In many ways the top growth and root growth can mirror each other.

When you throw out your quad for sampling you take a wet weight and then determine a dry weight.

Take a subsample from the quad and take a root weight by digging the root mass in 15 cm and wash out the soil take a root fresh weight then dry weight then relate this as a proportionality of the top weights. once you calculate the proportionality I believe you can get the root weight from the top weight.

These proportion might already exist in some data bases.

Dear Ali. This is the normal practice when soil sampling is done either before trial initiation, during plant growth stage or after harvesting at each sampling point, surface organic litter should be cleared away before sampling. After sampling, roots are normally removed manually by hand before oven-drying  and grinding.

Drs.Hepperly and Getachew,I wish to add a few details to what you both mentioned.It is easy to wash roots in sandy or sandy loam  soil but very difficult in clay or clay loam soils. The main problem is to save the fine roots and root hairs during the washing process.After carefully removing the root system along with soil from the required depth they are placed in a tray fitted with a sieve.If the soil is in friable moisture condition a major portion of the soil can easily be removed.Then water jet can be applied carefully to wet the soil and remove the soil adhering to the roots.The detached roots can be collected on the sieve. Also the shoot portion can be kept until washing and after that the root portion can be detached. Then the root system is air- dried and subsequently oven dried at 60o C.Now  root scanners are available to measure several root parameters after washing the roots.

To get a good idea of proportionality could be done by using a bucket soak, adding a mild detergent and gently water jetting as you suggest. After a good idea of this proportionality was found it could be inferred and the root can be estimated from the top ground mass which is easily taken. Different plants have different allocations to their root systems but for a given plant it might not vary substantially under a specific protocol. Root scanners are an eloquent way to address with observation tubes to reveal the root proliferation. The difficulty of working with roots is a real issue and accounts for weakness of data in that area. Unfortunately or fortunately for us we do have constraints in what we are willing to do to understand anything.

I suggest that you soak each soil core in a solution of sodium hexametaphosphate (100 g L-1) for 16 h to disperse soil aggregates (Bolinder et al., 1997, 2002; Moroke et al., 2005; Thivierge et al., 2015). After that, it will be easier to separate roots from soil. Also, the size of the sieve that you will use while cleaning is of primary importance. It must be large enough that soil particles can pass, but not too large to avoid an excessive loss of fine roots. In Böhm (1979. Methods of studying root systems. Springer-Verlag, Berlin), it is stated that a 0.7 mm mesh-size sieve will keep losses under 10% of root mass, which is reasonable when roots are manually cleaned. 

 Suggestion of detergent/dispersing agent and sieve size are important for separating roots from attached/sticking soil and for avoiding loss of fine roots while washing.Good suggestions from Happerly and Marie-Noelle.

Colleagues at Kiel University used a dedicated device ('root washer' - Wurzelwaschanlage) that uses water and compressed air to gently separate soil from OM. Air bubbles penetrate the suspension, and the roots and other OM are collected on a sieve (mesh can be selected according to needs).

For roots length determination, manual counting seems quite efficient, as the time needed to clean the sample to make it fit for an automated scanner outweighs the time required for manual counting. Shape recognition software was also tried, but I don't know how successful it was.

Suggest you try to download the PhD thesis of Ralf Mette, who used that technology extensively; AND - no point to invent the wheel again:

https://www.researchgate.net/publication/35177802_Ertragsstruktur_und_Mineralstoffaufnahme_von_Mais_und_Hafer_im_Einflussbereich_von_Wallhecken_Auswirkungen_auf_den_Mineralstoff-_und_Wasserhaushalt_unter_besonderer_Berucksichtigung_der_Wurzelokologie

Good luck with your research!  

Article Ertragsstruktur und Mineralstoffaufnahme von Mais und Hafer ...

Dear all,

We developed and tested a method to differentiate plant roots from debris by using scanned images and analyzing them with the free software ImageJ. Other software exist, but we looked for a free-license alternative. Although the cleaner (i.e. from debris, litter, etc.) the better, the method showed a good correspondence with the traditional intersection method of Newman. You can see the publication in the link and document attached. I hope you find it useful.

Article Winter cereal root growth and aboveground–belowground biomas...

Dr.Daniel has attached good paper which provides reference to widely cited Newman method and improved method of Bohm(1979).Dr.Daniel and coworkers used the root scanner.As mentioned in the paper root scanner is costly instrument but whoever can afford they can use it .Our institute(IISS ) scientists have  used root scanner successfully. Root-washer  described by Dr.Joska appears to a good tool to wash and collect roots.

Dear Thivierge and all, idea of using dispersing agent for separating root and soil is a great. But I think we also have to consider its effect on cation (Ca, Mg, K) leakage from the root if we are going to analyze them in the root. Is the concentration (100g NaHMP/L water) optimum to avoid such leakage?