Between the cuantitative variables we have: total height of the tree, diameter of the crown, the height of the bole (or trunk), the length of the crown over the street. the angle between the vertical and the trunk, etc.

Between the qualitative variables we have: The Genus sp. The sanitary situation of the leaves, the sanitary situation of the tree. type of sidewalk, etc.

This sounds like an interesting project. 

You could use your qualitative variables for stratification, and the quantitative variables from the past census as regressor data for the more current sample variables of interest. Model-based estimation, ie, prediction, in such cases, if well stratified, I found to work very well in industrial/business applications, even for cutoff sampling. Since you have random sampling here, it should work as well, or better. The model-based classical ratio estimator is very robust for many uses in survey statistics and simple econometric applications. Here you could consider multiple regression if you do not have collinearity issues. Perhaps just one or two good regressors, for each current variable being estimated, may be best. An advantage here with prediction is that even if some estimates you decide to make were not really randomly collected in your design, and/or you have additional data collected on one variable, while really trying to collect data on another, it is still useful. 

You have an unusual problem with time frames here. Often, in official statistics, there is a periodic, but infrequent census, perhaps annually taken, and a more frequent sample, say monthly, of at least some of the same data elements (variables of interest). We know that these data collections are not instantaneous, and that things change during these data collections, and that the sample results represent a point in time that is no longer current when evaluated, but this is usually not such a serious problem. In your case, it is. This would be a source of measurement error, and that is often not considered as much as it should be. In particular, here your regressor data (from the census) was collected over several years. One generally considers the regressor data without its own error term (though I think Wayne Fuller looked at this a good deal), and the regressor data would be expected to represent one instance in time per stratum/model application. Managing your data may be no small task. If you know that one section of trees were all examined in a short time period, and you know which trees in the corresponding sample are from that section, then that could be added to your qualitative variables in determining the stratifications to use. 

Note that the variance of the prediction error is available in statistical software. For example, in SAS PROC REG, the square root of that is STDI.   To obtain the estimated variannce or estimated standard error of the prediction error for totals is slightly more involved, but obtainable. For example, "Option #1" in the paper at the link below, page 10, I think, for the classical ratio estimator, can use statistical packages to calculate part of the estimator, or it can all be done by a few summations. That us, if you have something you want to total. Otherwise, you might just be interested in b and V(b). 

Sorry if I just repeated too many things that you already know and tried.  I hope your project is going well.  

Article Some Proposed Optional Estimators for Totals and their Relat...

Guillermo -

By the way, I should mention this possibility: model-assisted design-based sampling and estimation. 

If you have a census of the population to supply regressor (auxiliary) data, and a current randomized sampling design, you might want to look into a book such as the following, for next time anyway:

Särndal, CE, Swensson, B. and Wretman, J. (1992), Model Assisted Survey Sampling, Springer-Verlang.     

Cheers - Jim

Also, there is some good information on cluster sampling which  you may like to see in Cochran, W.G(1977), Sampling Techniques, 3rd ed., John Wiley & Sons.  -  Cochran does a good job in general.  -  I found material around pages 249 and 257 to be quite interesting.  -  Cluster sampling may be found in a number of sources, but Cochran is still a good one. 

... 

If you are looking at stratified random sampling, remember that optional allocations are for estimating at the more aggregate level.  If your strata are say, species, and you want to report by species, then these aren't really only strata, but subpopulations for which you also want to report.  They can still be treated as strata when estimating at the more aggregate level, but you also need adequate sampling for each subpopulation.  

...  

If you are looking into count data, perhaps the following might possibly be of some interest: 

https://onlinecourses.science.psu.edu/stat504/node/57

Dear James

Thank you, very much for all your input.

This proyect was a very interesting and large one, and I took in account your first inputs, but it was interrupted after my inform of 2014, because the lack of interest of the municipal authorities who were changed at the end of the last year.

If the new authorities decide to resume the project, I would gladly try new approaches such as those suggested by you. I know well Cochran (1977) but I have not Särndal, CE, Swensson, B. and Wretman, J. (1992), and I will try to the get it.

It sounds like a great project.  I hope it is resumed.  -  Best wishes.  

Querido Guillermo:

Te sugiero usar estos estimadores basados en el estadístico media-de-las-razones, alternativos al estimador de razón pero además insesgados y con la característica añadida de que en algunos casos se puede estimar la varianza del estimador de modo insesgado.

Guillermo -

Whether you want to use a mean of ratios or a classical ratio estimator depends upon the degree of heteroscedasticity present, and you need a good 'size' measure, x.  Further, see Särndal, C.-E., Swensson, B. and Wretman, J. (1992), (SSW), Model Assisted Survey Sampling,

Springer-Verlag, page 254* regarding optimal probability of selection for given degrees of heteroscedasticity. 

Those two alternatives represent the two extremes in heteroscedasticity to be expected in many situations.  (See Brewer, K.R.W. (2002), Combined survey sampling inference: Weighing Basu's elephants,

Arnold: London and Oxford University Press, page 111*.) 

Thus the mean of ratios estimator goes well with probability proportional to size sampling, and is most appropriate with a high degree of heteroscedasticity.  For the classic example of this "alternative ratio estimator," (SSW), see "On a population sample for Greece," by Raymond J Jessen, Richard H Blythe Jr, Oscar Kempthorne, W Edwards Deming, Journal of the American Statistical Association 42 (239), 357-384, 1947,

 https://scholar.google.com/scholar?hl=en&as_sdt=0%2C47&q=greek+population+survey+deming&btnG=#d=gs_qabs&p=&u=%23p%3DSJjYL5OHsGkJ

The coefficient of heteroscedasticity can be estimated from the data.  One method is the Iterated Reweighted Least Squares (IRLS) method.  Also, see https://www.researchgate.net/publication/263032446_Weighting_in_Regression_for_Use_in_Survey_Methodology.

Experimenting to see which sampling and estimation methodologies work best could be very interesting. Note that nonsampling error may have varying impacts.

........................................

*Noted in https://www.researchgate.net/publication/319434665_Comparison_of_Model-Based_to_Design-Based_Ratio_Estimators

Cheers - Jim

Without reflecting on the comprehensive comments provided by James Knaub, I might add separate work by the US Forestry Statistical Service on systematic forest sampling - their experience over many years - building on the framework mentioned above. More specifically work by Dr Raymond Czaplewski and colleagues involving use of clustered area sampling combined with model assistance and the application of a multivariate Kalman filter. While this is used at a continent scale, it may be interesting to read it in the context of your municipal example.

Ray's monograph: Czaplewski, R. L. 2010 Complex Sample Survey Estimation in Static State-Space. Fort Collins: Gen. Tech. Rep. RMRS-GTR-239, US Dept of Agriculture Forest Service: is one place to start, but there are more recent publications. You may also want to try out an application he developed and is currently under test: Czaplewski, R. L. 2017 KFz Algorithm: A Sequential Kalman Filter Estimator for Large State and Measurement Vectors and Rank Deficient Innovation Covariance Matrix. Preprint. Available at http://www.environmetrika.org/publications-1.html

I got a file error when I tried to open the file above, but that might just be my phone. At any rate, note that Raymond Czaplewski is also on ResearchGate, so there may be more literature available there.

Thank you Stephen, James and Mariano for your important input!

Guillermo -

If you keep this to just randomly sampling clusters that you then census, that would seem to be a straightforward case of single-stage cluster sampling, covered, for example, in Chapter 9 of Sampling Techniques, 3rd ed, 1977, Wiley, by W.G. Cochran.  The elements of each cluster, instead of being data taken from discrete numbers of people or stores or manufacturers, could be subunits of a plot, as in the case of crops, or maybe they could be individual trees. 

In sections 9.3 through 9.7, pages 238 - 247, Cochran  discussed cluster size in relation to variance.  Of special interest, please note section 9.5, "Variance Functions."  [The value of g there (2 gamma in other contexts) is often around 1.5 (and generally between 1 and 2).]  You might find that part of Cochran's book to be of particular interest. 

You have to consider sample size in terms of both how large the clusters should be, and how many clusters you want.  As you can see in the above noted sections of Cochran's book, this can offer you some complex considerations.  Other textbooks, such as (a) Sampling, 3rd ed, Steven K. Thompson, 2012, Wiley, (b) Sampling: Design and Analysis, 2nd ed, Sharon Lohr, 2010, Brooks/Cole, and (c) Model Assisted Survey Sampling, Särndal, Swensson, and Wretman, 1992, Springer-Verlang, like Cochran's Sampling Techniques, show the breakdown of variance into between-primary-unit, and within-primary-unit considerations.  However, Cochran's section 9.5, "Variance Functions," shows harder to find references to earlier work, relating the within cluster variance to the size of the cluster. That might be of substantial interest to your project.  Empirical work with agricultural surveys, noted on page 243 in Cochran, relates to heteroscedasticity as explained in https://www.researchgate.net/publication/320853387_Essential_Heteroscedasticity, which expands on a concept in a book by Ken Brewer, as referenced in that paper. 

As noted above, you have to consider sample size in terms of both how large the clusters should be (say, plot size), and how many clusters (plots) you want to sample.  That can be an interesting challenge.  A pilot study might help a great deal, I think.

Cheers - Jim