# In Situ Determination of Density Profiles in Complex Strata Using the Nuclear Density Cone Penetrometer

Authors: Rui Jia; Huayang Lei; Wenjun Zhang; Haizuo Zhou
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

The nuclear density cone penetrometer (ND-CP) is anin situ testing device that can provide continuous data on soil density, but it measures the composite density (ρc) of the soil within a spheroid centered at the midpoint between the gamma ray source and the detector. A theoretical model for predicting ρc of the ND-CP is proposed, and equations for calculating ρc are derived when the ND-CP penetrates into strata with different functions of density distributions. The calculated ρc profiles provide a good fit to the laboratory-measured ρc profiles by the ND-CP, indicating that the proposed theoretical model can be used to calculate the ρc within the spheroid. The comparisons of the density (ρ) and ρc values show that the ρc profiles are considerably different from the actualρ profiles at the boundaries of each stratum where density changes suddenly. Therefore, a method for deducing the actualρ profiles from the measured ρc profiles in complex strata is proposed, and this method is used to determine the actualρ profiles from the ND-CP measured ρc profiles in the field. The research results would be beneficial for thein situ determination of the density profile in complex strata using the ND-CP.


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## Body

## 1. Introduction

Soil density is defined as the mass of the soil per unit of volume. It depends on the density of the solid constituents, the porosity of the aggregate, and the degree of saturation [1, 2]. Soil density measurement is important because it determines the degree of compactness as a measure of soil structure and is used for calculating the pore space of soils [3]. The density of soil is an indispensable parameter used for the classification of stratigraphy, identification of soft interlayers, determination of the relative density of sand, assessment of the effect of ground improvement, and calculation of a series of soil physical indexes [4, 5]. Furthermore, the density of bed sediments is an important parameter used for predicting sediment transport, determining nautical depth, and planning dredging projects [6].There are mainly two types of methods available for soil density measurement. One is the long-established direct method, which involves the measurement of the sample mass and volume. The other is the indirect nondestructive method, which involves measuring the attenuation or scattering of gamma radiation by the soil and determining the soil density from the intensities of the original and the attenuated gamma radiation. The nondestructive method is equal in accuracy to the direct method of density determination and is simpler and quicker to use, especially where measurements at different depths are required [7]. It can be classified into the backscatter type and direct transmission type. In backscatter gauges, the gamma radiation source and detector are lead-shielded and located on the soil surface or are lowered into an access hole in the soil. The gamma rays emitted from the source enter into and interact with the soil and attenuate by scattering. In this process, some of the rays that are scattered towards the detector are counted. In transmission gauges, the gamma radiation source emits rays that enter into and interact with the soil, and some of the rays are scattered away from the detector. The rays passing through the soil are detected and counted by the detector [8].The use of gamma radiation to measure density is a geophysical technique that was first applied in oil well logging [9]. Subsequently, this technique has been used for a number of years in geotechnical engineering to measure the wet density of soils [10–14]. With recent advances in electronics and the miniaturization of various electronic sensors and devices, the original bulky nuclear density gauge has been developed and incorporated into the piezocone penetrometer, which is the most versatile tool currently available forin situ soil investigation [15, 16]; thus, the nuclear density cone penetrometer (ND-CP) was developed [17–19]. The ND-CP has been used in soil investigations, and it can provide reliable soil density measurements [20–23].In the ND-CP, the density is calculated by the count rate ratio (Rρ) of the gamma rays detected by the detector to the gamma rays emitted from the source. Thus, the ND-CP measures the composite density (ρc) of the soil within a spheroid centered at the midpoint between the gamma ray source and the detector [24]. Therefore, the measured ρc is not the actualρ of the soil at a certain depth in the ground. There may be a considerable difference between the measured ρc profile and the actualρ profile at the boundaries of each stratum, where the density changes suddenly, e.g., in layered strata, strata with soft interlayers, and bed sediments. Therefore, it is necessary to study how to determine the actualρ profiles in complex strata from the measured ρc profiles by the ND-CP.In this study, the density measurement principle and the density measurement volume of the ND-CP are described first. Then, a theoretical model for predictingρc is proposed when the ND-CP penetrates into strata with different density distribution functions. The theoretical model is verified by comparing the calculated ρc profiles by the theoretical model with the measured ρc profiles by the ND-CP in the laboratory. Then, a method for deducing the actualρ profile from the measured ρc profile is proposed based on the difference between theρ profiles and the ρc profiles in complex strata. Finally, the proposed method is used to determine the actualρ profiles from ND-CP measured ρc profiles in the field.


## 2. Measurement Principle of Nuclear Density Cone Penetrometer

### 2.1. Nuclear Density Cone Penetrometer

Figure1 shows the ND-CP. The ND-CP mainly consists of a standard piezocone penetrometer and a nuclear densitometer. The piezocone penetrometer is identical to the standard electronic cone. The nuclear densitometer mainly consists of a gamma ray source and a gamma ray detector. The diameters of the piezocone penetrometer and the nuclear densitometer are 35.7 mm and 48.6 mm, respectively. The distances of the cone resistance sensor, the pore pressure sensor, the sleeve friction sensor, the inclinometer sensor, the gamma ray source, the density count gamma ray detector, and the background count gamma ray detector from the apex of the cone are 0 m, 0.04 m, 0.11 m, 0.5 m, 0.721 m, 0.986 m, and 1.4535 m, respectively [24].Figure 1
Nuclear density cone penetrometer.


### 2.2. Density Measurement Principle

Gamma rays are a by-product of the natural decay of some radionuclides. The density measurement principle of the ND-CP is based on the absorption of gamma rays (interaction between the rays and the soil that reduces the intensity of the rays). The photons of the gamma rays can interact with the electrons of the soil. Three different interaction processes, namely, the photoelectric effect, Compton scattering, and pair production, are observed. The photoelectric effect is the interaction of a gamma photon with an electron in the inner shell of the atom. The gamma photon is completely absorbed, and the electron uses the additional energy to leave its atom as a free electron. The Compton effect is the scattering of a gamma photon by an inner shell electron, whereby only part of the photon energy is transferred to the electron. The energy and direction of the gamma photon thus change, and the electron leaves its electronic shell. Pair production is a process in which a gamma ray of sufficient energy is converted into an electron and a positron. Energies of gamma rays range from approximately 10 keV to 10 MeV. The photoelectric effect generally predominates at lower ray energies, the Compton effect predominates when the energy of the ray ranges from 1 MeV to 5 MeV, and pair production occurs at ray energies of several MeVs [25].The principle of the soil density measurement in the ND-CP utilizes the Compton scattering of gamma rays. When the ND-CP penetrates the ground, the gamma rays emitted from the source pass through the soil particles in the ground, interact with the SiO2 in the soil particles, collide with the electrons in the outer shell, and scatter repeatedly with the atomic electrons in the soil (Figure 2). In this process, some of the gamma rays are absorbed by the soil, while others reach the detector. The absorption depends on the density of the soil. The higher the density is, the more the electrons are available within a given volume to interact with the gamma rays, and the more gamma rays are absorbed. Hence, the soil density can be determined from the intensities of gamma rays emitted from the source and detected by the detector.Figure 2
Schematic diagram of Compton scattering of gamma rays in the soil (modified fromhttp://www.soilandrock.co.jp/).The count rate cannot be directly used to calculate density because the radioactive materials decay exponentially with time. The decay rate is characterized by the half-life of a radioisotope, which is defined as the time taken for half the original number of radioactive nuclei to decay. The gamma ray source used in the ND-CP is the radioactive isotope cesium-137, which has a half-life of approximately 30 years. Therefore, the measured count rate must be normalized by the standard count rate to account for the radioactive decay. In addition, to obtain the accurate absorption of gamma radiation in the soil, the amount of natural gamma rays in the ground must be measured and subtracted from the total count rate detected by the detector. The count rate ratio (Rρ), which is used to calculate the soil density, is defined as follows [24]:(1)Rρ=DCR-BCRSCRwhereDCR is the density count rate, which is measured by the density count gamma ray detector;BCR is the background count rate, which is measured by the background count gamma ray detector; andSCR is the standard count rate, which is measured in an inactive material under controlled conditions before eachin situ investigation [26]. The calibration equation between soil density and Rρ is as follows [24]:(2)Rρ=Aρ2-Bρ+Cwhereρ is density andA,B, andC are 0.6264, 4.0954, and 6.8422, respectively, for density values ranging from 1.0 g/cm3 to 2.2 g/cm3. Theoretically, the ND-CP can be used to measure the density of soil up to 2.2 g/cm3. However, similar to the piezocone penetrometer, the ND-CP is applicable to general clay, silt, and sand, especially to loose soil and mucky soil below the groundwater level, for which undisturbed samples are difficult to obtain.


### 2.3. Density Measurement Volume

The absorption of gamma rays and the measuredDCR depend on the density of the soil within a spheroid between the gamma ray source and the detector, as shown in Figure 1. The measurement volume is an oblate spheroid centered at the midpoint between the gamma ray source and detector (Figure 3). The length of semiaxisb, which is the distance from the center to the pole along the symmetry axis, is half of the gamma ray source-detector separation distance (2b) of the ND-CP. In this study, the value of 2b is 26.5 cm. The length of semiaxisa, which is the equatorial radius of the spheroid, decreases with an increasing density of the surrounding soil [10]. The lengths ofa are approximately 24 cm, 20 cm, and 17 cm in water, clay, and sand, respectively [24]. Notably, the measuredDCR is more sensitive to the density of the soil closer to the center of the spheroid.Figure 3
Density measurement volume of the ND-CP.The density measured by the ND-CP is the composite density (ρc) of the soil within a spheroid, as shown in Figure 3. Thus, the measured density (ρc) is not the actual density (ρ) at a certain depth. If the density changes suddenly with depth, e.g., as in layered soils, the measured ρc profile will be different from the actual ρ profile.


## 2.1. Nuclear Density Cone Penetrometer

Figure1 shows the ND-CP. The ND-CP mainly consists of a standard piezocone penetrometer and a nuclear densitometer. The piezocone penetrometer is identical to the standard electronic cone. The nuclear densitometer mainly consists of a gamma ray source and a gamma ray detector. The diameters of the piezocone penetrometer and the nuclear densitometer are 35.7 mm and 48.6 mm, respectively. The distances of the cone resistance sensor, the pore pressure sensor, the sleeve friction sensor, the inclinometer sensor, the gamma ray source, the density count gamma ray detector, and the background count gamma ray detector from the apex of the cone are 0 m, 0.04 m, 0.11 m, 0.5 m, 0.721 m, 0.986 m, and 1.4535 m, respectively [24].Figure 1
Nuclear density cone penetrometer.


## 2.2. Density Measurement Principle

Gamma rays are a by-product of the natural decay of some radionuclides. The density measurement principle of the ND-CP is based on the absorption of gamma rays (interaction between the rays and the soil that reduces the intensity of the rays). The photons of the gamma rays can interact with the electrons of the soil. Three different interaction processes, namely, the photoelectric effect, Compton scattering, and pair production, are observed. The photoelectric effect is the interaction of a gamma photon with an electron in the inner shell of the atom. The gamma photon is completely absorbed, and the electron uses the additional energy to leave its atom as a free electron. The Compton effect is the scattering of a gamma photon by an inner shell electron, whereby only part of the photon energy is transferred to the electron. The energy and direction of the gamma photon thus change, and the electron leaves its electronic shell. Pair production is a process in which a gamma ray of sufficient energy is converted into an electron and a positron. Energies of gamma rays range from approximately 10 keV to 10 MeV. The photoelectric effect generally predominates at lower ray energies, the Compton effect predominates when the energy of the ray ranges from 1 MeV to 5 MeV, and pair production occurs at ray energies of several MeVs [25].The principle of the soil density measurement in the ND-CP utilizes the Compton scattering of gamma rays. When the ND-CP penetrates the ground, the gamma rays emitted from the source pass through the soil particles in the ground, interact with the SiO2 in the soil particles, collide with the electrons in the outer shell, and scatter repeatedly with the atomic electrons in the soil (Figure 2). In this process, some of the gamma rays are absorbed by the soil, while others reach the detector. The absorption depends on the density of the soil. The higher the density is, the more the electrons are available within a given volume to interact with the gamma rays, and the more gamma rays are absorbed. Hence, the soil density can be determined from the intensities of gamma rays emitted from the source and detected by the detector.Figure 2
Schematic diagram of Compton scattering of gamma rays in the soil (modified fromhttp://www.soilandrock.co.jp/).The count rate cannot be directly used to calculate density because the radioactive materials decay exponentially with time. The decay rate is characterized by the half-life of a radioisotope, which is defined as the time taken for half the original number of radioactive nuclei to decay. The gamma ray source used in the ND-CP is the radioactive isotope cesium-137, which has a half-life of approximately 30 years. Therefore, the measured count rate must be normalized by the standard count rate to account for the radioactive decay. In addition, to obtain the accurate absorption of gamma radiation in the soil, the amount of natural gamma rays in the ground must be measured and subtracted from the total count rate detected by the detector. The count rate ratio (Rρ), which is used to calculate the soil density, is defined as follows [24]:(1)Rρ=DCR-BCRSCRwhereDCR is the density count rate, which is measured by the density count gamma ray detector;BCR is the background count rate, which is measured by the background count gamma ray detector; andSCR is the standard count rate, which is measured in an inactive material under controlled conditions before eachin situ investigation [26]. The calibration equation between soil density and Rρ is as follows [24]:(2)Rρ=Aρ2-Bρ+Cwhereρ is density andA,B, andC are 0.6264, 4.0954, and 6.8422, respectively, for density values ranging from 1.0 g/cm3 to 2.2 g/cm3. Theoretically, the ND-CP can be used to measure the density of soil up to 2.2 g/cm3. However, similar to the piezocone penetrometer, the ND-CP is applicable to general clay, silt, and sand, especially to loose soil and mucky soil below the groundwater level, for which undisturbed samples are difficult to obtain.


## 2.3. Density Measurement Volume

The absorption of gamma rays and the measuredDCR depend on the density of the soil within a spheroid between the gamma ray source and the detector, as shown in Figure 1. The measurement volume is an oblate spheroid centered at the midpoint between the gamma ray source and detector (Figure 3). The length of semiaxisb, which is the distance from the center to the pole along the symmetry axis, is half of the gamma ray source-detector separation distance (2b) of the ND-CP. In this study, the value of 2b is 26.5 cm. The length of semiaxisa, which is the equatorial radius of the spheroid, decreases with an increasing density of the surrounding soil [10]. The lengths ofa are approximately 24 cm, 20 cm, and 17 cm in water, clay, and sand, respectively [24]. Notably, the measuredDCR is more sensitive to the density of the soil closer to the center of the spheroid.Figure 3
Density measurement volume of the ND-CP.The density measured by the ND-CP is the composite density (ρc) of the soil within a spheroid, as shown in Figure 3. Thus, the measured density (ρc) is not the actual density (ρ) at a certain depth. If the density changes suddenly with depth, e.g., as in layered soils, the measured ρc profile will be different from the actual ρ profile.


## 3. Theoretical Model for Calculating Composite Density

### 3.1. Theoretical Model for Predicting Count Rate Ratio

A mathematical model is proposed for the ND-CP to predict the count rate ratio (Rρ) when the ND-CP spans two or more strata. Then, the composite density (ρc) within the measurement spheroid can be calculated by (2). Figure 4 shows the spheroid coordinates for the theoretical model used to calculate Rρ [24]. The adopted assumptions are as follows.Figure 4
Spheroid coordinates for the theoretical model to calculate the count rate ratio.( 1 ) It is assumed that the scattering and attenuation of gamma rays occur over a spheroid volume; thus, the volume affecting the measured Rρ is a spheroid [27]. The center of the spheroid is at the midpoint between the source and the detector. The vertical section of the spheroid is an ellipse, and the horizontal section of the spheroid is a circle.( 2 ) The spheroid volume is constant. The length of semiaxisb is equal to half of the gamma ray source-detector separation distance (26.5 cm) of the ND-CP. The length of semiaxisadecreases with increasing density of the surrounding soil [10]. For simplification, it is assumed to be a constant.( 3 ) All parts of the spheroid contribute equally to the scattering and attenuation of radiation, and the contribution of the count rate ratio from a small volume (ΔRρ) in the spheroid to the total count rate ratio (Rρ) is proportional to its volume. This assumption is not strictly true but does not lead to a large error [28].( 4 ) The calibration equation between Rρ and density is defined by (2) for the soils with densities ranging from 1.0 g/cm3 to 2.2 g/cm3.The volume of the spheroid is as follows:(3)V=43πa2bThe vertical cross-section of the spheroid in thex-z plane is an ellipse, and its equation is(4)x2a2+z-zib2=1wherea is the length of the major semiaxis,b is the length of the minor semiaxis,z is the depth, and zi is the depth corresponding to the midpoint of the ellipse. The horizontal cross-section of the spheroid at any point is a circle of radiusx, where 0<x<a. A small volume of the spheroid ∂V is(5)∂V=πx2δzwhere δz is the differential ofz. Thex calculated by (4) can be substituted into (5) to give(6)∂V=πa2b2b2-z-zi2δzThe assumption is that the contribution of the small volume ∂Rρ to Rρ is proportional to its volume:(7)∂RρRρ=∂VVSubstituting (2), (3), and (6) into (7) gives(8)∂Rρ=3Aρ2-Bρ+C4b3b2-z-zi2δzAssuming the density ρ is a function of depth ρ=ρ(z), integrating (8) over the spheroid volume gives Rρ:(9)Rρ=34b3∫Aρ2z-Bρz+Cb2-z-zi2δz


### 3.2. Equations for Calculating the Count Rate Ratio

If the density functionρ(z) of the strata is known, (9) can be used to calculate the count rate ratio (Rρ) when the ND-CP penetrates into the strata. The probable density profiles in the ground are described by constant, linear, parabolic, and square root functions (Figure 5), the equations of which are as follows:(10)ρz=ρ0(11)ρz=ρ1+β1z-z1(12)ρz=α1z-z12+ρ1(13)ρz=α2z-z11/2+ρ1Figure 5
Assumed density profiles in the ground: (a) constant; (b) linear; (c) parabolic; (d) square root.
(a) (b) (c) (d)If the density and depth at the top of the stratum (ρ1,z1) and at the bottom of the stratum (ρ2,z2) are known, the values of β1, α1, and α2 in (11), (12), and (13) can be determined as follows:(14)β1=ρ2-ρ1z2-z1(15)α1=ρ2-ρ1z2-z12(16)α2=ρ2-ρ1z2-z10.5Eqs. (17), (18), (19), and (20) can be used to calculate Rρ when the density functions are constant, linear, parabolic, and square root, respectively.(17)Rρ=3Aρ02-Bρ0+C4b3∫b2-z-zi2δz(18)Rρ=34b3∫Aρ1+βz-z12-Bρ1+βz-z1+Cb2-z-zi2δz(19)Rρ=34b3∫Aα1z-z12+ρ12-Bα1z-z12+ρ1+Cb2-z-zi2δz(20)Rρ=34b3∫Aα2z-z11/2+ρ12-Bα2z-z11/2+ρ1+Cb2-z-zi2δzThe solutions for (17) to (20) are derived in (21) to (24), respectively.(21)Rρ=34b3Aρ02-Bρ0+C-z33+ziz2+b2-zi2zz1z2(22)Rρ=34b3-Aβ2z55+-2Aρ1β+2Aziβ2+2Aβ2z1+Bβz44+b2Aβ2-Aρ12+4Aziρ1β-Azi2β2+2Aρ1βz1-Aβ2z12-4Aziβ2z1+Bρ1-Bβz1-2Bziβ-Cz33+2b2Aβρ1-2b2Aβ2z1-2Aρ1βzi2+2Aziρ12-4Aziρ1βz1+2Aziβ2z12+2Azi2β2z1-b2Bβ+Bzi2β-2Bziρ1+2Bziβz1+2Czi2z2+b2Aρ12-2b2Aρ1βz1+b2Aβ2z12-Azi2ρ12+2Aρ1βz1zi2-Azi2β2z12-b2Bρ1+b2Bβz1+Bzi2ρ1-Bzi2βz1+Cb2-Czi2zz1z2(23)Rρ=34b3-Aα12z77+4Aα12z1+2Aziα12z66+b2Aα12-6Aα12z12-2Aα1ρ1-8Aziα12z1-Aα12zi2+Bα1z55+-4b2Aα12z1+4Aα12z13+4Aα1ρ1z1+12Aziα12z12+4Aziα1ρ1+4Azi2α12z1-2Bz1α1-2Bziα1z44+6b2Aα12z12+2b2Aα1ρ1-Aα12z14-2Aα1ρ1z12-Aρ12-8Aziα12z13-8Aziα1ρ1z1-6Azi2α12z12-2Azi2α1ρ1-b2Bα1+Bz12α1+Bρ1+4Bziz1α1+Bzi2α1-Cz33+-4b2Aα1ρ1z1-4b2Aα12z13+2Aziα12z14+2Aziρ12+4Aziα1ρ1z12+4Azi2α12z13+4Azi2α1ρ1z1+2b2Bz1α1-2Bziz12α1-2Bziρ1-2Bzi2z1α1+2Cziz22+2b2Aα1ρ1z12+b2Aρ12+b2Aα12z14-Azi2α12z14-2Azi2α1ρ1z12-Azi2ρ12-b2Bz12α1-b2Bρ1+Bzi2z12α1+Bzi2ρ1+Cb2-Czi2zz1z2(24)Rρ=34b3-14Aα22z-z14+27Bα2-2Aρ1α2z-z17/2+13-2Aα22z1-ziz-z13-13Aρ12-Bρ1z-zi3-13Cz3+252Bα2z1-zi-4Aρ1α2z1-ziz-z15/2+12b2Aα22-Aα22z1-zi2z-z12+Cziz2+232b2Aρ1α2-2Aρ1α2z1-zi2-b2Bα2+Bα2z1-zi2z-z13/2+b2Aρ12-b2Bρ1+Cb2-zi2zz1z2If the gamma ray source and detector are in strata that can be described by one function of density,Rρ is calculated as an integral of that function. If the gamma ray source and detector are in strata described by two or more functions, Rρ is calculated as a piecewise integration of those functions. Then, the composite density (ρc), which is referenced to the midpoint of the spheroid, can be calculated by (25). Finally, the ρc profile can be obtained as the midpoint of the spheroid moves with depth along with the penetration of the ND-CP.(25)ρc=4.0954-2.5056Rρ-0.37151.2528


### 3.3. Verification of the Theoretical Model

The ND-CP was used to measure the density profiles of two kinds of layered soils by Karthikeyan and Tan in the laboratory [29]. The experimental setup consists of a loading frame and a cubic steel tank with dimensions of 1.5 m×1.5 m×1.5 m. The loading frame is placed on top of the steel tank to push the ND-CP using a hydraulic piston. The ND-CP measurement and the depth meter readings were recorded through a data logger. Figure 6 shows schematic configurations for layered soils in the steel tank. One configuration is with a top layer of water and bottom layer of soil (Figure 6(a)), and the other configuration is with a top layer of kaolin clay slurry and bottom layer of soil (Figure 6(b)). The measured ρc profiles do not show a sharp transition at the interface of the water and soil layers (Figure 6(a)) or at the interface of the slurry and soil layers (Figure 6(b)). The change in density from the water/slurry to the soil is extended over a distance equal to the source-detector separation (2b). This transition occurs as the gamma ray source enters the soil layer and continues until the gamma ray detector enters the soil layer. The actualρ profile shows a step change in density, while the measured ρc profile shows a continuous change in density. The change pattern in the measured ρc profile indicates that the ND-CP is penetrating the interface of two layers with different density values. The ρc profiles predicted by the theoretical model are also shown in the figure, and these predicted results show a close agreement with the measured ρc profiles, suggesting that the theoretical model is able to describe the signature of the ρc profiles measured by the ND-CP. The theoretical model will be used to investigate the differences between the actualρ profiles in the ground and the measured ρc profiles by the ND-CP.Figure 6
Verification of the theoretical model with laboratory test results.
(a) (b)


## 3.1. Theoretical Model for Predicting Count Rate Ratio

A mathematical model is proposed for the ND-CP to predict the count rate ratio (Rρ) when the ND-CP spans two or more strata. Then, the composite density (ρc) within the measurement spheroid can be calculated by (2). Figure 4 shows the spheroid coordinates for the theoretical model used to calculate Rρ [24]. The adopted assumptions are as follows.Figure 4
Spheroid coordinates for the theoretical model to calculate the count rate ratio.( 1 ) It is assumed that the scattering and attenuation of gamma rays occur over a spheroid volume; thus, the volume affecting the measured Rρ is a spheroid [27]. The center of the spheroid is at the midpoint between the source and the detector. The vertical section of the spheroid is an ellipse, and the horizontal section of the spheroid is a circle.( 2 ) The spheroid volume is constant. The length of semiaxisb is equal to half of the gamma ray source-detector separation distance (26.5 cm) of the ND-CP. The length of semiaxisadecreases with increasing density of the surrounding soil [10]. For simplification, it is assumed to be a constant.( 3 ) All parts of the spheroid contribute equally to the scattering and attenuation of radiation, and the contribution of the count rate ratio from a small volume (ΔRρ) in the spheroid to the total count rate ratio (Rρ) is proportional to its volume. This assumption is not strictly true but does not lead to a large error [28].( 4 ) The calibration equation between Rρ and density is defined by (2) for the soils with densities ranging from 1.0 g/cm3 to 2.2 g/cm3.The volume of the spheroid is as follows:(3)V=43πa2bThe vertical cross-section of the spheroid in thex-z plane is an ellipse, and its equation is(4)x2a2+z-zib2=1wherea is the length of the major semiaxis,b is the length of the minor semiaxis,z is the depth, and zi is the depth corresponding to the midpoint of the ellipse. The horizontal cross-section of the spheroid at any point is a circle of radiusx, where 0<x<a. A small volume of the spheroid ∂V is(5)∂V=πx2δzwhere δz is the differential ofz. Thex calculated by (4) can be substituted into (5) to give(6)∂V=πa2b2b2-z-zi2δzThe assumption is that the contribution of the small volume ∂Rρ to Rρ is proportional to its volume:(7)∂RρRρ=∂VVSubstituting (2), (3), and (6) into (7) gives(8)∂Rρ=3Aρ2-Bρ+C4b3b2-z-zi2δzAssuming the density ρ is a function of depth ρ=ρ(z), integrating (8) over the spheroid volume gives Rρ:(9)Rρ=34b3∫Aρ2z-Bρz+Cb2-z-zi2δz


## 3.2. Equations for Calculating the Count Rate Ratio

If the density functionρ(z) of the strata is known, (9) can be used to calculate the count rate ratio (Rρ) when the ND-CP penetrates into the strata. The probable density profiles in the ground are described by constant, linear, parabolic, and square root functions (Figure 5), the equations of which are as follows:(10)ρz=ρ0(11)ρz=ρ1+β1z-z1(12)ρz=α1z-z12+ρ1(13)ρz=α2z-z11/2+ρ1Figure 5
Assumed density profiles in the ground: (a) constant; (b) linear; (c) parabolic; (d) square root.
(a) (b) (c) (d)If the density and depth at the top of the stratum (ρ1,z1) and at the bottom of the stratum (ρ2,z2) are known, the values of β1, α1, and α2 in (11), (12), and (13) can be determined as follows:(14)β1=ρ2-ρ1z2-z1(15)α1=ρ2-ρ1z2-z12(16)α2=ρ2-ρ1z2-z10.5Eqs. (17), (18), (19), and (20) can be used to calculate Rρ when the density functions are constant, linear, parabolic, and square root, respectively.(17)Rρ=3Aρ02-Bρ0+C4b3∫b2-z-zi2δz(18)Rρ=34b3∫Aρ1+βz-z12-Bρ1+βz-z1+Cb2-z-zi2δz(19)Rρ=34b3∫Aα1z-z12+ρ12-Bα1z-z12+ρ1+Cb2-z-zi2δz(20)Rρ=34b3∫Aα2z-z11/2+ρ12-Bα2z-z11/2+ρ1+Cb2-z-zi2δzThe solutions for (17) to (20) are derived in (21) to (24), respectively.(21)Rρ=34b3Aρ02-Bρ0+C-z33+ziz2+b2-zi2zz1z2(22)Rρ=34b3-Aβ2z55+-2Aρ1β+2Aziβ2+2Aβ2z1+Bβz44+b2Aβ2-Aρ12+4Aziρ1β-Azi2β2+2Aρ1βz1-Aβ2z12-4Aziβ2z1+Bρ1-Bβz1-2Bziβ-Cz33+2b2Aβρ1-2b2Aβ2z1-2Aρ1βzi2+2Aziρ12-4Aziρ1βz1+2Aziβ2z12+2Azi2β2z1-b2Bβ+Bzi2β-2Bziρ1+2Bziβz1+2Czi2z2+b2Aρ12-2b2Aρ1βz1+b2Aβ2z12-Azi2ρ12+2Aρ1βz1zi2-Azi2β2z12-b2Bρ1+b2Bβz1+Bzi2ρ1-Bzi2βz1+Cb2-Czi2zz1z2(23)Rρ=34b3-Aα12z77+4Aα12z1+2Aziα12z66+b2Aα12-6Aα12z12-2Aα1ρ1-8Aziα12z1-Aα12zi2+Bα1z55+-4b2Aα12z1+4Aα12z13+4Aα1ρ1z1+12Aziα12z12+4Aziα1ρ1+4Azi2α12z1-2Bz1α1-2Bziα1z44+6b2Aα12z12+2b2Aα1ρ1-Aα12z14-2Aα1ρ1z12-Aρ12-8Aziα12z13-8Aziα1ρ1z1-6Azi2α12z12-2Azi2α1ρ1-b2Bα1+Bz12α1+Bρ1+4Bziz1α1+Bzi2α1-Cz33+-4b2Aα1ρ1z1-4b2Aα12z13+2Aziα12z14+2Aziρ12+4Aziα1ρ1z12+4Azi2α12z13+4Azi2α1ρ1z1+2b2Bz1α1-2Bziz12α1-2Bziρ1-2Bzi2z1α1+2Cziz22+2b2Aα1ρ1z12+b2Aρ12+b2Aα12z14-Azi2α12z14-2Azi2α1ρ1z12-Azi2ρ12-b2Bz12α1-b2Bρ1+Bzi2z12α1+Bzi2ρ1+Cb2-Czi2zz1z2(24)Rρ=34b3-14Aα22z-z14+27Bα2-2Aρ1α2z-z17/2+13-2Aα22z1-ziz-z13-13Aρ12-Bρ1z-zi3-13Cz3+252Bα2z1-zi-4Aρ1α2z1-ziz-z15/2+12b2Aα22-Aα22z1-zi2z-z12+Cziz2+232b2Aρ1α2-2Aρ1α2z1-zi2-b2Bα2+Bα2z1-zi2z-z13/2+b2Aρ12-b2Bρ1+Cb2-zi2zz1z2If the gamma ray source and detector are in strata that can be described by one function of density,Rρ is calculated as an integral of that function. If the gamma ray source and detector are in strata described by two or more functions, Rρ is calculated as a piecewise integration of those functions. Then, the composite density (ρc), which is referenced to the midpoint of the spheroid, can be calculated by (25). Finally, the ρc profile can be obtained as the midpoint of the spheroid moves with depth along with the penetration of the ND-CP.(25)ρc=4.0954-2.5056Rρ-0.37151.2528


## 3.3. Verification of the Theoretical Model

The ND-CP was used to measure the density profiles of two kinds of layered soils by Karthikeyan and Tan in the laboratory [29]. The experimental setup consists of a loading frame and a cubic steel tank with dimensions of 1.5 m×1.5 m×1.5 m. The loading frame is placed on top of the steel tank to push the ND-CP using a hydraulic piston. The ND-CP measurement and the depth meter readings were recorded through a data logger. Figure 6 shows schematic configurations for layered soils in the steel tank. One configuration is with a top layer of water and bottom layer of soil (Figure 6(a)), and the other configuration is with a top layer of kaolin clay slurry and bottom layer of soil (Figure 6(b)). The measured ρc profiles do not show a sharp transition at the interface of the water and soil layers (Figure 6(a)) or at the interface of the slurry and soil layers (Figure 6(b)). The change in density from the water/slurry to the soil is extended over a distance equal to the source-detector separation (2b). This transition occurs as the gamma ray source enters the soil layer and continues until the gamma ray detector enters the soil layer. The actualρ profile shows a step change in density, while the measured ρc profile shows a continuous change in density. The change pattern in the measured ρc profile indicates that the ND-CP is penetrating the interface of two layers with different density values. The ρc profiles predicted by the theoretical model are also shown in the figure, and these predicted results show a close agreement with the measured ρc profiles, suggesting that the theoretical model is able to describe the signature of the ρc profiles measured by the ND-CP. The theoretical model will be used to investigate the differences between the actualρ profiles in the ground and the measured ρc profiles by the ND-CP.Figure 6
Verification of the theoretical model with laboratory test results.
(a) (b)


## 4. Method for Deducing Actualρ Profile

### 4.1. Comparison ofρand ρc Profiles

The density profiles of complex strata, which are common in engineering practice, are assumed in Figure7. Figure 7(a) shows the density profiles of layered strata. The density of the first layer is 1.6 g/cm3, and the densities of the second layer are 1.5 g/cm3, 1.7 g/cm3, or 1.8 g/cm3. Figure 7(b) shows the density profiles of strata with soft interlayers. The density of the strata is 1.7 g/cm3. Three different soft interlayers are assumed. For the first case, the soft interlayer has a density of 1.3 g/cm3 and a thickness of 0.15 m; for the second case, the density is 1.5 g/cm3 and the thickness is 0.15 m; for the third case, the density is 1.3 g/cm3 and the thickness is 0.3 m. Figure 7(c) shows the density profiles of bed sediments. The density of the water is 1.0 g/cm3, and the density of the bed mud increases with depth in a linear fashion. Three different layers of fluid mud are assumed. For the first case, the fluid mud has a maximum density of 1.2 g/cm3 and thickness of 0.2 m; for the second case, the maximum density is 1.3 g/cm3 and the thickness is 0.2 m; for the third case, the maximum density is 1.15 g/cm3 and the thickness is 0.25 m.Figure 7
Comparison of assumedρ and calculated ρc profiles.
(a)
Layered strata (b)
Strata with soft interlayers (c)
Bed sedimentsTheρc profiles are calculated by the theoretical model when the ND-CP penetrates into the assumed complex strata (Figure 7), and the results are compared with theρ profiles. The ρc profiles are clearly different from theρ profiles near the interface of the layered strata, near the soft interlayer, and near the upper and lower boundaries of the fluid mud. The differences between ρc andρ near the interface of the layered strata in Figure 7(a) are explained as follows. (1) When both the gamma ray source and detector are in the top layer, ρc is equal to the density of the top layer. (2) Once the gamma ray source enters the bottom layer, ρc will be the weighted average density of the two layers, whileρ is still the density of the first layer. (3) When the midpoint of the gamma ray source and detector enters the bottom layer,ρ will change to the density of the bottom layer, while ρc is still the weighted average density of the two layers because the gamma ray detector is still in the first layer. (4) When the gamma ray detector also enters the bottom layer, ρc is equal toρof the bottom layer. Similar to the profiles for the case in which the ND-CP passes through the interface of layered strata, the ρc profiles are different from theρ profiles when the ND-CP passes through a soft interlayer and the fluid mud. Therefore, the ρc profile measured by the ND-CP must be interpreted, and the actualρ profile must be deduced from the measured ρc profile.


### 4.2. Method for Deducing Actualρ Profile from Measured ρc Profile

A method for deducing the actualρ profile from the measured ρc profile is proposed as follows:( 1 ) Determining the depths of and densities at the boundaries of each stratum based on the features of the measured ρc profile. (2) Assuming the probable density function of each stratum and the probableρ profile of the strata. (3) Back-calculating the ρc profile by using the theoretical model for the case of the ND-CP passing through layered strata. (4) Comparing the calculated ρc profile with the measured ρc profile and adjusting the assumedρ profile until the calculated ρc profile matches the measured ρc profile well. (5) Determining the actualρ profile to be the assumedρ profile when the calculated ρc profile fits the measured ρc profile.For the layered strata, as shown in Figure7(a), the measured ρc profile can be divided into three sections, namely, the first constant section, the second curved section, and the third constant section. The measured ρc values are different from the actualρ values in the range of 2b, which is the distance between the gamma ray source and detector. From the ρc profile, the boundary between layered strata can be determined, i.e., the midpoint of the second curved section of the ρc profile. The actualρ value of the top layer is assumed to be the measured ρc value of the first constant section, and the actualρ value of the bottom layer is assumed to be the measured ρc value of the third constant section. The ρc profile is back-calculated assuming that the ND-CP passes through the layered strata. If the calculated ρc profile matches well with the measured ρc profile, the actualρ profile of the layered strata was determined.For the strata with a soft interlayer, as shown in Figure7(b), the measured ρc profile can be divided into three sections, namely, the first constant section, the second curved section, and the third constant section. The measured ρc values are different from the actualρ values near the soft interlayer (2b plus the thickness of the soft interlayer). From the ρc profile, the upper and lower boundaries of the soft interlayer can be determined, that is, the depth of the start point of the second curved section plusb and the depth of the end point of the second curved section minusb. The actualρ of the top layer is assumed to be the measured ρc of the first constant section, and the actualρ of the bottom layer is assumed to be the measured ρc of the third constant section. The actualρ of the soft interlayer can be assumed based on the minimum ρc in the second curved section. The ρc profile is back-calculated assuming that the ND-CP passes through the strata with a soft interlayer. The actualρ profile of strata with a soft interlayer is determined when the calculated ρc profile matches the measured ρc profile well.For bed sediments, as shown in Figure7(c), the measured ρc profile can be divided into three sections, namely, the first constant section, the second curved section, and the third linear section. The measured ρc values are different from the actualρ values near the upper and lower boundaries of the fluid mud. From the ρc profile, the upper and lower boundaries of the fluid mud can be determined; that is, the depth of the point from which density increases plusb and the depth of the point from which density slightly increases linearly minusb. The actualρof the water layer is assumed to be the measured ρc of the first constant section. The slope of the actualρ profile of the bed mud layer is assumed to be the slope of the measured ρc profile of the third linear section. The actualρ distribution of the fluid mud layer can be assumed to be a parabolic, linear, or square root function. The ρc profile is back-calculated assuming that the ND-CP passes through bed sediments. The actualρ profile of bed sediments can be determined when the calculated ρc profile fits the measured ρc profile.


## 4.1. Comparison ofρand ρc Profiles

The density profiles of complex strata, which are common in engineering practice, are assumed in Figure7. Figure 7(a) shows the density profiles of layered strata. The density of the first layer is 1.6 g/cm3, and the densities of the second layer are 1.5 g/cm3, 1.7 g/cm3, or 1.8 g/cm3. Figure 7(b) shows the density profiles of strata with soft interlayers. The density of the strata is 1.7 g/cm3. Three different soft interlayers are assumed. For the first case, the soft interlayer has a density of 1.3 g/cm3 and a thickness of 0.15 m; for the second case, the density is 1.5 g/cm3 and the thickness is 0.15 m; for the third case, the density is 1.3 g/cm3 and the thickness is 0.3 m. Figure 7(c) shows the density profiles of bed sediments. The density of the water is 1.0 g/cm3, and the density of the bed mud increases with depth in a linear fashion. Three different layers of fluid mud are assumed. For the first case, the fluid mud has a maximum density of 1.2 g/cm3 and thickness of 0.2 m; for the second case, the maximum density is 1.3 g/cm3 and the thickness is 0.2 m; for the third case, the maximum density is 1.15 g/cm3 and the thickness is 0.25 m.Figure 7
Comparison of assumedρ and calculated ρc profiles.
(a)
Layered strata (b)
Strata with soft interlayers (c)
Bed sedimentsTheρc profiles are calculated by the theoretical model when the ND-CP penetrates into the assumed complex strata (Figure 7), and the results are compared with theρ profiles. The ρc profiles are clearly different from theρ profiles near the interface of the layered strata, near the soft interlayer, and near the upper and lower boundaries of the fluid mud. The differences between ρc andρ near the interface of the layered strata in Figure 7(a) are explained as follows. (1) When both the gamma ray source and detector are in the top layer, ρc is equal to the density of the top layer. (2) Once the gamma ray source enters the bottom layer, ρc will be the weighted average density of the two layers, whileρ is still the density of the first layer. (3) When the midpoint of the gamma ray source and detector enters the bottom layer,ρ will change to the density of the bottom layer, while ρc is still the weighted average density of the two layers because the gamma ray detector is still in the first layer. (4) When the gamma ray detector also enters the bottom layer, ρc is equal toρof the bottom layer. Similar to the profiles for the case in which the ND-CP passes through the interface of layered strata, the ρc profiles are different from theρ profiles when the ND-CP passes through a soft interlayer and the fluid mud. Therefore, the ρc profile measured by the ND-CP must be interpreted, and the actualρ profile must be deduced from the measured ρc profile.


## 4.2. Method for Deducing Actualρ Profile from Measured ρc Profile

A method for deducing the actualρ profile from the measured ρc profile is proposed as follows:( 1 ) Determining the depths of and densities at the boundaries of each stratum based on the features of the measured ρc profile. (2) Assuming the probable density function of each stratum and the probableρ profile of the strata. (3) Back-calculating the ρc profile by using the theoretical model for the case of the ND-CP passing through layered strata. (4) Comparing the calculated ρc profile with the measured ρc profile and adjusting the assumedρ profile until the calculated ρc profile matches the measured ρc profile well. (5) Determining the actualρ profile to be the assumedρ profile when the calculated ρc profile fits the measured ρc profile.For the layered strata, as shown in Figure7(a), the measured ρc profile can be divided into three sections, namely, the first constant section, the second curved section, and the third constant section. The measured ρc values are different from the actualρ values in the range of 2b, which is the distance between the gamma ray source and detector. From the ρc profile, the boundary between layered strata can be determined, i.e., the midpoint of the second curved section of the ρc profile. The actualρ value of the top layer is assumed to be the measured ρc value of the first constant section, and the actualρ value of the bottom layer is assumed to be the measured ρc value of the third constant section. The ρc profile is back-calculated assuming that the ND-CP passes through the layered strata. If the calculated ρc profile matches well with the measured ρc profile, the actualρ profile of the layered strata was determined.For the strata with a soft interlayer, as shown in Figure7(b), the measured ρc profile can be divided into three sections, namely, the first constant section, the second curved section, and the third constant section. The measured ρc values are different from the actualρ values near the soft interlayer (2b plus the thickness of the soft interlayer). From the ρc profile, the upper and lower boundaries of the soft interlayer can be determined, that is, the depth of the start point of the second curved section plusb and the depth of the end point of the second curved section minusb. The actualρ of the top layer is assumed to be the measured ρc of the first constant section, and the actualρ of the bottom layer is assumed to be the measured ρc of the third constant section. The actualρ of the soft interlayer can be assumed based on the minimum ρc in the second curved section. The ρc profile is back-calculated assuming that the ND-CP passes through the strata with a soft interlayer. The actualρ profile of strata with a soft interlayer is determined when the calculated ρc profile matches the measured ρc profile well.For bed sediments, as shown in Figure7(c), the measured ρc profile can be divided into three sections, namely, the first constant section, the second curved section, and the third linear section. The measured ρc values are different from the actualρ values near the upper and lower boundaries of the fluid mud. From the ρc profile, the upper and lower boundaries of the fluid mud can be determined; that is, the depth of the point from which density increases plusb and the depth of the point from which density slightly increases linearly minusb. The actualρof the water layer is assumed to be the measured ρc of the first constant section. The slope of the actualρ profile of the bed mud layer is assumed to be the slope of the measured ρc profile of the third linear section. The actualρ distribution of the fluid mud layer can be assumed to be a parabolic, linear, or square root function. The ρc profile is back-calculated assuming that the ND-CP passes through bed sediments. The actualρ profile of bed sediments can be determined when the calculated ρc profile fits the measured ρc profile.


## 5. An Example of Practical Application

### 5.1. ND-CP Measuredρc Profile

The density profiles of bed sediments at Isahaya Bay, Japan, were investigated using the ND-CP. The details on the equipment and methods used in this investigation can be found in a prior publication [23]. The ND-CP measured density profiles at two locations (Figure 8) are used to illustrate how to use the proposed method to determine the actualρ profile from the measured ρc profile.Figure 8
Investigation locations.The measuredBCR andDCR and the calculated ρc values at locations A1 and A2 are shown in Figure 9.BCR is approximately zero in water and approximately 50 cps in bed mud.DCR in bed sediments is lower than that in water and decreases with increasing density. Rρ can be calculated according to the measuredBCR andDCR by using (1), and then the density can be calculated by using (25). The density profiles show similar transitions and vary significantly around the water/mud interface. The measured ρc profiles can be divided into three sections, namely, an initial constant section, a middle curved section, and a final linear section.Figure 9
Investigation results: (a) at location A1; (b) at location A2.
(a) (b)


### 5.2. Determination of the Actualρ Profile

There is an obvious difference between the actualρ and the measured ρc profiles near the upper and lower boundaries of the fluid mud layer, as shown in Figure 7(c). The theoretical model proposed in Section 3 and the method proposed in Section 4 were used to deduce the actualρ profiles from the measured ρc profiles as follows.( 1 ) For the measured ρc profile at location A1 (Figure 10(a)), the ρc at and depth of the intersection of the initial constant and middle curved sections (pointA) are 1.000 g/cm3 and 1.83 m, respectively. The ρc at and depth of the intersection of the middle curved and final linear sections (pointB) are 1.287 g/cm3 and 2.31 m, respectively. For the measured ρc profile at location A2 (Figure 11(a)), ρc and depth at pointA are 1.025 g/cm3 and 10.45 m, respectively. ρc and depth at pointB are 1.233 g/cm3 and 10.83 m, respectively.Figure 10
Determination of the actual density profile of bed sediments at location A1: (a) measuredρc profile; (b) assumedρ profile; (c) calculated ρc profile; (d) deducedρ profile.
(a) (b) (c) (d)Figure 11
Determination of the actual density profile of bed sediments at location A2: (a) measuredρc profile; (b) assumedρ profile; (c) calculated ρc profile; (d) deducedρ profile.
(a) (b) (c) (d)( 2 ) For the assumedρ profile at location A1 (Figure 10(b)), the depth of the upper boundary of the fluid mud at pointC is 1.9625 m, which is equal to the depth at pointA (1.83 m) plusb (0.1325 m). The depth of the lower boundary of the fluid mud at pointD is 2.1775 m, which is equal to the depth at pointB (2.31 m) minusb (0.1325 m).ρ at pointC is 1.000 g/cm3, which is equal to ρc at pointA.ρ at pointD is 1.286 g/cm3, which is equal to ρc at pointB minuskb (wherek is the slope of the final linear section). For the assumedρprofile at location A2 (Figure 11(b)), the depth at pointC is 10.5825 m, and the depth at pointD is 10.6975 m.ρ at pointC is 1.025 g/cm3, andρ at pointD is 1.222 g/cm3.( 3 ) Theρ function of the first section (water layer) is assumed to be constant, theρ function of the second section (fluid mud) is assumed to be parabolic, linear, or square root, and theρ function of the third section (bed mud) is assumed to be linear. Theρ of the first section is assumed to be 1.000 g/cm3 (freshwater) and 1.025 g/cm3 (seawater) forρ profiles at locations A1 and A2, respectively. The parabolic, linear, and square root density functions for the second section can be determined according to the density and depth values at pointC and pointD by (11) to (16). The slope of the third section is assumed to be 0.005 and 0.08, respectively, for theρprofiles at locations A1 and A2 based on the measured ρc values at the final linear sections.( 4 ) The ρc profiles are calculated by (21) to (25), assuming the ND-CP passes through the strata with the assumedρ profiles in Figures 10(b) and 11(b). The calculated ρc profiles at locations A1 and A2 are shown in Figures 10(c) and 11(c), respectively. The calculated ρc profiles clearly vary when the assumedρfunction of the second section varies (among parabolic, linear, or square root functions). Therefore, the calculated ρc profile can be changed by adjusting the assumedρ profile.( 5 ) The calculated ρc profiles for the different assumedρ profiles are compared with the measured ρc profiles. The calculated ρc profiles match the measured ρc profiles well if the density function of the second section is assumed to be a square root function [24]. The deducedρ profiles are also shown in Figures 10(d) and 11(d). Theρvalues are clearly different from the ρc values around the fluid mud layer. Therefore, it is suggested that the deducedρ profile from the measured ρc profile should be adopted to predict sediment transport, determine nautical depth, and plan dredging projects.


## 5.1. ND-CP Measuredρc Profile

The density profiles of bed sediments at Isahaya Bay, Japan, were investigated using the ND-CP. The details on the equipment and methods used in this investigation can be found in a prior publication [23]. The ND-CP measured density profiles at two locations (Figure 8) are used to illustrate how to use the proposed method to determine the actualρ profile from the measured ρc profile.Figure 8
Investigation locations.The measuredBCR andDCR and the calculated ρc values at locations A1 and A2 are shown in Figure 9.BCR is approximately zero in water and approximately 50 cps in bed mud.DCR in bed sediments is lower than that in water and decreases with increasing density. Rρ can be calculated according to the measuredBCR andDCR by using (1), and then the density can be calculated by using (25). The density profiles show similar transitions and vary significantly around the water/mud interface. The measured ρc profiles can be divided into three sections, namely, an initial constant section, a middle curved section, and a final linear section.Figure 9
Investigation results: (a) at location A1; (b) at location A2.
(a) (b)


## 5.2. Determination of the Actualρ Profile

There is an obvious difference between the actualρ and the measured ρc profiles near the upper and lower boundaries of the fluid mud layer, as shown in Figure 7(c). The theoretical model proposed in Section 3 and the method proposed in Section 4 were used to deduce the actualρ profiles from the measured ρc profiles as follows.( 1 ) For the measured ρc profile at location A1 (Figure 10(a)), the ρc at and depth of the intersection of the initial constant and middle curved sections (pointA) are 1.000 g/cm3 and 1.83 m, respectively. The ρc at and depth of the intersection of the middle curved and final linear sections (pointB) are 1.287 g/cm3 and 2.31 m, respectively. For the measured ρc profile at location A2 (Figure 11(a)), ρc and depth at pointA are 1.025 g/cm3 and 10.45 m, respectively. ρc and depth at pointB are 1.233 g/cm3 and 10.83 m, respectively.Figure 10
Determination of the actual density profile of bed sediments at location A1: (a) measuredρc profile; (b) assumedρ profile; (c) calculated ρc profile; (d) deducedρ profile.
(a) (b) (c) (d)Figure 11
Determination of the actual density profile of bed sediments at location A2: (a) measuredρc profile; (b) assumedρ profile; (c) calculated ρc profile; (d) deducedρ profile.
(a) (b) (c) (d)( 2 ) For the assumedρ profile at location A1 (Figure 10(b)), the depth of the upper boundary of the fluid mud at pointC is 1.9625 m, which is equal to the depth at pointA (1.83 m) plusb (0.1325 m). The depth of the lower boundary of the fluid mud at pointD is 2.1775 m, which is equal to the depth at pointB (2.31 m) minusb (0.1325 m).ρ at pointC is 1.000 g/cm3, which is equal to ρc at pointA.ρ at pointD is 1.286 g/cm3, which is equal to ρc at pointB minuskb (wherek is the slope of the final linear section). For the assumedρprofile at location A2 (Figure 11(b)), the depth at pointC is 10.5825 m, and the depth at pointD is 10.6975 m.ρ at pointC is 1.025 g/cm3, andρ at pointD is 1.222 g/cm3.( 3 ) Theρ function of the first section (water layer) is assumed to be constant, theρ function of the second section (fluid mud) is assumed to be parabolic, linear, or square root, and theρ function of the third section (bed mud) is assumed to be linear. Theρ of the first section is assumed to be 1.000 g/cm3 (freshwater) and 1.025 g/cm3 (seawater) forρ profiles at locations A1 and A2, respectively. The parabolic, linear, and square root density functions for the second section can be determined according to the density and depth values at pointC and pointD by (11) to (16). The slope of the third section is assumed to be 0.005 and 0.08, respectively, for theρprofiles at locations A1 and A2 based on the measured ρc values at the final linear sections.( 4 ) The ρc profiles are calculated by (21) to (25), assuming the ND-CP passes through the strata with the assumedρ profiles in Figures 10(b) and 11(b). The calculated ρc profiles at locations A1 and A2 are shown in Figures 10(c) and 11(c), respectively. The calculated ρc profiles clearly vary when the assumedρfunction of the second section varies (among parabolic, linear, or square root functions). Therefore, the calculated ρc profile can be changed by adjusting the assumedρ profile.( 5 ) The calculated ρc profiles for the different assumedρ profiles are compared with the measured ρc profiles. The calculated ρc profiles match the measured ρc profiles well if the density function of the second section is assumed to be a square root function [24]. The deducedρ profiles are also shown in Figures 10(d) and 11(d). Theρvalues are clearly different from the ρc values around the fluid mud layer. Therefore, it is suggested that the deducedρ profile from the measured ρc profile should be adopted to predict sediment transport, determine nautical depth, and plan dredging projects.


## 6. Summary and Conclusions

The nuclear density cone penetrometer (ND-CP) can be used forin situ investigation on the soil density, while it measures the composite density (ρc) of the soil within a spheroid centered at the midpoint between the gamma ray source and the detector. A theoretical model for calculating ρc and a method for deducing theρ profile from the measured ρc profile were proposed in this study. The following conclusions can be drawn from this study.( 1 ) A theoretical model for predicting the count rate ratio (Rρ) of the ND-CP is proposed, and equations for calculating Rρ are derived for the case of the ND-CP penetrating into strata with different density distribution functions. The calculated ρc profiles exhibit a good fit to the laboratory-measured ρc profiles by the ND-CP, indicating that the proposed theoretical model can be used to calculate the ρc within the spheroid. A comparison of theρ and ρc values shows that the ρc profiles are considerably different from the actualρ profiles at the boundaries of each stratum where the density changes suddenly.( 2 ) A method for deducing the actualρ profile from the measured ρc profile is proposed according to the differences between theρ and ρc profiles. The depth of and density at the boundaries of each stratum are first determined based on the features of the measured ρc profile. Then, theρ profile of the strata is assumed, and the ρc profile is back-calculated. Finally, the calculated ρc profile is compared with the measured ρc profile, and the actualρ profile is identified when the calculated ρc profile fits the measured ρc profile.( 3 ) The proposed model and method are used to determine the actualρ profiles from measured ρc profiles of bed sediments in the field. The actualρvalues are clearly different from the ρc values around the fluid mud layer. It is suggested that the deducedρ profile should be adopted to predict sediment transport, determine nautical depth, and plan dredging projects. The findings in this study would be beneficial for thein situ determination of density profiles using the ND-CP. Further research should be conducted to verify the proposed model and method in determining the actualρ profiles from the measured ρc profiles by the ND-CP in the field of geotechnical and pavement engineering.



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*Source: 5804271-2019-06-18.xml*