## 3 Results
### Structure Characterisation  
Having completed ambient vibration testing on each of the afore-mentioned towers, their natural frequencies were calculated in the laboratory. The measurements were taken in the frequency domain and an example of the results is illustrated in Figure 19.  
Figure 20 shows the comparison made between the analytically estimated natural frequency and the empirical natural frequency identified through ambient vibration testing. In order to express correlation with the geometric characteristics of the towers under assessment, both measurements in this figure are compared to their slenderness.  
Figure 19: Frequency domain measurements from ambient vibration tests performed at (**a**) Sal tower (Casares, Malaga) and (**b**) Guadalmansa tower (Estepona, Malaga).  
Figure 20: Comparison between the estimated natural frequency (blue), measured frequency (orange), and slenderness (green) for each tower.  
Comparative analysis of the results indicates there is no direct correlation between the estimation made through empirical formulation and the results obtained from the tests. General analysis shows that, for 20% of the towers, the measurements are similar, while dispersion above the standard error is found in the rest of the towers. At a deeper level, both the estimated historical period of the construction and its morphology were assessed. Thus, for cube-shaped towers with a slenderness of less than 2.0, the estimated frequency is higher than the measured frequency, except in the Duque tower. The ratio is maintained in cylindrical towers with no exceptions. However, that is not the case for the Moya tower (horseshoe-shaped), due to a fact that relates directly to the massiveness of its construction, where the surface area of the walls on Level 0 is 72% of the total figure and 57% on Level 1.  
When slenderness is compared with the estimated natural frequency (Figure 21a), it is possible to see how the curve varies with the values obtained, which is logical given that the values for frequency were attained by applying the empirical Equation (2). On the other hand, when the ratio between slenderness and the measured natural frequency is compared (Figure 21b), the curve is seen to adapt less that would appear to be related to the state of conservation of the structures under assessment, which include towers with significant structural damage. In fact, structures whose measured natural frequency is lower than would normally be expected given their slenderness tend to suffer from structural damage. Therefore, towers which reveal significant dispersion below the curve imply a greater presence of structural damage that influences their frequency and, therefore, their level of bracing. On the other hand, if dispersion occurs in the upper part of the curve, the structure has a satisfactory safety index against horizontal loads, most likely related to earlier restoration measures in which the original wall structure has been consolidated or reinforced.  
Based on the above analysis, the towers of Urique, Esteril, Manganeto, Lagos, Duque and, to a lesser extent, Tramores do not reach the required level of stiffness in respect of their slenderness, which implies that the wall structure has significant damage affecting its bracing. Of the afore-mentioned towers, Duque is worth highlighting in that it has the greatest variation between its measured natural frequency and what could be expected given its slenderness. Intervention to reinforce its strength structure would at least need to attain a natural frequency over 4.5-5 Hz, thus achieving suitable adjustment to the curve proposed in Figure 21b. In any case, both figures clearly show that the lower the natural frequency, the higher the slenderness of the structure, and therefore that both factors are inversely proportional.