Understanding and managing identification uncertainty of close modes in operational modal analysis

Close modes are much more difficult to identify than well-separated modes and their identification (ID) results often have significantly larger uncertainty or variability. The situation becomes even more challenging in operational modal analysis (OMA), which is currently the most economically viable means for obtaining in-situ dynamic properties of large civil structures and where ID uncertainty management is most needed. To understand ID uncertainty and manage it in field test planning, this work develops the ‘uncertainty law’ for close modes, i.e., closed form analytical expressions for the remaining uncertainty of modal parameters identified using output-only ambient vibration data. The expressions reveal a fundamental definition that quantifies ‘how close is close’ and demystify the roles of various governing factors. The results are verified with synthetic, laboratory and field data. Statistics of governing factors from field data reveal OMA challenges in different situations, now accountable within a coherent probabilistic framework. Recommendations are made for planning ambient vibration tests taking close modes into account. Up to modelling assumptions and the use of probability, the uncertainty law dictates the achievable precision of modal properties regardless of the ID algorithm used. The mathematical theory behind the results in this paper is presented in a companion paper.


Introduction
The modal properties of a structure include primarily the natural frequencies, mode shapes and damping ratios. They are the interface between the physical properties (e.g., stiffness and mass) and response of a structure. Modal identification (ID) aims at back-calculating the modal properties from vibration data. It provides vital information for understanding the as-built characteristics of a structure without tracing back to physical properties whose identification is less well-defined and can be much more challenging depending on the complexity of structural model used. Modal ID is demanded for many downstream applications, e.g., vibration diagnosis, control, model updating [1] and structural health monitoring [2][3] [4]. A comprehensive report on structural system identification of constructed facilities can be found in [5].
The nature of input loading and whether it can be controlled or known (measured) during the test govern the choice of ID algorithm and the achievable ID precision. A traditional means is 'experimental modal analysis' (EMA) [6] [7] where the input is controlled to achieve a good signal-tonoise (s/n) ratio for modal ID. Generating or controlling the input to dominate response is expensive and typically impossible for large structures where the ambient response from fixtures and environment is already difficult to beat. Operational modal analysis (OMA) [8] [9][10] aims at modal ID using 'output-only' vibration data without knowing the input excitations. It significantly improves feasibility and implementation economy, showing great promise for regular practice in the near future. In OMA, the unknown input is typically modelled by a stochastic process with constant spectral properties, e.g., white or band-limited white within the resonance bands of interest. This allows the spectral characteristics of the measured response to be governed by modal properties of interest, making them identifiable and distinguishable from the loading. For a similar reason, vibration modes are intuitively more distinguishable (hence identified) when their frequencies are 'well-separated' than 'closely-spaced'.
There is currently no formal quantitative definition for close modes, but qualitatively their frequencies are so close that their resonance bands overlap, e.g., visually in the power spectral density (PSD) or singular value (SV) spectrum of data [11]. Compared to well-separated modes, close modes are not as common but they do occur and carry significance. They most typically occur in various forms of tower with two or more horizontal axes of symmetry, e.g., tall buildings [12] [13], telecommunication (guyed) masts and freestanding lattice towers [14], cylindrical chimneys [15] [16], space launchers [17] and lighthouses [18]. For tall buildings the stiffness and mass properties along two horizontal principal directions can be very similar by design. For the other structures symmetry and resultant close modes are a natural consequence of the structural form adopted to fulfil their spatial (mode shape) properties by matrix-decomposition of the PSD matrix (or variant), except for uncorrelated modal forces and orthogonal measured mode shapes. Reference [26] discussed a potential over-estimation of damping ratio in FDD (Frequency Domain Decomposition) due to leakage in the estimated PSD. An enhanced PSD through modal filtering was proposed in [27] to improve the estimation of frequencies and damping ratios of close modes, although issues still remain for mode shapes. Bayesian Operational Modal Analysis (BAYOMA) methodology operating in the frequency domain and applicable for close modes has been developed [28]; see [10] for a monograph. The linear algebra and programming effort is much more involved than the wellseparated mode counterpart [29]. See also [30] for a recent development based on expectationmaximisation algorithm that shows promise for simpler algorithm and computer-coding. Time domain algorithms are less directly affected by the presence of close modes. Examples with close modes can be found in [31] [32] for NeXT-ERA (Natural Excitation Eigen-Realisation Technique) and [33][34] [35] for SSI (Stochastic Subspace Identification).
Regardless of ID method, it is commonly perceived that the identification error or uncertainty associated with close modes is significantly higher than well-separated modes, although there is no full account on the actual mechanism or quantification. An empirical study in [36] reveals that the quality of ID results generally deteriorates when the modes are 'close' in the sense that the fractional difference of frequencies normalised by damping ratio is small. This is also evident in a parametric study based on synthetic data identified by BAYOMA [37]. See also [38] and other references mentioned herein.
As part of a research campaign to understand and manage ID uncertainty in OMA, this work develops closed form expressions, referred as 'uncertainty law', that explicitly relate the ID uncertainty of close modes in OMA to test configuration. This contributes to a significant advancement beyond previous work for well-separated modes [39]. For the complexity of the theory involved and to facilitate reading and appreciation of significance, this work is presented in two companion papers. This paper presents the key results and implications, followed by verifications with synthetic, laboratory and field data; and finally recommendations for planning field vibration tests. The mathematical theory is presented in the companion paper [40]. A Bayesian modal ID approach based on the Fast Fourier Transform (FFT) on the resonance band of close modes, i.e., the same context of BAYOMA, is assumed in the derivation. Up to the same (conventional) modelling assumptions, the expressions for ID uncertainty dictate the achievable precision of any other methods because there is a 1-1 correspondence between the time domain data and its FFT (so no loss of information); and probabilistic information in data has been processed in a consistent manner following rigorously Bayes' rules.

Wideband uncertainty law (key theoretical results)
We first summarise the assumptions and key theoretical findings on ID uncertainty of close modes.
They are proven in the companion paper and will be discussed qualitatively in Section 3. Consider two classically damped modes (  'data' in the Bayes' theorem for modal identification; κ is a dimensionless 'bandwidth factor'. This is illustrated in Figure 1. We assume that 1 | | < χ (imperfect modal force coherence) and 1 | | < ρ (linearly independent mode shapes; = ρ modal assurance criterion), for otherwise the problem degenerates and requires a separate formal analysis. Effectively, this work assumes that the subject close modes can be 'detected', e.g., from observation of multiple lines displaying dynamic amplification, as illustrated in Figure 1. The general question of detecting (close) modes is related to, e.g., whether the modes are well excited beyond noise level and whether the measured DOFs allow the mode shapes to be distinguished. While the question of detecting modes is important and is often addressed empirically, its theoretical treatment is out of the present scope.

Figure 1 Schematic diagram showing the theoretical singular value spectrum of ambient data on a resonance band with two close modes
In the above context, we have obtained analytical expressions for the 'remaining', i.e., ID uncertainty, of the natural frequencies, damping ratios and mode shapes identified based on information from data through its FFT in the resonance band. The results are collectively referred as 'uncertainty law' (for close modes). ID uncertainty is quantified in terms of a coefficient of variation (c.o.v. = standard deviation/mean). The expressions relate the ID uncertainty to the 'true' modal properties that are assumed to have given rise to the data. They are 'asymptotic expressions' in the sense that they have been derived for long data ( ; see later). These assumptions, except for wide resonance band, were adopted in previous studies of well-separated modes [39]. Empirically, one may think of, e.g.,

5
> κ and s/n ratio >100 meeting these requirements. Wide band is the condition under which we are currently able to obtain mathematically rigorous and insightful expressions for ID uncertainty of close modes. A mathematically rigorous theory that accounts for the effect of finite bandwidth and s/n ratio has not been developed but these are addressed by empirical factors; see Table 1 in Section 4. Although this work assumes acceleration data in its development, it is also applicable to other data types (e.g., velocity, displacement) provided that the s/n ratio is defined in a consistent manner with the data type; see Section 6.6 in the companion paper.
In reality, the noise PSDs of different channels are never the same but the uncertainty law is robust to this modelling error unless the PSDs differ by orders of magnitude. This is because the uncertainty of the noise PSDs is asymptotically uncorrelated from the remaining parameters. When applying the uncertainty law for cases with a large channel noise disparity, one may use a value of e S with a representative order of magnitude, e.g., the geometric mean. The effect of 'leakage', i.e., smearing of energy over neighbouring frequencies in FFT, is neglected in the scope of uncertainty law because it is asymptotically small for long data.

Mode shape
Mode shape uncertainty is most intriguing, revealing all governing factors and so it is presented first.
As a background, it was found in a recent study [41] that for close modes there are two types of ID uncertainties: one (Type 1) orthogonal to the mode shape subspace (MSS) spanned by the mode shapes, and the other (Type 2) within the MSS. Type 1 was found in well-separated modes but Type 2 is unique to close modes. See Figure 2 for an illustration with 3 = n measured DOFs. It was shown that Type 1 and Type 2 uncertainties are asymptotically uncorrelated (a nice result but not trivial) and so the total variance is simply the sum of their individual variances. Mode shape is a vectorvalued quantity subjected to norm constraint. Its uncertainty can be measured by the 'mode shape c.o.v.', defined as the square root sum of eigenvalues of the mode shape covariance matrix (see Section 11.3 of [10]). For small uncertainty it can be interpreted as the mean value of the hyperangle the uncertain mode shape makes with its mean position.   Table 1 for empirical factors to account for bandwidth and s/n ratio ′ are the c.o.v.s of Type 1 and Type 2, respectively; '~' is to be read mathematically as 'asymptotic to', i.e., the ratio of the two sides tends to 1 under the stated asymptotic conditions, i.e., long data, high s/n ratio, etc. In this work we show (see Section 9 of companion paper) that Type 1 mode shape c.o.v. is given by a product of factors: -small for high s/n ratio -vanish for noiseless data Type 2 (within MSS) -not depend on s/n ratio -prevail even for noiseless data where the influences due to the different factors have been indicated and will be discussed later in Section 3; i γ ′ is the modal s/n ratio defined previously for well-separated modes [39]; a dimensionless data duration as a multiple of natural period; 2 q will be described shortly. On the other hand, Type 2 mode shape c.o.v. is given by a product of factors, all except one different from Type 1 (see Section 6.4 of companion paper for proof): where j refers to the index of the other mode, i.e., is a 'coherence factor' carrying the amplification due to modal force coherence; are 'modal entangling factors' induced by the following 'disparity' parameters that quantify how the two modes differ (in addition to modal force) in damping, frequency and in an overall sense, respectively: The modal entangling factors in (5) and (6) are not intuitive but they carry the mechanism by which frequency and damping disparities mix together with modal force coherence to affect ID uncertainty. See Figure 6 later for a geometric interpretation. The definitions of the above parameters are motivated from the analytical expressions of the c.o.v.s, i.e., they carry fundamental significance instead of being empirically defined. See Table 7 of the companion paper for a summary.

Qualitative analysis and insights
As uncertainty law, (1), (8) and (9) give the leading order value of the remaining uncertainty about the modal properties identified from ambient vibration data under test configuration and environment quantified by various parameters in the formulae. The uncertainty law involves a combination of Bayesian and frequentist concept. The value calculated from the formula is not exactly the value of 'posterior' (i.e., given data) uncertainty in a Bayesian context; it cannot be, since such value should depend on data. However, assuming that the data indeed obeys modelling assumptions and results from some 'true' parameter values (as appearing in the formulae; a frequentist assumption) then for long data, high s/n ratio and wide band their ratio will tend to 1.
Similar to the Laws of Large Numbers in statistics, this is only a theoretical statement because in reality data never obeys modelling assumptions perfectly and 'true' parameter values need not exist in the real world. In a Bayesian perspective, the belief of true parameter values is referred as 'mindprojection fallacy' [42] [43]. However, it is this type of statement that serves the purpose of understanding and managing uncertainty before data is available.

Uncertainty law and Fisher Information Matrix
How is the uncertainty law derived? In the context just mentioned, for long data the posterior covariance matrix (Bayesian) of modal parameters is equal to the inverse of the 'Fisher Information Matrix' (FIM, frequentist). The FIM is a real-symmetric matrix with dimension equal to the number of parameters to be identified in the problem. For the present OMA problem where the FFTs of data are asymptotically independent with a joint complex Gaussian distribution, standard results (e.g., Section 9.4 of [10]) show that the entry in the FIM corresponding to modal parameters x and y is where ) (⋅ tr denotes the 'trace' (i.e., sum of diagonal entries) of the argument matrix; k E is the theoretical data PSD matrix (evaluated at true parameter values) at FFT frequency k f and the sum is over all frequencies in the selected band; see equation (1) for details in the companion paper [40].
This 'exact' FIM is applicable for general situations, i.e., even under non-asymptotic situations of low s/n ratio, limited band, etc. However, it does not offer any insights to serve the purpose of uncertainty law. The formulae in (1), (8) and (9) involve a tremendous effort to obtain the diagonal entries of the inverse of FIM in analytical form, though under asymptotic conditions that have been discovered to allow this possibility which should not be taken for granted. See the companion paper for derivation details.

What difference do close modes make?
For instructional purpose it is useful to review the uncertainty law for well-separated modes so that we can see what difference the close mode problem makes and what factors matter. For wellseparated modes identified with a wide resonance band, the c.o.v.s are given by [39] ) (8) and (9)  So what difference does it make when two modes are close rather than well-separated? For frequencies and damping ratios, (8) and (9)  ). This is a case where the modes are perceived to be clearly distinguishable; and where the 'operational deflection shapes' obtained by matrix decomposition of the data PSD matrix coincide with the physical mode shapes i φ . According which still depends on the disparities in frequencies, damping ratios and modal force PSDs.
Essentially, once we allow the mode shapes to 'trade' within their MSS to 'fit' the data, there is always a component of uncertainty within the MSS that will not vanish even for noiseless data. Such uncertainty is amplified when the subject mode has a lower PSD than the other mode ( or when the two modes get closer (smaller i d ). In Sections 3.3 to 3.5 to follow, we discuss systematically the effect of modal disparity (i.e., how modes differ) and modal force coherence.

Modal disparity
One basic question in the study of close modes is 'How close is close?' Equation (3) reveals that for ID uncertainty the fundamental definition that measures the difference of modes in an overall sense is reflect the difference in frequencies and damping ratios, respectively.
'Disparity' is used as a new term in this work to describe these parameters as they are not simple difference of modal properties. The parameter i e is often used to measure the difference in frequencies, e.g., if the frequency of Mode 2 is at the half-power frequency of Mode 1. The presence of i c reminds that the difference in damping ratios does make the modes different. For close modes it is approximately the fractional difference between the damping ratios, i.e., The theory shows that Pythagoras theorem applies to encapsulate the effect of difference in frequencies and damping ratios in i d on ID uncertainty, which is a nice result but hardly trivial. Figure 4 illustrates how two modes with different disparities may appear by plotting their dynamic amplification factors (between modal force and modal acceleration), assuming identical damping ratios. These plots are akin to PSDs of ambient data. They suggest that a disparity of the order of 1 may be considered close while a disparity of 10 is clearly well-separated. A disparity of 0.5 is considered very close. It does happen in field cases; see Table 3 later.
, the same for well-separated modes (Type 1 uncertainty). Intuitively, decreasing damping increases the modal s/n ratio and frequency disparity, which reduces Type 1 and Type 2 mode shape uncertainties, respectively.

Coherence of modal forces
The coherence χ between modal forces is their correlation in the frequency domain. From first principles, if the modal force coherence is zero then the FFTs of the two modal responses will be uncorrelated. Modal force coherence mixes with the disparity parameters to affect the ID uncertainty of all modal properties in a non-trivial manner through the coherence factors i (4), (10) and (11) (6), though in a somewhat non-trivial manner.

Bounds on coherence factors
For mode shapes, if increasing with | | χ but otherwise this is generally not the case. It is shown in Section 10 (appendix) The lower bound of 1 implies that modal force coherence χ always amplifies Type 2 mode shape uncertainty. For frequencies and damping ratios, the coherence factors i f Q and i Q ζ in (10) and (11) also depend on the phase angles ψ , φ and i φ through the factors i f R and i R ζ in (12) and (13). Such dependence is trigonometric in nature and is of less significance than those on 2 q and | | χ . It is shown in Section 11 (appendix) that Substituting into (10) or (11), and simplifying gives Simpler (but looser) bounds that depend only on | | χ can be obtained by further taking 0 2 = q on the lower bound and 1 2 = q on the upper bound:

Effect of disparity on modal entangling factors i q
The effect of disparity parameters ( i e , i c ) on the modal entangling factors i q is obscured by their relationship with another two entangling factors i g in (6), on which i q in (5) depends. Generally, increasing disparity reduces the magnitude of i q and hence the influence of coherence, which is intuitively correct. It can be shown by direct algebra that the following identity holds:

MAC effect
From first glance the effect of MAC ρ on ID uncertainty is somewhat counter-intuitive. Equation (8) and (9) (8) and (9) only give the leading order uncertainty. Further evidence reveals that the first order uncertainty does deteriorate with increased MAC through a s/n ratio discounted by ) 1 ( 2 ρ − ; see (26) and Table 1 in Section 4 later. On the other hand, a higher MAC means that the two mode shapes are closer to each other, which means that there is a lesser extent to which they can 'trade' (by rotating towards one another) to 'fit' data (give similar likelihood), and hence smaller uncertainty.
Note that the case 1 = ρ is excluded from discussion because then the problem degenerates and the present theory does not apply.

Zero disparity and mode shape identifiability
One important implication from the uncertainty law is that the c.o.v.s of frequency and damping ratios in (8) and (9)  , and so the scaled FFT of data becomes  The ideal scenario of zero disparity is discussed here to illustrate identifiability, although it is almost impossible in reality because it is very sensitive to structural configuration. Equation (3) shows that as long as disparity is non-zero it is still possible to identify mode shapes but the required data length can be significantly longer than that for well-separated modes when disparity is small (even for noiseless data). This governs the achievable identification precision of close modes. It should be noted that the issue of disparity discussed here is related to the temporal/frequency rather than spatial aspect of response/data. It does not have a direct linkage with observability that is often discussed in the system identification literature.

Bandwidth and s/n ratio effect
The results in Section 2 assume that the resonance band for modal ID is sufficiently wide, in the sense that Finite bandwidth and s/n ratio encountered in reality do make a difference to ID uncertainty especially when they are not wide/high. Addressing these two issues in a mathematically rigorous manner is another challenge that is left for the future. In this work, 'correction factors' κ A and γ A in Table 1 In Table 1, are respectively the bandwidth factor and modal s/n ratio for the subject mode with natural frequency f , damping ratio ζ and modal force PSD S (mode index omitted for simplicity); and Check that these mode shapes have unit norm and their MAC is ρ , which is set to be 0.5. The data has a duration of 1000 sec and a high s/n ratio of Figure 7 shows the values (cross) based on the empirical correction factors in Table 1 versus the 'exact' values based on the inverse of the exact form of Fisher Information Matrix (FIM) evaluated at the 'true' parameter values assumed here, i.e., (14). The latter involves no assumption on bandwidth or s/n ratio, although it does not yield any insight because of its implicit form. The high s/n ratio and wide band asymptotic values (blue dots) are also shown for comparison. They are based on (1), (8) and (9)

Verification and applications
In this section the uncertainty law of close modes developed in this work is investigated with synthetic, laboratory and field data. Six cases are considered and summarised in Table 2; see also  Table 1 in the real setting. They also allow us to develop insights into the mechanism that gives rise to the ID uncertainty by investigating the statistics of governing factors. Laboratory data was collected with piezoelectric accelerometers with channel noise in the order of 10 micro-/√ . Field data was collected with servo-accelerometers with channel noise in the order of 1 micro-/√ .   Figure 11 and Figure 12 Building B in [12] 30 min./set x 72 set = 36 h at 50Hz 3 DOFs, xyz near core on roof; see Fig.3(b) in [11] 300m+ tall commercial building, very close modes, during typhoon, low to high amplitude = κ 2 -20+, s/n ratio = 50 -10 4 + Figure 13 and Figure 14 Eddystone Lighthouse [18] 10 min./set x 60 sets = 10 h at 128Hz 4 DOFs, xy at two levels between 1/2 and 2/3 height; see Fig.8 in [18] 49m tower, Helipad acts like a TMD, complicated wind/seawave environment = κ 5 -20, s/n ratio = 200 -10 4 + Figure 15 and Figure 16 Jiangyin Bridge [44] 3200sec/set x 13 sets = 11.6h at 25.6Hz Rugeley Chimney [16] 1200sec/set x 138 sets = 46h at 8Hz 4 DOFs, xy (radial and tangential) at two levels, near top and 1/4 height; see Fig.3 in [16] 183m chimney with TMD, under normal wind condition, apparently significant deviation from classical damping = κ 3 -20+, s/n ratio = 1000 -10 4 + Figure 19 and Figure 20

Synthetic data
The synthetic data features a moderately high s/n ratio (>1000), wide band ( 10 > κ ) and long data (about 1000 natural periods in each set). One hundred data sets with different modal properties are randomly generated to cover a variety of scenarios. Figure 10 In Figure 10 1), (8) and (9)) modified by the factors in Table 1 to account for finite bandwidth and s/n ratio. Their values are shown as crosses ('x') in Figure 10(a)-(c). They represent the best effort of this work to explain the ID uncertainty of close modes. They agree with the red circles, effectively verifying the mathematical correctness of the wide band law.
As a remark, if the data used is long and it is indeed distributed as the same likelihood function as in BAYOMA/FIM, i.e., no modelling error (as is possible for synthetic data here), the BAYOMA value (xaxis) will theoretically converge (in a statistical sense) to the exact FIM value (y-axis, red circle). In this sense the exact FIM value is the closest analytical value one can get to match the BAYOMA value; see [45] for a further discussion. However, this convergence is only a theoretical statement which can at best be expected from synthetic data because no model is perfect for experimental data. This aspect of convergence is only relevant in the verification of mathematical correctness (at the research stage) of the exact FIM or uncertainty law where synthetic data must be used. It is irrelevant to the intended application of uncertainty law, however, which is to understand and manage ID uncertainty for planning tests where no data is available.  (7). In all the six cases considered here as well as in typical applications, Type 1 mode shape uncertainty (i.e., orthogonal to both mode shapes; the only type for well-separated modes) is negligible and so the mode shape c.o.v. shown in the plots is effectively of Type 2, i.e., with uncertain directions within the subspace spanned by the two mode shapes. The points exhibit a general decreasing trend, which is consistent with . The scatter is due to variations in other properties among the data sets, e.g., modal force coherence and MAC.  (21). This is demonstrated in the plot. Finally, Figure 10(f) shows the values of | | χ and ρ (MAC) among the data sets. For the synthetic data sets here they are distributed uniformly merely because of the way they are generated. For the laboratory and field cases later they reflect statistics in the corresponding situations.

Laboratory and field data
We now discuss the results of the laboratory and field data in a collective manner w.r.t. different aspects. Figure 11, Figure 13, Figure 15, Figure 17 and Figure 19 show the spectra (PSD and SV) of a typical data set in each case. The results analogous to Figure 10 are summarised in Figure 12, Figure   14, Figure 16, Figure 18 and Figure 20.
The cases collectively cover low to high s/n ratios, from a few tens to over ten thousand. The laboratory shear frame is intended to provide an experimental case under controlled environment.
Rugeley Chimney provides a case with obvious violation of modelling error, i.e., non-classical damping due to tuned mass damper (TMD). Modal ID of the field structures has been studied previously; see references in the first column of Table 2. The current investigation provides an opportunity to understand their ID uncertainties. The tall building, lighthouse and chimney have close fundamental modes that govern their vibration response, giving compelling reasons for their proper identification and understanding. The lighthouse data is unconventional; obtaining it is a challenge in itself.
On the verification side, in the plots (a)-(c) of Figure 12, Figure 14, Figure 16, Figure 18 and Figure 20, the crosses roughly match with the red circles, suggesting that the proposed formulae (wide band expressions with empirical factors) can give a good match with what can be best achieved (exact FIM). Outliers do exist, e.g., for laboratory frame (one point in Figure 12(c)). The amount of scattering in the crosses and circles about the 1:1 line is similar in all cases except for Rugeley Chimney, which is a special case with modelling error to be discussed later. Similar to the case of synthetic data, the green dots (well-separated modes law) in plot (c) fall below the 1:1 line by orders of magnitude, showing that they fail to explain the ID uncertainty of mode shapes of close modes.
They perform better on the frequencies and damping ratios (plots (b) and (c)), typically attaining the right order of magnitude. Table 3 gives a summary of the statistics of the identified (MPV) damping ratio, disparity, coherence and MAC between the two modes in each band. It can be examined together with plots (d)-(f) in Figure 12, Figure 14, Figure 16, Figure 18 and Figure 20. Damping ratio ranges from 0.5% to a few percent, which is typical. Rugeley Chimney is an exception, with values of 10% for some data sets that can be potentially erroneous because of modelling error (see Section 5.3). The laboratory frame was designed to have identical stiffness along the two horizontal axes, although the ID results indicated that the field structures have even lower disparity. Disparity has a mean value of the order of 1 and a low value around 0.5. Data sets with low disparity are associated with high mode shape c.o.v. and can present challenge for modal ID. See for example the left end of Figure 14(d) for the tall building and Figure 16 for the lighthouse. Coherence and MAC are typically not high, except for a small number of data sets that can give values as high as 0.85, e.g., the tall building. Those are associated with a high mode shape c.o.v., however.

Effect of modelling error
Uncertainty law is intended to explain ID uncertainty assuming that the data behaves according to modelling assumptions. Logically when there is significant deviation from modelling assumptions it need not serve the intended purpose. Rugeley Chimney had a TMD installed at the top when the data was collected. It provides a case with apparent modelling error regarding classical damping. The PSD and SV spectra in Figure 19 have a hump on the left side of the natural frequencies, which is judged to be attributed to the action of the TMD. The presence and extent of the hump change from one data set to another, presumably as the TMD action changes in direction and extent. Ideally the TMD introduces two additional modes (along two horizontal directions) to the structure but since the DOFs at the TMD are not measured (as is typical), it is often not possible to distinguish the TMD modes. Thus, only the two modes with very close frequencies are identified in the band assuming classical damping (as in BAYOMA). This clearly induces modelling error, although the effect is unknown. The crosses and red circles in Figure 20 Figure 12, Figure 14, Figure 16 and Figure 18. This is especially so for the damping ratio and is believed to be associated with modelling error.

Practical implication and recommendation
Well-separated modes are conventional subjects in modal ID. A logical way to think about the implications of uncertainty law of close modes developed in this work is to see what concepts or requirements need to be adjusted/introduced beyond those already in place for well-separated modes [39]. This is how the c.o.v.s of frequencies and damping ratios in (8) and (9) have been written. Close modes bring in the coherence factors i f Q and i Q ζ in (10) and (11). It is more useful to think of the coherence factors in terms of the bounds in (21); see also Figure 5. Further correction to capture the effect of bandwidth and s/n ratio is needed. This can be done using the empirical factors in Table 1, where the modal s/n ratio i γ ′ ′ is equal to the old one for well-separated modes Close modes bring additional uncertain dimensions to mode shapes and this overturns our intuition about the governing uncertainty accumulated for well-separated modes. Mode shape uncertainty is no longer negligible. It can even render the problem unidentifiable. For well-separated modes it is always orthogonal to the identified mode shape direction (Type 1, see (2) and Figure 2) and is negligible for high s/n ratio. For close modes, Type 1 uncertainty remains to be negligible for high s/n ratio, but the additional non-vanishing uncertainty (Type 2, see (3)) smearing between mode shapes is of the same order of magnitude as or even larger than damping uncertainty. Based on (3) ) and coherence ( i Q φ ). It is useful to think of i Q φ in terms of its upper bound, which coincides with those of frequency and damping in Figure 5.
Accordingly, doubling the c.o.v. will account for the effect of coherence in most cases.

Planning for well-separated modes -what we already knew
Uncertainty law for well-separated modes was developed in [39] to allow one to manage quantitatively the ID uncertainty. In this case damping uncertainty is the governing factor and its c.o.v. is given by where ζ is the damping ratio (mode number omitted), c N is the dimensionless duration as a multiple of natural periods, e.g., a duration of 100 sec for a 2Hz mode gives

Planning for close modes -what we did not know
Regardless of whether one has planned for close modes in ambient vibration tests, they can be encountered and present challenge to modal analysts. The knowledge generated in this work allows one to plan with a strong scientific basis. To cater for close modes, both damping and mode shape uncertainty need to be assessed. Equations (3) and (9) offer insights but their direct use requires too much detail and is not suitable for planning. Simple provisions are recommended here as an extension of those for well-separated modes. The damping c.o.v. can still be assessed using (29) but now with two modifications: 1) the value should be amplified by ζ Q in (11) (omitted mode index i ) to account for coherence; and 2) the s/n ratio for evaluating γ A and κ A (via κ ) in Table 1 should be (26). The data duration (as a multiple of natural periods) required to achieve a c.o.v. of ζ δ in the damping ratio is given by Without specific information on | | χ and ρ , their choice is a compromise between practicality and conservatism. Figure 5 suggests that taking 4 = ζ Q will be conservative for | | χ up to 0.85, but this implies an inflation of four times in the data length compared to that without coherence effect.
More practical solution may be achieved at the expense of slightly reduced conservatism, e.g., taking 2 = ζ Q will allow for | | χ up to 0.7. Remarkably, taking to allow for | | χ up 0.5, essentially because the bounding curve in Figure 5 is flat for small | | χ . See

Example
Consider ambient vibration test planning where the data duration is often governed by the precision in the damping ratio of the mode with the lowest frequency. We shall first assume that the mode is well-separated and then see later the additional duration required to allow for the possibility of close modes. For the purpose of determining the data length, assume a damping ratio of 1% and a required c.o.v. of 30%, which represents a moderate precision. The first term in (30) gives the minimum required duration as  natural periods. Using more or better sensors will not reduce this duration significantly because the s/n ratio is already high enough. The above procedure was proposed and related issues were discussed previously in the work of uncertainty law for wellseparated modes [39], which may be consulted for further details and examples.
What provision should be made to account for the possibility of two close modes? In the first place, one should examine the adequacy of senor locations (or directions if uniaxial) for distinguishing the mode shapes of the two potentially close modes, so that the singular value spectrum of data will have two significant lines displaying dynamic amplification, reflecting a two-dimensional mode shape subspace. The following discussion assumes that the mode shapes are distinguishable; otherwise it is out of the scope of this work. Taking nominally  (taken to be the same as damping c.o.v.), which is marginally acceptable; see Figure 3. Better quality in the mode shape with a c.o.v. of 10% will require 9 3 2 = times as much the data length, i.e., 3159 9 351 = × natural periods. Depending on the natural period, this duration may be too long to be practical or it may incur significant modelling error in data stationarity or time invariance in structural properties. Here we see that for close modes it is not practical to demand the same level of mode shape precision as for well-separated modes (c.o.v. often below 1%). There is also less room for conservatism, e.g., allowing for modal coherence up to 0.85 will require a further inflation in data

Conclusions
This work has made discoveries that allow one to understand the identification (ID) uncertainty of close modes in operational modal analysis (OMA) and manage it in ambient vibration tests. The asymptotic formulae for ID uncertainty reveal explicitly the effect of governing factors including the disparities in frequencies, damping ratios, modal force PSDs and their coherence, and mode shapes; see (8), (9) and (1). Mode shape uncertainty is most intriguing, extending into dimensions unique to close modes and prevailing even with noiseless data, therefore posing a new precision limit on OMA distinguished from the previously found limit for well-separated modes.
The mathematical theory for the uncertainty law of close modes is much more complicated than that for well-separated modes; see the companion paper [40]. The ID uncertainty admits a remarkably simple and insightful mathematical form when the resonance band containing information for identification is sufficiently wide. It has not been possible to derive mathematically rigorous formulae to capture the effects of finite bandwidth and signal-to-noise ratio. Instead, they are addressed by empirical correction factors; see Table 1. The investigation with field data reveals the source and mechanism of close mode uncertainty under various field situations. Modal disparities are of the order of 1 with low values around 0.5; see Table 3 for other statistics. In addition to damping, planning field tests considering close modes also involves managing mode shape uncertainty; see Section 6 for simple recommendations.
Some remarks are in order. Uncertainty laws are intended for understanding achievable precision limits and managing ID uncertainty in test planning where data is not available. When data is available the ID uncertainty should be calculated by a modal ID algorithm (e.g., BAYOMA) based on the particular data set. Uncertainty laws are developed based on the same set of modelling assumptions in the modal ID algorithm (except for the asymptotic conditions), and so they do not reflect modelling error. The latter should be judged or controlled by other means, e.g., avoiding excessively long time windows to justify stationarity. Although acceleration data is often referred, this work is also applicable to other data types (e.g., velocity, displacement) provided that the s/n ratio is defined in a consistent manner; see Section 6.6 in the companion paper.

Acknowledgements
This work is part of a research project funded by the UK Engineering and Physical Sciences Research Council (EPSRC) on "Uncertainty quantification and management in ambient modal identification" (grant EP/N017897/1 and EP/N017803) to understand ID uncertainty and provide a strong scientific basis for implementing and planning ambient vibration tests; and STORMLAMP (grant EP/N022947/1 and EP/N022955/1) that obtained the lighthouse data. Part of the work was performed when the first and third author (SKA & BBL) were with the University of Liverpool.

Appendix. Supplementary data
The research materials supporting this publication can be accessed by contacting ivanau@ntu.edu.sg.