trunk weight distribution. TRF can be measured on small plants with dynamometer and a winch, or extrapolated from partial angles (http://joa.isa-arbor.com/request.asp?JournalID=1&ArticleID=3039&Type=2) Failure occurs usually at the maximal drag near 2/3 of trunk height. A sample for further readings is http://edis.ifas.ufl.edu/fr010

We have tested ground penetrating radar in urban streets and pulled >200 trees to failure in pulling tests. As in other studies, root location bears little or no information on tree stability. Root plate tilt measured non-destructively with low loads simulating wind correlated well with load at failure. So static pulling tests together with estimates of wind load in storms can be used to assess the safety of urban trees.

I have learned a lot from your solutions. Thank so much for all your sharing. I start on finding out what I expect. Special thank to Steffen Rust!  

For both depth and root width it will certainly require some  level of excavation and then the root width can be measured by using venier calipers depending on the kind of trees being measured. It is also possible that from tree height and breast height measurements such measurements can be inferred. It may be important also to specify the root sizes with respect to depth of occurrence. in this case you may use a soil profile approach close to your selected trees.