As previously outlined, CPT soil classifications are carried out by utilizing SBT charts where measured q_c and f_s values are empirically related to type of soil. The popular SBT chart developed by Robertson et al. [2] is based on q_t and friction ratio, R_f. Measured cone bearing and sleeve values are blurred/averaged. It is required to apply optimal estimation algorithms so that the effect of blurring/averaging is minimized.

The cone tip resistance measured at a particular depth is affected by the values above and below the depth of interest which results in an averaging or blurring of the true values (q_v) values [8] [9] [10] [11]. This phenomenon is especially of concern when mapping thin soil layers which is critical for liquefaction assessment. Mathematically the measured cone tip resistance q_c is described as [9] [10] [11].

q c ( d ) = ∑ j = 1 60 × ( C d Δ ) w c ( j ) × q v ( Δ q c + j ) + v ( d )

Δ q c = ( d − Δ w c ) , Δ w c = 30 × ( d c Δ ) (1)

where

d: the cone depth

d_c: the cone tip diameter

Δ: the q_c sampling rate

q_c(d): the measured cone penetration tip resistance

q_v(d): the true cone penetration tip resistance

w_c(d): the q_v(d) averaging function

v(d): additive noise, generally taken to be white with a Gaussian pdf

In Equation (1) it assumed that w_c averages q_t over 60 cone diameters centered at the cone tip. Boulanger and DeJong [8] outline how to calculate w_c below (after correcting the equation for w₁ [9] ):

w c = w 1 w 2 ∑ w 1 w 2 (2a)

w 1 = C 1 1 + | ( z ′ z ′ 50 ) m z | (2b)

w 2 = 2 1 + ( q t , z ′ q t , z ′ = 0 ) m q (2c)

where

w₁: accounts for the relative influence of any soil decreasing with increasing distance from the cone tip.

w₂: adjusts the relative influence that soils away from the cone tip will have on the penetration resistance based on whether those soils are stronger or weaker.

z ′ : the depth relative to the cone tip normalized by the cone diameter.

z ′ 50 : the normalized depth relative to the cone tip where w₁ = 0.5C₁.

C₁: equal to unity for points below the cone tip, and linearly reduces to a value of 0.5 for points located more than 4 cone diameters above the cone tip.

m_z: exponent that adjusts the variation of w₁ with z ′ .

m_q: exponent that adjusts the variation of w₂ with ( q v , z ′ q v , z ′ = 0 ) .

Boulanger and DeJong [9] provide a thorough outline and review on the setting of the parameters given in Equation (2) based upon extensive research and modelling. In general terms, soils in front of the cone tip have a greater influence on penetration resistance than the soils behind the cone tip. In the subsequently outlined test bed simulations the parameters in Equation (2) are set identical to those outlined by Boulanger and DeJong. In this case, exponents m_q = 2 and m_z = 3.

Baziw and Verbeek [9] [10] [11] developed an algorithm to optimally obtain true q_v cone bearing estimates from blurred measurements q_c. The initial algorithm developed by Baziw and Verbeek [9] [10] (the so called q_cHMM-IFM) combined a Bayesian recursive estimation (BRE) Hidden Markov Model (HMM) filter with Iterative Forward Modelling (IFM) parameter estimation in a smoother formulation. In recent modifications and enhancements of the q_cHMM [11] it was possible to drop the IFM portion of the algorithm. This was predominantly accomplished by refining the HMM filter parameters.

In CPT, sleeve friction is the measure of the average skin friction as the probe is advanced through the soil. Figure 4 outlines typical sleeve friction resistance and distribution generated by an algorithm (ABAQUS) which implements a Finite Element Model (FEM) designed for modelling large displacements such as those generated during CPT [12]. Figure 4(a) illustrates typical FEM sleeve friction resistance at the center of the sleeve. The high frequency fluctuations shown in Figure 4(a) are a type of measurement noise generated by the FEM algorithm due to the mesh size, the contact interface, and the parameter of soil and steel.

Figure 4(b) illustrates the FEM distribution of resistance along the length of the sleeve. In Figure 4(b) the sleeve friction close to cone tip is nearly 0 MPa and gradually increases to the uniform value of 0.029 MPa at approximately 30 mm from the bottom of the shaft for the case ϕ = 34˚ and σ ′ v o = 0.05 MPa. Susila and Hryciw [12] state that non-uniform sleeve friction distribution has been confirmed by Kiousis et al. [13]. Kiousis et al. state that there is a very thin separation between soil and cone shaft for approximately 35 mm above the upper end of the cone tip.

The sleeve friction distribution illustrated in Figure 4(b) can be thought of as a Sleeve Friction Weighting Function (SFWF) where various values of sleeve friction along the shaft (due to varying soils) are weighted to give a final measured value assumed to occur at the center of the shaft. The distribution illustrated in Figure 4(b) is mathematically approximated by Equation (3). Figure 5 illustrates the implementation of Equation (3).

SFWF ( z * ) = 1     for     z * > 30   mm (3a)

SFWF ( z * ) = a b s ( z * − 30 ) 3 / 30 3     for     z ′ ≤ 30   mm (3b)

where

z * = the distance from bottom of sleeve

SFWF = Sleeve Friction Weighting Function

The weighting of the true sleeve friction values by the SFWF coupled with blurring of the q_v values can result in significant distortions in the calculated friction ratio R_f. Figure 6 illustrates a simulation of cone bearing, sleeve friction and friction ratio (it assumed that both q_c and f_c have been corrected for pore pressure). In Figure 6, the true values of q_v, fv and R_fv are red traces while the corresponding measured values are the black traces. Table 1 outlines the corresponding soil behavior types (based on the SBT chart of Figure 3) for the test bed simulation illustrated in Figure 6.

The sleeve friction measurements f_t were generated from the true sleeve friction values f_v by implementing Equation (4) outlined below.

f t ( i ) = ∑ j = 1 L * SFWF ( j ) × f v ( i − l * + j ) (4)

where

Δ: sleeve friction sampling rate

L: sleeve friction shaft length weaker

L*: L/Δ

l: l/2

l*: l/Δ

When implementing Equation (4), Δ is initially set to a 1 mm sampling rate. The simulated date sets are then obtained by extracting data from the 1 mm sampling rate data sets at the user specified rate. This is done so that the true in-situ measurement conditions are simulated.

As is illustrated in Figure 6(c) there are significant distortions in the simulated measured friction ratios based upon the measured cone bearing and sleeve friction values. This leads to uncertainties in soil classifications. This paper outlines an optimal sleeve friction estimation algorithm. The sleeve friction optimal estimation implemented in conjunction with the q_cHMM algorithm facilitates obtaining accurate soil classification estimates.

The OSFE-HMM algorithm utilizes a posteriori information from the q_cHMM algorithm so that the solution space is reduced. The q_cHMM algorithm facilities quantifying the soil layering (i.e., layer interfaces). This soil layering information is inputted into the OSFE-HMM algorithm. Soil layering can readily be quantified based upon estimated q_v values. Figure 7 illustrates Figure 6(a) where the soil layers (L₁ to L_N) are identified by blue lines and were determined from the output from the q_cHMM algorithm (red lines). Each of these soil layers has an associated sleeve friction values f_v1 to f_vN which needs to be estimated.

The second component of the OSFE-HMM algorithm implements Iterative Forward Modelling (IFM) to estimate the sleeve friction values f_v1 to f_vN. IFM is a parameter estimation technique which is based upon iteratively adjusting the parameters until a user specified cost function is minimized. The desired parameter estimates are defined as those which minimize the user specified cost function. The IFM technique which is utilized within the OSFE-HMM algorithm is the downhill simplex method (DSM) originally developed by Nelder and Mead

[14]. The DSM in multidimensions has the important property of not requiring derivatives of function evaluations and it can minimize nonlinear-functions of more than one independent variable. A simplex defines the most elementary geometric figure of a given dimension: a line in one dimension, the triangle in two dimensions, the tetrahedron in three, etc.; therefore, in an N-dimensional space, the simplex is a geometric figure that consists of N + 1 fully interconnected vertices. The DSM has been used in a variety of scientific applications such as obtaining seismic source locations [15] and blind seismic deconvolution [16].

The DSM starts at N + 1 vertices that form the initial simplex. The initial simplex vertices are chosen so that the simplex occupies a good portion of the solution space. In addition, it is also required that a scalar cost function be specified at each vertex of the simplex. The DSM searches for the minimum of the costs function by taking a series of steps, each time moving a point in the simplex away from where the cost function is largest. The simplex moves in space by variously reflecting, expanding, contracting, or shrinking. The simplex size is continuously changed and mostly diminished, so that finally it is small enough to contain the minimum with the desired accuracy.

For the OSFE-HMM algorithm, the IFM cost function to be minimized is the RMS difference between the measured sleeve friction values and synthetic sleeve friction measurements generated by implementing Equation (4) with the estimated sleeve friction values f_v1 to f_vN used as input. As with the test bed simulation, when implementing Equation (4) in OSFE-HMM algorithm, Δ is initially set to a 1 mm sampling rate. The synthetic sleeve friction measurements are then obtained by extracting data from the 1 mm sampling rate data sets at the user specified rate. This is done so that the true in-situ measurement conditions are replicated.

The performance of the OSFE-IFM algorithm was evaluated by processing the challenging test bed simulation illustrated in Figure 6. Figure 6 illustrates a highly variable CPT profile where it assumed that both the measured q_c and f_c have been corrected for pore pressure. In Figure 6, the true values of q_v, f_v and R_fv are red traces while the corresponding measured values are the black traces.

Figure 8 illustrates the output from the q_cHMM algorithm. In Figure 8, it is shown the test bed specified true q_v values (red line), derived measured q_t values (black line) and estimated q ′ v values from the qcHMM algorithm (blue line). As is illustrated in Figure 8, the estimated q ′ v values are nearly identical to the true q_v values.

Figure 9 illustrates the output from the OSFE-IFM algorithm after processing the measured f_t sleeve values shown in Figure 6(b). In Figure 9, it is shown the test bed specified true f_v values (red line), derived measured f_t values (black line) and estimated f ′ v values from the OSFE-IFM algorithm (blue line). As is illustrated in Figure 9, the estimated f ′ v values are very close to the true f_v values.

Figure 10 illustrates the friction ratio output obtained from implementation of the q_cHMM and OSFE-IFM algorithms where R ′ f v values are derived from q ′ v and f ′ v estimates. In Figure 10 the test bed specified R_fv values, measured

R_ft values and estimated R ′ f v values are identified by red, black and blue lines, respectively. As is illustrated in Figure 10, the estimated R ′ f v values are very close to the true R_fv values.